Echoes of the early universe. This microwave map of the whole sky was created using data from the Cosmic Background Explorer (COBE) satellite. It shows minute variations (between blue and pink) in the cosmic microwave radiation some 300 000 years after the big bang, and may explain the origin of the `lumpiness' of the universe.
The Universe and Cosmology (1 of 6)
The study of the universe, its overall structure and origin, is known as cosmology. In the 17th century, the universe was thought to be static, infinite and unchanging. Modern cosmology can be traced back to the 1920s when the American astronomer Edwin Hubble, using observations made by Vesto Slipher in 1912, showed that the space between galaxies is increasing and the universe is therefore expanding.
There are several theories describing the origin and future of the universe. Among these, the big-bang theory is the most widely accepted. Opinions differ, however, as to the future of the universe. Some think that it will continue to expand for ever, whereas others believe that it will eventually end by collapsing in on itself in a big crunch.
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p004-2
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The Universe and Cosmology (page 2)
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The big bang theory, is demonstrated in this animation. First matter collapses into the center, then it explodes and expands throughout the universe. Scientific opinions differ on this theory and this is one of many interpretations.
The Universe and Cosmology (2 of 6)
The big bang
The universe is thought to have originated between 15 000 and 20 000 million years ago in a cataclysmic event known as the big bang. Theoretical models of the big bang suggest that events in the early history of the universe occurred very rapidly.
At the beginning of time, the universe comprised a mixture of different subatomic particles, including electrons, positrons, neutrinos and antineutrinos, together with photons of radiation. The temperature was 100 000 million deg C (180 000 million deg F) and its density 4000 million times that of water.
One second later, the temperature has dropped to 10 000 million deg C (18 000 million deg F). Matter is spreading out and the density of the universe has fallen to 400 000 times that of water. Heavier particles, protons and neutrons, begin to form.
Fourteen seconds later the temperature has dropped to 3000 million deg C (5400 million deg F). Oppositely charged positrons and electrons are annihilating each other and liberating energy. Stable nuclei of helium consisting of two protons and two neutrons begin to form.
Three minutes after the creation of the universe, the temperature has fallen to 900 million deg C (1620 million deg F). This is cool enough for deuterium nuclei consisting of one proton and one neutron to form.
Thirty minutes later, the temperature is 300 million deg C (540 million deg F). Very few of the original particles remain, most of the electrons and protons having been annihilated by their antiparticles, positrons and antiprotons. Many of the remaining protons and neutrons have combined to form hydrogen and helium nuclei and the density of the universe is about one tenth that of water. Expansion of the universe continues and the hydrogen and helium begins to form into stars and galaxies.
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Big bang theory
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p004-3
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The Universe and Cosmology (page 3)
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The Copernican universe. Nikolaus Copernicus (1473-1543) correctly argued that the planets orbit the Sun, but he mistakenly thought the Sun was at the center of the universe. In fact the universe has no center.
The Universe and Cosmology (3 of 6)
The 3 K microwave background
Astronomers can detect an `echo' from the big bang in the form of microwave radiation. The existence of the microwave background was predicted by George Gamow in 1948 and found by Penzias and Wilson in 1965. The radiation has a maximum intensity at a wavelength of 2.5 mm (0.1 in) and represents a temperature of 3 K (-270 deg C or -454 deg F). In the vicinity of the Solar System, the radiation appears to have equal intensity in all directions.
The apparent uniformity of the background radiation provided a problem for big-bang theorists, in that it posed the question of how the universe became as irregular (`lumpy') as it is, with clusters of galaxies in some areas, and empty space in others. A possible answer came in 1992, when data from the COBE satellite showed minute differences in temperature (+ and - 0.27 millikelvins) in the background radiation. These have been interpreted as evidence of infinitesimal density fluctuations, which in turn would have led to local gravitational effects in the expanding fireball. With the beginnings of gravitational instability in certain regions, matter would begin to coalesce, eventually giving rise to protogalaxies.
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p004-4
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The Universe and Cosmology (page 4)
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The expanding universe . The speed (indicated by the length of arrow) at which a galaxy is moving away from the observer becomes greater the further the galaxy is from the observer. Wherever the observer is in the universe, all other galaxies are seen to be receding.
The Universe and Cosmology (4 of 6)
Red shift
In 1868, the English amateur astronomer Sir William Huggins (1824-1910) noticed that lines in the spectra of certain stars were displaced towards the red end of the spectrum. Huggins realized that this was due to the Doppler effect, which had been discovered in 1842. Just as the noise from a moving vehicle will appear to change pitch as it passes, the color of light from a star will change in wavelength as the star moves towards us, or away from us. Stars moving away from the Earth have their light moved towards the red end of the spectrum (red shift), while those moving towards us exhibit a shift towards the blue end.
Hubble's law
In 1929, Edwin Hubble (1889-1953) - who also worked on the classification of galaxies - analyzed the red shifts of a number of galaxies. He found that the speed at which a galaxy is moving away from us is proportional to its distance - i.e. the more distant a galaxy, the faster it is receding. This principle was formulated as Hubble's law, which can be written in the form:
speed = H x distance, where H is the Hubble constant.
Various values for the Hubble constant have been proposed, but the generally accepted value is 56 km (35 mi) per second per megaparsec (a megaparsec is 3.26 million light years). Thus a galaxy that is receding from the Earth at 56 km/sec will be 326 000 light years distant.
tant.
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Expanding Universe
**++*
p004-5
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The Universe and Cosmology (page 5)
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The Cosmic Distance Scale.
The Universe and Cosmology (5 of 6)
The future of the universe
At present the universe is still expanding, but whether or not this will continue for ever depends upon the amount of matter it contains. One possible ending for the universe is the big crunch. The galaxies and other matter may be moving apart, but their motion is restrained by their mutual gravitational attraction. If there is sufficient matter in the universe, gravity will eventually win and begin pulling the galaxies together again, causing the universe to experience a reverse of the big bang - the big crunch.
What will follow the big crunch is hard to imagine. One possibility is that a new universe will come into being, perhaps containing completely different types of particles from our present universe. The cyclic theory suggests that the universe may continue alternately to expand and collapse.
However, it may be that there is not enough matter in the universe for the big crunch to happen. If this is the case, the universe will continue to expand for ever. Although this means there may never be an `edge' to the universe, there is bound to be an end to the observable universe. Hubble's law states that the speed of recession of a galaxy is proportional to its distance. A galaxy which is far enough away to be traveling at the speed of light will no longer be visible and this will therefore mark the end of the universe we can see. The end of the observable universe lies at a distance of between 15 000 and 20 000 million light years.
The steady-state theory
Another cosmological model, which is no longer generally accepted, is the steady-state theory. This supposes that the universe has always existed and will always continue to exist.
The theory was first proposed in 1948 by a group of Cambridge astronomers and popularized by Sir Fred Hoyle (1915- ). However, among many other objections, the theory offers no satisfactory explanation for the 3 K microwave background radiation.
* TIME
* STARS AND GALAXIES
* THE HISTORY OF ASTRONOMY
* NEWTON AND FORCE
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* OPTICS
* ATOMS AND SUBATOMIC PARTICLES
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The Universe and Cosmology (page 6)
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Definition of the parsec. If the change in angle between the two observations of the star is 1 second of arc, then d = 1 parsec (equivalent to 3.26 light years). In fact, no star is as close to the Solar System as 1 parsec. Note that the angle in the diagram has been exaggerated for the sake of clarity.
The Universe and Cosmology (6 of 6)
ASTRONOMICAL DISTANCES
The light year is a unit used to measure great distances, and is equal to the distance traveled by light in one year. Light (in a vacuum) travels at 300 000 km/sec (186 000 mi/sec), and so a light year is approximately 9 461 000 million km (5 875 000 million mi).
Distances to nearby stars can be measured by the parallax method. Any object, when viewed from two different vantage points, will appear to move against a background of more distant objects. This apparent change in position is called the parallax, and is measured as an angle. Thus if a nearby star is viewed from the Earth at intervals of six months, the Earth will have moved from one side of its orbit to the other and the star will seem to move against the background of more distant stars. The diameter of the Earth's orbit is known, so the distance of the star can be calculated.
The parallax method leads to the definition of the parsec, which is the distance at which an object would exhibit a parallax of one second of arc (i.e. 1/3600 of a degree). One parsec is 3.26 light years, so with the exception of the Sun, no stars are as close as one parsec.
An Aztec calendar woodcut. In the inner circle are the symbols of the twenty Aztec days, each of which is associated with a particular god. The Aztec year of 365 days was made up of 18 months of 20 days each, with an additional 5 unlucky days. At the center of the woodcut is the date on which the Aztecs believed the world would be destroyed by an earthquake, together with the dates on which they believed similar destructions had occurred previously.
Time (1 of 3)
Time forms the basis of many scientific laws, but time itself is very difficult to define. Time, like distance, separates objects and events and for this reason can be regarded as the fourth dimension. However, time cannot be measured directly. We must make do with measuring the way in which the passage of time affects things.
Time always moves forwards. This is demonstrated by the fact that there are many processes that once done cannot be undone. Although we measure time as if it passes at a regular rate, time can appear to move at different rates depending upon what one is doing. Someone enjoying themselves may find that time appears to pass very quickly. Conversely, a person laboring at a monotonous job may find that time appears to pass slowly. This is known as subjective time.
Time systems
The Earth's orbit is not circular but elliptical, so the Sun does not appear to move against the stars at a constant speed. Most everyday time systems are therefore based on a hypothetical `mean Sun', which is taken to travel at a constant speed equal to the average speed of the actual Sun.
A day is the time taken for the Earth to turn once on its axis. A sidereal day is reckoned with reference to the stars and is the time taken between successive passes of the observer's meridian by the same star. (The meridian is an imaginary line from due north to due south running through a point directly above the observer.) One sidereal day is 23 hours 56 minutes 4 seconds. A solar day is calculated with respect to the mean Sun. The mean solar day is 24 hours long.
A year is the time taken for the Earth to complete one orbit of the Sun. The Earth's true revolution period is 365 days 6 hours 9 minutes 10 seconds, and this is known as a sidereal year. However, the direction in which the Earth's axis points is changing due to an effect known as precession. The north celestial pole now lies near the star Polaris in the constellation Ursa Minor, thus Polaris is also known as the Pole Star. By the year AD 14 000, the Earth's axis will point in a different direction and the bright star Vega in Lyra will be near the pole. This effect also means that the position of the Sun's apparent path across the sky is changing with respect to the stars. A tropical year compensates for the effects of precession and is 365 days 5 hours 48 minutes 45 seconds long. It is the tropical year which is used as the basis for developing a calendar.
The SI unit of time is the second, which was originally defined as 1/86 400 of the mean solar day. However, as we have seen, the Earth is not a very good timekeeper, so scientists no longer use it to define the fundamental unit of time. The
second is now defined as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of a cesium-133 atom.
Greenwich Mean Time (GMT) is the local time at Greenwich, England. The Greenwich Meridian is the line of 0 deg longitude which passes through Greenwich Observatory. The mean Sun crosses the Greenwich Meridian at midday GMT. Also known as Universal Time (UT), GMT is used as a standard reference time throughout the world. Sidereal time literally means `star time'. It is reckoned with reference to the stars and not the Sun.
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* QUANTUM THEORY AND RELATIVITY
* THE REVOLUTIONARY CALENDAR
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Time (page 2)
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Sidereal and solar days. In traveling through distance A the Earth rotates once in relation to more distant stars, so completing one sidereal day (23 hours 56 minutes 4 seconds). To complete a mean solar day, with the Sun in the same position in the sky as it was 24 hours before, the Earth has to turn approximately 1 deg more, and in so doing travels the additional distance B.
(Diagram not to scale.)
` N Time (2 of 3)
Calendars
The Earth takes 365.2422 days to complete one orbit of the Sun, which makes planning a calendar rather difficult, as the extra 0.2422 days per year have to be accounted for. The first person to attempt to do this was Julius Caesar (100-44 BC), who commissioned the astronomer Sosigenes to produce what became known as the Julian calendar. In order to compensate for the errors that had accumulated over previous years, Caesar decreed that the first year of his new calendar would be 445 days long. Thus 46 BC became known as `The Year of Confusion'. Julius Caesar also introduced the idea of leap years to compensate for the extra 0.2422 days per year. However, the original Julian calendar had a leap year every third year - 0.0911 days or 2 hours 11 minutes too much.
The calendar was further refined by the Emperor Augustus (63 BC-AD 14), who in 8 BC revised the frequency to one leap year every fourth year. This gives an `extra' 0.2500 days per year and reduces the error to 0.0078 days (i.e. 11 minutes) per year. By the 14th century, the extra 11 minutes per year had accumulated to a total error of 10 days. Pope Gregory XIII (1572-85) therefore introduced the Gregorian calendar, decreeing that 4 October 1582 should be followed by 15 October. He also decided that the century years would not be leap years, unless divisible by 400. Thus, for example, 1800 and 1900 were not leap years, but 2000 will be. The Gregorian calendar allows an extra 0.2425 days per year and is in error by just 0.0003 days per year. It will therefore be many centuries before the calendar will need to be revised again.
When Britain and its colonies, long antagonistic to anything emanating from Rome, eventually adopted the Gregorian calendar in 1752, riots broke out with mobs chanting, `Give us back our eleven days!' Russia did not adopt the Gregorian calendar until after the 1917 Revolution, by which time there was a 13-day lag. This explains why the October Revolution (25 October 1917) is now commemorated in November (7 November).
The Muslim calendar has either 354 or 355 days in a year. The Jewish calendar employs a year that varies from a minimum of 353 days to a maximum of 385 days. The Chinese used to have a calendar based on a 60-year cycle. Although it was banned in China in 1930, the Chinese calendar is still used in parts of Southeast Asia.
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* QUANTUM THEORY AND RELATIVITY
* THE REVOLUTIONARY CALENDAR
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Time (page 3)
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The equipment responsible for creating the shortest `slice' of time ever - a pulse of light from a laser lasting only 30 femto seconds. A femto second is 10 to the power of -15 seconds (0.000000000000001 or a millionth billionth of a second). Scientists use such pulses of light as `stopwatches' to study subtle physical and chemical changes, such as how electrons move in semiconductor materials. In one second, a pulse of light can travel almost to the Moon, but in 30 femto seconds it travels no further than one third of the thickness of a human hair.
Time (3 of 3)
Measuring time
The earliest device for measuring time was the sundial, which can be traced back to the Middle East c. 3500 BC. A sundial comprises a rod or plate called a gnomon that casts a shadow on a disc; where the shadow points indicates the position of the Sun and hence the time of day.
Mechanical clocks, driven by falling weights, appeared in the 14th century, and the first mechanical watches, driven by a coiled mainspring, in the 16th century. The first pendulum clock was invented by Christiaan Huygens (1629- 95), a Dutch physicist, in the middle of the 17th century.
Pendulum clocks could not be used on board ships owing to the vessel's motion. In 1714 the British Longitude Board offered a prize for the development of a marine chronometer, as being able to tell the time accurately is vital to navigation. Fourteen years later the English clockmaker John Harrison (1693-1776) set to work on the problem, producing his first marine chronometer in 1735.
The first quartz clock, operated by the vibrations of a quartz crystal when an electrical voltage is applied, appeared in 1929. The quartz clock is accurate to within one second in ten years. This was followed in 1948 by the atomic clock, which depends on the natural vibrations of atoms. The most accurate modern atomic clocks are accurate to one second in 1.7 million years.
THE DAYS OF THE WEEK
English name Named after
Sunday The Sun
Monday The Moon
Tuesday Tiw, the Anglo-Saxon counterpart of
the Nordic god Tyr, son of Odin
Wednesday Woden, the Anglo-Saxon
counterpart of Odin, the
Nordic god of war
Thursday Thor, the Nordic god of thunder,
eldest son of Odin
Friday Frigg, the Nordic goddess of love,
wife of Odin
Saturday Saturn, Roman god of agriculture
and vegetationnnnnnnnnnn
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* QUANTUM THEORY AND RELATIVITY
* THE REVOLUTIONARY CALENDAR
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ftsTitleOverride
Stars and Galaxies (page 1)
ftsTitle
The Orion Nebula (M42), as seen through a large telescope sensitive to infra-red.
Stars and Galaxies (1 of 4)
A galaxy is a system of many thousands of millions of stars, together with interstellar gas and dust. Many galaxies are spiral in shape, while others can be spherical, elliptical or irregular. Telescopes have revealed the existence of about 1000 million galaxies, although apart from our own galaxy, only three can be clearly seen with the naked eye.
Stars - of which our Sun is an example - are accretions of gas that radiate energy produced by nuclear-fusion reactions. They range in mass from about 0.06 to 100 solar masses, one solar mass being equivalent to the mass of the Sun. The properties of a star and the manner in which it evolves depend principally on its mass.
Stars are formed within clouds of dust and gas called nebulae. Patches of gas and dust inside a nebula collapse under gravity forming dark regions called protostars. As the protostars continue to collapse, they become denser and hotter. Eventually, they may become hot enough for nuclear-fusion reactions to start and thus turn into stars.
The position of our Sun in the Galaxy, shown schematically.
Stars and Galaxies (2 of 4)
Galaxies
The American astronomer Edwin Hubble devised a system for classifying galaxies that is still in use. He grouped galaxies into three basic categories: elliptical, spiral and irregular. Elliptical galaxies range from the spherical EO type to the very flattened E7. Spiral galaxies are labeled Sa, Sb or Sc, depending upon how tightly wound the arms are. Some spirals appear to have their arms coming from the ends of a central bar and these barred spirals are designated SBa, SBb or SBc. Irregular galaxies are those whose shape is neither spiral nor elliptical.
Sometimes known as the Milky Way, our own galaxy - `the Galaxy' - contains about 10 000 million stars. It is an ordinary spiral galaxy and the Sun is situated in one of the spiral arms. The diameter of the Galaxy is about 100 000 light years and the Sun is some 30 000 light years from the center. The nearest star to the Sun, Proxima Centauri, is 4.2 light years distant. The Galaxy is rotating and the Sun takes 225 million years to complete one revolution. This is sometimes called a cosmic year.
Some galaxies are extremely active and emit vast amounts of radiation. One such galaxy is the powerful radio source Centaurus A. Quasars are very distant and immensely bright objects, which are thought to represent the nuclei of active galaxies. They may be powered by massive central black holes. The most distant quasar yet detected, PKS 2000-330, is 13 000 million light years from the Earth.
Binary, multiple and variable stars
The majority of stars - over 75% - are members of binary or multiple star systems. Binary stars consist of two stars each orbiting around their common center of gravity. An eclipsing binary can occur where one component of the system periodically obscures, and is obscured by, the other (as seen from Earth). This leads to a reduction in the light intensity seen from Earth - which is how binary stars were first discovered. Some stars are actually complex multiple stars. For example, the `star' Castor in the constellation of Gemini has six individual components.
Most stars are of constant brightness, but some - variable stars - brighten and fade. The variability can be caused by a line-of-sight effect, as in eclipsing binaries. In other cases, changes in the star itself cause periodic increases and reductions of energy output. Variable stars can have periods ranging from a few hours to several years.
Magnitude
Magnitude is a measure of a star's `brightness'. Apparent magnitude indicates how bright a star appears to the naked eye. Paradoxically, the lower the magnitude the brighter the star. Magnitude is measured on a logarithmic scale, taking as its basis the fact that a difference of 5 in magnitude is equivalent to a factor of 100 in brightness. On this basis, a star of magnitude +1 is 2.512 times brighter than a star of +2, 2.512 to the power of 2 (= 6.310) times brighter than a star of +3, and 2.512 to the power of 5 (= 100) times brighter than a star of +6.
The limit of naked-eye visibility depends upon how clear the sky is, but the faintest stars that can be seen on a really clear night are about magnitude +6. The world's largest telescopes can detect objects as faint as magnitude +27. Very bright objects can have negative magnitudes: the planet Venus can reach -4.4, the full Moon -12.0 and the Sun -26.8.
The nearer a star is, the brighter it will appear. Different stars lie at different distances, so apparent magnitude does not measure the true brightness of a star. Absolute magnitude compensates for a star's distance by calculating its apparent magnitude if it were placed at a distance of 32.6 light years (= 10 parsecs). For example, Sirius is a nearby star and has an apparent magnitude of -1.5. However, its absolute magnitude is +1.3. The Sun has an absolute magnitude of +4.8.
Color and temperature
The color of a star gives an indication of its temperature. Hot stars are blue, while cool stars are red. Stars are grouped into spectral types according to their temperatures.
Type Color Temperature
( deg C) ( deg F)
O Blue 25 000-40 000 45 000-75 000
B Blue 11 000-25 000 20 800-45 000
A Blue-white 7500-11 000 13 500-20 000
F White 6000-7500 10 800-13 500
G Yellow 5000-6000 9000-10 800
K Orange 3500-5000 6300-9000
M Red 3000-3500 5400-6300
Each spectral type is further subdivided on a scale 0-9. The Sun is classified as G2.
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE HISTORY OF ASTRONOMY
* ENERGY: COAL, OIL AND NUCLEAR
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ftsTitleOverride
Stars and Galaxies (page 3)
ftsTitle
The Hertzsprung-Russell Diagram for classifying stars.
Stars and Galaxies (3 of 4)
The Hertzsprung-Russell diagram
Stars can be arranged on a diagram that plots their absolute magnitudes against their spectral types. This is known as a Hertzsprung-Russell diagram, after the Danish astronomer Ejnar Hertzsprung (1873-1967) and Henry Norris Russell (1877-1957), an American. Most stars fit into a diagonal band (called the main sequence) across the diagram.
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE HISTORY OF ASTRONOMY
* ENERGY: COAL, OIL AND NUCLEAR
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Stars and Galaxies (page 4)
ftsTitle
The relative sizes of different types of stars. Typical red giants are 100 times the size of the Sun, which in turn is 100 times the size of a white dwarf. White dwarfs are 1000 times larger than neutron stars, which typically have a diameter of 10-20 km (6-12 mi). Red supergiants may be 5 times larger than a typical red giant.
Stars and Galaxies (4 of 4)
Stellar evolution and black holes
The manner in which a star evolves depends upon its mass. Protostars with mass less then 0.06 of the Sun will never become hot enough for nuclear reactions to start. Those with mass between 0.06 and 1.4 solar masses quickly move on to the main sequence and can remain there for at least 10 000 million years. When the available hydrogen is used up, the core contracts, which increases its temperature to 100 million deg C (180 million deg F). This produces conditions in which helium can begin a fusion reaction and the star expands to become a red giant. Finally, the outer layers of the star are expelled, forming a planetary nebula. The core then shrinks to become a small white dwarf star.
Stars of between 1.4 and 4.2 solar masses evolve more quickly and die younger. They remain on the main sequence for about one million years before the red giant phase begins. The temperature continues to increase as even heavier elements are synthesized until iron is produced at the temperature of 700 mil lion deg C (1260 million deg F). The star is then disrupted in a huge supernova explosion producing a vast expanding cloud of dust and gas. At the center of the cloud a small neutron star will remain. This rotates very rapidly and is incredibly dense: 1 cm3 (0.061 cu in) of neutron-star material has a mass of about 250 million tons.
The evolution of more massive stars is stranger still. They may end their lives by producing a black hole - an object so dense that not even light can escape. The only means of detecting a black hole is by observing its gravitational effects on other objects. The X-ray source Cygnus X-1 may comprise a giant star and a black hole. Material would be pulled away from the star by the black hole and heated - giving off X-rays as it is pulled in.
Type Color Temperature
( degC) ( deg F)
O Blue 25 000-40 000 45 000-75 000
B Blue 11 000-25 000 20 800-45 000
A Blue-white 7500-11 000 13 500-20 000
F White 6000-7500 10 800-13 500
G Yellow 5000-6000 9000-10 800
K Orange 3500-5000 6300-9000
M Red 3000-3500 5400-6300
* THE UNIVERSE AND COSMOLOGY
* THE SUN AND THE SOLAR SYSTEM
* THE HISTORY OF ASTRONOMY
* ENERGY: COAL, OIL AND NUCLEAR
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The Sun and the Solar System (page 1)
ftsTitle
The surface of the Sun, seen here in a false- color extreme ultra- violet image, taken by the Skylab space station in 1973. At the top a huge solar prominence - some 500 000 km (300 000 mi) in height - leaps up through the Sun's atmosphere.
The Sun and the Solar System (1 of 6)
The Sun, the principal source of light and heat for planet Earth, is a very ordinary star, situated near the edge of a spiral arm about 30 000 light years from the center of the Galaxy. The Sun is the center of the Solar System, which includes at least nine major planets and their satellites, together with interplanetary material and thousands of minor planets, comets and meteoroids.
The Sun mainly consists of the gases hydrogen and helium. At its center is a vast nuclear reactor whose temperature is at least 14 million deg C (25 million deg F). The Sun produces energy by nuclear fusion, a process in which hydrogen is converted into helium.
The Sun is losing mass at a rate of 4 million tons per second, but its total mass is 2 x 10 to the power of 27 tons, which is 330 000 times that of the Earth and 745 times that of all the planets put together. The diameter of the Sun is 1 392 000 km (863 000 mi) - 109 times greater than that of the Earth - and its volume is 1 300 000 times that of the Earth.
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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The Sun and the Solar System (page 2)
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Solar prominences are masses of glowing hydrogen that penetrate the Sun's atmosphere. They are quite spectacular, releasing brilliant outbursts of light, charged particles and radiation. These energy releases can interfere with radio, satellite and other communications.
The Sun and the Solar System (2 of 6)
The Sun's surface and atmosphere
When the image of the Sun is projected through a telescope, dark patches, called sunspots, can often be seen. Although they look black, sunspots are actually quite bright - they only appear dark by contrast with the surrounding brighter areas. Sunspots are about 2000 deg C (3600 deg F) cooler than other parts of the Sun's surface. The number of sunspots that can be seen on the Sun varies over an 11-year cycle. At the maximum of the cycle it is possible to see many sunspot groups, whereas at the minimum of the cycle no spots may be seen for several days.
The bright surface of the Sun is called the photosphere. A closer view of the photo-sphere shows that it consists of millions of granules, each of which is several hundred kilometers in diameter. The surface of the Sun is constantly changing its appearance, with individual granules persisting for about 10 minutes. Rising up from the photosphere are huge jets of gas called spicules. These can reach 15 000 km (9000 mi) in diameter, but last for just a few minutes.
The part of the Sun that lies above the photosphere is called the chromosphere. It is red in color and consists mainly of hydrogen gas. It is normally impossible to see with the naked eye, owing to the proximity of the much brighter photo-sphere. However, during a total eclipse of the Sun, when the photosphere is obscured by the Moon, the chromosphere can be seen.
Masses of glowing hydrogen called prominences are sometimes ejected from the chromosphere. These penetrate the outermost part of the Sun's atmosphere. Prominences average 100 000 km (60 000 mi) in length. There are two types of prominences - active and quiescent. Active prominences are violent and short-lived phenomena, whereas quiescent prominences are much calmer and may persist for several weeks.
The outermost layer of the solar atmosphere is the corona, which consists of thin hydrogen gas at high temperature. Solar flares are another category of brilliant outbursts in the solar atmosphere. They are often associated with sunspot groups and can reach maximum brightness in just a few minutes. They are essentially magnetic phenomena and send out large amounts of charged particles and radiation.
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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The Sun and the Solar System (page 3)
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A display of aurora, which take many forms, including curtains, arcs, rays, bands, and fan-shaped coronas. The arc is the most stable form of aurora, and it can persist for hours without noticeable variation.
The Sun and the Solar System (3 of 6)
The solar wind and aurorae
The Sun is constantly sending out a stream of charged particles into space. This is known as the solar wind. The strength of the solar wind is not constant, but changes with the activity of the Sun. Near the peak of the sunspot cycle, the solar wind is at its strongest.
The Earth has a strong magnetic field, which traps ionized particles from the solar wind in the upper atmosphere. These regions are the Van Allen Zones, two belts which extend from 1000 to 5000 km (620 to 3100 mi) and from 15 000 to 25 000 km (9300 to 15 500 mi) above the equator.
The solar wind also interacts with the Earth's magnetic field to produce brilliant displays of light called aurorae - the aurora borealis (or `northern lights') in the northern hemisphere and the aurora australis in the southern. Aurorae are formed by the charged particles in the solar wind interacting with gases in the Earth's atmosphere at a height of about 100 km (60 mi), causing them to emit visible light. This can be seen from the ground as an ever-changing pattern of white or multicolored lights. The charged particles are attracted towards the Earth's magnetic poles and aurorae are therefore best seen from high latitudes. The most spectacular displays occur about 24 hours after major solar-flare activity.
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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The Sun and the Solar System (page 4)
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The Sun and the Solar System (4 of 6)
The Solar System
The Solar System consists of the Sun together with a large number of bodies and matter that is gravitationally bound to it. These include the planets, their satellites, minor planets, comets, meteoroids and interplanetary gas and dust.
There are nine known planets, all going round the Sun in elliptical orbits. In order of mean distance from the Sun, the planets are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Of these, Pluto has a rather eccentric orbit so that although its average distance from the Sun is greater than that of Neptune, part of its orbit brings it closer to the Sun than Neptune.
Thus from 1979 to 1999 Neptune holds the title of the outermost planet. However, in 1992 the discovery was announced of a body even further from the Sun. This body, which has a diameter of 200 km (120 mi), is thought to be an asteroid or comet. It has been designated 1992 QB1, and its mean distance from the Sun may be as much as 8800 million km (5400 million mi).
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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The Sun and the Solar System (page 5)
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The Sun and the Solar System (5 of 6)
The Solar System is generally thought to have formed 4600 million years ago by accretion (cumulative coming together) from the solar nebula - a spinning cloud of gas and dust that also gave birth to the Sun. Gravity was the dominant force during the formation of the Solar System, and at some stage nuclei developed within the solar nebula that eventually accreted into the planets we now know. The fact that the planets all orbit the Sun in the same direction is thought to be a relic from the rotation of the original solar nebula.
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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p010-6
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The Sun and the Solar System (page 6)
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Halley's Comet, photographed in 1910. Comets probably consist of a small nucleus of ice and dust. When a comet approaches the Sun, part of the nucleus vaporizes to form a luminous cloud (the coma) and the tail, which always points away from the Sun. Although the nucleus may only be a few km in diameter, comas may have diameters up to 1 million km (620 000 mi).
The Sun and the Solar System (6 of 6)
COMETS
Comets can best be described as `dirty snowballs'. They are thought to originate in a region known as the Oort Cloud, about one light year from the Sun.
Sometimes comets are perturbed from the Oort Cloud and swing in towards the Sun. The gravitational attraction of a planet may trap a comet into a closed but highly elliptical orbit, which will periodically bring it close to the Sun. The best-known example is Halley's Comet, which has a period of 76 years. Halley's Comet is named in honor of the English astronomer Edmond Halley (1656-1742), who successfully predicted the comet's return in 1758.
Other comets may reach open parabolic or hyperbolic orbits. These will swing past the Sun just once and then be lost from the Solar System for ever.
Comets may have on occasion collided with the Earth. On 30 June 1908 a great explosion occurred in the sparsely populated Tunguska area of Siberia, flattening trees over a wide area. The object that caused this enormous devastation may have been the nucleus of a small comet, rather than a meteorite, which would have left a crater.
* STARS AND GALAXIES
* THE INNER PLANETS
* THE OUTER PLANETS
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The Inner Planets (page 1)
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A photomosaic of Mercury, made up from photographs taken by Mariner 10 in 1974. The image has been tinted to approximate the visual appearance of the planet.
The Inner Planets (1 of 5)
The Inner Planets
The four inner members of the Sun's family are relatively small, rocky planets. The Earth is a member of this group and they are therefore often known as the terrestrial planets. Despite this initial similarity, the four terrestrial planets are very different worlds. Mercury and Venus are inhospitably hot, whereas for much of the year Mars is bitterly cold.
Until the 1960s, little was known about the Earth's nearest neighbors in space. However, since the advent of the space age, unmanned craft have visited all members of the inner Solar System. Mariner 10 has flown by Mercury and many different spacecraft have flown by, orbited and landed on Venus and Mars.
Mercury
Mercury's proximity to the Sun makes it a difficult planet to see, as it only appears low in the west after sunset, or low in the east before sunrise. The first telescopic observations were made from Danzig by Johannes Hevelius (1611-87), who saw that the planet shows phases like the Moon.
Following a series of observations be ginning in 1881, the Italian Giovanni Schiaparelli (1835-1910) reported markings on the surface of Mercury, leading him to conclude that it had a rotation period of 88 days. We now know that Schiaparelli was wrong. Modern observations have shown that Mercury rotates once every 59 days, which is two thirds as long as a Mercury `year'. Mercury therefore rotates on its axis three times for every two orbits of the Sun.
Almost all of our information about Mercury comes from Mariner 10, the only spacecraft to have visited the planet. The pictures it returned showed a barren, rocky world, covered in craters, some of which are over 200 km (120 mi) in diameter. To all intents and purposes, Mercury does not have an atmosphere: its atmospheric pressure is one thousandth of a millionth of a millionth of the pressure on the Earth.
* TIME
* THE SUN AND THE SOLAR SYSTEM
* THE OUTER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
* THE EARTH'S STRUCTURE AND ATMOSPHERE
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p012-2
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The Inner Planets (page 2)
ftsTitle
Venus is seen revolving. However, due to its dense atmosphere the surface of the planet can not be seen. Venus is the closest planet to the Earth, but revolves in a direction opposite to that of the Sun, the Earth, and other planets.
The Inner Planets (2 of 5)
Venus
Venus is the Earth's nearest neighbor and is often the brightest object in the night sky, apart from the Moon. Like Mercury, Venus can only be seen with the naked eye in the morning or evening. Venus, like Mercury, also shows phases, which binoculars or a small telescope will reveal.
Venus is covered by a dense atmosphere, so telescopes cannot show any surface detail. In 1962 astronomers managed to bounce radio waves off the planet and these showed that Venus was rotating backwards (i.e. in the opposite direction to the Sun and nearly all the other planets) once every 243 days.
Venus's atmosphere is composed mainly of carbon dioxide and has clouds of sulfuric acid floating in it. Carbon dioxide acts rather like the glass in a greenhouse, letting in the energy from the Sun, but without allowing much heat to escape. The upper clouds race around the planet once every four days - much faster than the planet itself is turning.
The first spacecraft to transmit from the surface of Venus was the Soviet Venera 7, which found a temperature of 470 deg C (878 deg F) and an atmospheric pressure 90 times that on Earth. The later probes, Veneras 9, 10, 13 and 14, returned pictures from the surface of Venus.
In 1978, the American Pioneer-Venus spacecraft was put into orbit around the planet to start making a map of the surface, using radar to penetrate Venus's thick clouds. This revealed a complex surface with low-lying plains, upland areas, volcanoes and rift valleys. The American Magellan spacecraft, launched in 1989, will produce a more detailed radar map of the planet's surface. Mission planners hope to detect features as small as 250 m (820 ft) across and the probe will study 92% of the planet.
The Earth
The Earth is the largest of the inner planets and the only planet able to support life. Its atmosphere consists mainly of nitrogen (78%) and oxygen (21%). Two thirds of its surface is covered by water, which has an average depth of 3700 m (12 140 ft). The land rises above the oceans to an average height of 860 m (2800 ft). More detailed information on the Earth's interior, atmosphere and magnetic field will be found on pp. 76-7.
* TIME
* THE SUN AND THE SOLAR SYSTEM
* THE OUTER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
* THE EARTH'S STRUCTURE AND ATMOSPHERE
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p012-3
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The Inner Planets (page 3)
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In 1969 man landed on the moon. For the first time, we had a close-up view of this terrestrial body, allowing us to inspect the moon and collect rock samples. These rocks turned out to be 3700 million years old - as old as the oldest rocks on Earth.
The Inner Planets (3 of 5)
The Moon
The Moon, which maintains a mean distance from the Earth of 384 400 km (238 700 mi), is the Earth's largest satellite and has a mass 1/81 of that of the Earth. The Moon has a diameter of 3476 km (2159 mi), making it larger than the planet Pluto. The Moon orbits the Earth once every 27.3 days in synchronous rotation - i.e. it keeps the same face towards the Earth.
Surface features include craters formed by meteoritic bombardment, mountains and broad plains, which in the past were mistakenly named `seas' or maria (Latin, singular mare). The temperature on the lunar surface ranges from -180 deg C (-292 deg F) to +110 deg C (+200 deg F).
It was not until October 1959 that the Soviet probe Luna 3 returned the first pictures from the far side of the Moon - which turned out to be much the same as the near side, except for the absence of maria. When men first landed on the Moon in 1969 they found rocks that were 3700 million years old - as old as some of the oldest rocks found on the Earth.
As seen from the Earth, the Moon passes through a series of phases every 27.3 days - waxing from new Moon, through first quarter, to full Moon, then waning to last quarter and new Moon again.
* TIME
* THE SUN AND THE SOLAR SYSTEM
* THE OUTER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
* THE EARTH'S STRUCTURE AND ATMOSPHERE
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The Inner Planets (page 4)
ftsTitle
A view of the surface of Mars, processed by computer from images sent back by the Viking spacecraft and from topographic maps. In the left foreground is the giant Valles Marineris canyon system, which is over 3000 km (1860 mi) long and up to 8 km (5 mi) deep. In the background are the three Tharsis volcanoes. The canyon system was formed by a combination of geological faulting, landslides, and erosion by wind and water.
The Inner Planets (4 of 5)
Mars is the fourth planet from the Sun. It is the most hospitable planet other than the Earth, having a thin carbon dioxide atmosphere. Early observers, including Schiaparelli, believed they saw canals and vegetation on Mars, but modern observations have shown that these do not exist.
In 1965 the American Mariner 4 space craft flew by Mars and returned pictures showing a barren, cratered surface, massive extinct volcanoes and deep chasms. The largest volcano on Mars is Olympus Mons, which rises 22 km (14 mi) above the surrounding plains. Later probes, including Mariner 9 and Vikings 1 and 2 (the latter two successfully sending back pictures from the planet's surface) revealed other features that may have been caused by running water at some time in the past history of the planet.
Mars has two small satellites, Phobos and Deimos, which were mapped by Vikings 1 and 2 in 1976. In 1989 Phobos was further studied by the Soviet Phobos 2.
The minor planets
The minor planets, sometimes known as asteroids, comprise several thousand objects, most of which orbit between Mars and Jupiter (the asteroid belt). The largest of the minor planets is Ceres, which has a diameter of 940 km (584 mi). Once thought to be the residue of a planet broken up by the gravitational pull of Jupiter, most astronomers now think that the minor planets represent a class of primitive objects which were `left over' during the formation of the Solar System, due to Jupiter's disruptive pull.
* TIME
* THE SUN AND THE SOLAR SYSTEM
* THE OUTER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
* THE EARTH'S STRUCTURE AND ATMOSPHERE
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ftsTitleOverride
The Inner Planets (page 5)
ftsTitle
The phases of the Moon. The drawings in the outer circle show how the Moon is viewed from the Earth at its various phases. The Moon goes through the complete cycle of phases every 27.3 days.
The Inner Planets (5 of 5)
Meteors and meteorites
The Sun is also orbited by millions of minute particles called meteoroids. They are about the size of grains of sand and are therefore too small to be seen in space. When a meteoroid enters the Earth's atmosphere, it is heated by friction and destroyed. As this happens, the air glows, producing the effect we see as a meteor or `shooting star'. Over 40 million meteoroids enter the atmosphere every day.
Larger bodies may survive and reach the Earth intact. These are called meteorites. Sometimes a meteorite may produce a crater - one of the best preserved of such craters is in Arizona and measures 1265 m (4150 ft) in diameter.
THE INNER PLANETS:
BASIC STATISTICS
MERCURY
Diameter: 4880 km / 3032 mi
(0.38 x Earth)
Mass: 0.555 x Earth
Average temperature: 420 deg C / 790 deg F (day) -180 deg C /-290 deg F (night)
Rotation period: 59 Earth days *
Tilt of axis: 2deg
Average distance from Sun:
57 900 000 km / 36 000 000 mi
(0.387 x Earth)
Length of year: 88 Earth days
Number of known moons: none
* The length of a solar day (sunrise to sunrise) on Mercury is 176 Earth days.
VENUS
Diameter: 12 103 km / 7520 mi
(0.95 x Earth)
Mass: 0.81 x Earth
Average temperature: 464 deg C / 867 deg F
Rotation period: 243 Earth days *
Tilt of axis: 178deg
Average distance from Sun: 108 200 000 km / 67 200 000 mi
(0.723 x Earth)
Length of year: 225 Earth days
Number of known moons: none
* The length of a solar day on Venus is 116 Earth days.
EARTH
Diameter: 12 756 km / 7921 mi
Mass: 1.00 x Earth
Average temperature: 15 deg C / 59 deg F
Rotation period: 24 h
Tilt of axis: 23.5deg
Average distance from Sun: 149 600 000 km / 92 900 000 mi
Length of year: 365 days
Number of known moons: 1
Diameter: 6780 km / 4213 mi
(0.53 x Earth)
Mass: 0.11 x Earth
Average temperature: -53 deg C /-63 deg F
Rotation period: 24 h 37 m
Tilt of axis: 24deg
Average distance from Sun: 227 900 000 km / 141 500 000 mi
(1.523 x Earth)
Length of year: 687 Earth days
Number of known moons: 2
* TIME
* THE SUN AND THE SOLAR SYSTEM
* THE OUTER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
* THE EARTH'S STRUCTURE AND ATMOSPHERE
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ftsTitleOverride
The Outer Planets (page 1)
ftsTitle
Volcanic plumes on Io, one of Jupiter's moons. These various views were taken using different filters. Io is continuously rocked by voilent volcanic activity, and matter may be thrown up to 250 km (155 mi) above the surface.
The Outer Planets (1 of 6)
The outer planets are very different from the inner planets. They are much further away from the Sun and, with the exception of Pluto, are much larger than the inner planets. Jupiter, Saturn, Uranus and Neptune are giant `gas' planets without solid surfaces.
Much of our knowledge of the outer planets has been gained from the American space probes Pioneer 10, Pioneer 11, Voyager 1 and Voyager 2, the last of which visited Jupiter, Saturn, Uranus and Neptune in turn from 1979 to 1989.
Jupiter
Jupiter is the largest planet in the Solar System. It appears very bright to the naked eye and can outshine everything in the sky except the Sun, the Moon, Venus and (very occasionally) Mars.
Through a telescope, several belts or bands can be seen in Jupiter's atmosphere. The planet's rapid rotation rate of 9 hours 55 minutes throws the equator outwards, producing a distinct `squashed' appearance. One of the most prominent features is the Great Red Spot, which may have been seen as long ago as 1664. Modern observations of Jupiter have been made by four spacecraft, Pioneers 10 and 11 and Voyagers 1 and 2; pictures from these show that the Red Spot is a whirling storm in Jupiter's atmosphere.
At the center of Jupiter is a rocky core. Above this are layers of metallic hydrogen (i.e. so cold that it is solid) and liquid hydrogen. Jupiter's upper atmosphere is mainly hydrogen and helium, but also contains small amounts of many different gases, including methane, ammonia, ethane, acetylene, water vapor, phosphine, carbon monoxide and germanium tetrahydride.
Jupiter is known to have at least 16 satellites. Four of these, Io, Europa, Ganymede and Callisto, were seen by Galileo with his early telescope in 1610 and they are often called the `Galilean Satellites' in his honor. The remaining satellites are small objects - the outermost group revolving around the planet in a retrograde direction. They are probably asteroids that were captured by Jupiter's immense gravitational pull. Voyager 1 also discovered a very faint ring around Jupiter (see below), which Voyager 2 was able to study in more detail.
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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The Outer Planets (page 2)
ftsTitle
Saturn and its rings, photographed from a distance of 13.9 million km (8.6 million mi) as Voyager 2 was approaching the planet at a speed of about 42 000 km/h (26 000 mph). The shadow cast by the rings can clearly be seen in the planet's equatorial region.
The Outer Planets (2 of 6)
Saturn
The next planet out from the Sun is Saturn, with its magnificent system of rings. In 1610 Galileo became the first person to look at Saturn through a telescope. He saw the rings, but could not understand what they were, at first thinking that Saturn was a `triple planet'. It was not until 1659 that the Dutch physicist Christiaan Huygens (1629-95) realized Saturn's true nature. Most of our current information about Saturn has come from the three spacecraft that have flown by the planet: Pioneer 11 in 1979, Voyager 1 in 1980 and Voyager 2 in 1981.
If we could cut a slice out of Saturn we would see a small rocky core at the center. Above this is a region of metallic hydrogen, followed by a deep ocean of liquid hydrogen. The outside layer of the planet is made of hydrogen gas, together with some helium. Tiny amounts of the gases methane, ammonia, ethane and phosphine have also been detected. Saturn has at least 18 satellites, including Titan, which has a dense nitrogen atmosphere.
Uranus
Uranus was discovered in March 1781 by Sir William Herschel (1738-1822), an amateur astronomer born in Germany but living in England. Herschel was making a routine survey of the sky when he came across an object that did not look like a star. At first he thought he had found a comet. The new object was watched care fully over the following months so that its orbit could be calculated. Once this was done, astronomers realized that Herschel had found a new planet.
When the sky is very dark and very clear, Uranus can just be seen with the naked eye. However, the planet is always extremely faint and the observer has to know exactly where to look. A telescope will show Uranus as a small disc. Most of our information about Uranus was sent back by Voyager 2, which flew past the planet in January 1986.
One of the strangest things about Uranus is that it orbits the Sun tipped on its side, which means that the calendar on the planet must be very odd indeed. At present the planet's north pole is pointing towards the Sun and anyone above the north pole of Uranus would have been in sunlight since 1966 and will not see the Sun set until 2007. This will be followed by 42 years of darkness while the south pole points towards the Sun. Although the south pole had been in darkness for 20 years, Voyager 2 found that it is slightly warmer than the north pole - which must give rise to some very peculiar weather.
Uranus has an atmosphere of hydrogen and helium, which surrounds a layer of water, methane and ammonia ices. At the center of Uranus is a rocky core. Before Voyager 2 flew past the planet, Uranus was known to have 5 satellites. Voyager found 10 more bringing the total to 15.
Uranus also has a system of rings. These were discovered in 1977 when the planet passed between a star and the Earth. Astronomers saw the light from the star flash on and off as each of the rings passed in front of it. Nine rings were eventually detected from Earth, but Voyager found two more.
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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ftsTitleOverride
The Outer Planets (page 3)
ftsTitle
Neptune and its Great Dark Spot photographed through colored filters by Voyager 2 in August 1989 from a distance of 6.1 million km (3.8 million mi). The Great Dark Spot is a giant storm system the size of the Earth.
The Outer Planets (3 of 6)
Neptune
Neptune is so distant that it can never be seen with the naked eye and even a small telescope will only show it as a tiny disc. Larger telescopes show that the planet is bluish-green in color, but very few markings can be seen. Our first good look at Neptune came in August 1989, when the Voyager 2 spacecraft flew past the planet. Voyager revealed Neptune's most prominent feature, the Great Dark Spot. This is a massive storm system in the planet's atmosphere and is similar to Jupiter's Great Red Spot. Small wispy white clouds of methane ice can also be seen circling the planet.
The outside layer of Neptune is made of the gases hydrogen and helium. Beneath this comes a layer of ice. The ice is made of frozen methane and ammonia, as well as frozen water. At the center of the planet is a rocky core.
Neptune was discovered thanks to the mathematical calculations of two men. They were an Englishman, John Couch Adams (1819-92), and Urbain Le Verrier (1811-77) from France. Adams and Le Verrier had been looking at the movements of Uranus and had noticed that it was behaving in a very strange manner. Sometimes it would speed up and move quickly along its orbit, whereas at other times it would slow down. The two men both thought that Uranus was being pulled by the gravity of another planet, so they set out to calculate where the `hidden planet' must be. Both men worked on their own, but they eventually arrived at the same answer. Le Verrier sent details of his calculations to the observatory in Berlin. The German astronomers Johann Galle (1812-1910) and Heinrich d'Arrest (1822-75) looked for the planet in September 1846 and found it almost immediately.
Neptune is now known to have eight satellites and a system of four rings Triton is the largest of Neptune's satellites, with a diameter of 2270 km (1410 mi) - slightly smaller than that of the Earth's Moon. Triton has an atmosphere and its surface is covered with a frozen mixture of nitrogen and methane. the planet's north pole is pointing towards the Sun and anyone above the north pole of Uranus would have been in sunlight since 1966 and will not see the Sun set until 2007. This will be followed by 42 years of darkness while the south pole points towards the Sun. Although the south pole had been in darkness for 20 years, Voyager 2 found that it is slightly warmer than the north pole - which must give rise to some very peculiar weather.
Uranus has an atmosphere of hydrogen and helium, which surrounds a layer of water, methane and ammonia ices. At the center of Uranus is a rocky core. Before Voyager 2 flew past the planet, Uranus was known to have 5 satellites. Voyager found 10 more bringing the total to 15.
Uranus also has a system of rings. These were discovered in 1977 when the planet passed between a star and the Earth. Astronomers saw the light from the star flash on and off as each of the rings passed in front of it. Nine rings were eventually detected from Earth, but Voyager found two more.
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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p014-4
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The Outer Planets (page 4)
ftsTitle
Triton, Neptune's largest moon, photographed by Voyager 2. At the bottom is the south polar cap.
The Outer Planets (4 of 6)
Pluto
Although Pluto is normally the most distant member of the Solar System, it has a highly eccentric orbit that sometimes carries it inside that of Neptune. Thus for 20 years from 1979 to 1999 Neptune is temporarily the most distant planet.
Pluto is a very small, icy planet. Its satellite, Charon, is very large by comparison, having 10% of Pluto's mass - compared to the Moon, which has only 1.2% of the Earth's mass. Together, Pluto and Charon virtually form a twin-planet system. Little is known about the nature of Pluto, but evidence of a thin methane atmosphere has been detected.
Planetary rings
The gas planets Jupiter, Saturn, Uranus and Neptune all possess systems of rings. Of these, Saturn's is by far the most spectacular.
Rings are made from millions of small particles of ice and dust. Rings can have diameters of thousands of kilometers, but they are typically less than one kilometer thick. The mechanics of ring systems are not fully understood, but, especially in the case of Saturn, the ring particles appear to be kept in place by tiny `shepherd' satellites.
strange manner. Sometimes it would speed up and move quickly along its orbit, whereas at other times it would slow down. The two men both thought that Uranus was being pulled by the gravity of another planet, so they set out to calculate where the `hidden planet' must be. Both men worked on their own, but they eventually arrived at the same answer. Le Verrier sent details of his calculations to the observatory in Berlin. The German astronomers Johann Galle (1812-1910) and Heinrich d'Arrest (1822-75) looked for the planet in September 1846 and found it almost immediately.
Neptune is now known to have eight satellites and a system of four rings Triton is the largest of Neptune's satellites, with a diameter of 2270 km (1410 mi) - slightly smaller than that of the Earth's Moon. Triton has an atmosphere and its surface is covered with a frozen mixture of nitrogen and methane.
Pluto
Although Pluto is normally the most distant member of the Solar System, it has a highly eccentric orbit that sometimes carries it inside that of Neptune. Thus for 20 years from 1979 to 1999 Neptune is temporarily the most distant planet.
Pluto is a very small, icy planet. Its satellite, Charon, is very large by comparison, having 10% of Pluto's mass - compared to the Moon, which has only 1.2% of the Earth's mass. Together, Pluto and Charon virtually form a twin-planet system. Little is known about the nature of Pluto, but evidence of a thin methane atmosphere has been detected.
Planetary rings
The gas planets Jupiter, Saturn, Uranus and Neptune all possess systems of rings. Of these, Saturn's is by far the most spectacular.
Rings are made from millions of small particles of ice and dust. Rings can have diameters of thousands of kilometers, but they are typically less than one kilometer thick. The mechanics of ring systems are not fully understood, but, especially in the case of Saturn, the ring particles appear to be kept in place by tiny `shepherd' satellites.
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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ftsTitleOverride
The Outer Planets (page 5)
ftsTitle
Jupiter rotation as seen from one of its satellite, Io. Io is one of 16 satellites and one of four large enough to be seen by Gallileo in 1610. The red spot is a whirling wind storm in Jupiter's atmosphere.
The Outer Planets (5 of 6)
THE OUTER PLANETS: BASIC STATISTICS
JUPITER
Diameter: 142 800 km / 88 700 mi
(11.0 x Earth)
Mass: 318 x Earth
Average temperature: -150 deg C /-238 deg F
Rotation period: 9 h 55 m
Tilt of axis: 3deg
Average distance from Sun: 778 300 000 km / 483 300 000 mi
(5.202 x Earth)
Length of year: 11.9 Earth years
Number of known moons: 16
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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Jupiter rotation
**++*
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ftsTitleOverride
The Outer Planets (page 6)
ftsTitle
An approach of Saturn and its rings that shows the detail that in 1610 Gallileo could not see with his telescope. Gallileo mistakenly believed Saturn to be three planetary bodies. Here the rings are clearly visible and astoundingly beautiful.
The Outer Planets (6 of 6)
SATURN
Diameter: 120 000 km / 74 000 mi
(9.41 x Earth)
Mass: 95 x Earth
Average temperature: -180 deg C /-292 deg F
Rotation period: 10 h 40 m
Tilt of axis: 27deg
Average distance from Sun: 1 427 000 000 km / 886 000 000 mi
(9.538 x Earth)
Length of year: 29.5 Earth years
Number of known moons: 18
URANUS
Diameter: 52 400 km / 32 500 mi
(4.11 x Earth)
Mass: 15 x Earth
Average temperature: -210 deg C /-346 deg F
Rotation period: 17 h 14 m
Tilt of axis: 98deg
Average distance from Sun: 2 869 600 000 km / 1 782 000 000 mi (19.181 x Earth)
Length of year: 84.0 Earth years
Number of known moons: 15
NEPTUNE
Diameter: 49 400 km / 30 700 mi
(3.87 x Earth)
Mass: 17 x Earth
Average temperature: -225 deg C /-373 deg F
Rotation period: 16 h 3 m
Tilt of axis: 29deg
Average distance from Sun:
4 496 700 000 km / 2 792 500 000 mi
(30.058 x Earth)
Length of year: 164.8 Earth years
Number of known moons: 8
PLUTO
Diameter: 1100 km / 680 mi
(0.09 x Earth)
Mass: 0.002 x Earth
Average temperature: -220 deg C /-364 deg F
Rotation period: 6 d 9 h
Tilt of axis: 50deg
Average distance from Sun: 5 900 000 000 km / 3 700 000 000 mi
(39.44 x Earth)
Length of year: 248 Earth years
Number of known moons: 1
* THE SUN AND THE SOLAR SYSTEM
* THE INNER PLANETS
* THE HISTORY OF ASTRONOMY
* SPACE EXPLORATION
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Saturn approach
**++*
p016-1
ftsTitleOverride
The History of Astronomy (page 1)
ftsTitle
Tycho Brahe (1546-1601), the great Danish astronomer, in his observatory on the island of Hven. Tycho was the last of the great astronomers to rely on the naked eye, and he improved every previous astronomical observation. Among other things he calculated the length of the year to an accuracy of less than one second.
The History of Astronomy (1 of 3)
Mankind has always been fascinated by the stars. The history of astronomy stretches back in time to the beginnings of civilization. Prehistoric structures such as the stone circles at Stonehenge and Avebury in England are aligned on astronomical principles, so the civilizations that built them must have had some astronomical knowledge. The earliest recorded astronomical observations were made by the Chinese, the Egyptians and the Babylonians, who, however, made no real effort to interpret what they saw.
Many important astronomical observations were made by the Greeks during a period of roughly 800 years from 600 BC to AD 200. No major breakthroughs were then made until the 16th century.
The early astronomers
Thales of Miletus (640-560 BC) was the first of the great Greek philosophers. Though he believed the Earth to be flat, he initiated serious astronomical observation. Thales is accredited with predicting an eclipse in 585 BC. A solar eclipse occurs when the Moon passes between the Earth and the Sun, and a lunar eclipse is caused by the Moon passing into the Earth's shadow.
The Greek astronomer Aristarchus of Samos (c. 310-250 BC) was one of the first people to state that the Earth turns on its axis and orbits the Sun. He attempted to measure the relative distances of the Sun and Moon, arriving at a value of between 18 and 20. Modern measurements have shown that the true value is about 390.
Eratosthenes (276-196 BC) was a librarian in the Greek city of Alexandria in Egypt. He devised an experiment for measuring the circumference of the Earth, based on the observation that the Sun shone directly down a well in Aswan at midday on Midsummer's Day. Eratosthenes found that the angle of the Sun at the same time in Alexandria, 800 km (497 mi) north of Aswan, was about 1/50 of a circle. He therefore deduced that the distance from Alexandria to Aswan was 1/50 of the circumference of the Earth, which he calculated to be 40 000 km (24 840 mi). This is in close agreement with the modern value of 40 007 km (24 844 mi).
Claudius Ptolemaeus or Ptolemy (c. AD 120-180) was an Alexandrian scholar, whose Almagest was regarded as a standard text until the 16th century. In the Ptolemaic system, the Earth was stationary at the center of the Universe, with the Sun, planets, Moon and stars revolving around it; their paths were small circles, whose centers moved along larger circles.
Nikolaus Copernicus (1473-1543) was a Polish astronomer who argued that the Sun, not the Earth, is at the center of the Universe. His theories were published in 1543 in the book, Concerning the Movement of the Heavenly Bodies. Al though Copernicus was correct in believing that the planets orbited the Sun and not the Earth, he mistakenly thought that the Sun marked the center of the Universe and that the planets moved in perfect circles.
The Danish astronomer Tycho Brahe (1546-1601) had intended to become a lawyer, but changed his mind upon witnessing a solar eclipse in 1560. In 1572 he witnessed the supernova in Cassiopeia and wrote a book about it. From 1576 to 1596 he worked at Uraniborg on the Danish island of Hven, compiling an accurate star catalogue.
The German mathematician Johannes Kepler (1571-1630) was the first person to show that the planets move around the Sun in elliptical orbits. He was originally an assistant to Tycho Brahe and his work was based on Tycho's observations. Kepler devised three important laws of planetary motion that are still in use today:
1. The motion of a planet around the Sun describes an ellipse, with the Sun at one of the foci (i.e. centers).
2. A line joining the center of the Sun with the center of a planet sweeps out equal areas in equal times.
3. The square of a planet's orbital period is proportional to the cube of its mean distance from the Sun.
The Italian Galileo Galilei (1564-1642) developed the astronomical telescope and used it to discover the four main satellites of Jupiter. He also observed the stars of the Milky Way and craters on the Moon, and found that the planet Venus showed phases. Galileo's book, Two Chief Systems of the World, published in 1632, backed up the theory of a Sun-centered Solar System, but was banned by the Roman Catholic Church as this theory went against Church dogma, which stated that the Earth was the center of the Universe. He was forced to retract his views by the Inquisition and placed under house arrest.
The laws of motion derived by the English scientist Sir Isaac Newton (1642-1727) are fundamental to our understanding of the physical world and are also of enormous importance to astronomers. Newton's discovery of gravity is also essential to all subsequent astronomical theory.
Several later astronomers are mentioned on previous pages in relation to their particular discoveries.
* SPACE EXPLORATION
* NEWTON AND FORCE
* QUANTUM THEORY AND RELATIVITY
* OPTICS
* THE HISTORY OF SCIENCE
* SEEING THE INVISIBLE
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The History of Astronomy (page 2)
ftsTitle
An Islamic astrolabe from the 10th century. It was used for casting horoscopes and for determining the direction of Mecca.
The History of Astronomy (2 of 3)
Early instruments and observations
One of the first astronomical instruments was the astrolabe, a circular disc marked off in degrees along its rim, which was used to measure the altitudes of stars and planets.
The quadrant was another device for measuring the positions of celestial objects. As its name implies, the quadrant is a 90 deg arc, together with a pointer for sighting stars. Before the advent of the telescope, astronomers would use astrolabes and quadrants for measuring stellar positions.
One of the earliest good star charts was the Uranometria, published by the Bavarian Johann Bayer (1572-1625) in 1603. Bayer introduced the system of identifying individual stars within a constellation by letters of the Greek alpha bet.
The Lovell Telescope at Jodrell Bank (near Manchester, England) was the world's first giant-dish radio telescope. Since its construction in 1957, it has provided a vast amount of data from the furthest reaches of space.
The History of Astronom (3 of 3)
Optical telescopes
The optical telescope was invented by the Dutch spectacle maker, Hans Lippershey (died c. 1619), but its first practical form was developed by Galileo. There are two principal types: refracting telescopes, which employ lenses, and reflecting telescopes, which use mirrors.
The two largest optical telescopes are at Zelenchuskaya in the USSR, with a 6.0 m (19 ft 8 in) diameter mirror, and the Hale Telescope at Mount Palomar, California, which has a 5.0 m (16 ft 5 in) diameter mirror. Future developments are likely to include telescopes employing several mirrors or a multi-faceted mirror controlled by computer. These would be able to detect far fainter objects than is possible with conventional optical telescopes.
Radio astronomy
In 1931 the American radio engineer Karl Jansky (1905-1949) used an improvised aerial to detect radio emissions from the Milky Way. This marked the beginnings of radio astronomy which has made possible such exciting developments as the discovery of quasars.
The most famous steerable radio telescope is the 75 m (250 ft) diameter Lovell Telescope at Jodrell Bank, England. The world's largest radio telescope, the 300 m (1000 ft) diameter dish at Arecibo in Puerto Rico, is built into a natural hollow in the ground.
Many astronomical bodies also emit gamma rays and X-rays, which can give us information about the physical properties of those bodies. However, these radiations are difficult to study, because they do not pass through the Earth's atmosphere. Most observations are therefore confined to spacecraft.
Modern astronomy now covers the whole of the electromagnetic spectrum, and observations are also being made at infrared, ultraviolet and microwave wave lengths.
Is There Life Elsewhere?
The acronym SETI stands for Search for Extra-Terrestrial Intelligence. The search began in earnest with Project Ozma in 1960, when a 26 m (85 ft) diameter radio telescope at Green Bank, West Virginia, USA, was used to search for signals from the nearby stars Tau Ceti and Epsilon Eridani. Nothing was found. Since Project Ozma, several further attempts have been made to detect radio signals produced by other intelligent civilizations, but without success.
It is possible to represent the probable number of advanced civilizations in the Galaxy by a mathematical equation. This was originally done by Frank Drake of Cornell University in the USA, and the equation thus bears his name.
Drake's equation states that the number (N) of advanced technical civilizations in the Galaxy can be expressed as follows: N = N* x fp x ne x fl x fi x fc x fL where: N* is the number of stars in the Galaxy; fp is the fraction of stars that have planetary systems; ne is the number of Earth-like planets in every star system (again, this will be a fraction, in that it is thought that most stars do not have Earth-like planets); fl is the fraction of suitable planets on which life actually arises; fi is the fraction of inhabited planets on which intelligent life evolves; fc is the fraction of planets inhabited by intelligent beings that attempt to communicate; and fL is the fraction of the planet's life for which the civilization survives.
A major problem with the Drake equation is that some of the terms are very difficult to estimate. Different estimates can yield wildly different results. These range from mankind being the only intelligent civilization in the Galaxy, to our being one among many millions of intelligent Galactic life forms.
The Lunar Roving Vehicle or `moon buggy' in action among the foot hills of the Lunar Apennines. The LRV was taken to the Moon by Apollo 15 in 1971
Space Exploration (1 of 5)
The development of space technology has been remarkably rapid. Less than 10 years after Yuri Gagarin became the first person in space, Neil Armstrong set foot upon the Moon. During the 21st century, space travel is likely to become routine. Permanent space stations will operate in Earth orbit, bases will be built on the Moon and man will begin to explore the planet Mars.
The theory of rocketry was developed at the beginning of the 20th century by the Russian physicist Konstantin Tsiolkovsky (1857-1935). He produced designs for multistage liquid-fueled rockets decades before such vehicles were actually built. Tsiolkovsky also wrote about space suits, satellites and colonizing the Solar system. The inscription on his tombstone reads `Mankind will not remain tied to Earth forever.'
The pioneers of rocketry
The first successful liquid-propellant rocket was launched by the American physicist Robert Goddard (1882-1945) in 1926. By the mid-1930s, Goddard had perfected rockets that could travel to an altitude of several kilometers.
Born in Germany, Wernher von Braun (1912-77) helped to develop the V-2 rocket during World War II. He surrendered to the Americans in 1945 and led the team that launched Explorer 1 in 1958, the first American artificial satellite. He then turned his attention to the Apollo program, which landed a man on the Moon in 1969.
The first artificial satellites
The first object successfully launched into space was the Soviet Sputnik 1, which lifted off on 4 October 1957. The satellite measured temperatures and electron densities, before burning up as it re-entered the atmosphere on 4 January 1958.
The dog Laika became the first living creature in space following her launch aboard Sputnik 2 on 3 November 1957. Laika spent 10 days in orbit, but died when her oxygen supply was exhausted.
Man in space
The era of manned spaceflight began on 12 April 1961, when the Soviet cosmonaut Yuri Gagarin (1934-68) was launched aboard Vostok 1. His spacecraft completed a single orbit of the Earth in a flight lasting 90 minutes. Gagarin landed by parachute, having been ejected from the capsule during its descent.
The USA became the second country to put a man into orbit when John Glenn (1921- ) was launched aboard his Friendship 7 capsule on 20 February 1962.
Following the success of Gagarin's flight, President Kennedy announced that the USA intended to place a man on the Moon by the end of the decade. Thus the Apollo program was born, which used the massive Saturn V rockets. The project reached a successful climax on 20 July 1969 when Neil Armstrong (1930- ) and Edwin `Buzz' Aldrin (1930- ) landed their Apollo 11 lunar module Eagle at the Sea of Tranquillity. The Apollo Moon-landing program ended in 1972, five more successful missions having landed 10 more men on the Moon.
Space stations
Space stations are primarily used for scientific research, but have also been used to test the ability of humans to endure long periods of weightlessness in preparation for interplanetary flight. The first space station was the Soviet Union's Salyut 1, which was launched on 19 April 1971. This was followed by six further stations in the Salyut series, before the larger Mir space station was launched on 20 February 1986. Two cosmonauts spent a year aboard Mir from 21 December 1987 to 21 December 1988.
The American Skylab space station was launched on 14 May 1973 and was subsequently visited by three crews, the last of which stayed for 84 days.
* THE INNER PLANETS
* THE OUTER PLANETS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENGINES
* SEEING THE INVISIBLE
* AIRCRAFT
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Space Exploration (page 2)
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The Saturn V rocket is launched, the vehicle that carried the first astronauts to the moon. This NASA program was called Apollo, and ended after successfully landing a total 10 men on the moon.
Space Exploration (2 of 5)
Man in space
The era of manned spaceflight began on 12 April 1961, when the Soviet cosmonaut Yuri Gagarin (1934-68) was launched aboard Vostok 1. His spacecraft completed a single orbit of the Earth in a flight lasting 90 minutes. Gagarin landed by parachute, having been ejected from the capsule during its descent.
The USA became the second country to put a man into orbit when John Glenn (1921- ) was launched aboard his Friendship 7 capsule on 20 February 1962.
Following the success of Gagarin's flight, President Kennedy announced that the USA intended to place a man on the Moon by the end of the decade. Thus the Apollo program was born, which used the massive Saturn V rockets. The project reached a successful climax on 20 July 1969 when Neil Armstrong (1930- ) and Edwin `Buzz' Aldrin (1930- ) landed their Apollo 11 lunar module Eagle at the Sea of Tranquillity. The Apollo Moon-landing program ended in 1972, five more successful missions having landed 10 more men on the Moon.
Space stations
Space stations are primarily used for scientific research, but have also been used to test the ability of humans to endure long periods of weightlessness in preparation for interplanetary flight. The first space station was the Soviet Union's Salyut 1, which was launched on 19 April 1971. This was followed by six further stations in the Salyut series, before the larger Mir space station was launched on 20 February 1986. Two cosmonauts spent a year aboard Mir from 21 December 1987 to 21 December 1988.
The American Skylab space station was launched on 14 May 1973 and was subsequently visited by three crews, the last of which stayed for 84 days.
* THE INNER PLANETS
* THE OUTER PLANETS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENGINES
* SEEING THE INVISIBLE
* AIRCRAFT
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Saturn V rocket is launched
**++*
p018-3
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Space Exploration (page 3)
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Floating in space. On 7 February 1984 Captain Bruce McCandless left the US space shuttle Challenger to make the first untethered `float' in space.
Space Exploration (3 of 5)
Space shuttles
Unlike earlier spacecraft, space shuttles are reusable. The main vehicle is winged like an airplane, but is launched into orbit by booster rockets that are then discarded. The main vehicle can subsequently land like a conventional glider.
The American space shuttle made its debut on 12 April 1981, with the launch of the orbiter Columbia. The program came to an abrupt halt on 28 January 1986, 73 seconds after the 25th shuttle launch. A leak from a rocket booster caused an explosion that destroyed the Challenger orbiter and killed its crew of seven. Shuttle operations resumed on 29 September 1988, when Discovery was launched on the 26th shuttle mission.
The Soviet Union's reusable spacecraft is the VKK, which stands for Vosdushno Kosmicheski Korabl (`airborne space craft'). The first VKK to be launched was Buran (`snowstorm'), which completed two orbits of the Earth on 15 November 1988. Although designed to carry a crew, Buran's first flight was unmanned.
Unmanned probes
Much of our knowledge of the Solar System has come from unmanned probes. These have now returned data from every known planet except Pluto. Spacecraft have landed on Venus and Mars and probes are either planned, or on course, to enter orbit around Jupiter and Saturn.
* THE INNER PLANETS
* THE OUTER PLANETS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENGINES
* SEEING THE INVISIBLE
* AIRCRAFT
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Space Exploration (page 4)
ftsTitle
The space shuttle is silhouetted by the rotating Earth, while an astronaut does an EVA (a space walk) outside to build portions of a space station.
Space Exploration (4 of 5)
The future
The rocket is the space launcher of the 20th century, but the spaceplane will be the launcher of the 21st century. Space planes will be fully reusable and able to take off as well as land from runways like conventional airplanes. One such space plane was the British Hotol project, abandoned in 1992 owing to lack of funding. Other designs for spaceplanes are being studied by the USA (the X-30 project), Japan and Germany.
The early years of the 21st century are likely to see a return to the Moon. How-ever, unlike the Apollo missions, the next time human beings venture to the Moon they will be equipped to stay and establish a permanent base. Following the return to the Moon, a manned flight to Mars is likely to be undertaken. The collapse of the Soviet economy and the break up of the USSR (1991) raised doubts concerning further projects, although Russian missions have taken place.
* THE INNER PLANETS
* THE OUTER PLANETS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENGINES
* SEEING THE INVISIBLE
* AIRCRAFT
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Space Exploration (page 5)
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Space Exploration (5 of 5)
The Uses of Space Technology
Many artificial satellites are used for communications. Comsats, as they are sometimes known, are often placed in geostationary orbit, 36900 km (22900 mi) above the equator. Satellites in this orbit travel at the same speed as the Earth rotates and thus appear to remain fixed in the sky.
Weather satellites operate either in geostationary orbit, or polar orbit. A polar orbit carries a satellite over the North and South Poles, passing over a different strip of the Earth on every orbit. Such satellites can survey the entire planet every 24 hours.
Earth-resources satellites, such as those in the US Landsat series, can be used to prospect for new mineral resources, check the spread of diseases in crops and monitor pollution.
Space provides a good vantage point for astronomers, whose instruments can examine the universe from above the distortions of the atmosphere.
The Hubble Space Telescope, launched
by space shuttle in 1990, is designed to detect fainter objects than any telescope on the ground, although it has suffered technical problems.
The movement of a space station cancels out the effect of gravity, making astronauts and their equipment weightless. These conditions can be used for manufacturing new materials such as perfect crystals.
The military also make use of space. Satellites can detect ground detail far more effectively than conventional aircraft. Spy satellites can detect objects as small as individual vehicles and
people. The US Strategic Defense Initiative ('Star Wars') program
aimed to provide a space-based laser defense system against nuclear weapons. However, many scientists doubted the feasibility and desirability of the project and since the end of the Cold War it has been abandoned.
* THE INNER PLANETS
* THE OUTER PLANETS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENGINES
* SEEING THE INVISIBLE
* AIRCRAFT
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Motion and Force (page 1)
ftsTitle
1. Displacement. Anna walks from A to X, then to Y, then to B. AB is her displacement, and the arrow shows the direction of the displacement.
Motion and Force (1 of 5)
Physics is the study of the basic laws that govern matter. Mechanics is the branch of physics that describes the movement or motion of objects, ranging in scale from a planet to the smallest particle within an atom. Sir Isaac Newton developed a theory of mechanics that has proved highly successful in describing most types of motion, and his work has been acclaimed as one of the greatest advances in the history of science.
The Newtonian approach, although valid for velocities and dimensions within normal experience, has been shown to fail for velocities approaching the speed of light and for dimensions on a subatomic scale. Newton's discoveries are therefore considered to be a special case within a more general theory.
* THE HISTORY OF ASTRONOMY
* FORCES AFFECTING SOLIDS AND FLUIDS
* THERMODYNAMICS
* QUANTUM THEORY AND RELATIVITY
* THE HISTORY OF SCIENCE
* FUNCTIONS, GRAPHS AND CHANGE
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ids and 6
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Motion and Force (page 2)
ftsTitle
2. Instantaneous velocity v can be expressed as a horizontal component and a vertical component. The velocity at point P is v. It has a horizontal component v cos q and a vertical component v sin q.
Motion and Force (2 of 5)
Motion
When a body is in motion it can be thought of as moving in space and time. If the body moves from one position to another, the straight line joining its starting point to its finishing point is its displacement. This has both magnitude and direction, and is therefore said to be a vector quantity. The motion is linear.
The rate at which a body moves, in a straight line or rectilinearly, is its velocity. Again, this has magnitude and direction and is a vector quantity. In contrast, the speed, which has magnitude, but is not considered to be in any particular direction, is a scalar quantity. The average velocity of the body during this rectilinear motion is defined as the change in displacement divided by the total time taken. Its dimensions are therefore length divided by time, and are given in meters per second (m s-1). The instantaneous velocity (the velocity of any instant) at any point is the rate of change of velocity at that point.
If the body moves with a changing velocity, then the rate of change of the velocity is the acceleration. This is defined as the change in velocity in a given time interval. Its dimensions are velocity divided by time, and are given in metros per second per second (m s-2). When a body moves with uniform acceleration (uniformly accelerated motion), the displacement, velocity and acceleration are related. These relationships are described in the kinematic equations, sometimes called the laws of uniformly accelerated motion. Kinematics is the study of bodies in motion, ignoring masses and forces.
To use the equations to solve a problem in kinematics it is necessary to identify the information given in the problem, then to identify which of the four equations can be manipulated to give the answer required.
Galileo Galilei (1564-1642) was an Italian physicist and astronomer who investigated the motion of objects falling freely in air. He believed that all objects falling freely towards the Earth have the same downward acceleration. This is called the acceleration due to gravity or the gravitational acceleration. Near the surface of the Earth it is 9.80 m s-2, but there are small variations in its value depending upon latitude and elevation. In the idealized situation, air resistance is neglected, although in a practical experiment it would have to be considered. In a demonstration on the Moon in August 1971, an American astronaut showed that, under conditions where air resistance was negligible, a feather and a hammer, released at the same time from the same height, would fall side by side. They landed on the lunar surface together.
When real motion is considered, both the magnitude and the direction of the velocity have to be investigated. A golf ball, hit upwards, will return to the ground. During flight its velocity will change in both magnitude and direction. In this case, instead of average velocity, the instantaneous velocities have to be evaluated. The velocity can, at any instant, be considered to be acting in two directions, vertical and horizontal. Then the velocity at that instant can be separated into a vertical and a horizontal component. Each component can be considered as being uniformly accelerated rectilinear motion, so the kinematic equations can be applied in each direction. Then the instantaneous velocity and position at any point of the flight can be calculated.
* THE HISTORY OF ASTRONOMY
* FORCES AFFECTING SOLIDS AND FLUIDS
* THERMODYNAMICS
* QUANTUM THEORY AND RELATIVITY
* THE HISTORY OF SCIENCE
* FUNCTIONS, GRAPHS AND CHANGE
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Motion and Force (page 3)
ftsTitle
3. Centripetal acceleration. At point P the body is moving with instantaneous velocity v. The centripetal acceleration along PO is v2/r2 (where r is the radius of the circle), and this force prevents the body from moving in a straight line along PV.
Motion and Force (3 of 5)
Circular motion
If a body moves in a circular path at constant speed its direction of motion (and therefore its velocity) will be changing continuously. Since the velocity is changing, the body must have acceleration, which is also changing continuously. Thus the laws of uniformly accelerated motion do not apply. The acceleration of a body moving in a circular path is called the centripetal (`center-seeking') acceleration. This is directed inward, towards the center of the circle.
* THE HISTORY OF ASTRONOMY
* FORCES AFFECTING SOLIDS AND FLUIDS
* THERMODYNAMICS
* QUANTUM THEORY AND RELATIVITY
* THE HISTORY OF SCIENCE
* FUNCTIONS, GRAPHS AND CHANGE
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p020-4
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Motion and Force (page 4)
ftsTitle
A gyroscope in low gravity demonstrates the laws of physics outside the force of gravity. It clearly shows Newton's first law which states that a body will remain at rest or at a constant speed unless it is acted upon by another force. The gyroscope tends to move in a single direction until the astronaut nudges it with the string.
Motion and Force (4 of 5)
Newton's laws of motion
Newton's laws of motion state relationships between the acceleration of a body and the forces acting on it. A force is something that causes a change in the rate of change of velocity of an object.
Newton's first law states that a body will remain at rest or traveling in a straight line at constant speed unless it is acted upon by an external force. Notice that the force has to be an external one. In general, a body does not exert a force upon itself.
The tendency of a body to remain at rest or moving with constant velocity is called the inertia of the body. The inertia is related to the mass, which is the amount of substance in the body. The unit of mass is the kilogram (kg).
Newton's second law states that the resultant force exerted on a body is directly proportional to the acceleration produced by the force. The unit of force is the newton (N), which is defined as the force that, acting on a body of mass 1 kg, produces an acceleration of 1 m s-2.
The mass of a body is often confused with its weight. The mass is the amount of matter in the body, whereas the weight is the gravitational force acting on the body, and varies with location. The unit of weight is the newton. Thus a body will have the same mass on the Moon as on Earth, but its weight on the Moon will be less than on Earth since the gravitational force on the Moon is approximately one sixth of that on Earth. The same person, stepping on a set of compression scales at the bottom of a mountain and then at the top, would weigh less at the top because of the slight decrease in the gravitational force, which results from the slight increase in distance from the center of the Earth.
Newton expressed his second law by stating that the force acting on a body is equal to the rate of change in its `quantity of motion', which is now called momentum. The momentum of a body is defined as the product of its mass and velocity.
Newton's third law states that a single isolated force cannot exist on its own: there is always a resulting `mirror-image' force. In Newton's words, `To every action there is always opposed an equal reaction.' This means that, because any two masses exert on each other a mutual gravitational attraction, the Earth is always attracted towards a ball as much as the ball is attracted towards the Earth. Because of the huge difference in their sizes, however, the observable result is the downward acceleration of the ball.
The principle of the conservation of momentum follows from this third law. This states that, when two bodies interact, the total momentum before impact is the same as the total momentum after impact. Thus the total of the components of momentum in any direction before and after the interaction are equal.
In an accelerating or non-inertial frame of reference, Newton's second law will not work unless some fictitious force is introduced. For example, passengers on a circus merry-go-round feel as if they are being forced outward when the machine is operating. This is ascribed to a `center-fleeing' or `centrifugal force'. The passengers experience this because they are moving within the system; they are within an accelerating frame of reference. To an observer on the ground it appears that the passengers on the ride should fly off at a tangent to the circular motion unless there were a force keeping them aboard. This is the centripetal force and is experienced as the friction between each passenger and the seat. If a passenger were to fall off, it would be because the centripetal force was not strong enough, not because the `centrifugal force' was too great.
* THE HISTORY OF ASTRONOMY
* FORCES AFFECTING SOLIDS AND FLUIDS
* THERMODYNAMICS
* QUANTUM THEORY AND RELATIVITY
* THE HISTORY OF SCIENCE
* FUNCTIONS, GRAPHS AND CHANGE
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p020-5
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Motion and Force (page 5)
ftsTitle
A pen and magnet are dropped in low gravity. They float in mid air and are attracted to each other due to magnetic forces.
Motion and Force (5 of 5)
Gravitation
Gravitational force is one of the four fundamental forces that occur in nature. The others are electromagnetic force, and the strong and the weak nuclear forces. The electromagnetic and weak forces have recently been shown to be part of an electro-weak force.
Gravitational force is the mutual force of attraction between masses. The gravitational force is much weaker than the other forces mentioned above. However, this long-range force should not be thought of as a weak force. An object resting on a table is acted on by the gravitational force of the whole Earth - a significant force. The almost equal force exerted by the table is the result of short-range forces exerted by molecules on its surface.
Newton's law of gravitation was first described in his Philosophiae Naturalis Principia Mathematica (`The Mathematical Principles of Natural Philosophy'), which he wrote in 1687. Newton used the notion of a particle, by which he meant a body so small that its dimensions are negligible compared to other distances. He stated that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is represented by G and is known as the gravitational constant. The law is an `inverse-square law', since the magnitude of the force is inversely proportional to the square of the distance between the masses. A similar inverse-square law applies for the force between two electric charges.
Newtonian mechanics were so successful that a mechanistic belief developed in which it was thought that with the know ledge of Newton's laws (and later those of electromagnetism) it would be possible to predict the future of the Universe if the positions, velocities and accelerations of all particles at any one instant were known. Later the quantum theory and the Heisenberg Uncertainty Principle confounded this belief by predicting the fundamental impossibility of making simultaneous measurements of the position and velocity of a particle with infinite accuracy.
THE KINEMATIC
EQUATIONS
For a body moving in a straight line with uniformly accelerated motion:
1. v = u + at
2. s = ut + 1/2 at<2>
3. v <2> = u <2> + 2as
4. s = 1/2 t (u + v)
where s = displacement
t = time
u = initial or starting velocity
v = velocity after time t
a = acceleration
NEWTON'S LAWS OF
MOTION
1.A body will remain at rest or traveling in a straight line at constant speed unless it is acted upon by an external force.
2.The resultant force exerted on a body is directly proportional to the acceleration produced by the force.
F = ma where F is the force exerted, m is the mass of the body a is the acceleration v1 is the initial velocity v2 is the final velocity 3.To every action there is an equal and opposite reaction.
NEWTON'S LAW OF
GRAVITATION
Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
F = G m1 m2
-----------
x to the power of 2
where F is the force m1, m2 are the masses x is the distance between the particles.
* THE HISTORY OF ASTRONOMY
* FORCES AFFECTING SOLIDS AND FLUIDS
* THERMODYNAMICS
* QUANTUM THEORY AND RELATIVITY
* THE HISTORY OF SCIENCE
* FUNCTIONS, GRAPHS AND CHANGE
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p022-1
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Forces affecting Solids and Fluids (page 1)
ftsTitle
1. Equilibrium. The net force on the body is zero. The mass is balanced by the reaction force: R = mg (mg is the gravitational force acting on the mass). The men are pulling with equal force: F = F. The body will not move.
Forces affecting Solids and Fluids (1 of 9)
In addition to the fundamental forces, other forces such as frictional, elastic and viscous forces may be encountered. Because of their different natures, solids and fluids appear in some ways to react differently to similar applied forces.
When forces are applied to solids they tend to resist. Friction inhibits displacement, but is overcome after a certain limit. Bodies may be deformed by tensions. Fluids, although lacking definite shape, are held together by internal forces. They exert pressure on the walls of the containing vessel. Fluids - by definition - have a tendency to flow; this may be greater in some substances than in others and is governed by the viscosity of the fluid.
Statics
Newton's first law, stated for a single particle, can also apply to real bodies that have definite sizes and shapes and consist of many particles. Such a body may be in equilibrium, which means it is at rest or moving with constant velocity in a straight line. This means that it is acted on by zero net force, and that it has no tendency to rotate.
A body is acted on by zero net force if the total or resultant of all the forces acting on it is zero - i.e. all the forces cancel each other out. If the body is at rest it is in static equilibrium. Studies of such conditions are important in the design of bridges, dams and buildings.
* MOTION AND FORCE
* THERMODYNAMICS
* HOW CARS WORK
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p022-2
ftsTitleOverride
Forces affecting Solids and Fluids (page 2)
ftsTitle
2. Torque. Torque or moment of a force = force x perpendicular distance = Fd.
Forces affecting Solids and Fluids (2 of 9)
Forces involved in rotation
Torque (or moment of a force) measures the tendency of a force to cause the body to rotate. In this case the force causes angular acceleration, which is the rate of change of angular momentum of the body. Torque is defined as the product of the force acting on a body and the perpendicular distance from the axis of the rotation of the body to the line of action of the force. Torque has units of force x distance, usually expressed as newton meters (N m).
Torque is increased if either the force or the perpendicular distance is increased. If a wedge is used to keep a door open, it has maximum effect if it is placed on the floor as far from the hinge as possible.
* MOTION AND FORCE
* THERMODYNAMICS
* HOW CARS WORK
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Forces affecting Solids and Fluids (page 3)
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3. A couple. The total turning force acting on the wing nut is 2Fd.
Forces affecting Solids and Fluids (3 of 9)
When a body is acted on by two equal and opposite forces, not in the same line, then the result is a couple, which has a constant turning moment about any axis perpendicular to the plane in which they act. When the total or net torque on a body is zero about any axis, the body is in equilibrium. A body is in stable equilibrium if a small linear displacement causes a force to act on the body to return it to its previous position, or an angular displacement causes a couple to act to bring it back to its previous position, called the equilibrium position.
* MOTION AND FORCE
* THERMODYNAMICS
* HOW CARS WORK
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Forces affecting Solids and Fluids (page 4)
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4. Center of gravity. A racing car has a very low center of gravity and will remain stable even on a slope. A loaded truck will have a high center of gravity and will topple over if driven on too steep a slope.
Forces affecting Solids and Fluids (4 of 9)
The center of mass of a body is a point, normally within the body, such that the net resultant force produces an acceleration at this point, as though all the mass of the body were concentrated there. For bodies of certain shapes this point may lie outside the object.
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Forces affecting Solids and Fluids (page 5)
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5. Center of mass. In the left-hand picture the person is in a stable state because the perpendicular through his center of mass falls between the legs of the chair. In the center picture, although the chair is tipped, the perpendicular still falls within the safe area between the legs of the chair. In the right-hand picture the perpendicular falls outside the legs of the chair. The person sitting on the chair is in trouble.
Forces affecting Solids and Fluids (5 of 9)
If a uniform gravitational field is present, the center of gravity coincides with the center of mass. Thus all the weight can be considered to act at this single point. The stability of an object is helped by keeping the center of gravity as low as possible. A racing car is low-slung to improve stability.
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Forces affecting Solids and Fluids (page 6)
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6. Friction. If one were to enlarge a section of two apparently smooth surfaces, roughnesses would become apparent - so explaining why friction occurs between two surfaces.
Forces affecting Solids and Fluids (6 of 9)
Friction
Sliding friction occurs when a solid body slides on a rough surface. Its progress is hindered by an interaction of the surface of the solid with the surface it is moving on. This is called a kinetic frictional force.
Another type of friction is called static friction. Before the object moves, the resultant force acting on it must be zero. The frictional force acting between the object and the surface on which it rests cannot exceed its limiting value. Thus, when the other forces acting on the object, against friction, exceed this value the object is caused to accelerate. The limiting or maximum value of the frictional force occurs when the stationary object acted on by the resultant force is just about to slip.
Both these types of friction involve interaction with a solid surface. The frictional forces depend on the two contacting surfaces and in particular on the presence of any surface contaminants. The friction between metal surfaces is largely due to adhesion, shearing and deformation within and around the regions of real contact. Energy is dissipated in friction and appears as internal energy, which can be observed as heat. Thus car brakes heat up when used to slow a vehicle. The results of friction may be reduced by the use of lubricants between the surfaces in contact. This is one function of the oil used in car engines. A further type of friction is rolling friction, which occurs when a wheel rolls. Energy is dissipated through the system, because of imperfect elasticity. This effect does not depend upon surfaces and is unaffected by lubrication.
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Forces affecting Solids and Fluids (page 7)
ftsTitle
7. Elasticity and plastic flow. Some bodies behave elastically when small forces are applied, but above a critical point they experience plastic flow, i.e. they are irreversibly extended.
Forces affecting Solids and Fluids (7 of 9)
Elasticity
Elasticity deals with deformations that disappear when the external applied forces are removed. Most bodies may be deformed by the action of external forces and behave elastically for small deformations.
Strain is a measure of the amount of deformation. Stress is a quantity proportional to the force causing the deformation. Its value at any point is given by the magnitude of the force acting at that point divided by the area over which it acts.
It is found that for small stresses the stress is proportional to the strain. The constant of proportionality is called the elastic modulus and it varies according to the material and the type of deformation. Young's modulus refers to changes in length of a material under the action of an applied force. The shear modulus relates to another type of deformation - that of planes in a solid sliding past each other. The third modulus is called the bulk modulus and characterizes the behavior of a substance subject to a uniform volume comparison.
A special example of deformation is the extension or elongation of a spring by an applied force. Hooke's law, formulated by the English scientist Robert Hooke (1635-1703), states that, for small forces, the extension is proportional to the applied force. Thus a spring balance will have a uniform scale for the measurement of various weights. In scientific terms, spring steel - which returns to its initial state readily - is almost perfectly elastic. In contrast, a soft rubber ball dropped on hard ground bounces to only about half its initial height, demonstrating imperfect elasticity.
Some bodies behave elastically for low values of stress, but above a critical level they behave in a perfectly viscous manner and `flow' like thick treacle, with irreversible deformation. This is called plastic flow.
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Forces: Elasticity and plastic flow
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Forces affecting Solids and Fluids (page 8)
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8. Hydraulic brakes: a simple hydraulic system.
Forces affecting Solids and Fluids (8 of 9)
Fluids at rest - hydrostatics
Pressure is defined as the perpendicular or normal force per unit area of a plane surface in the fluid, and its unit is the pascal (Pa), equivalent to 1 newton per square meter (N m2). At all points in the fluid at the same depth the pressure is the same. The pressure depends only on depth in an enclosed fluid, and is independent of cross-sectional area. In the hydraulic brakes of a car, a force is applied by the foot pedal to a small piston. The pressure is transmitted via the hydraulic fluid to a larger piston connected to the brake. In this way the force applied to the brake is magnified by comparison with the force applied to the pedal.
Atmospheric pressure may be measured using a barometer. At sea level, it is equivalent to the weight of a column of mercury about 0.76 m high, which is about
1.01 x 105 Pa. It varies by up to about 5%, depending on the weather systems passing overhead.
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Forces: Hydralic brakes
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Forces affecting Solids and Fluids (page 9)
ftsTitle
9. Archimedes' principle. Total upward force is equal to the weight of fluid displaced.
Forces affecting Solids and Fluids (9 of 9)
The buoyancy force was described by the Greek mathematician and physicist Archimedes (287-212 BC). Archimedes' principle states that an object placed in a fluid is buoyed up by a force equal to the weight of fluid displaced by the body. It was the realization of this principle that - according to legend - prompted Archimedes to leap out of his bath with shouts of 'Eeureka! Eureaka!' ('I have found it! I have found it!').
A body with density greater than that of the fluid will sink, because the fluid it displaces weighs less than it does itself. A body with density less than that of the fluid will float.
A submarine varies its density by flooding ballast tanks with sea water or emptying them; this enables it to dive or rise to the surface.
Surface tension
Surface tension occurs at an interface between a liquid and either a gas or a solid. Molecules in a liquid exert forces on other molecules. At the surface there is asymmetry in these forces, resulting in surface tension. Thus falling rainwater coalesces into spherical drops.
Surface tension may be reduced by the presence of a detergent, which acts as a wetting agent, spreading out the liquid over the solid surface. In the human lung, fluid on the alveoli surfaces contains a detergent to prevent collapse of the lung during breathing; its absence, particularly in babies, is fatal.
VISCOSITY AND TURBULENCE
Viscosity relates to the internal friction in the flow of a fluid - how adjacent layers in the fluid exert retarding forces on each other. This arises from cohesion of the molecules in the fluid. In a solid the deformation of adjacent layers is usually elastic. In a fluid, however, there is no permanent resistance to change of shape; the layers can slide past each other, with continuous displacement of these layers. Fluids are described as newtonian if they obey Newton's law that the ratio of the applied stress to the rate of shearing has a constant value. This is not true for many substances. Some paints, for example, do not have constant values for the coefficient of viscosity; as the paint is stirred it flows more easily and the coefficient is diminished. Molten lava is another non-newtonian fluid.
If adjacent layers flow smoothly past each other the steady flow is described as laminar flow. If the flow velocity is increased the flow may become disordered with irregular and random motions called turbulence. Smoke rising from a cigarette starts with smooth laminar flow but soon breaks into turbulent flow with the formation of eddies. Reynold's number is used to predict the onset of turbulence. It is defined as: Re = speed x density x dimension viscosity or, alternatively, as the ratio of the inertial force to the viscous force: Re = inertial force viscous force This is a pure ratio so has no units. It is a characteristic of the system and the dimension may be the diameter of a pipe or the radius of a ballbearing. Viscosity is relevant for small values of Reynold's number. Above a certain value, turbulence is likely to break out. Thus, for the fall of a very small raindrop, resistance is viscous and is proportional to the product of the viscosity of air, the radius of the raindrop and its speed. For a large raindrop, the resistance is proportional to the product of the density of air, the square of the radius of the raindrop and the square of its speed. Sometimes when special smooth surfaces are involved the onset of turbulence is delayed.
The study of viscosity and turbulence is important in understanding problems such as the flow of arterial blood around the body. For larger arteries, turbulence will be a major consideration.
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Thermodynamics (page 1)
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1. Work. Work done on a body by a constant force is the product of the magnitude of the force and the displacement of the body as a result of the action of the force: W=Fd.
Thermodynamics (1 of 6)
Thermodynamics is the study of heat and temperature. Heat is a form of energy, and the temperature of a substance is a measure of its internal energy. One fundamental principle in the study of thermodynamics is the conservation of energy. This theory was developed in the late 19th century by about a dozen scientists, including James Joule (1818-89), a brewery-owner from the north of England, and Baron Herman von Helmholtz (1821-94), a German physiologist. Although there seemed to be plenty of evidence in the world that energy was not conserved, this important principle was eventually established.
Much of the energy that seems to be lost in typical interactions - such as a box sliding across a floor - is converted into internal energy; in the case of the sliding box, this is the kinetic energy gained by the atoms and molecules within the box and the floor as they interact and are pulled from their equilibrium positions. The name given to the energy in the form of hidden motion of atoms and molecules is thermal energy. Strictly speaking, heat is transferred between two bodies as a result of a change in temperature, although the term `heat' is commonly used for the thermal energy as well. Processes that turn kinetic energy, which is the organized energy of a moving body, into thermal energy, which is the disorganized energy due to the motion of atoms, include friction and viscosity. In a steam engine, heat is turned into work.
* MOTION AND FORCE
* FORCES AFFECTING SOLIDS AND FLUIDS
* THE HISTORY OF SCIENCE
* ENERGY
* ENGINES
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Thermodynamics (page 2)
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2. Work done. A exerts a force F on B and as a result B moves to position B with displacement d at angle q to the line of F. Work W = Fd cos q.
Thermodynamics (2 of 6)
Work and energy
When a force acts on a body, causing acceleration in the direction of the force, work is done. The work done on a body by a constant force is defined as the product of the magnitude of the force and the consequent displacement of the body in the direction of the force
The unit of work is the joule (sometimes referred to as the newton meter). A joule (J) is defined as the work done on a body when it is displaced 1 meter as the result of the action of a force of 1 newton acting in the direction of motion: 1 J = 1 N m The result may be expressed more generally.
Energy is the capacity of a body to do work. The total energy stored in a closed system - one in which no external forces are experienced - remains constant, however it may be transformed. This is the principle of conservation of energy. It may take the form of mechanical energy (kinetic or potential; see below), electrical energy, chemical energy, or heat energy. There are also other forms of energy, including gravitational, magnetic, the energy of electromagnetic radiation, and the energy of matter.
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Thermodynamics (page 3)
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3. Kinetic energy. Kinetic energy is equal to half the product of the mass and the square of the velocity: Ek = 1/2 mv2.
Thermodynamics (3 of 6)
The kinetic energy of a body is the energy it has because it is moving. It is equal to half the product of the mass and the square of the velocity.
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Thermodynamics (page 4)
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4. Potential energy. The potential energy of a body is the product of its mass, its height above the ground, and the acceleration due to gravity: Ep = mgh.
Thermodynamics (4 of 6)
Alternatively, a body may have potential energy. In contrast to kinetic energy, which is dependent upon velocity, potential energy is dependent upon position. The gravitational potential energy of a body of mass m at a height h above the ground is mgh, where g is the acceleration due to gravity. This gravitational potential energy is equal to the work that the Earth's gravitational field will do on the body as it moves to ground level.
Potential energy can be converted into kinetic energy or it can be used to do work. It acts as a store of energy. If a body moves upward against the gravitational force, work is done on it and there is an increase in gravitational potential energy.
Temperature
Temperature is a measure of the internal energy or `hotness' of a body, not the heat of the body. Thermometers are used to measure temperature. They may be based on the change in volume of a liquid (as in a mercury thermometer), the change in length of a strip of metal (as used in many thermometers), or the change in electrical resistance of a conductor. Other parameters may also be involved in measuring temperature.
The thermodynamic temperature scale - also known as the kelvin scale or the ideal gas scale - is based on a unit called the kelvin (K); the scale is used in both practical and theoretical physics. The scale is named after the Scottish physicist William Thomson, later Lord Kelvin (1824-1907), who did important work in thermodynamics and electricity. An ideal gas is one that would obey the ideal gas law perfectly. In fact no gas is ideal, but most behave sufficiently closely that the ideal gas law can be used in calculations. At ordinary temperatures and pressures, dry air can be considered as a very good approximation to an ideal gas.
On the kelvin scale the freezing point of water is 273.15 K (0 deg C or 32 deg F) and its boiling point is 373.15 K (100 deg C or 212 deg F): one degree kelvin is equal in magnitude to one degree on the Celsius scale. The temperature of 0 (zero) K is known as absolute zero. At this temperature, for an ideal gas, the volume would be infinitely large and the pressure zero.
Heat and internal energy
The molecular energy (kinetic and potential) within a body is called internal energy. When this energy is transferred from a place of high energy to one of lower energy, it is described as a flow of heat.
If two bodies of different temperatures are placed in thermal contact with each other, after a time they are found both to be at the same temperature. Energy is transferred from the warmer to the colder body, until both are at a new equilibrium temperature. Heat is a form of energy, and heat flow is a transfer of energy resulting from differences in temperature.
The unit of internal energy and heat is the joule, as defined above. Units used previously include the calorie, which is equivalent to 4.2 joules and is defined as the heat required to raise the temperature of 1 gram of water from 14.5 deg C (58.1 deg F) to 15.5 deg C (59.9 deg F). The unit used by nutritionists is actually the kilocalorie, or Calorie (= 1000 calories).
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Thermodynamics (page 5)
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5. The ideal gas law. This combines Boyle's law and Charles' law.
Thermodynamics (5 of 6)
The kinetic theory of gases
The kinetic theory of gases takes Newton's laws and applies them statistically to a group of molecules. It treats a gas as if it were made up of extremely small - dimensionless - particles, all in constant random motion. It is based on an ideal gas.
One conclusion is that the pressure and volume of such a gas are related to the average kinetic energy for each molecule. The kinetic theory explains that pressure in a gas is due to the impact of the molecules on the containing walls around the gas. There is an equation that relates the pressure, temperature and volume of an ideal gas - the ideal gas law.
The temperature of an ideal gas is a measure of the average molecular kinetic energies. At a higher temperature the mean speed of the molecules is increased. For air at room temperature and atmospheric pressure the mean speed is about 500 m s to the power of -1 (about 1800 km/h or 1100 mph, the velocity of a rifle bullet).
The internal energy of a gas is associated with the motion of its molecules and their potential energy. For a gas that is more complex than one with monatomic (i.e. single-atom) molecules, account has to be taken of energies associated with the rotation and vibration of its molecules, as well as their speed.
A thermally isolated system is one that neither receives nor transmits transfer of heat, although the temperature within the system may vary. If mechanical or electrical work is performed on a thermally isolated system, its internal energy increases. James Joule observed the effects of doing measured amounts of work on insulated bodies (thermally isolated systems). He discovered an equivalence relation between the amount of work done (W) and the heat gained (Q): W = JQ The constant J was described by Joule as the mechanical equivalent of heat.
Laws of thermodynamics
The first law of thermodynamics is a development of the law of conservation of energy, which states that in any interaction, energy is neither created nor destroyed. The first law states that if, during an interaction, a quantity of heat is absorbed by a body, it is equal to the sum of the increase in internal energy of the body and any external work done by the body.
The increase in internal energy will be made up of an increase in the kinetic energy of the molecules in the body and an increase in their potential energy, since work will have been done against intermolecular forces as the body expands.
The change in internal energy of a body thus depends only on its initial and final states. The change may be the result of an increase in energy in any form - thermal, mechanical, gravitational, etc. Another statement of this law is that it is possible to convert work totally into heat.
The second law of thermodynamics states that the converse is not true. There are several ways in which the second law may be stated but, essentially, it means that heat cannot itself flow from a cold object to a hot object. Thus the law shows that certain processes may only operate in one direction.
The second law was established after work by a French engineer, Sadi Carnot (1796-1832), who was trying to build the most efficient engine. His ideal engine - the Carnot engine - established an upper limit for the efficiency with which thermal energy could be converted into mechanical energy. Real engines fall short of this ideal efficiency because of losses due to friction and heat conduction. As the temperature of the sink (the place where energy is removed from the system) in a working engine is near room temperature, the amount of work that can be done is restricted by the relatively small temperature difference. This limits the efficiency of most steam engines to about 30-40%. Thus it makes sense to use the vast amounts of waste heat from electrical power stations for heating purposes rather than allow it to be lost in cooling towers.
Entropy
Entropy is a parameter used in statistical mechanics to describe the disorder or chaos of a system. A highly disordered state is one in which molecules move haphazardly in all directions, with many different velocities. An alternative form of the second law of thermodynamics is that the entropy of the universe never decreases. It follows from this analysis that the universe is moving through increasing disorder towards thermal equilibrium. Therefore the universe cannot have existed for ever, otherwise it would have reached this equilibrium state al ready.
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Thermodynamics (page 6)
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6. Latent heat. The effect of latent heat on climate.
Thermodynamics (6 of 6)
Latent heat
When heat flows between a body and its surroundings there is usually a change in the temperature of the body, as well as changes in internal energies. This is not so when a change of form occurs, as from solid to liquid or from liquid to gas. This is called a phase change and involves a change in the internal energy of the body only.
The amount of heat needed to make the change of phase is called the hidden or latent heat. To change water at 100 deg C (212 deg F) to water vapor requires nearly seven times as much heat (latent heat of vaporization) as to change ice to water (latent heat of fusion). This varies for water at different temperatures - more heat is required to change it to water vapor at 80 deg C (176 deg F), less at 110 deg C (230 deg F). In each case the attractive forces binding the water molecules together must be loosened or broken.
Latent heat has an important effect on climate. A similar cycle takes place in a heat pump or refrigerator.
HEAT TRANSFER CONDUCTION
Heat conduction occurs when kinetic and molecular energy is passed from one molecule to another. Metals are good conductors of heat because of electrons that transport energy through the material. Air is a poor conductor in comparison. Thus a string vest keeps its wearer warm by trapping air and so preventing the conduction of heat outwards from the body.
CONVECTION
Heat convection results from the motion of the heated substance. Warm air is less dense than cold air and so, according to Archimedes' principle, it rises. Convection is the main mechanism for mixing the atmosphere and diluting pollutants emitted into the air.
RADIATION
Radiation is the third process for heat transfer. All bodies radiate energy in the form of electromagnetic waves. This radiation may pass across a vacuum, and thus the Earth receives energy radiated from the Sun. A body remains at a constant temperature when it both radiates and receives energy at the same rate.
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Quantum Theory and Relativity (page 1)
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1. An electromagnectic wave traveling along Ox, made of up of electrical and magnetic fields that point along Oy and Oz respectively and oscillate rapidly.
Quantum Theory and Relativity (1 of 4)
Three of the most important theories of the 20th century are the quantum theory and the theories of special and general relativity. When special relativity is combined with the full quantum theory and with electromagnetism, almost all of the physical world is described by it. The most important application is in the theory of subatomic particles. General relativity is as yet not fully combined with quantum theory and is a theory of gravity and cosmology.
The physical world is not as simple as the theories of Newton supposed, although such views are appropriate simplifications for large objects moving relatively slowly with respect to the observer. Quantum mechanics is the only correct description of effects on an atomic scale, and special relativity must be used when speeds approaching the speed of light, with respect to the observer, are involved.
The development of quantum theory
At the very beginning of the 20th century scientists such as the German physicist Max Planck (1858-1947) discovered that the theories of classical physics were not sufficient to explain certain phenomena on the subatomic scale, particularly in the field of electromagnetic radiation and the study of light waves. Their work resulted in the development of the quantum theory, which states that nothing can be measured or observed without disturbing it: the observer can affect the outcome of the effect being measured.
The Scottish physicist James Clerk Maxwell (1831-79) had developed a theory about the electromagnetic wave nature of light, and this was crucial to the development of quantum theory. Maxwell showed that at any point on a beam of light there is a magnetic field and an electric field that are perpendicular to each other and to the direction of the light beam. The fields oscillate millions of times every second, forming a wave pattern (as shown in diagram 1).
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Quantum Theory and Relativity (page 2)
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2. Interference. The waves passing through slits A and B and reaching the screen C will be either in phase or out of phase and will either reinforce or cancel each other. The result is a series of light and dark bands on the screen. Reinforcement occurs when the path difference is a whole number of wavelengths.
Quantum Theory and Relativity (2 of 4)
Photons
If light is directed onto a piece of metal in a vacuum, electrons are knocked from the surface of the metal. This is the photoelectric effect. For light of a given wavelength, the number of electrons emitted per second increases with the intensity of the light, although the energies of the electrons are independent of the wavelength.
This discovery led the German physicist Albert Einstein (1879-1955) to deduce that the energy in a light beam exists in small discrete packets called photons or quanta. These can be detected in experiments in which light is allowed to fall on a detector, usually photographic film. This has led to the theory of the dual nature of light, which behaves as a wave during interference experiments (see diagram 2) but as a stream of particles during the photoelectric effect. Further work on this phenomenon has led to the acceptance of wave-particle duality, which is a fundamental principle in quantum physics. The way a system is described depends upon the apparatus with which it is interacting: light behaves as a wave when it passes through slits in an interference experiment, but as a stream of particles when it hits a detector (see diagram 3).
m 3).
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Quantum Theory and Relativity (page 3)
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3. The photon nature of light. The results of two-slit interference after the passage of 50,200 and 2000 photons have passed through. The characteristic pattern is only observed after many photons have passed. The initial results appear random.
Quantum Theory and Relativity (3 of 4)
Uncertainty
Werner Karl Heisenberg (1901-76), a German physicist, interpreted wave-particle duality differently. He proposed that when a beam of light is directed at a screen with two slits, the interference pattern formed exists only if we do not know which slit the photon passed through. If we make an additional measurement and determine which slit was traversed, we destroy the interference pattern. Heisenberg showed that it was impossible to measure position and momentum simultaneously with infinite accuracy; he expressed his findings in the uncertainty principle named after him. This changed the thinking about the precision with which simultaneous measurements of two physical quantities can be made.
Particles
Matter is made up of vast numbers of very small particles. The behavior of these particles cannot be described by the theories of classical physics, since there is no equivalence to subatomic particles in everyday mechanics. Thus it is not helpful to discuss the behavior of electrons in atoms in terms of tiny `planets' orbiting a `sun'. Louis Victor de Broglie (1892-1987), the French physicist, suggested that if light waves can behave like particles, then particles might in certain circumstances behave like waves. Later experiments confirmed that under appropriate conditions particles can exhibit wave phenomena.
Atomic energy levels
Quantum systems are described by a mathematical equation known as the Schrodinger equation after the Austrian physicist Erwin Schrodinger (1887-1961), who first formulated it. In situations such as where a negatively charged electron is bound to the positively charged nucleus of an atom, the Schrodinger equation has solutions for only discrete or quantized allowed values of the energy of the electron. The energy of an electron in an atom cannot take a lower value than the least of the allowed values - the ground state - so the electron cannot fall into the nucleus. If an atom, through the interaction of forces on it, is excited into an allowed state of energy that is higher than the ground state, it can emit a photon and jump into the ground state The energy of the photon is equal to the difference in energy levels of the two states. The energy of the photon is related to the wavelength of the light wave associated with it - thus light can be emitted by atoms only at particular wavelengths.
Quantum mechanics
Quantum mechanics is the study of the observable behavior of particles. This includes electromagnetic radiation in all its details. In particular, it is the only appropriate theory for describing the effects that occur on an atomic scale.
Quantum mechanics deals exclusively with what can be observed, and does not attempt to describe what is happening in between measurements. This is not true of classical theories, which are essentially complete descriptions of what is occurring whether or not attempts are made to measure it. In quantum mechanics the experimenter is directly included in the theory. Quantum mechanics predicts all the possible results of making a measurement, but it does not say which one will occur when an experiment is actually carried out. All that can be known is the probability of something being seen. In some experiments one event is very much more likely than any other, therefore most of the time this is what will be found, but sometimes one of the less probable events will occur. It is impossible to predict which will occur; the only way to find out is by making the appropriate measurement. For example, in an isotope of the element americium, 19% of the nuclei decay purely by alpha-particle emission and 81% decay by alpha emission followed by photon emission. For any individual americium nucleus it is not possible to say which decay will occur, only what will be observed on average. In some experiments the same event can occur in different ways. What is measured depends on whether it is known which of the possible paths was taken. Thus any additional knowledge, which can only be gained by making an additional measurement, changes the outcome of the first experiment.
ment.
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Quantum Theory and Relativity (page 4)
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4. Relative time. Supernova A occurs 120 light years from Earth (i.e. light takes 120 years to travel from Supernova A to Earth), and Supernova B is 200 light years from Earth. From Earth the two events are observed simultaneously in 1920, even though Supernova B occurred 80 years before Supernova A. From Planet X, in contrast, the two events are not observed simultaneously. Planet X is only 40 light years from Supernova B, but 150 light years from Supernova A. Therefore, on Planet X, Supernova B is observed in 1760, while Supernova A is not seen until 1950.
Quantum Theory and Relativity (4 of 4)
Special relativity
Inertial frames. Physical laws such as Newton's laws of mechanics are stated with respect to some frame of reference that allows physical quantities such as velocity and acceleration to be defined. A frame of reference is called inertial if it is unaccelerated and it does not contain a gravitational field.
Einstein's relativity principle. In 1905 Einstein stated that all inertial frames are equally good for carrying out experiments. This assumption, coupled with the evidence that the speed of light is the same in all frames, led Einstein to develop the theory of special relativity. This theory has been extensively tested using particle accelerators, where electrons or protons travel at speeds within a fraction of 1% of the speed of light. The masses of such particles measured by an observer in the laboratory in which the particles are traveling are higher than the masses measured by an observer at rest with respect to the particles.
Time. The classical view of time is that if two events take place simultaneously with reference to one frame then they must also occur simultaneously within another frame. In terms of special relativity, however, two events that occur simultaneously in one frame may not be seen as simultaneous in another frame moving relative to the first (see diagram 4). The sequence of cause and effect in related events is not, however, affected. Light plays a special role in synchronizing clocks in different frames because it has the same speed in all frames. In the classical view all observers have the same time scale, whereas in special relativity every inertial observer requires an individual time scale.
Space-time. An important feature of special relativity is that time and space have to be considered as unified and not as two separate things. This means that time is related to the frame of reference in which it is being measured. This is a different view of space to that of Newton.
Length contraction. The equations of special relativity lead to the very simple prediction that the length of a moving body in the direction of its motion measured in another frame is reduced by a factor dependent on its velocity with respect to the observer. What this means is that a car traveling very fast on a motorway would be measured by a stationary observer to be slightly shorter and heavier than usual, although the driver would not determine any difference. The length of a body is greatest when it is measured in a frame traveling with the body; as the speed of the body relative to the frame of reference approaches the speed of light the measured length approaches zero.
Time dilation. A similar effect happens to moving clocks (which can be any regularly occurring phenomenon, such as the vibration of atoms - the basis of atomic clocks - or the decay of particles). A clock moving with a uniform velocity in one frame is measured as running slow in another frame. Its fastest rate is in its own frame, and at speeds - relative to the observer - approaching the speed of light the clock rate approaches zero.
Paradox of reality? Both of the above effects of length and time contraction have been the inspiration of numerous `paradoxes' (and some science-fiction writing), and have been criticized on such grounds. But this simply goes to show that our `common-sense' view of the world is rooted in frames that travel with respect to the observer at tiny speeds compared to that of light, and is just as inappropriate in describing these phenomena as it is in describing the quantum effects of the atomic world. Time dilation has been measured experimentally both with decaying particles and with actual macroscopic clocks. In all cases the effects predicted by special relativity were encountered.
General relativity
This is an extension of the theory of special relativity to include gravitational fields and accelerating reference frames. Gravitational fields arise because of the distortions of space-time in the vicinity of large masses, and space-time is no longer thought of as having an existence in dependent of the mass in the universe. Rather, space-time, mass and gravity are interdependent. The concept of `curved space-time' was put forward by Einstein in his general theory of relativity. The motion of astronomical bodies is controlled by this deformation or curvature of space and time close to large masses. Light is also bent by the gravitational fields of large masses. Light rays have been observed to bend as they pass close to the Sun, so providing experimental verification of Einstein's theories.
THE SPEED OF LIGHT
All experimental evidence suggests that the speed of light in a vacuum is a constant independent of the speed of the observer. The speed of light (c) is approximately 3 x 10 to the power of 8 m s-1(300 000 km or 186 000 mi per second).
The most important feature is that although this is an enormous speed (a typical aircraft travels at about 250 m s-1 ), it is finite. It is also a limiting speed - nothing material travels faster than light (or electromagnetic radiation in general) in a vacuum.
E = mc to the power of 2
The gain in mass that occurs in a body moving at high speed led Einstein to conclude that the energy (E) and mass (m) of a body are equivalent. He derived a formula to relate them - the well-known E = mc to the power of 2, where c is the speed of light. Perhaps the most elegant verification of the theory of equivalence is shown by the annihilation at rest of an electron and a positron into two gamma rays, each with an energy equal to the particle rest mass.
mass.
* TIME
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* ELECTROMAGNETISM
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* ELEMENTS AND THE PERIODIC TABLE
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Wave Theory (page 1)
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1. A longitudinal wave in a 'slinky' spring.
Wave Theory (1 of 11)
Water waves are a phenomenon that can be seen, and the effects of sound waves are sensed directly by the ear. Some of the waves in the electromagnetic spectrum can also be sensed by the body: light waves by the eye, and the heating effect of infrared by the skin. There are other electromagnetic waves, however, that cannot be experienced directly through any of the human senses, and even infrared can generally only be observed using specialized detectors.
Wave phenomena are found in all areas of physics, and similar mathematical equations may be used in each application. Some of the general principles of wave motions are explored here.
Wave types and characteristics
A traveling wave is a disturbance that moves or propagates from one point to another. Mechanical waves are traveling waves that propagate through a material - as, for example, happens when a metal rod is tapped at one end with a hammer. An initial disturbance at a particular place in a material will cause a force to be exerted on adjacent parts of the material. An elastic force then acts to restore the material to its equilibrium position. In so doing, it compresses the adjacent particles and so the disturbance moves outward from the source. In attempting to return to their original positions, the particles overshoot, so that at a particular point a rarefaction (or stretching) follows a compression (or squeezing). The passage of the wave is observed as variations in the pressure about the equilibrium position or by the speed of oscillations. This change is described as oscillatory (like a pendulum) or periodic.
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Waves: A longitudinal wave
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2. A transverse wave in a 'slinky' spring.
Wave Theory (2 of 11)
There are two main types of periodic oscillation - transverse and longitudinal. In transverse waves the vibrations are perpendicular to the direction of travel. In longitudinal waves the vibrations are parallel to the direction of travel. Sound waves are alternate compressions and rarefactions of whatever material through which they are traveling, and the waves are longitudinal.
Water waves may be produced by the wind or some other disturbance. The particles move in vertical circles so there are both transverse and longitudinal displacements. The motion causes the familiar wave profile with narrow peaks and broad troughs.
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Wave Theory (page 3)
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3. Amplitude is the maximum displacement from the equilibrium position.
4. Wave attenuation. Energy is lost as a wave travels through a medium-- the amplitude is reduced, and the wave is said to be attenuated.
Wave Theory (3 of 11)
Wave motions transfer energy - for example, sound waves, seismic waves and water waves transfer mechanical energy. However, energy is lost as the wave passes through a medium. The amplitude diminishes and the wave is said to be at tenuated. There are two distinct processes - spreading and absorption. In many cases there is little or no absorption - electromagnetic radiation from the Sun travels through space without any absorption at all, but planets that are more distant than the Earth receive less radiation because it is spreading over a larger area and so the intensity< (the ratio of power to area) decreases according to an inverse-square law. The same applies to sound in the atmosphere. In some cases, however, energy is absorbed in a medium, as, for example, when light enters and exposes a photographic film, or when X-rays enter flesh. For homogeneous radiation, absorption is exponential, for example if half the radiation goes through 1 mm of absorber, a quarter would go through 2 mm and an eighth through 3 mm.
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Wave Theory (page 4)
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5. Frequency. Frequency = cycles per second. 1 cycle per second = 1 hertz (Hz).
6. Wavelength. Wavelength is the distance between two successive points along a wave with similar amplitudes.
Wave Theory (4 of 11)
The frequency (f) of the wave motion is defined as the number of complete oscillations or cycles per second. The unit of frequency is the hertz (Hz), named after the German physicist Heinrich Rudolf Hertz: 1 hertz = 1 cycle per second. The amplitude is the maximum displacement from the equilibrium position. The wavelength () is the distance between two successive peaks (or troughs) in the wave. The speed of propagation (v) of the compressions, or phase speed of the wave, is equal to the product of the frequency and the wavelength:v = f ().
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Wave Theory (page 5)
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7. Spherical and plane wavefronts. Wavefronts propagating outwards from point source O will be spherical in a three-dimensional context (such a light waves propagating from the Sun) or circular in a two-dimensional context (such as water waves propagating from a dropped pebble). Once far enough from from the source, such wavefronts can for most practical purposes be considered as straight lines - plane wavefronts - much in the same way that the curvature of the Earth is not noticeable to someone standing on it.
Wave Theory (5 of 11)
Waves originating from a point source will propagate outwards, in all directions, forming wavefronts; these wavefronts will be circular or spherical if propagating through a homogeneous medium. When the distance of a wavefront from the source is great, then it can be considered as a plane wavefront.
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8. Reflection of plane waves at a plane surface. The waves are parallel as they approach XY and after they are reflected. AN is the normal to XY at A. i is the angle of incidence of the wave as it meets XY. The angle of reflection is r, and i = r.
9. Waves reflected at a curved surface. Waves behave in the same way as light reflected in a concave mirror. S is the principal focus of the surface A.
Wave Theory (6 of 11)
Reflection and refraction
Reflection of plane waves at a plane surface are as shown in diagram
8. The angle between the direction of the wavefront and the normal (i.e. a line perpendicular to the plane surface) is the angle of incidence (i). The angle between the reflected wave and the normal to the plane surface is the angle of reflection (r), and these angles i and r are equal. The behavior of waves reflected at curved surfaces is shown in diagram 9.
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Wave Theory (page 7)
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10. Refraction of a plane wavefront. MAN is the normal to XY; i is the angle of incidence; r is the angle of refraction. The waves are parallel after refraction.
Wave Theory (7 of 11)
If a wave travels from one medium to another, the direction of propagation is changed or `bent'; the wave is said to be refracted. The wave will travel in medium 1 with velocity v, and come upon the surface of medium 2 with angle of incidence i. Then the wave will be refracted, as in diagram 10, and r is the angle of refraction. The new velocity will be v2, which will be less than v1 if medium 2 is more dense than medium 1, but greater than v1 if medium is less dense. The velocities are related by: v1 v2 = sin i sin r and the ratio of sin i /sin r is a constant. This constant is the refractive index of medium 2 with respect to medium
1. This relationship was formulated by the Dutch astronomer Willebrord Snell (1591-1626) and is known as Snell's Law.
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Wave Theory (page 8)
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11. Constructive interference results in the effect of the waves being combined.
12. Destructive interference results in the waves cancelling eacht other out.
Wave Theory (8 of 11)
Interference
If several waves are traveling through a medium, the resultant at any point and time is the vector sum of the amplitudes of the individual waves. This is known as the superposition principle. Two or more waves combining together in this way exhibit the phenomenon of interference. If the resultant wave amplitude is greater than those of the individual waves then constructive interference is taking place; if it is less, destructive interference occurs. If two sound waves of slightly different frequencies and equal amplitudes are played together (for example two tuning forks), then the resulting sound has what is called varying amplitude. These varying amplitudes are called beats and their frequency is the beat frequency. This frequency is equal to the difference between the frequencies of the two original notes. Listening for beats is an aid to tuning musical instruments: the closer the beats, the more nearly in tune is the instrument.
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Wave Theory (page 9)
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3. Modulation methods, as used in radio transmission.
Wave Theory (9 of 11)
Amplitude and frequency modulation
Radio waves can be used to carry sound waves by superimposing the pattern of the sound wave onto the radio wave. This is called modulation, and is one of the basic forms of radio transmission. There are two ways of modulating radio waves. Amplitude modulation (AM) is the form most commonly used. The amplitude of the radio carrier wave is made to vary with the amplitude of the sound signal. For frequency modulation (FM) the frequency of the carrier wave is made to vary so that the variations are in step with the changes in amplitude of the sound signal.
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Wave Theory (page 10)
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A slinky performs demonstrating various wave patterns. NASA astronauts produce longitudinal and transverse waves by using a slinky in low gravity. The waves, without the sagging caused by gravity, are clearly visible.
Wave Theory (10 of 11)
Standing or stationary waves
These are the result of confining waves in a specific region. When a traveling wave, such as a wave propagating along a guitar string towards a bridge, reaches the support, the string must be almost at rest. A force is exerted on the support that then reacts by setting up a reflected wave traveling back along the string. This wave has the same frequency and wavelength as the source wave. At certain frequencies the two waves, traveling in opposite directions, interfere to produce a stationary- or standing-wave pattern. Each pattern or mode of vibration corresponds to a particular frequency.
The standing wave may be transverse, as on a plucked violin string, or longitudinal, as in the air in an organ pipe. The positions of maximum and minimum amplitude are called antinodes and nodes respectively. At antinodes the interference is constructive. At nodes it is destructive.
If a periodic force is applied to a system with frequency at or near to the natural frequency of the system, then the resulting amplitude of vibration is much greater than for other frequencies. These natural frequencies are called resonant frequencies. When a driving frequency equals the resonant frequency then maximum amplitude is obtained.
The natural frequency of objects can be used destructively. High winds can cause suspension bridges to reach their natural frequency and vibrate, sometimes resulting in the destruction of the bridge. Soldiers marching in formation need to break step when crossing bridges in case they achieve the natural frequency of the structure and cause it to disintegrate.
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Waves: A slinky
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p028-11
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Wave Theory (page 11)
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14. Diffraction of waves passing through a small gap.
Wave Theory (11 of 11)
Diffraction
Waves will usually proceed in a straight line through a uniform medium. However, when they pass through a slit with width comparable to their wavelength, they spread out, i.e. they are diffracted. Thus waves are able to bend round corners. For a sound wave of 256 Hz the wavelength is about
1.3 m (4 1/4 ft), comparable with the dimensions of open doors or windows.
If a beam of light is shone through a wide single slit onto a screen that is close to the slit, then a bright and clear image of the slit is seen. As the slit is narrowed there comes a point where the image does not continue getting thinner. Instead, a diffraction pattern of light and dark fringes is seen.
Huygens' principle was proposed in 1676 by the Dutch physicist Christiaan Huygens (1629-95) to explain the laws of reflection and refraction. He postulated that light was a wave motion. Each point on a wavefront becomes a new or secondary source. The new wavefront is the surface that touches all the wavefronts from the secondary sources. Diffraction describes the interference effects observed between light derived from a continuous portion of a wavefront, such as that at a narrow slit. The work of the British physician and physicist Thomas Young (1773-1829) and others eventually supported Huygens' theory.
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Acoustics (page 1)
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1. Seismic surveying relies on the variation of seismic velocities in different rocks; this causes some layers to reflect the waves more strongly than others.
Acoustics (1 of 4)
The range of frequencies for which sound waves are audible to humans is from 20 to 20 000 Hz (i.e. vibrations or cycles per second) - the higher the frequency, the higher the pitch. In music, the A above middle C is internationally standardized at 440 Hz. For orchestral instruments, the frequencies range between 6272 Hz achieved on a handbell, and 16.4 Hz on a sub-contrabass clarinet. On a standard piano (and a violin) the highest note is 4186 Hz. For the organ, the highest frequency is 12 544 Hz and the lowest is 8.12 Hz, using pipes of 1.9 cm ( 3/4 in) and 19.5 m (64 ft) respectively.
Frequencies that are lower than the human audible range are referred to as infrasonic, and those above as ultrasonic. Many mammals such as dolphins and bats have sensitive hearing in the ultrasonic range, and they use high-pitched squeaks for echolocation. Large animals such as whales and elephants use frequencies in the infra-sonic range to communicate over long distances. It is thought that migrating birds can detect infrasonic sounds produced by various natural features, and that they use the distinctive sounds produced by particular features as aids to navigation.
The velocity of sound
Sound shares the general characteristics of other wave forms. Sound waves are longitudinal compressions (squeezings) and rarefactions (stretchings) of the medium through which they are traveling, and are produced by a vibrating object.
If a sound wave is traveling in any medium then the pressure variations formed along its path cause strains as a result of the applied stresses. The velocity of the sound is given by the square root of the appropriate elastic modulus divided by the density.
The velocity of sound - as with the velocity of other types of wave - differs in different media. In still air at 0 deg C, the velocity of sound is about 331 m s-1 (1191.6 km/h, or 740 mph). If the air temperature rises by 1 deg C, then the velocity of sound increases by about 0.6 m s-1. The velocity of sound in a metal such as steel is about 5060 m s-1. Sometimes, in a Western film, someone will put an ear to a railway line to listen for an oncoming train. This works be cause the sound wave travels much faster through the steel track than through the air.
In the ocean depths the combined effect of salinity, temperature and pressure results in a minimum velocity for sound. The channel that is centered around this minimum velocity at a depth of about 1000-1300 m (3300-4250 ft) allows sound waves, traveling at the minimum velocity, to propagate within it with relatively little loss over large horizontal distances. Signals have been transmitted in this way from Australia to Bermuda.
The fact that the velocity of sound varies in different media is one reason why seismic techniques can be used to probe layers of rock or minerals underground. Similarly ultrasonic scanning can be used in medicine - for example, in the imaging of a baby in its mother's womb. In each case variations in materials are shown up through variations in the time it takes sound waves to travel to the detector.
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ftsTitleOverride
Acoustics (page 2)
ftsTitle
2. Sounds at night may under certain conditions be heard a long distance away, but not in the intervening area. At night the air nearer the ground is often cooler than the air higher up. Cool air is denser than warm air, so sound waves traveling upwards through a less and less dense medium will be refracted (bent) at increasingly large angles until at a certain critical angle of incidence the angle of refraction becomes 90 degrees to the vertical; above this critical angle of incidence, the waves are reflected back downwards. This is called total internal reflection.
Acoustics (2 of 4)
Refraction of sound
At night the air near the ground is often colder than the air higher up, as the Earth cools after sunset. Thus a sound wave moving upward will be slowly bent back towards the horizontal as it meets warmer layers of air. Eventually it will be reflected back downwards. Under these circumstances sound can be heard over long distances. This phenomenon is explained by Snell's law of refraction; layers of air at different temperatures act as different media through which sound travels at different velocities. During World War I the guns at the front in northern France could sometimes be heard in southern England, although not in the intervening area. This was a significant piece of evidence for the existence of the stratosphere, that part of the atmosphere above the troposphere (the lowest layer); in the middle and upper stratosphere the temperature increases with altitude.
The ear is an extremely sensitive detector. Its threshold of hearing corresponds to an intensity of sound of 10-12 watts per square meter (W m to the power of 2): this is a measure of the energy impinging on the ear, and is known as the threshold intensity. The loudest tolerable sound is about 1 W m to the power of 2. This range is enormous, and so a logarithmic scale, to the base 10, is used. The original unit was the bel, named after Alexander Graham Bell (1847-1922), the Scottish inventor of the telephone.
The bel is graduated using a logarithmic scale, but as the bel is rather a large unit, the decibel (db) is more normally used (1 bel = 10 db). If threshold intensity is at 0 db, then a sound at ten times threshold intensity is 10 db, one at a hundred times threshold intensity is 20 db, one at a thousand times threshold intensity is 30 db, and so on. This means that the value of 1 W m to the power of 2 is at 120 db above threshold.
The ear canal resonates slightly to sounds with frequencies of about 3200 Hz. The human ear is most sensitive in the range 2500-4000 Hz. Even then, only about 10% of the population can hear a 0 db sound and then only in the 2500-4000 Hz region. The response of the ear is not linear, i.e. there is no direct relationship with the intensity of the sound it detects. Sensitivity is related to frequency: it de creases strongly at the lowest audible frequencies, but less so at the highest.
The audible range of the normal human ear varies with age. It is usually about 20-20 000 Hz in the mid-teens. For someone 40 years old, the upper limit is more likely to be 12 000-14 000 Hz. At the lower hearing threshold, the pressure fluctuations from the sound wave are about 3 x 10-10 of atmospheric pressure. The eardrum (called the tympanic membrane) vibrates at very low velocities - about 10 cm (4 in) per year. This may seem strange, given that it can vibrate at frequencies of up to 20 000 Hz (cycles per second); however, the low velocity is explained by the fact that the detected displacement of the air molecules adjacent to the eardrum each time it moves is less than the typical atomic radius (about 10-10 m). The human ear is an astonishingly sensitive detector and so it is not surprising that constant overload will bring deterioration in its performance.
Characteristics of tones
There are three main characteristics of the notes played by musical instruments. Loudness would seem to be the most simple, but it is complicated by the non-linear response of the ear. At 100 Hz and 10 000 Hz the hearing threshold is about 40 db compared to the 0 db at 2500-4000 Hz. Thus the concept of loudness is not dependent just on the energy reaching the ear, but also on frequency.
Pitch is closely related to frequency. If the frequency of vibration is doubled the pitch rises by one octave. In general, the higher the frequency the higher the pitch.
Sounds created by musical instruments are not simple waveforms, but are the result of several waves combining. This complexity results in the tone quality or timbre of a note played by a particular musical instrument. Even a `pure' note may contain many waves of different frequencies. These frequencies are harmonics or multiples of the fundamental or lowest frequency, which has 2 nodes and 1 antinode, and is called the first harmonic. The second harmonic has 3 nodes and 2 antinodes. The wavelength is halved and the frequency is doubled. The third harmonic has 4 nodes and 3 antinodes. The wavelength is one third of the original wavelength, and the frequency has tripled. Different instruments emphasize different harmonics. Musical synthesizers are able to mimic instruments by mixing the appropriate harmonics electronically at various amplitudes.
The greater the plane's velocity exceeds the velocity of sound, the narrower the angle of the cone. If an airplane is particularly long, a double boom may be heard, as both the front and tail of the plane generate shockfronts.
Acoustics (4 of 4)
THE DOPPLER EFFECT
The Doppler effect (or Doppler shift) - first described by the Austrian physicist C.J. Doppler (1803-53) - is valid for all waves. It is more often noticed in acoustics and is particularly noticeable in the sirens used for emergency-service vehicles. The intensity and pitch of the siren seems to rise as the vehicle is approaching, then diminishes as it moves away.
This is explained by the fact that, as an observer moves towards a sound source, the pressure oscillations are encountered more frequently than if the observer were stationary. Thus the source seems to be emitting at a higher frequency. Conversely, if the observer is moving away from the source, the frequency seems to decrease. It also applies if the source is moving and the observer is stationary.
Horseshoe bats emit beams of sound at constant frequency; the frequency from different species of horseshoe bat varies in the range of about 40-120 kilohertz (kHz: 1 kHz = 1000 Hz). Applying the Doppler effect, the frequency of the echo returning to their ears after reflection from an object depends on whether the bat and object are getting closer together or moving apart. The bats seem able to observe Doppler shifts of less than 1%. The nostrils are spaced a quarter of a wavelength apart so that along the normal - or perpendicular - to a line between the nostrils the waves are in phase and thus of maximum total amplitude. Thus the emitted sound is beamed straight in front.
The Doppler effect is also observed in optics, and has proved to be of profound significance in our understanding of the universe. If a stellar spectrum is compared with an arc or spark spectrum of an element present in the star, then its spectral lines may be displaced. A shift to the red end of the spectrum - to the longer wavelengths - means that the star is moving away from the Earth. The US astronomer Edwin Hubble (1889-1953) studied the red shift in various galaxies, and determined that their velocity of recession was proportional to their distance from the Earth. He thus confirmed that the universe appears to be expanding.
SONIC BOOMS
If the velocity of the source of the sound wave is greater than the velocity of sound - 300 m s-1 (1080 km/h, or 670 mph) in the upper troposphere (up to 10 km / 6 mi above the Earth's surface) - then the wavefront produced is not spherical but conical. Different wave crests bunch together, forming a shock wave. A supersonic plane (traveling faster than the speed of sound) produces such a shockfront, causing a loud bang (a sonic boom) and large pressure variations.
* WAVE THEORY
* OPTICS
* EARTHQUAKES
* SEEING AND HEARING
* MEDICAL TECHNOLOGY
* SEEING THE INVISIBLE
* WHAT IS MUSIC?
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p032-1
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Optics (page 1)
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1. Reflection and refraction.
Optics (1 of 5)
Optics is the branch of physics that deals with the high-frequency electromagnetic waves that we call light. Optics is concerned with the way in which light propagates from sources to detectors via intermediate lenses, mirrors and other modifying elements. The electromagnetic spectrum includes a wide range of waves in addition to light, light being that small part of the spectrum that can be detected by the human eye.
This region, with wavelengths from 700 manometers (nm; 1 nm = 10-9 m) in the red region to 400 nm in the violet, is extended for practical optical systems into the ultraviolet and the mid-infrared regions. For many purposes light can be treated as a classical wave phenomenon, but some effects can only be described by using the full quantum theory.
A beam of light may be considered to be made up of many rays, all traveling outwards from the source. This approach is used in ray diagrams. In geometric simplifications, as in the diagrams used here, rays of light are drawn as straight lines. The wavelength and amplitude of light waves are very short compared to the other dimensions of the systems. The basic concept is very simple: light travels in straight lines unless it is reflected by a mirror or refracted by a lens or prism.
A point source of light emits rays in all directions. For an isolated point source in a vacuum the geometric wavefront will be a sphere. The variation of the speed of light in different materials must be taken into account - the speed of light (as of other electromagnetic waves) in a vacuum is 3 x 10 to the power of 8 m s-1 (300 000 km or 186 000 miles per second), but it travels more slowly through other media. Light waves have transverse magnetic and electric fields.
Reflection and refraction
Light is reflected and refracted (i.e. bent) in the same way as other waves. Diagram 1 shows a monochromatic (single-color) beam of light falling or incident upon a transparent material such as a block of glass. Angle i is the angle of incidence of the beam. Part of the beam is reflected at an angle t, the angle of reflection; and part is transmitted according to the law of refraction, and r is the angle of refraction.
Snell's law of refraction can be stated as: n1 sin i = n2 sin r where n1 and n2 are the refractive indices of the materials (the sine of an angle is explained on pp. 62-3). Basically, the refractive index of a material determines how much it will refract light.
The refractive index of a material is often expressed relative to another material. If no other material is quoted, the refractive index is assumed to be relative to air. The refractive index of a medium can also be derived as the ratio of the speed of light in a vacuum to the speed of light in the medium. The refractive index for a typical optical glass is 1.6, whereas the refractive index of diamond is about 2.4 in visible light.
* THE HISTORY OF ASTRONOMY
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* ELECTROMAGNETISM
* ATOMS AND SUBATOMIC PARTICLES
* PHOTOGRAPHY AND FILM
* RADIO, TELEVISION AND VIDEO
* TELECOMMUNICATIONS
* SEEING THE INVISIBLE
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Optics (page 2)
ftsTitle
2. Total internal reflection. In the lower diagram, the angle of incidence (i) has become so large that the ray is not refracted but is reflected back into medium 1.
Optics (2 of 5)
The prism
The refractive index of optical glasses is not constant for light of all frequencies. It is greater at the violet end and less at the red end of the spectrum. This means that a beam of light containing a mixture of different frequencies, for example sunlight, will leave a prism with the different frequencies bent by different amounts.
A prism is a block of glass with a triangular cross-section; it is used to deviate a beam of light by refraction. A beam of white light will be split into its component monochromatic colored lights - from red to violet - which will form the familiar rainbow effect. Any light can be split up in this way; the display of separated wave- lengths is called the spectrum of the original beam. The effect of prisms on light has been well known for many centuries. Newton used this effect, called dispersion, to produce and study the spectrum of sunlight. Under the right conditions dispersion occurring in spherical raindrops in the atmosphere produces a rainbow.
Total internal reflection
When light travels from one medium to another less dense medium it is deviated or turned away from the normal - perpendicular to the interface at the point of incidence. This means the angle of refraction (r) is greater than the angle of incidence (i). When the angle of refraction is less than 90 deg , some of the incident light will be refracted and some will be reflected. If the angle of incidence in creases, the angle of refraction will in crease more. It is possible to increase the angle of incidence to such a value that eventually the refracted ray disappears and all the light is reflected. This is known as total internal reflection.
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ftsTitleOverride
Optics (page 3)
ftsTitle
3. Properties of lenses. The lens at the top is convex (i.e. it is thicker in the middle), while the one below it is concave (i.e. it is thinner in the middle).
Optics (3 of 5)
The lens
A lens is a piece of transparent material made in a simple geometric shape. Usually at least one surface is spherical, and often both are. In diagram 3 the features of lenses are described. Under appropriate conditions a lens will produce an image of an object by refraction of light. It does this by bending rays of light from the object.
Some rays are refracted more than others, depending how they arrive at the surface of the lens. The lens affects the velocity of the rays, since light travels more slowly in a dense medium such as the lens than in a less dense medium such as air. In this way, the expanding geometric wavefront that is generated by the object is changed into a wavefront which, for a convex or converging lens, converges to a point behind the lens. If the object is located a long way from the lens (strictly an infinite distance, but a star is an excellent approximation for practical purposes) this point is known as the rear focal point or principal focus of the lens. Notice that a lens has two principal foci - one on each side. The distance between the optical center of the lens and the principal focus is the focal length (f). If a point source of light is placed at the principal focus of the convex lens, the rays of light will be refracted to form a parallel beam.
Because of the effects of dispersion, the distance from the lens at which red light and blue light from an object will be focused will be different. This can be demonstrated in the color fringes that can be seen in simple hand magnifiers (small magnifying glasses). Such fringes are unacceptable in, for example, camera lenses. A lens made from two different types of glass can be made to bring two colors to exactly the same focus with only a very small variation for other frequencies. Such a lens is called achromatic. Single-element lenses are, therefore, only used for simple applications. Lenses for cameras, binoculars, telescopes and microscopes are made with many elements, with different curvatures. These lenses are made from glasses with different refractive indices and dispersions. The additional elements allow the lens designer greatly to reduce the faults or aberrations of the lens.
Mirrors
Mirrors are reflecting optical elements. Plane mirrors are used to deviate light beams without dispersion or to reverse or invert images. Curved mirrors, which usually have spherical or parabolic surfaces, can form images, and are often used in illumination systems such as car headlamps.
Mirrors can be coated with metals such as aluminum or silver, which have high reflectance for visible light (or gold for the infrared). Alternatively, they may be coated with many thin layers of non-metallic materials for very high reflectances over a more restricted range of frequencies. A freshly coated aluminum mirror will reflect about 90% of visible light. Special mirrors, such as those used in lasers, can reflect over 99.7% of the light at one frequency.
Mirrors do not introduce any chromatic aberrations into optical systems. Those with large diameters are also lighter than glass lenses of equivalent size. For these reasons they are always used as the primary reflectors of large astronomical telescopes.
* THE HISTORY OF ASTRONOMY
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* ELECTROMAGNETISM
* ATOMS AND SUBATOMIC PARTICLES
* PHOTOGRAPHY AND FILM
* RADIO, TELEVISION AND VIDEO
* TELECOMMUNICATIONS
* SEEING THE INVISIBLE
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ftsTitleOverride
Optics (page 4)
ftsTitle
4. A rudimentary microscope. A small object placed close to the front focal point of the objective is greatly magnified. The eye views the intermediate image at infinity through the eyepiece.
5. The astronomical telescope magnifies the angular deviation light rays from an infinitely distant object. To the viewer this makes the star appear much closer.
Optics (4 of 5)
The microscope and the telescope
The microscope is a device for making very small objects visible. It was probably invented by a Dutch spectacle-maker, Zacharias Janssen (1580-1638), in 1609. Essentially, it is an elaboration of the simple magnifying glass. The objective - a lens with short focal length - is used to form a highly magnified image of a small object placed close to its focal point. This can be viewed directly, by means of another lens called the eyepiece. It can also be recorded directly on film or viewed via a video camera.
The telescope is used to form an enlarged image of an infinitely distant object, and the enlarged image is viewed by the observer by means of an eyepiece. The term `infinite' is used relatively in this context: compared with the length of the telescope, the distance of the object can be considered as infinite. Telescopes are often made with reflecting mirrors instead of glass lenses, as large lenses sag under their own weight, thereby introducing distortions into the image. The primary mirror is often a large concave paraboloid.
Fiber optics
Light can be transmitted over great distances by the use of flexible glass fibers. These fibers are usually each less than 1 mm (1/25 in) in diameter, and can be used singly or in bunches. Each fiber consists of a small core surrounded by a layer of `cladding' glass with a slightly lower refractive index. Certain rays experience total internal reflection, and this, coupled with the very low absorption of modern silica glasses, allows light to travel very long distances with little reduction in intensity. Fiber optics provide the basis of endoscopy, a medical diagnostic technique and are also used extensively in telecommunications, as light in a fiber optic cable can carry more digital (on or off) signals with less loss of intensity than a copper wire carrying electrical digital signals.
* THE HISTORY OF ASTRONOMY
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* ELECTROMAGNETISM
* ATOMS AND SUBATOMIC PARTICLES
* PHOTOGRAPHY AND FILM
* RADIO, TELEVISION AND VIDEO
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ftsTitleOverride
Optics (page 5)
ftsTitle
Recording a hologram. When the light beams from sources such as lasers overlap they produce interference fringes due to the wave nature of light. A hologram is produced by recording the fringes from the interference of two beams of laser light. The reference beam (RB) falls directly onto the film, while the object beam (OB) is reflected from the object.
Replaying a hologram. When the hologram is replayed by shining a beam of laser light onto it, the light is diffracted in such a way that it appears to come from the position of the original object. The image can be viewed from a range of angles and is a true three-dimensional reconstruction of the object.
Optics (5 of 5)
LASERS
The term `laser' is derived from the technical name for the process - Light Amplification by Stimulated Emission of Radiation. Stimulated emission is the emission of a photon - a particle of light.
When an amplifying material, such as a gas, crystal or liquid, is placed between appropriate mirrors, photons from a light beam repeatedly pass through it stimulating more photons and thus increasing their number with each pass. The additional photons all have the same frequency, phase and direction. One of the mirrors is made so that a small amount of light passes through it; this is the external laser beam, which can be continuous or pulsed. This beam can be focused onto very small areas and the intensity - the ratio of power to area - can be very great, enabling some lasers to burn through thick metal plates. Lasers have a wide variety of uses, for example in surveying, communications and eye surgery.
HOLOGRAMS
A hologram is a `three-dimensional' or stereoscopic image formed by two beams of light. Holography differs from conventional photography in that both the amplitude of the light and its phase (a measure of the relative distance the light has traveled from the object) are recorded on the film. It has many scientific uses as well as the more familiar display holograms now being used, for example, on credit cards.
* THE HISTORY OF ASTRONOMY
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* ELECTROMAGNETISM
* ATOMS AND SUBATOMIC PARTICLES
* PHOTOGRAPHY AND FILM
* RADIO, TELEVISION AND VIDEO
* TELECOMMUNICATIONS
* SEEING THE INVISIBLE
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p034-1
ftsTitleOverride
Electromagnetism (page 1)
ftsTitle
1. The magnetic field around a bar magnet can be plotted using a small compass, or by scattering iron filings on a sheet of paper placed above it.
Electro-magnetism (1 of 5)
Electromagnetism is the study of effects caused by stationary and moving electric charges. Electricity and magnetism were originally observed separately, but in the 19th century, scientists began to investigate their interaction. This work resulted in a theory that electricity and magnetism were both manifestations of a single force, the electromagnetic force.
The electromagnetic force is one of the fundamental forces of nature, the others being gravitational force and the strong and weak nuclear forces. Recently the electromagnetic and weak forces have been shown to be manifestations of an electro-weak force. Magnetism has been known about since ancient times, but it was not until the late 18th century that the electric force was identified - by the French physicist Charles Augustin de Coulomb (1736- 1806).
Magnetism
Metallic ores with magnetic properties were being used around 500 BC as compasses. It is now known that the Earth itself has magnetic properties Investigation of the properties of magnetic materials gave birth to the concept of magnetic fields, showing the force one magnet exerts on another. These lines of force can be demonstrated by means of small plotting compasses or iron filings (see diagram 1). An important feature of a magnet is that it has two poles, one of which is attracted to the Earth's magnetic north pole, while the other is attracted to the south pole. Conventionally, the north-seeking end of a magnet is called its north pole, and the other is the south pole. Magnets are identified by the fact that unlike or opposite poles (i.e. north and south) attract each other, while like poles (north and north, or south and south) repel each other.
Magnetic effects are now known to be caused by moving electric charges. Atomic electrons are in motion, and thus all atoms exhibit magnetic fields.
* STARS AND GALAXIES
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* OPTICS
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* THE EARTH'S STRUCTURE
* MEDICAL TECHNOLOGY
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ftsTitleOverride
Electromagnetism (page 2)
ftsTitle
2. Electric fields
Electro-magnetism (2 of 5)
Static electric charges
In dry weather, a woolen sweater being pulled off over the hair of the wearer may crackle; sparks may even be seen. This is caused by an electric charge, which is the result of electrons being pulled from one surface to the other. Objects can gain an electric charge by being rubbed against another material.
Experiment has shown that there are two types of charge. These are now associated with the negative and positive charges on electrons and protons respectively. Similar electric charges (i.e. two positives, or two negatives) repel each other and unlike charges (i.e. a positive and a negative) attract. (Note that the terms `positive' and `negative' are merely conventions for opposite properties.) No smaller charge than that of the electron has been detected (but see quarks).
The force of repulsion or attraction is known as the electric force. It is described by Coulomb's law, an inverse-square law similar to the law for the gravitational force. Coulomb's law states that the attractive or repulsive force (F) between two point (or spherically symmetrical) charges is given by: where k is a constant, Q1 and Q2 are the magnitudes of the charges, and r is the distance between them. The force acts along the direction of r. The unit of charge is called a coulomb (C) and is the quantity of electric charge carried past a given point in 1 second by a current of 1 ampere (see below).
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* STARS AND GALAXIES
* QUANTUM THEORY AND RELATIVITY
* WAVE THEORY
* OPTICS
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* THE EARTH'S STRUCTURE
* MEDICAL TECHNOLOGY
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Electromagnetism (page 3)
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3. How lightning is caused. In the 1st stage, a net charge collects on the top of the cloud, equal in polarity to the charge collecting on the surface of the earth. A net charge of opposite polarity collects at the base of the cloud. In the 2nd stage, if the cloud has become very large, discharges start to take place. A 'leader' (invisible to the naked eye) opens an ionized channel through the air
which will allow electricity to follow. In the 3rd stage, the lightning strikes, following along the path made by the leader.
Electro-magnetism (3 of 5)
Electric field
Arrows can be plotted to show the magnitude and direction of the magnetic force that acts at points around a magnet (see diagram 1), or the electric force that acts on a unit charge at each point. In the latter case, such a map (see diagram 2) would show the distribution of the electric field intensity. It is measured in terms of a force per unit charge, or newtons per coulomb.
In the same way that a mass may have gravitational potential energy because of its position, so a charge can have electrical potential energy. This potential per unit charge is measured in volts (V), named after the Italian physicist Alessandro Volta (1745-1827). The volt may be defined as follows: if one joule is required to move 1 coulomb of electric charge between two points, then the potential difference between the points is 1 joule per coulomb = 1 volt.
The electrical potential may vary with distance. This change may be measured in volts per meter (V m-1). The Earth's surface is negatively charged with an average electric field over the whole of its surface of about 120 V m-1. In the presence of thunderclouds or where the air is highly polluted the field may be much greater. This field is maintained partially by thunderstorms, which transfer negative charge to the Earth (see diagram 3). Dry air can only allow an electric field of 3 x 106 V m-1 to build up before there is a sudden breakdown - a lightning flash. If water droplets are present, then the value is lower, perhaps 1 x 106 V m-1.
Electric current, conductors and insulators
Electric current consists of a flow of electrons, usually through a material but also through a vacuum, as in a cathode-ray tube in a TV set. Current flows when there is a potential difference or voltage (see above) between two ends of a conductor (see below). Conventional current flows from the positive terminal to the negative terminal. However, electron flow is in fact from negative to positive.
For measurement purposes, an electric current is defined as the rate of flow of charge. The unit of electric current is the ampere (A), often abbreviated to amp: 1 ampere = 1 coulomb per second.
The ampere is named after the French physicist Andru Marie Ampere (1775-1836), who pioneered work on electricity and magnetism.
A material that will allow an electric current to flow through it is a conductor. The best conductors are metals. A material that will not allow an electric current to flow is an insulator. Effective insulators include rubber, plastic and porcelain.
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Lightning causesick
**++*
p034-4
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Electromagnetism (page 4)
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4. Oersted's discovery. When a current flows through a wire, magnetic compasses on a plane at right angles to the wire will be deflected until they are tangential to a circle drawn around the wire.
Electro-magnetism (4 of 5)
Electromagnetic fields
In 1820 the Danish physicist Hans Christiaan Oersted (1777-1851) discovered that a copper wire bearing an electric current caused a pivoted magnetic needle to be deflected until it was tangential to a circle drawn around the wire (see diagram 4). This was the first connection to be established between the electrical and magnetic forces. Oersted's work was developed by the Frenchmen Jean-Baptiste Biot (1774-1862) and Fulix Savart (1791-1841), who showed that the field strength of a current flowing in a straight wire varied with the distance from the wire. Biot and Savart were able to find a law relating the current in a small part of the conductor to the magnetic field. Ampere, at about the same time, found a more fundamental relationship between the current in a wire and the magnetic field about it.
We now believe that the Earth's magnetic field is generated by the motion of charged particles in the liquid iron part of the core. This is known as the dynamo theory.
From Newton's third law and Oersted's observation it might be expected that a magnetic field can exert a force on a moving charge. This is observed if a magnet is brought up close to a cathode-ray tube in a TV set. The beam of electrons moving from the cathode to the screen is deflected. The force acts in a direction perpendicular to both the magnetic field and the direction of electron flow. If the magnetic field is perpendicular to the direction of the electrons, then the force has its maximum value. This is the second way in which the electric and magnetic properties are linked.
Electromagnetic induction
The next advance came in 1831, when the English physicist Michael Faraday (1791-1867) found that an electric current could be induced in a wire by another, changing current in a second wire. Faraday published his findings before the US physicist Joseph Henry (1797-1878), who had first made the same discovery. Faraday showed that the magnetic field at the wire had to be changing for an electric current to be produced. This may be done by changing the current in a second wire, by moving a magnet relative to the wire, or by moving the wire relative to a magnet. This last technique is that employed in a dynamo generator, which maintains an electric current when it is driven mechanically. An electric motor uses the reverse process, being driven by electricity to provide a mechanical result.
Maxwell's theory
The work of the Scottish physicist James Clerk Maxwell (1831-79) on electromagnetism is of immense importance for physics. It united the separate concepts of electricity and magnetism in terms of a new electromagnetic force. Maxwell ex tended the ideas of Ampre, then in 1864 he proposed that a magnetic field could also be caused by a changing electric field. Thus, when either an electric or magnetic field is changing, a field of the other type is induced. Maxwell predicted that electrical oscillations would generate electromagnetic waves, and he derived a formula giving the speed in terms of electric and magnetic quantities. When these quantities were measured he calculated the speed and found that it was equal to the speed of light in a vacuum. This suggested that light might be electromagnetic in nature - a theory that was later confirmed in various ways. Thus, when an electric current in a wire changes, electromagnetic waves are generated, which will be propagated with a velocity equal to that of light.
The electric and magnetic field components in electromagnetic waves are perpendicular to each other and to the direction of propagation. The existence of electromagnetic waves was demonstrated experimentally in 1887 by the German physicist Heinrich Rudolf Hertz (1857-94) - who also gave his name to the unit of frequency. In his laboratory, Hertz transmitted and detected electromagnetic waves, and he was able to verify that their velocity was close to the speed of light.
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Electromagnetism (page 5)
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Electro-magnetism (5 of 5)
THE ELECTROMAGNETIC SPECTRUM
Prior to Maxwell's discoveries it had been known that light was a wave motion, although the type of wave motion had not been identified. Maxwell was able to show that the oscillations were of the electric and magnetic field. Hertz's waves had a wavelength of about 60 cm; thus they were of much longer wavelength than light waves. Nowadays we recognize a spectrum of electromagnetic radiation that extends from about 10-15 m to 109 m. It is subdivided into smaller, sometimes overlapping, ranges. The extension of astronomical observations from visible to other electromagnetic wavelengths has revolutionized our knowledge of the universe.
Radio waves have a large range of wavelengths - from a few millimeters up to several kilometers.
Microwaves are radio waves with shorter wavelengths, between 1 mm and 30 cm. They are used in radar and microwave ovens.
Infrared waves of different wavelengths are radiated by bodies at different temperatures. (Bodies at higher temperatures radiate either visible or ultraviolet waves.) The Earth and its atmosphere, at a mean temperature of 250 K (-23 deg C or -9.4 deg F) radiates infrared waves with wavelengths centered at about 10 micrometers ( m) or 10-5 m (1 m = 10-6 m).
Visible waves have wavelengths of 400-700 manometers (nm; 1 nm = 10-9 m). The peak of the solar radiation (temperature of about 6000 K / 6270 deg C / 11 323 deg F) is at a wavelength of about 550 nm, where the human eye is at its most sensitive. Ultraviolet waves have wavelengths from about 380 nm down to 60 nm. The radiation from hotter stars (above 25 000 K / 25 000 deg C / 45 000 deg F is shifted towards the violet and ultraviolet parts of the spectrum. X-rays have wavelengths from about 10 nm down to 10-4 nm.
Gamma rays have wavelengths less than 10-11 m. They are emitted by certain radio active nuclei and in the course of some nuclear reactions.
Note that the cosmic rays continually bombarding the Earth from outer space are not electromagnetic waves, but high-speed protons and x-particles (i.e. nuclei of hydrogen and helium atoms) together with some heavier nuclei.
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Electricity in Action (page 1)
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1. A simple cell. The lamp lights but soon goes out because bubbles of hydrogen cling to the copper electrode, thus decreasing the output of the cell. This is known as polarizing. The zinc electrode is eventually eaten away.
Electricity in Action (1 of 6)
There have been several key advances in the application of electricity towards developing our civilization. The first two were the dynamo and the electric motor. The dynamo provided a way of producing electricity in large quantities, and the electric motor provided a way of converting electric current into mechanical work. >>The evolution of electromagnetic theory provided the basis for the modern communications industry through radio and television, while the miniaturization of electronic components using semiconductor materials enabled powerful computers to be built for control purposes and to handle large amounts of information.
Batteries and cells
Electric current is the flow of electrons through a conductor. The first source of a steady electric current was demonstrated by the Italian physicist Alessandro Volta (1745-1827) in 1800. His original voltaic pile used chemical energy to produce an electric current. The pile consisted of a series of pairs of metal plates (one of silver and one of zinc) piled on top of each other, each pair sandwiching a piece of cloth soaked in a dilute acid solution (see also p.51).
The same principle is still used today (see diagram 1). The plates are called electrodes and must be made of dissimilar metals. Alternatively, one may be made of carbon. The positive electrode - the one from which electrons flow inside the cell - is called the anode. The negative electrode is the cathode. The acid solution is called the electrolyte and in a dry cell is absorbed into a paste (see diagram 2).
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Electricity in Action (page 2)
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2. A dry cell, the basis for modern batteries.
Electricity in Action (2 of 6)
A single cell can normally produce only a small voltage, but a number of them connected in a series (positive to negative) will give a higher voltage. A series of cells connected in this way is called a battery. Some batteries, known as accumulators, are designed so that they can be `recharged' by the passage of an electric current back through them. Similar principles as those used in cells are used in electrolysis and electroplating.
* QUANTUM THEORY AND RELATIVITY
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Electricity in Action (page 3)
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3. Connecting circuits
Electricity in Action (3 of 6)
Circuitry
A circuit is a complete conductive path between positive and negative terminals; conventionally current flows from positive to negative, although the direction of electron flow is actually from negative to positive. When electrical components such as bulbs and switches are joined end to end the arrangement is a series connection. When they are connected side by side, this is called parallel connection (see diagram 3).
Resistance
When an electric current passes through a conductor there is a force that acts to reduce or resist the flow. This is called the resistance and is dependent upon the nature of the conductor and its dimensions. The unit of resistance is the ohm (), named after the German physicist Georg Simon Ohm (1787-1854). He discovered a relationship between the current (I ), voltage (V ) and resistance (R) in a conductor: V = IR.
This is known as Ohm's law.
Power
Power is the rate at which a body or system does work (work and its unit, the joule, are defined on p. 24). The power in an electric conductor is measured in watts (W), named after the British engineer James Watt (1736-1819). One watt is one joule per second, or the energy used per second by a current of one amp flowing between two points with a potential difference of one volt (volts and amps are defined on p. 34). In an electric conductor the power (W ) is the product of the current (I ) and the voltage (V ).
Lighting, heating and fuses
A light bulb consists of a glass envelope containing an inert (`noble') gas, usually argon, at low pressure. The bulb has two electrodes connected internally by a filament - a fine coiled tungsten wire of high resistance. The passage of a suit able electric current through the filament will raise its temperature sufficiently to make it glow white hot (2500 deg C / 4500 deg F). The inert gas prevents the filament from evaporating. The efficiency of filament lamps is low. Gas discharge lamps are much more efficient. They consist of a glass tube with electrodes sealed into each end. The tube is filled with a gas such as neon, sodium or mercury vapor, which can be excited (see discontinuity, pp. 26-7) to emit light by the application of a high voltage to the electrodes.
When electrons pass through a wire they cause the atoms in it to vibrate and generate heat - the greater the resistance, the greater the heat generated. This effect is used in electric heating devices. An electric radiant heater glows red hot. The temperature reached by using a special tough resistance wire is 900 deg C (1650 deg F). The connecting wires are of low resistance and stay cool.
If a small resistance consisting of wire with a low melting point is connected in a circuit the amount of current that can flow will be limited by that resistance. If too much current flows the resistance will overheat and melt, breaking the circuit. This resistance is called a fuse and can be used as a device to protect circuits from current overload.
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p036-4
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Electricity in Action (page 4)
ftsTitle
4. A simple generator. As the coiled wire rotates within the magnetic field, an electrical current is induced within the circuit, illuminating the lamp. This simple device shows the basic principle by which all electricity is generated.
Electricity in Action (4 of 6)
Alternating current and direct current
There are two types of current electricity. The type produced by a battery is direct current (DC), in which there is a constant flow of electrons in one direction. The type used in most electrical appliances is alternating current (AC), in which the direction of flow of electrons alternates. The frequency of alternating current can vary over an enormous range. The electric mains operate at 50 Hz (cycles per second) in the UK and Europe, and at 60 Hz in the USA. Most of today's electricity is produced by AC generators. These were developed following Faraday's discovery of the induction of a current in a circuit as a result of a changing magnetic field.
Generators and motors
A dynamo (see diagram 4) is an electrical current generator, consisting of a coil that is rotated in a magnetic field by some external means. The source of the rotation may be a turbine in which blades are moved by the passage through them of water, as in a hydroelectric plant, or steam, produced from a boiler heated by nuclear fission or by burning fossil fuels. Wind turbines spin as a result of the passage of air through the large rotors. Different types of generator produce either AC or DC current, while alternators (used to charge car batteries) produce AC current that is then rectified to DC current using semiconductor diodes (see below).
An electric motor is a similar device to a generator, but works in reverse. An electric current is applied to the coil windings, causing rotation of the armature, which consists of a shaft on which are mounted electromagnet windings.
Electron emission
If the filament of a light bulb is heated, the energy of some of the electrons in the filament is greatly increased by thermal motion, although the average increase for all the electrons is very small. If their energy reaches an adequate level, many are able to escape; this process is called thermionic emission. If another electrode is put in the evacuated bulb and placed at a higher potential than the filament, this will act as an anode and will attract electrons towards it. A current will then flow in an external circuit; the device thus formed is called a diode (see diagram 5). If a third electrode in the shape of a grid is placed in the tube between the filament and the anode, then the anode current is so sensitive to changes in the grid voltage that the whole device, called a triode, can act as an amplifier (see diagram 6).
The photoelectric effect occurs when light of a sufficiently high frequency shines onto a metal, causing electrons to be emitted from its surface.
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Electric, A simple generator
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p036-5
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Electricity in Action (page 5)
ftsTitle
5. A diode. Electrons emitted by the heated cathode flow through the vacuum to the anode. A diode allows passage of electricity in one direction only.
6. A triode. The potential on the grid controls the flow of the electrons between the cathode and anode. It can act as a switch or an amplifier.
Electricity in Action (5 of 6)
Conductors and semiconductors
A metal consists of an array of positive ions in a `sea' of free electrons. The electrons move randomly with mean speeds of around 10 to the power of 6 m s-1. When a potential difference is applied across a metal a small drift velocity is added. The metal atoms are thought to give up one or more electrons, which can then migrate freely through the material. These electrons move in a zigzag manner along a conductor. As a result their typical velocity, called the drift velocity, is small, in the order of 10-4 m s-1. Thus it would take more than an hour to move one meter. Note that the electric signals that drive the electrons travel with a speed in the order of 108 m s-1 in some circuits.
This classical picture of electron conduction explains some but not all conduction phenomena. For these a quantum mechanical model is required. This model explains the basis of semiconductors, which now play such an important part in electronics.
Metals are good conductors of electricity because there are always many unoccupied quantum states into which electrons can move. Non-metallic solids and liquids have nearly all their quantum states occupied by electrons, so it is difficult to produce large currents. If the numbers of unoccupied states and of electrons free to move into them are small the material is an insulator. If there are more free electrons and unoccupied states the substance is called a semiconductor.
Semiconductors have a charge-carrier density that lies between those of conductors and insulators. Two metal-like elements, silicon and germanium are the two semiconductors used most frequently. These may be `doped' with an impurity to modify their conduction behavior - n-type doping increases the number of free electrons, p-type increases the number of unoccupied states. If the doping results in the charge carriers being negative electrons, then the result is an n-type semiconductor. If electron deficiencies or holes are the charge carriers, then the result is a p-type semiconductor.
Most semiconductor devices are made from materials that are partly p-type and partly n-type. The boundary between them is known as a p-n junction. Such a device, called a semiconductor diode, will act as a rectifier, a device used to convert alternating current to direct current.
In some materials, such as gallium arsenide, a p-n junction will emit light when ever an electric current passes through it. This device is called a light-emitting diode. These are used in digital displays in clocks and radios. The light is emitted when an electron and hole meet at the junction and annihilate each other - they cancel each other out.
Solar cells
The photovoltaic effect occurs when light is absorbed by a p-n or n-p junction. Electrons are liberated at the junction by an incident photon and diffuse through the n-type region. The hole drifts through the p-type layer until it recombines with an electron flowing round the external circuit. The first practical photovoltaic device - called a solar cell - was made in 1954. In essence a solar cell is a light-emitting diode acting in reverse - it converts light into electric current, which is the basis of solar power.
Transistors
A transistor consists of semiconductor material in n-p-n or p-n-p form. The middle part is the base and the ends are called the emitter and collector. An integrated circuit consists of many transistors, rectifiers or other components embedded in a chip of silicon.
Superconductivity
Superconductivity was discovered by the Dutch physicist Kamerlingh Onnes (1853-1926) in 1911. Below a certain critical temperature, various metals show zero resistance to current flow. Once a current is started in a closed circuit, it keeps flowing as long as the circuit is kept cold. The critical temperature for aluminum is 1.19 K(-272 deg C / -457 deg F), and similar values hold for other metals. Some alloys have higher critical temperatures. Up to 1986 the highest transition temperature known was about -248 deg C (-414 deg F). More recently a new class of copper oxide and other materials have shown superconductivity up to at least -148 deg C (-234 deg F). These developments promise enormous savings in energy.
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p036-6
ftsTitleOverride
Electricity in Action (page 6)
ftsTitle
Electricity in Action (6 of 6)
TRANSFORMERS
If two insulated coils of wire are wound on the same soft iron core and an alternating current is passed through one of the coils, a current will be induced in the other coil. The ratio of the numbers of turns on the input coil (N1) and the output coil (N2) will determine the ratio of the output voltage (V2) to the input voltage (V1). The relationship is:
V2 N2
-- = --
V1 N1
In this way transformers can either step voltage up or step it down. Note that they have the reverse effect on current. This principle is used for efficient long-distance power transmission.
* QUANTUM THEORY AND RELATIVITY
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p038-1
ftsTitleOverride
Atoms and Subatomic Particles (page 1)
ftsTitle
1. The Rutherford model of atomic structure. Negatively charged electrons orbit a positively charged nucleus.
Atoms and Subatomic Particles (1 of 5)
Of the fundamental forces that are important in the natural world, the gravitational force is the dominant long-range force when the motion of planets and other celestial bodies is considered. When the smallest entities are investigated, the other fundamental forces - the electromagnetic force, the strong force (which holds together the atomic nucleus) and the weak force (which is involved in nuclear decay) - become important.
The word atom is derived from an ancient Greek word for a particle of matter so small it cannot be split up. In his atomic theory of 1803, the British chemist John Dalton (1766-1844) defined the atom as the smallest particle of an element that retained its chemical properties. Various phenomena could be explained using this hypothesis - which still holds good today.
Atomic structure
However, no physical description of the atom was available until after the discovery of the electron in 1897 by the British physicist J.J. Thompson (1856-1940). The nuclear atom was proposed by the English physicist Ernest Rutherford (1871-1937) in 1911. His model consists of a small but dense central nucleus, which is positively charged, orbited by negatively charged electrons. The nucleus contains over 99.9% of the mass of the atom, but its diameter is of the order of 10-15 m - compared to the much larger size (about 10-10 m) of the atom.
The electron was first recognized by its behavior as a particle. In 1923 a wave-particle duality for atomic particles - analogous to the concept of the wave-particle duality of light proposed by the French physicist Louis Victor de Broglie (1892-1987) - was put forward. The wavelength of a particle would be equal to the Planck constant divided by its momentum. As the wavelength is dependent on momentum it can take any value. For an electron the wavelength can be of the order of the atomic diameter. This led to the development of the electron microscope. At suitable energy levels the wavelength of electrons and neutrons can be equivalent to the atomic spacing in solids. Thus a crystal can be used as a diffraction grating (as for X-rays). This has led to a better understanding of the way in which the electrons orbit the atomic nucleus.
* QUANTUM THEORY AND RELATIVITY
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* ENERGY 1
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p038-2
ftsTitleOverride
Atoms and Subatomic Particles (page 2)
ftsTitle
2. The Rutherford-Bohr model of atomic structure. The number of electrons orbiting the nucleus is equal to the number of positively charged protons within the nucleus. The number of electrons within each shell is also limited
no more than 2 in the first shell, 8 in the second, 18 in the third, etc.
Atoms and Subatomic Particles (2 of 5)
The Danish physicist Niels Bohr (1885-1962) had suggested that electrons were allowed to move in circular orbits or shells around the nucleus, but that only certain orbits were allowable. This theory was able to explain many of the features of the spectrum of light emitted by excited hydrogen atoms. The wavelengths of the spectral lines are related to the energy levels of the allowed orbits. The wave theory of the electron provided a reason for the allowed orbits. These would be those whose circumference was a multiple of the electron's wavelength. When Rutherford showed experimentally that an atom must consist of a small nucleus surrounded by electrons, there was a fundamental problem. To avoid collapsing into the nucleus, the electrons would have to move in orbits - as Bohr had proposed. This means that they must have continuous acceleration towards the nucleus. But, according to the electromagnetic theory, an accelerated charge must radiate energy, so no permanent orbit could exist. Bohr therefore argued that energy could not be lost continuously but only in quanta (discrete amounts) equivalent to the difference in energies between allowed orbits. Thus light would be emitted when an electron jumps from one allowed level to another of lower energy.
* QUANTUM THEORY AND RELATIVITY
* ELECTROMAGNETISM
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENERGY 1
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p038-3
ftsTitleOverride
Atoms and Subatomic Particles (page 3)
ftsTitle
3. How atoms emit light
Atoms and Subatomic Particles (3 of 5)
Nuclear structure
With the exception of the hydrogen atom, which only contains one proton, atomic nuclei contain a mixture of protons and neutrons, collectively known as nucleons. The proton carries a positive charge, equal in magnitude to that of the negatively charged electron. The neutron is of similar size but is electrically neutral. Each has a mass about 1836 times that of the electron (which has a rest mass of 9.11 x 10-31 kg). The protons and neutrons in the atomic nucleus are held tightly together by the strong nuclear force, which overcomes the much weaker electromagnetic force of repulsion between positively charged protons.
The mass of a nucleus is always less than the sum of the masses of its constituent nucleons. This is explained using the relationship derived by Einstein. If the nucleus is to be separated into protons and neutrons then the strong nuclear force needs to be overcome and energy has to be supplied to the nucleus - from an external source - to break it up. This energy is called the binding energy and is related to the mass defect (the difference between the masses of the nucleus and its component parts). Those nuclei with large binding energies per nucleon are most stable; these have about 50-75 nucleons in the nucleus.
* QUANTUM THEORY AND RELATIVITY
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p038-4
ftsTitleOverride
Atoms and Subatomic Particles (page 4)
ftsTitle
4. Nuclear fission. A neutron bombards the uranium-235 nucleus, causing it to split and release energy when the strong nuclear force is broken. Two lighter nuclei are formed and these are also radioactive. The neutrons released may bombard and split other nuclei
further fission can take place. A chain reaction will be set up if the mass of uranium-235 is above a certain level
the critical mass.
Atoms and Subatomic Particles (4 of 5)
Fission and fusion
Nuclear power comes from either of two processes - fission and fusion, which are both forms of nuclear reaction. In the fission process a large nucleus, such as uranium-235 (235 U), splits to form two smaller nuclei that have greater binding energies than the original uranium. Thus energy is given out in the process. Fission is used in nuclear reactors and in atomic weapons. There are other isotopes in addition to uranium-235, such as plutonium-239, that give rise to fission. In the fusion process, two light nuclei fuse together to form two particles, one larger and one smaller than the original nuclei. Usually one of them is sufficiently strongly bound to give a great release of energy. The fusion of hydrogen to form helium is a power source in stars such as the Sun, although the solar fusion process differs in detail from the simpler process described. Nuclear fusion is the basis of the hydrogen bomb, and research is continuing into the possible use of fusion in power generation.
Radioactivity
Radiation - either as a spontaneous emission of particles or as an electromagnetic wave - may occur from certain sub stances. This is radioactivity. The three types of radiation are from alpha decay, beta decay and gamma decay.
Alpha ( ) decay produces nuclei of helium that each contain two neutrons and two protons. They are called alpha-particles and are formed in spontaneous decay of the parent nucleus. Thus uranium-238 decays to thorium-234 with emission of an alpha-particle.
In beta ( ) decay the emitted particles are either electrons or positrons (identical to the electron but with a positive charge). The parent nucleus retains the same number of nucleons but its charge varies by plus or minus 1. In these processes another kind of particle - either a neutrino or an antineutrino - is produced. The neutrino has no charge (the word means 'little neutral one') and a mass that - if it could be measured at rest - would probably be zero. The relativistic mass can, however, be significant, as the speed - with respect to any observer - is that of electromagnetic radiation.
In gamma ( ) decay high-energy photons may be produced in a process of radioactive decay if the resultant nucleus jumps from an excited energy state to a lower energy state. The rate at which radioactive decay takes place depends only on the number of radioactive nuclei that are present. Thus the half-life, or the time taken for half a given number of radioactive nuclei to decay, is characteristic for that type of nucleus. The isotope carbon-14 has a half-life of 5730 years, and measurement of its decay is used in carbon-dating of organic material. Decay can result in a series of new elements being produced, each of which may in its turn decay until a stable state is achieved.
* QUANTUM THEORY AND RELATIVITY
* ELECTROMAGNETISM
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENERGY 1
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Atoms and Subatomic Particles (page 5)
ftsTitle
5. Nuclear fusion occurs when two small nuclei collide and combine, breaking the weak nuclear force and releasing energy. The reaction shown involves nuclei of deuterium and tritium (isotopes of hydrogen) combining to produce helium (a waste product), a neutron, and released energy. This type of reaction releases considerably more energy than a fission process for a given mass of material. However, the neutrons released have to be contained or controlled in some way.
Atoms and Subatomic Particles (5 of 5)
Nuclear particles
Over 200 elementary particles are now known. They may be divided into two types: hadrons and leptons. Hadrons (from the Greek for 'bulky') are heavy particles that are affected by the strong force. Leptons (from the Greek for 'small') are generally light particles, such as electrons and neutrinos, that are not subject to the strong force. A further very important distinction is that between fermions (Fermi-Dirac particles) and bosons (Bose-Einstein particles).Fermions have a permanent existence, whereas bosons can be produced and destroyed freely, provided the laws of conservation of charge and of mechanics are obeyed. Leptons are fermions. Every type of particle is thought to have a companion antiparticle, that is, a particle with the same mass but opposite in some other characteristic such as charge. Thus the positron with positive charge is the antiparticle of the negatively charged electron. Some particles such as the photon may be their own antiparticles. Whilst the leptons are thought to be fundamental particles, the hadrons are thought to be made up of quarks (a word borrowed from James Joyce's novel Finnegans Wake). Quarks may have fractional electrical charge. It is probable that free quarks do not exist. If three quarks combine, the resulting hadron is called a baryon; if a quark and an antiquark combine the result is called a meson. A meson is a boson; it is a short-lived particle that jumps between protons and neutrons, thus holding them together. In the same way that Mendeleyev's table of chemical elements predicted new elements such as gallium and germanium that were subsequently discovered, so a pattern of hadrons may be drawn up based on combinations of different types of quark. This pattern is called the eight-fold way - a term borrowed from Buddhism. It predicted the existence of the omega-particle (-particle), the discovery of which in 1963 helped to validate the theory. There are believed to be six types or flavors of quark - up, down, charmed, strange, top and bottom. Evidence for the existence of all except the top quark is now available.
Quarks carry electrical charge and another type of charge called color. The force associated with the color charge binds the quarks together and is thought to be the source of the strong force binding the hadrons together. Thus the color force is the more fundamental force. The weak force is associated with the radio active beta-decay of some nuclei. It has been shown - in the theory of the electroweak force - that the electromagnetic and weak forces are linked. This theory predicted the existence of the W and Z deg particles, which were discovered at the CERN nuclear accelerator at Geneva during 1982-83.
Nuclear accelerators
Accelerators are large machines that accelerate particle beams to very high speeds, so enabling research into particle physics. Electric fields are used to accelerate the particles, either in a straight line (linear accelerator) or in a circle (cyclotron, synchrotron or synchrocyclotron). Powerful magnetic fields are used to guide the beams. Energy levels of the particles may be as high as several hundred giga electronvolts. An electronvolt (eV) is the increase in energy of an electron when it undergoes a rise in potential of 1 volt: 1 eV = 1.6 x 10-19 joules (J). Nuclear accelerators have provided experimental evidence for the existence of numerous subatomic particles predicted in theory.
ISOTOPES AND NUCLEAR NOTATION
It is possible for atoms of the same element to contain equal numbers of protons but different numbers of neutrons in their nuclei - these different atoms are called isotopes. Isotopes of an element contain the same nuclear charge, and their chemical properties are identical, but they display different physical properties. An isotope may be represented in various ways, such as uranium-235, U-235 or 235U. A special notation is used to show the numbers of protons and neutrons in a nucleus. 226 Ra 88 defines a radium nucleus with 88 protons and 138 neutrons, making a total of 226 nucleons. Similarly, 14 C 6 is a carbon nucleus with 6 protons and 8 neutrons, making a total of 14 nucleons.
* QUANTUM THEORY AND RELATIVITY
* ELECTROMAGNETISM
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* NUCLEAR ARMAMENT AND DISARMAMENT
* ENERGY 1
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ftsTitleOverride
What is Chemistry? (page 1)
ftsTitle
The discovery of alcoholic fermentation is lost in the mists of time. Noah's first task after the Flood was to plant a vineyard (Genesis 9.20). The alcohol present in beers, wines and spirits is ethanol. The art of distilling alcohol from fermented juices represents an important early example of a chemical separation technique.
What is Chemistry? (1 of 6)
Alchemy, from which modern chemistry derives its name, probably had its origins in the region of Khimi, 'the land of black earth', in the Nile Delta. It was here, more than 4000 years ago, that it was first discovered that the action of heat on minerals could result in the isolation of metals and glasses with useful properties - and which could therefore be sold at a profit. The practice of alchemy spread throughout the Arab world and into Asia, gaining from the Chinese the secret of making gunpowder in the process.
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What is Chemistry? (page 2)
ftsTitle
In 1763 a clergyman in Chipping Norton, England, described the effect of willow bark for the cure of 'agues' (fevers). In Naples in 1838 the first chemical synthesis of salicylic acid was achieved
the precursor of aspirin (acetylsalicylic acid). Large-scale production of aspirin has continued since 1900; it is still the most widely used analgesic (pain-killer).
What is Chemistry? (2 of 6)
One of the aims of alchemy was the transmutation of metals: alchemists strove for a 'philosopher's stone' that could be used to convert easily corrupted 'base' metals such as iron, copper and lead into the 'noble' metal gold, which retained its luster and its commercial value. They thought that the philosopher's stone would also be the 'elixir' of immortality - that it would confer eternal health on those who possessed it. Much experimentation followed, which - al though not leading to the desired ends - led to the development of techniques that formed the basis of modern chemistry.
Alchemy became associated with mystical practices and ideas, but from the 12th century the availability of Arab writings on alchemy gradually led to the study of chemical processes using more rational techniques and ideas - although many of the original aims were retained. Indeed, even Sir Isaac Newton experimented with the transmutation of base metals into gold - relevant research, given that he was Master of the Royal Mint!
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What is Chemistry? (page 3)
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Nitrous oxide or laughing gas is a sweet-smelling, colorless, non-flammable gas, which has been used as anesthetic. Its popular name is derived from the euphoric initial effects on inhalation; in the early 1800s, the English chemist Sir Humphrey Davy (1778-1829) used to invite his poet friends
Coleridge, Southey and Wordsworth
to experience its effects.
ence its effects.
What is Chemistry? (3 of 6)
The aims of modern chemistry
In modern chemistry, the philosopher's stone has been replaced by a fundamental belief in the importance of understanding the physical laws that govern the behavior of atoms and molecules. Such an understanding has resulted in the development of methods for converting cheaply avail able and naturally occurring minerals, gases and oils into substances that have high commercial or social value.
During the last 150 years this approach has completely transformed our world. The discovery that iron could be made into steel by chemical means played a major part in the Industrial Revolution. In the 20th century, spectacular increases in the yields of cereals from an acre of farmland can be traced to the discovery in Germany in 1908 that nitrogen from air could be converted into ammonia fertilizers. Similarly, the greater understanding of the structures and reactions of carbon-based (organic) compounds has resulted in products such as medicines and synthetic fibers that affect all our lives.
The evolution of chemistry from small laboratories making new substances in tiny quantities to modern industrial processes producing millions of tons of chemicals brings its own problems. The rotten-egg smell of hydrogen sulfide in a school chemistry laboratory may be relatively harmless, but a leakage of a noxious gas, on a proportionate scale, from a chemical plant can represent a major health hazard. There is therefore a twofold responsibility in modern industrial chemistry - not only to produce the chemical products that an affluent society needs in ever-increasing quantities, but to do so in a way that does not lead to major local or global environmental effects.
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ftsTitleOverride
What is Chemistry? (page 4)
ftsTitle
Molecules such as bromoacetone react with water on the surface of the eye to produce acids that irritate the eye and cause tears to flow. Tear gas is a variant of this molecule.
| What is Chemistry? (4 of 6)
Elements and molecules
The structure of atoms serves as a convenient starting point for discussing chemical phenomena. In chemical processes, the nuclei of atoms remain unchanged - shattering at once the al-chemist's dream of transmuting elements. The great variety of known chemical compounds results from the different ways in which the electrons of atoms are able to interact either with atoms of the same kind or with atoms of a different kind. In an element, all the atoms are of the same kind, but the varying strengths of the interactions between the electrons in different types of atom means that elements have very different properties. For example, helium melts at -272 deg C (-458 deg F), whereas carbon in the form of diamond has a melting point of 3500 deg C (6332 deg F). This ability of electrons to interact between atoms is known as chemical bonding.
The elements nitrogen, oxygen, fluorine and chlorine form strong bonds, with two identical atoms linked together. They therefore exist at room temperature as gases, with pairs of linked atoms moving chaotically in space. Two or more atoms linked in this fashion are described as molecules, and a shorthand notation is used to describe their chemical identity. The atomic symbol for the element is used in conjunction with the number of atoms present to define the chemical formula of the molecule. The elements described above are therefore designated, respectively, by the formulae N2, O2, F2 and Cl2.
Other familiar elements, such as sulfur and phosphorus, form additional bonds to like atoms, and their formulae reflect this fact. Thus sulfur forms a ring of eight atoms and is described by the formula S8. As the number of atoms in the fundamental unit increases, the element is no longer a gas but becomes a solid with a low melting point; thus sulfur can be extracted from the Earth as a molten fluid.
Most elements do not form discrete molecular entities such as those described above, but have structures that are held together by chemical bonds in all directions. Most of the 109 known elements are metals, such as iron and copper, and have infinite structures of this kind. Such elements can no longer be given distinct molecular formulae and are therefore represented by the element symbol alone; thus iron, for example, is represented simply as Fe. For a more detailed discussion of the properties of metals.
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What is Chemistry? (page 5)
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North Sea gas contains methane, which
unlike coal gas
is odorless. Tiny amounts of ethanethiol
an evil-smelling chemical
are therefore added to aid its detection from open taps and gas leaks.
What is Chemistry? (5 of 6)
Chemical compounds
In chemical compounds, the atoms of more than one element come together to form either molecules or infinite structures. They are described by formulae similar to those given above for elements. For example, water has a finite structure based on one oxygen atom chemically bonded to two hydrogen atoms and is denoted by the formula H2O. Common salt (sodium chloride; NaCl) has sodium (Na) and chlorine (Cl) atoms linked together in an infinite three-dimensional lattice (similar to that formed by potassium chloride.
In a pure chemical compound, all the molecules have the same ratio of different atoms and behave in an identical chemical fashion. Thus a pure sample of water, for example, behaves identically to any other pure sample, however different their origins may be. Furthermore, the same ratios of atoms are retained irrespective of whether the compound is a solid, a liquid or a gas. For example, ice, water and water vapor all have molecules with the constitution H2O. The transformation of ice into water and then into water vapor by heating is not a chemical reaction, because the identities of the molecules do not change.
From the 109 chemical elements now known, more than 2 million chemical compounds have been made during the last 100 years. The chemist views chemistry as a set of molecular building blocks, constructing more and more complex and diverse molecular structures, the variety of which is limited only by his or her imagination. It is important to emphasize that the properties of a chemical compound are unique and not a sum of the properties of the individual elements from which it is made. For example, common salt does not have any properties remotely like those of metallic sodium, which catches fire on contact with water, or chlorine, which is a harmful yellow-green gas.
Although all compounds are unique, they can be classified into broad families based on common chemical properties. Acids, bases, salts, and oxidizing and reducing agents are examples of such families pp.. Classifications reflecting the atoms present are also useful for cataloguing purposes: for example, hydrides, chlorides and oxides indicate compounds containing hydrogen, chlorine and oxygen respectively. Another particularly important classification is that of organic compounds, which contain carbon and are not only important for life processes but make up many modern industrial chemicals such as plastics, paints and artificial fibers.
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What is Chemistry? (page 6)
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Exaltone was the first synthetic chemical to be used in the manufacture of perfumes. Now a major industry based on the production and use of chemicals with a wide range of smells. These chemicals obviate the need to extract tiny amounts of chemicals from animals, such as musk from deer and civettone from civets.
What is Chemistry? (6 of 6)
Mixtures
When elements or compounds are mixed together but not chemically bonded, they form a chemical mixture. A mixture can be of two solids (e.g. salt and sand), two liquids, two gases or permutations of these. A mixture can be separated into its pure chemical constituents by either chemical or physical means. For example, adding water to the sand-salt mixture dissolves the salt, leaving the sand in a pure state. The salt and water is itself a mixture described as a solution, from which the pure salt can be obtained by boiling off the water.
The modern-day chemist has many other techniques for separating mixtures, such as distillation, chromatography, crystallization and electrolysis. The petrochemical industry is a prime example of how this technology can be used to convert natural gas and crude oil into a range of useful commercial and domestic products.
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Elements and the Periodic Table (page 1)
ftsTitle
Abundance of the elements in the Earth's crust.
Elements and the Periodic Table (1 of 8)
The world we see around us is made up of a limited number of chemical elements. In the Earth's crust, there are 82 stable elements and a few unstable (radioactive) ones. Among the stable elements, there are some, such as oxygen and silicon, that are very abundant, while others - the metals ruthenium and rhodium, for example - are extremely rare. Indeed, 98% of the Earth's crust is made up of just eight elements, while the rest account for only 2%.
Each element is associated with a unique number, called its atomic number. This figure represents the number of protons (positively charged particles) in the nucleus of each atom of the element. Hydrogen has one proton, so it is the first and lightest of the elements and is placed first in the Periodic Table; helium has two protons, and thus is the second lightest element and is placed second in the Table; and so we continue through each of the elements, establishing their order in the Table according to their atomic numbers.
numbers.
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Elements and the Periodic Table (page 2)
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Elements and the Periodic Table (2 of 8)
The atomic number of bismuth is 83, and this number of protons represents the upper limit for a stable nucleus. Beyond 83, all elements are unstable, although their radioactive decay may be so slow that some of them, such as thorium and uranium, are found in large natural deposits. The largest atomic number so far observed is 109, but only a few atoms of this element have been made artificially, so little is known about it. Its name is unnilennium, meaning 'one-zero-nine'.
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Elements and the Periodic Table (page 3)
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Elements and the Periodic Table (3 of 8)
Groups and blocks
When an atom is electrically neutral, the number of electrons (negatively charged particles) circling the nucleus is the same as the number of (positive) protons in the nucleus. Thus, for example, an electrically neutral atom of calcium contains 20 protons and 20 electrons. While the atomic number identifies an atom and determines its order in the Periodic Table, it is these electrons surrounding the nucleus that determine how it behaves chemically. Electrons can be thought of as moving around the nucleus in certain fixed orbits or 'shells', the electrons in a particular shell being associated with a particular energy level. With regard to an atom's chemical behavior, it is the electrons in the outer shell that are most important, and it is these that fix the group position of the atom in the Table.
The major energy levels are numbered 1, 2, 3, etc., counting outwards from the nucleus. This number is called the principal quantum number, and is given the symbol n. Each energy level can hold only a certain number of electrons; the further out it is, the more it can accommodate. This capacity is related to the value of n: the maximum number of electrons each shell can hold is 2n to the power of 2. Thus the nearest shell to the nucleus can hold only 2 electrons (2 x 1 to the power of 2), the next 8 (2 x 2 to the power of 2), then 18, then 32, and so on.
Each principal energy level is divided into smaller sub-levels, called s, p, d and f, which hold a maximum of 2, 6, 10 and 14 electrons respectively. The first principal energy level thus contains only the s sub-level; the second contains the s and p sub-levels; and so on. It is these sub-levels that identify the main blocks of the Periodic Table: thus the s-block is made up of 2 columns or groups, the p-block of 6, the d-block of 10, and the f-block of 14.
ck of 14.
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ftsTitleOverride
Elements and the Periodic Table (page 4)
ftsTitle
Elements and the Periodic Table (4 of 8)
Group position and chemical reactivity
Hydrogen has one electron in the first principal energy level, while helium has two - the maximum capacity for this level. The possession of one extra electron may seem a trivial difference, but a world of difference separates hydrogen and helium; hydrogen is very reactive and forms compounds with many other elements; helium combines with nothing. These two elements are rather exceptional in all their chemical behavior and are given a small section of their own in the Table, above groups 17 and 18 of the p-block.
Hydrogen and helium are placed on the far right of the Table so that the latter falls in the same group (group 18) as other elements - the so-called noble gases - that have full outer shells. Thus below helium we find neon, another chemically unreactive gas, which has the second principal energy level filled and is said to have an electron configuration of 2.8. Just as we find hydrogen, a highly reactive element, immediately to the left of helium, so we find another reactive element - fluorine (configuration 2.7) - to the left of neon. Fluorine (like the other elements in group 17 - the halogens) is one electron short of a full outer shell. Fluorine's tendency to combine with other elements in order to achieve a full (and so stable) outer shell makes it one of the most reactive of all the elements - so reactive that it will even combine with the noble gases krypton and xenon.
* QUANTUM THEORY AND RELATIVITY
* ATOMS AND SUBATOMIC PARTICLES
* CHEMICAL BONDS
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ftsTitleOverride
Elements and the Periodic Table (page 5)
ftsTitle
Elements and the Periodic Table (5 of 8)
The noble gases, with their stable electron arrangement, make a natural break in the arrangement of the Periodic Table. After the p sub-shell has been filled, the next electron starts another shell further out from the nucleus. This lone electron makes the elements of group 1 - the alkali metals - highly reactive, be cause they tend to lose the extra electron in order to form a full outer shell. They are indeed so reactive that some of them, such as caesium, explode when dropped into water.
The groups of the Periodic Table are numbered 1 to 18, with the f-block not included. Members of the same group have the same number of electrons in the outer shell of the atom and consequently behave in a similar manner chemically. This fact is reflected in the composition of their chemical compounds (which can in turn be explained in terms of their oxidation states). Thus, for example, the formulae of the chlorides of sodium and potassium in group 1 are NaCl and KCl, while their oxides are Na2O and K2O.
As we go from left to right across the Table, we can see particular properties change in a regular fashion. It was this periodic rise and fall in such properties as density and atomic volume that led to the term 'Periodic Table'. In fact, members of the same group often bear only a superficial chemical resemblance to one another. The image and philosophy of the periodic Table is so powerful, however, that it remains the standard starting point for learning about chemistry.
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* QUANTUM THEORY AND RELATIVITY
* ATOMS AND SUBATOMIC PARTICLES
* CHEMICAL BONDS
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* METALS
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ftsTitleOverride
Elements and the Periodic Table (page 6)
ftsTitle
Elements and the Periodic Table (6 of 8)
Valency
Russian chemist Dmitri Ivanovich Mendeleyev, the discoverer of the Periodic Table, was aware from the work of other scientists that, when the atoms of elements combined, each one had a characteristic ability to combine with a particular number of those atoms, such as hydrogen, which could combine with only one other atom. Chlorine could combine with only one hydrogen atom; oxygen could combine with two; nitrogen could combine with three; and carbon could combine with four hydrogen atoms. This ability was called valency.
By about 1870 at least sixty-three elements were known and Mendeleyev was now able to arrange these in the order of their atomic weights. When he did so and considered the valencies of the atoms, he at once discovered an interesting fact. Lithium had a valency of one; the next element, beryllium, had a valency of two; the next, boron, had a valency of three; and the next, carbon, had a valency of four. After that was nitrogen with a valency of three; oxygen, with a valency of two; and fluorine, with a valency of one. This could have been a coincidence, but Mendeleyev found that the pattern of rise and fall of valencies - a periodic characteristic - held throughout the whole table. Mendeleyev also knew that the elements fell naturally into groups with similar chemical properties - groups like fluorine, chlorine, bromine and iodine (now called the halogens) or the metals lithium, sodium and potassium. When he arranged his list - still in order of atomic weight - in vertical columns of identical valencies, he found to his astonishment that all the elements in each column fell into such groups. Chlorine was under fluorine, bromine was under chlorine and iodine was under bromine. The table was thus aptly called the Periodic Table.
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ftsTitleOverride
Elements and the Periodic Table (page 7)
ftsTitle
Elements and the Periodic Table (7 of 8)
THE HISTORY OF THE PERIODIC TABLE
The discovery of the Periodic Table was made possible by an Italian chemist, Stanislao Cannizzaro (1826-1910), who in 1858 published a list of fixed atomic weights (now known as relative atomic masses) for the 60 elements that were then known. By arranging the elements in order of increasing atomic weight, a curious repetition of chemical properties at regular intervals was revealed. This was noticed in 1864 by the English chemist John Newlands (1838-98), but his 'law of octaves' brought him nothing but ridicule. It was left to the Russian chemist Dmitri Ivanovich Mendeleyev (1834-1907) to make essentially the same discovery five years later. What Mendeleyev did, however, was so much more impressive that he is rightly credited as the true discoverer of the Periodic Table.
While working on his Principles of Chemistry in 1869, Mendeleyev wrote the names and some of the main features of the elements on individual cards, to help establish a suitable order in which to discuss their chemistry. It was while arranging this pack of cards in different ways that he stumbled upon the pattern we now recognize as the Periodic Table. Mendeleyev laid out his cards in order of the atomic weights of the elements, placing together elements that formed similar oxides. By arranging similar elements in columns, he established the arrangement of the Table that has been followed ever since.
* QUANTUM THEORY AND RELATIVITY
* ATOMS AND SUBATOMIC PARTICLES
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* METALS
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Elements and the Periodic Table (page 8)
ftsTitle
Dmitri Ivanovich Mendeleyev
Elements and the Periodic Table (8 of 8)
THE HISTORY OF THE PERIODIC TABLE (continued)
Mendeleyev's genius lay in the fact that he recognized that there was an underlying order to the elements - he did not design the Periodic Table, he discovered it. If he was right, he knew that there should be places in his table for new elements. He was so confident in his discovery that he predicted the properties of these missing elements - and his predictions were subsequently shown to be accurate. In some cases, Mendeleyev also swapped the order of atomic weights, so that similar elements appeared in the same groups. This apparent anomaly was not explained until 1913, when the theory of isotopes was put forward.
Since 1869, when Mendeleyev published his table, a further 40 elements have been found or produced by nuclear reactions, and the Periodic Table has been redesigned to accommodate them. Mendeleyev lived long enough to learn of the discovery of the electron, but not long enough to know how the arrangement of electrons about the nucleus of the atom explains the structure of the Table.
* QUANTUM THEORY AND RELATIVITY
* ATOMS AND SUBATOMIC PARTICLES
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* METALS
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Chemical Bonds (page 1)
ftsTitle
Chemical Bonds (1 of 6)
Although there are only 109 known elements, there are millions of chemical substances found in nature or made artificially. These substances are not simply mixtures of two or more elements: they are specifically determined chemical compounds, formed by combining two or more elements together in a chemical reaction. The chemical 'glue' that holds these compounds together is known as chemical bonding.
The properties of compol cube containing just 27 ions.highly reactive, others inert; some are solids with high melting points, others are gases. Furthermore, the properties of a compound are generally very different from those of its constituent elements. To understand how and why these differences arise, we need to understand the different types of chemical bond.
Ionic bonding
The atoms of the element neon have a full outer shell of electrons, with the electron configuration 2.8. This arrangement is very stable and neon is not known to form chemical bonds with any other element. An atom of the element sodium (Na) has one more electron than neon (configuration 2.8.1), while an atom of the element fluorine (F) has one electron less (configuration 2.7). If an electron is transferred from a sodium atom to a fluorine atom, two species are produced with the same stable electron configuration as neon. Unlike neon, however, the species are charged and are known as ions. The sodium atom, having lost a (negative) electron, has a net positive charge and is known as a cation (written Na+), while the fluorine atom, having gained an electron, has a net negative charge and is called a fluoride anion (written F-).
When oppositely charged ions such as Na+ and F- are brought together, there is a strong attraction between them; a large amount of energy is released - the same amount of energy as would have to be supplied in order to separate the ions again. This force of attraction is called an ionic (or electrovalent) bond. The energy released more than compensates for the energy input required to transfer the electron from the sodium atom to the fluorine atom. Overall there is a net release of energy and a solid crystalline compound - sodium fluoride (NaF) - is formed. The structure of a similar ionic compound - potassium chloride (KCl) - is illustrated in the box.
Atoms that have two more electrons than the nearest noble gas (such as magnesium, configuration 2.8.2) or two less (such as oxygen, 2.6) also form ions having the noble-gas configuration by transfer of electrons - in this case Mg2+ and O2-. The ionic compound magnesium oxide (MgO) has the same arrangement of ions as NaF, but since the ions in MgO have a greater charge, there is a stronger force between them. Thus more energy must be supplied to overcome this force of attraction, and the melting point of MgO is higher than that of NaF. Al though the ions are fixed in position in the solid crystal, they become free to move when the solid is melted. As a liquid, therefore, the compound becomes electrolytic and is able to conduct electricity.
Many other more complex ionic structures are known. The formula of any ionic compound can be worked out by balancing the charges of its ions. For example, Mg2+ and F- form MgF2, while Na+ and O2- form Na2O.
m Na2O.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Bonds (page 2)
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/ / Chemical Bonds (2 of 6)
Covalent bonding
If we bring together two fluorine atoms, each with seven outer electrons (one less than neon), the formation of two ions with the noble-gas configuration is not possible by transfer of electrons. If, however, they share a pair of electrons - one from each atom - then both effectively achieve the noble-gas configuration and a stable molecule results: There is a force of attraction between the shared pair of electrons and both positive nuclei, and this is what is known as a covalent bond. The stronger the attraction of the nuclei for the shared pair, the stronger the bond.
An atom of oxygen, having two electrons less than neon, must form two covalent bonds to attain a share in eight electrons. For example, a molecule of water (H2O), consisting of two hydrogen atoms (H) and one oxygen atom (O), has two covalent O-H bonds. Another way for oxygen to achieve the stable noble-gas configuration is to form two bonds to the same atom. Thus two oxygen atoms bond covalently to one another by sharing two pairs of electrons. This is known as a double bond.
Like oxygen, sulfur (S) has six outer electrons and again needs to form two bonds to attain a share in eight electrons. There are two ways in which sulfur atoms join together - either in rings of eight atoms (S8) or in long chains of many atoms bonded together. The different forms in which elemental sulfur exist are known as allotropes; other elements found in allotropic forms include carbon (graphite, diamond and newly discovered buckminsterfullerene) and oxygen.
Atoms of nitrogen (N), containing five outer electrons, need to form three covalent bonds to attain a share in eight electrons. This may be done, for example, by forming one bond to each of three hydrogen atoms, to give ammonia (NH3). Another possibility is to form all three bonds to a second nitrogen atom, which produces a nitrogen molecule (N2), containing a triple covalent bond. The carbon atom (C), which has four outer electrons, needs to form four bonds to attain the noble-gas configuration. Thus a carbon atom forms one bond to each of four hydrogen atoms to give methane. Although carbon is not known to form a quadruple bond to another carbon atom, some other elements, such as the heavy metal rhenium, do form such quadruple bonds. (For more on carbon bonding.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Bonds (page 3)
ftsTitle
The allotropes of sulfur. One type of sulfur crystal, known as rhombic sulfur, contains rings of eight atoms.
Chemical Bonds (3 of 6)
Giant molecules
Although two carbon atoms do not form a quadruple bond to one another, carbon atoms can combine to form a giant crystal lattice in which each atom is bonded to four others by single covalent bonds. This is the structure of diamond, one of the allotropes of elemental carbon. Many other elements and compounds exist as giant covalent crystal lattices, including quartz, which is a form of silicon dioxide (SiO2). Crystals of these substances contain many millions of atoms held together by strong covalent bonds, so that a large amount of energy is needed to break them. Thus these substances all have high melting points and are hard solids.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Bonds (page 4)
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The allotropes of sulfur. When this is melted and poured into water, plastic sulfur is formed, containing long tangled zigzag chains of covalently bonded sulfur atoms.
Chemical Bonds (4 of 6)
Intermolecular forces
As we have seen, two neon atoms do not form covalent bonds with one another be cause of their full outer shells of electrons. There are, however, weak forces of attraction between two neon atoms. We know this because, when neon gas is compressed or cooled, it eventually turns into a liquid in which the atoms are weakly attracted to one another. These weak forces are called van der Waals forces and their strength depends on the size of the molecule.
Bromine (Br2) is made up of large covalently bonded molecules that have much stronger van der Waals forces between them than exist between atoms of neon. Thus at room temperature bromine exists as a mixture of liquid and vapor. However, the forces between the bromine molecules are much weaker than covalent bonds, so that - while it is easy to separate the bromine molecules from one another and vaporize the liquid - it requires much more energy to separate the bromine atoms by breaking the covalent bond between them.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Bonds (page 5)
ftsTitle
A snowflake, magnified to 20 times its actual size. Hydrogen bonding is responsible for the characteristic hexagonal symmetry of the ice crystals in a snowflake.
Chemical Bonds (5 of 6)
Hydrogen bonds
Some small molecules have much higher melting and boiling points than would be expected on the basis of their size. One such example is water (H2O), which has about the same mass as a neon atom but has a much higher melting point. There must therefore be unusually strong intermolecular forces between the water molecules. Although the oxygen and hydrogen atoms share a pair of electrons in a covalent bond, the oxygen atom exerts a stronger 'pull' on these electrons and so becomes electron-rich, leaving the hydrogen atom electron-poor. As a result, there is a force of attraction between hydrogen and oxygen atoms on neighboring molecules. This is known as hydrogen bonding.
As well as accounting for the surprisingly high melting point of water, hydrogen bonding is responsible for the rigid open structure of ice crystals and is very important in influencing the structures and properties of biological molecules. Although hydrogen bonds are stronger than van der Waals forces, they are still much weaker than covalent bonds.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Bonds (page 6)
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Chemical Bonds (6 of 6)
IONIC COMPOUNDS
In a crystal of potassium chloride (KCl), each K+ ion (represented here as a purple sphere) surrounds itself with as many Cl- ions (green spheres) as there is space for - which turns out to be six; in the same way, each Cl- ion is surrounded by six K+ ions. The ions are packed in a regular repeating manner, so that - even though the smallest crystal of KCl contains many millions of ions - it has the same cubic shape as a simple mode.
* ATOMS AND SUBATOMIC PARTICLES
* WHAT IS CHEMISTRY?
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL REACTIONS
* SMALL MOLECULES
* THE REACTIVITY SERIES
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Chemical Reactions (page 1)
ftsTitle
A precipitation reaction. The addition of colorless potassium iodide solution to colorless lead nitrate leads to the formation of an insoluble precipitate of yellow lead iodide; colorless potassium nitrate remains in solution.
Chemical Reactions (1 of 3)
Chemical reactions are the means by which new substances are formed from old ones. Among the chemical reactions occurring everywhere around us are the changes that take place when fuels are burnt, the industrial methods by which metals are extracted from their ores, and the processes controlling life itself. Since chemistry is centrally concerned with the means by which substances change, the study of reactions lies at the very heart of the subject.
During a chemical reaction, the atomic constituents of the substances that react together (the reactants) are rearranged to produce different substances (the products). Thus, for example, in the reaction of potassium (K) with water (H2O), potassium hydroxide (KOH) and hydrogen gas (H2) are formed. This information can be represented as a chemical equation. By convention the reactants appear on the left-hand side and the products on the right; letters may also be added after each chemical species to indicate its physical state - s means 'solid', l 'liquid', aq 'aqueous' (solution) and g 'gas': 2K(s) + 2H2O(l) 2KOH(aq) + H2(g)
An essential characteristic of chemical reactions is that there is an exchange of energy between the reacting system and the surroundings. So much heat is liberated during the reaction of potassium and water that the highly inflammable hydrogen gas frequently ignites above the molten metal.
Stoichiometry and the mole concept
According to the law of constant composition, matter cannot be created or destroyed during a chemical reaction. Thus in the reaction described above, the number of atoms of potassium, hydrogen and oxygen (calculated by multiplying each element in the equation by the numbers placed before the chemical formula) is the same before and after the reaction, and the equation is said to be balanced. The numerical proportions in which substances combine to form the products of a chemical reaction is de scribed as the reaction stoichiometry.
A balanced equation is thus a quantitative statement about the chemical reaction concerned. Such an equation (in conjunction with the mole concept; see below) enables us to predict how much product will be formed from a given mass of reactants. This provides valuable in formation that can be put to use, for example, in industrial production processes and in the analysis of chemical samples of unknown composition.
A mole is a measure of the amount of substance, based on the atomic theory of matter. A mole is defined as the number of carbon atoms in 12 grams of the isotope carbon-12 and has the colossal value of 6.022 x 10 to the power of 23. Every chemical compound has a fixed relative molecular mass or RMM (determined by the relative atomic masses of its constituent elements), so that molar quantities (the number of moles) of any substance can be found using simple arithmetic.
Reactions of acids and bases
According to the Br
nsted-Lowry definition, acids are substances that tend to donate protons (ionized hydrogen atoms, H+) to other molecules. For example, gaseous hydrogen chloride (HCl) readily dissolves in water to form hydrochloric acid, by donating a proton to a water molecule: HCl(g) + H2O(l) H3O+(aq) + Cl-(aq)
The products of the reaction are ions (electrically charged species): H3O+ (the hydronium ion) and Cl- (the chloride ion). Many non-metal oxides form acids when dissolved in water; for example, sulfur trioxide gas (SO3) dissolves in water to form sulfuric acid (H2SO4) - the reaction that occurs in the formation of acid rain.
By contrast, bases are defined as proton acceptors, capable of accepting protons from hydronium ions present in solution. A good example of a Br
nsted-Lowry base is the hydroxide ion, OH-(aq), which reacts with the hydronium ion to produce two molecules of water: H3O+(aq) + OH-(aq) 2H2O(l)
Examples of bases include sodium and potassium hydroxides (NaOH and KOH), which generate aqueous hydroxide ions in solution. Many metal oxides are also basic, such as calcium oxide (CaO; 'lime'), which reacts violently with water to form calcium hydroxide (Ca(OH)2; 'slaked lime'). Aqueous solutions of bases are known as alkalis.
Acids and bases can be detected by their effects on a class of natural dyes called indicators. The best-known indicator is litmus, a dye derived from lichen, which is turned red by acids and blue by bases.
Acids and bases react together to form compounds known as salts, which are neither acidic nor basic. For example, sodium hydroxide reacts with hydrochloric acid to form sodium chloride (common salt): NaOH(aq) + HCl(aq)
NaCl(aq) + H2O(l)
The end-points of such neutralization reactions can be determined visually by the choice of an appropriate indicator, which changes color when acid and base are exactly neutralized, i.e. when they have completely reacted. Neutralizations are of great importance in quantitative chemical analysis.
Precipitation reactions
Ionic compounds that dissolve in water produce electrolyte solutions. These consist of ions moving randomly throughout the solution; for example, sodium chloride in aqueous solution contains sodium (Na+) and chloride (Cl-) ions. These ions are responsible for the electrical conductivity of electrolytes.
Silver nitrate (AgNO3) is another ionic solid that dissolves readily in water, producing a colorless solution of aqueous silver (Ag+) and nitrate (NO3-) ions. If solutions of silver nitrate and sodium chloride are mixed, a white turbidity (cloudiness) instantly forms; this is due to the precipitation of fine particles of highly insoluble silver chloride. The precipitate gradually accumulates at the bottom of the vessel, leaving colorless sodium nitrate in solution: NaCl(aq) + AgNO3(aq)
AgCl(s) + NaNO3(aq)
The overall reaction is one in which ions are exchanged between partners. It is possible to predict the outcome of such precipitation reactions from a knowledge of the solubilities of the various species involved.
lved.
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Chemical Reactions (page 2)
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An oxidation reaction: magnesium metal burns in air to produce magnesium oxide. This common and important type of reaction includes combustion processes, the reactions that occur when metals corrode in air, and the biochemical reactions by which we obtain energy from food.
Chemical Reactions (2 of 3)
Electron-transfer reactions: oxidation and reduction
Magnesium metal (Mg) burns with an incandescent white flame in air because of a vigorous reaction with oxygen, forming magnesium oxide: Mg(s) + O2(g) 2MgO(s)
This is an example of the class of reactions known as oxidations, which include all combustion processes such as those occurring when fuels burn in air, as well as the reactions that cause metals to corrode in air. The product in this case is an ionic solid, which we could write more specifically as Mg to the power of 2 +O to the power of 2-. During the reaction, magnesium loses two electrons to form the cation Mg2+; the electrons are accepted by oxygen, which becomes the oxide anion O2- .
The transfer of electrons between chemical species is a common process in many chemical reactions, so the term oxidation has come to possess a wider meaning than that implying solely the addition of oxygen atoms to an element or compound. As in the case of the formation of magnesium oxide, oxidation means the loss of electrons by a compound; the opposite process, reduction, implies a gain of electrons. Thus magnesium is said to be oxidized to Mg2+, while oxygen is reduced to O2-; the overall reaction is described as a redox process. Many metals are extracted from their ores by reduction reactions.
If an electric current is passed through an electrolyte such as an aqueous solution of copper(II) chloride (CuCl2), a redox process known as electrolysis occurs. Positively charged Cu2+ ions are attracted to the negative electrode (the cathode), where they take up two electrons each and are thereby reduced to copper metal, which is deposited on the cathode. At the same time, the negatively charged Cl- ions are attracted to the positive electrode (the anode), where they give up their extra electrons (i.e. are oxidized) to form chlorine gas. Electrolysis is the basis of electroplating, in which a thin layer of metal, such as copper, tin, chromium or silver, is applied as a protective or decorative finish on cheaper or less durable materials. It is also used to purify metals and to extract reactive metals such as aluminum from their ores.
In a disproportionation reaction, a single chemical species is simultaneously oxidized and reduced. For example, copper(I) sulfate (Cu2SO4) dissolves in water to produce copper metal and copper(II) sulfate: Cu2SO4(aq) Cu(s) + CuSO4(aq)
Here one Cu+ ion is oxidized to Cu2+, while the second is reduced to Cu metal.
Reaction equilibria
All the reactions described so far have gone to completion, i.e. a fixed quantity of reactants is converted into a fixed quantity of products. However, in general such a state of affairs is more the exception than the rule. The end of a reaction occurs when there is no further change in the amount of products formed or reactants destroyed: this is the point at which a reaction is said to reach equilibrium. At equilibrium, there may be appreciable amounts of reactants still present. For example, when acetic acid is dissolved in water, it forms a weak acid, because at equilibrium there is only a low concentration of hydronium ions: CH3COOH(l) + H2O(l)
CH3CO2-(aq) + H3O+(aq)
The equilibrium, indicated by the two half-headed arrows, lies in favor of the reactants (in contrast to strong acids, such as hydrochloric acid, which are totally dissociated into ions). For the same reason, aqueous ammonia is a weak base, because at equilibrium there is a low concentration of hydroxide ions in the solution. Most of the solution consists of unreacted ammonia and water, in contrast to strong bases, which consist solely of ions.
The position of equilibrium in a chemical reaction (whether it favors reactants or products) depends in a detailed way on the thermodynamic properties of all the species involved.
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Chemical Reactions (page 3)
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The Belousov-Zhabotinsky reaction: a spectacular example of the complexity of many chemical reactions. The astonishing organization underlying such complex reactions is now recognized to be of crucial importance in the regulation of many processes in living systems.
Chemical Reactions (3 of 3)
Rates of chemical reactions
Very often it is important to know not only where the position of a chemical equilibrium lies but how fast it is reached. A graphic illustration is provided by a mixture of hydrogen gas and oxygen gas at room temperature. If undisturbed, the mixture does not react, but if a spark is passed through the gases, there is a violent explosion leading to the formation of water. Thus temperature is seen to exert a strong influence on the rate at which a reaction proceeds - the higher the temperature, the faster the reaction.
On the other hand, the same reaction can be made to proceed smoothly at room temperature by the addition of finely divided platinum metal, which acts as a catalyst. A catalyst is a substance that is not chemically transformed during a re action but whose presence serves to accelerate its rate. Catalysts play an extremely important role in many industrial processes, where they allow reactions to be carried out under conditions that would otherwise have to be much more severe.
Another factor influencing the rate of reaction is the intrinsic reactivity - i.e. willingness to undergo chemical reaction - of the chemical species involved. An example is provided by the reactions of the halogens - fluorine, chlorine, bromine and iodine - with hydrogen gas. The form of the reaction is identical for all the halogens, in each case producing the hydrogen halide (halogen-containing compound). However, the rate of the reaction decreases progressively down the series, fluorine reacting violently and iodine sluggishly at room temperature.
Complex reactions
Many chemical reactions, including those of the halogens with hydrogen gas, are actually much more complicated than their simple chemical equations suggest, and can be very sensitive to the precise conditions existing at the time of the reaction. One of the most fascinating illustrations of this is the Belousov-Zhabotinsky reaction, which involves a mixture of malonic acid, cerium(III) sulfate, potassium bromate and an iron(II) catalyst (ferroin) in aqueous acid solution. Depending on the conditions used, the reaction can display blue-red oscillations (chemical clock behavior) and/or spatial patterns (chemical waves), which spread in a regular manner through the solution. The red color is due to regions in which the reduced form Fe2+ predominates, while blue regions consist of oxidized Fe3+ ions.
Such remarkable organization, brought about by complex chemical (or biochemical) reactions in which vast numbers of molecules 'beat' in unison, is now recognized to be of fundamental importance for the control and regulation of many processes in living systems.
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Small Molecules (page 1)
ftsTitle
Dry ice vaporizing. A colorless gas at room temperature, carbon dioxide becomes a liquid under compression. If the liquid is allowed to expand rapidly to atmospheric pressure, it becomes cooler and freezes to dry ice
a white, snowlike solid. Dry ice sublimes (passes directly into vapor without melting) at
78.5
109.3
F); it is used to produce stage fog and as a refrigerant.
Small Molecules (1 of 3)
Although the Earth's atmosphere consists almost entirely of two gases - nitrogen and oxygen - a number of other gases are present at low concentration, together with varying amounts of water vapor. With the exception of the noble gases, most other components of air form part of natural cycles, each remaining in the atmosphere only for a limited time. Not only are these gases of major importance in relation to industrial processes that dominate economies throughout the world, but cyclical processes involving water, oxygen, carbon dioxide and nitrogen - together with solar radiation - are essential to plant and animal life.
Current interest in various atmospheric gases centers on the possible global effects of changes in their atmospheric concentration due to human activities. Increase in carbon dioxide may upset the heat balance at the Earth's surface, while the use of chlorofluorocarbons (CFCs) might result in depletion of the ozone layer, thereby allowing destructive high-energy solar radiation to reach the Earth's surface.
Although these small molecules are simple in the sense that they are composed of few atoms, their structures and - for those with three or more atoms - their shapes vary. In most cases, their atoms are held together in the molecule by two, four or six electrons, resulting in single, double or triple covalent bonds. Three of these molecules (nitric oxide, nitrogen dioxide and oxygen) are paramagnetic - i.e. attracted to a magnet, like iron - because of the number or arrangement of their electrons.
* WHAT IS CHEMISTRY?
* CHEMICAL BONDS
* THE EARTH'S STRUCTURE AND ATMOSPHERE
* THE HYDROLOGICAL CYCLE
* WEATHER
* PLANT PHYSIOLOGY
* THE BIOSPHERE
* THREATS TO THE ENVIRONMENT
Outline
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p048-2
ftsTitleOverride
Small Molecules (page 2)
ftsTitle
Small Molecules (2 of 3)
Hydrogen
Hydrogen (H2) is the simplest of all stable molecules, consisting of two protons and two electrons. It is a colorless, odorless gas and is lighter than air. The last of these properties led to its use in lifting airships, but this use was discontinued - because of its explosiveness when ignited - following the Hindenburg disaster in 1937. Most hydrogen is used on the site where it is produced, but it is also transported as compressed gas in steel cylinders and in liquid form at very low temperature.
Water
The total amount of water (H2O) on Earth is fixed, and most is recycled and re-used. The largest reservoirs are the oceans and open seas, followed by glaciers, ice caps and ground water. Very little is actually contained within living organisms, al though water is a major constituent of most life forms.
Water is one of the most remarkable of all small molecules. On the basis of its molecular weight (18), it should be a gas; its high boiling point (100 deg C / 212 deg F) is due to the interaction of water molecules with each other (hydrogen bonding), which effectively increases its molecular weight. Water is also unusual in that - as ice - it is less dense than the liquid at the same temperature.
Carbon dioxide and oxygen
Carbon dioxide (CO2) is a colorless gas with a slight odor and an acid taste. It is available as gas, as liquid and as the white solid known as 'dry ice'. Its cycle in nature is tied to that of oxygen, the relative levels of the two gases in the atmosphere (apart from human activity) being regulated by the photosynthetic activity of plants. It is produced on a vast scale, mostly as a by-product of other processes.
With the ever-increasing input of carbon dioxide to the atmosphere, due largely to the burning of fossil fuels and forests and the manufacture of cement, the natural 'sinks' for carbon dioxide - chiefly photosynthesis and transfer to the oceans - can no longer keep pace with the total input. If this imbalance continues, it is thought that levels will be reached where the infrared-absorbing properties of carbon dioxide will result in a progressive warming of the Earth's atmosphere, accompanied by melting of the polar ice and flooding of what is now dry land - the so-called greenhouse effect. This is perhaps too extreme and pessimistic a view: in the past there have been many warm interglacial periods due to factors not ascribable to human activity.
Oxygen (O2) is a highly reactive colorless, odorless and tasteless gas. At low temperature, it condenses to a pale blue liquid, slightly denser than water. Oxygen supports burning, causes rusting and is vital to both plant and animal respiration.
Ozone
Ozone (O3) is a highly toxic, unstable, colorless gas. Its primary importance stems from its formation in the stratosphere. In this layer of the atmosphere, temperature in creases with height, principally because of the reaction of high-energy ultraviolet solar radiation with oxygen. This can be expressed in the form of two chemical equations: O2 + sunlight 2 O (atoms)
O2 + O (atom) O3
Ozone in the stratosphere functions as a very effective filter for high-energy ultraviolet solar radiation. Radiation in this energy range is sufficiently high to break bonds between carbon and other atoms, making it lethal to all forms of life. It is currently thought that the introduction of CFCs (used in sprays and refrigerants) and the related 'halons' (used in fire extinguishers) may contribute to the partial or even total destruction of the ozone layer. These classes of compounds are highly volatile, chemically very stable and essentially insoluble in water, so that they are not washed out of the atmosphere by rain. When, by normal convection, they reach the stratosphere, they react destructively with ozone.
Carbon monoxide
Carbon monoxide (CO) is a colorless, odorless, toxic gas. The input to the atmosphere due to human activity is about 360 million tons per year, mostly from the incomplete combustion of fossil fuels. The natural input is some 10 times this figure and results from the partial oxidation of biologically produced methane. The background level of 0.1 parts per million (ppm) can rise to 20 ppm at a busy road intersection, and a five-minute cigarette gives an intake of 400 ppm.
Since the atmospheric level of carbon monoxide is not rising significantly, there must be effective sink processes, one being its oxidation in air to carbon dioxide. In addition, there are soil microorganisms that utilize carbon monoxide in photosynthesis.
* WHAT IS CHEMISTRY?
* CHEMICAL BONDS
* THE EARTH'S STRUCTURE AND ATMOSPHERE
* THE HYDROLOGICAL CYCLE
* WEATHER
* PLANT PHYSIOLOGY
* THE BIOSPHERE
* THREATS TO THE ENVIRONMENT
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Small Molecules (page 3)
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Small Molecules (3 of 3)
Nitrogen
Nitrogen (N2) is a colorless, odorless gas. Although very stable and chemically unreactive, it cycles both naturally and as a result of its use in the chemical industry. The natural cycle results from the ability of some types of bacteria and blue-green algae (in the presence of sunlight) to 'fix' nitrogen - i.e. to convert it into inorganic nitrogen compounds (ammonium and nitrate salts) that can be assimilated by plants. Since 1913 human activity has increasingly contributed to the cycling of nitrogen, because of the catalytic conversion of nitrogen into ammonia (used mainly in nitrate fertilizers, see below), which ultimately reverts to nitrogen gas.
Oxides of nitrogen
The presence of nitric oxide (NO) and nitrogen dioxide (NO2) at high levels in the atmosphere is closely connected with the internal-combustion engine. At the high temperature reached when petroleum and air ignite, nitrogen and oxygen combine to form nitric oxide, which slowly reacts with more oxygen to form nitrogen dioxide. Most internal-combustion engines also produce some unburnt or partially burnt fuel: in the presence of sunlight, this reacts with nitrogen dioxide by a sequence of fast reactions, forming organic peroxides, which are the harmful constituents of photo chemical smog (smoke plus fog).
Ammonia
Ammonia (NH3) is a colorless gas with a penetrating odor, and is less dense than air. It is highly soluble in water, giving an alkaline solution. World production is of the order of 100 million tons a year, most of which is converted into fertilizers (80%), plastics (9%) and explosives (4%).
Oxides of sulfur
Both sulfur dioxide (SO2) and sulfur trioxide (SO3) are pungent-smelling acidic gases, which are produced by volcanic action and - to the extent of some 150 million tons a year - by the burning of fossil fuels and smelting operations. The level of sulfur dioxide in unpolluted air is 0.002 parts per million (ppm), but in the 1952 London smog the levels rose to 1.54 ppm, accompanied by a dramatic increase in the death rate. In the atmosphere sulfur dioxide is slowly oxidized to sulfur trioxide, reactions that are catalysed by sunlight, water droplets and particulate matter in the air. Ultimately the latter is deposited as dilute sulfuric acid - acid rain.
* WHAT IS CHEMISTRY?
* CHEMICAL BONDS
* THE EARTH'S STRUCTURE AND ATMOSPHERE
* THE HYDROLOGICAL CYCLE
* WEATHER
* PLANT PHYSIOLOGY
* THE BIOSPHERE
* THREATS TO THE ENVIRONMENT
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Metals (page 1)
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Electrochemistry in industry: copper-plated sheets used for flexible electronic circuit boards are seen being removed from an electroplating bath. The bath is filled with an electrolyte of copper(II) sulfate solution; the object to be plated acts as the cathode, while the anode is made of copper metal. As an electric current passes through the solution, the copper anode `dissolves' and copper metal is deposited at the cathode.
Metals (1 of 4)
Metals are usually defined by their physical properties, such as strength, hardness, luster, conduction of heat and electricity, malleability and high melting point. They can also be characterized chemically as elements that dissolve (or whose oxides dissolve) in acids, usually to form positively charged ions (cations). By either definition, more than three quarters of the known elements can be classified as metals. They occupy all but the top right-hand corner of the Periodic Table, the remainder being non-metals. A few elements on the borderline, such as germanium, arsenic and antimony, have some of the properties of both metals and non-metals, and are often classed as metalloids.
Given such a large number of metals, it is not surprising that some of them have rather untypical properties. For instance, mercury is a liquid at room temperature, and - with the exception of lithium - all the alkali metals (see Periodic Table) melt below 100 deg C (212 deg F). The alkali metals are also quite soft - they can easily be cut with a knife - and extremely reactive: rubidium and cesium cannot be handled in air and may react explosively with water.
Occurrence
Most metals occur naturally as oxides, while some - mostly the heavier ones, such as mercury and lead - occur as sulfides. Only a few - the noble and coinage metals (see Periodic Table) - are found in the metallic state, being chemically the most inert metals. It is their chemical unreactiveness that makes them useful in coinage and jewelry, since they do not corrode.
A few metals do not occur naturally at all, because they are radioactive and have decayed away. Technetium and all the elements with higher atomic numbers than plutonium (Pu, 94; see Periodic Table) are made by the `modern alchemy' of nuclear reactors or accelerators, while promethium is found only in minute amounts as a product of the spontaneous fission of uranium. The very heaviest elements have been obtained only a few atoms at a time, and are intensely radioactive.
The discovery and extraction of metals
Artificial elements have of course been known only in modern times, since the 1940s or later. The discovery of most other metals was also comparatively recent: with the exception of zinc, platinum and the handful of metals known to the Ancients, all metals have been discovered since 1735. The only metals known in antiquity were copper, silver, gold, iron, tin, mercury and lead. Of these, it was not the most abundant - iron - that was discovered first: the Bronze Age came before the Iron Age. The reason for this is that it is easier to extract the metals used in bronze - copper and tin - from their minerals than it is to extract iron from its ores. The discovery of copper is thought to have been accidental: pieces of the metal ore used in fireplaces came into contact with the hot charcoal, so releasing the metal. Essentially the same process under controlled conditions (smelting) is used in modern blast furnaces. Any of the metals from manganese (Mn) to zinc (Zn) in the Periodic Table can be obtained by roasting their oxides with coke at temperatures of up to about 1600 deg C (2912 deg F).
The ores of the lighter, more reactive metals cannot be reduced by carbon at practical temperatures, because their atoms are more strongly bonded in the ore. These metals are usually obtained by electrolysis or by the reaction of their compounds with an even more reactive metal. For instance, the reduction of aluminum oxide with carbon requires a temperature in excess of 2000 deg C (3632 deg F); so electrolysis of a melt of aluminum oxide in a mixture of cryolite (a double fluoride of aluminum and sodium) and calcium fluoride at about 950 deg C (1742 deg F) is used. On the other hand, titanium is obtained by converting its oxide into the chloride, which is then reduced with elemental sodium or magnesium. These methods are rather expensive, but are justified by the usefulness of the metals obtained, which are both strong and light.
ight.
* ELECTROMAGNETISM
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* MINING, MINERALS AND METALS
* IRON AND STEEL
* HUMAN PREHISTORY
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ftsTitleOverride
Metals (page 2)
ftsTitle
Metal deformation occurs as defects in the crystalline structure move under a shearing stress (red arrows).
Alloying. Impurity atoms (blue) are the wrong size to fit into the metal's crystal lattice. They therefore tend to site themselves at defective points in the lattice, where they become immobile and thus 'pin" the defects in place.
Metals (2 of 4)
Conductivity
The conduction of heat and electricity that characterizes metals is due to their unique type of bonding. The solid metals behave as if they were composed of arrays of positively charged ions, with electrons free to move throughout the crystalline structure of the metal. This results in high electrical conductivity. The conduction of heat can also be seen in terms of the motion of electrons, which becomes faster as temperature rises. Since the electrons are mobile, the heat can be conducted readily through the solid.
The majority of metals are good conductors of electricity, but germanium and tin (in the form stable below 19 deg C / 64 deg F) are semiconductors.
Mechanical strength
Many metals are used because of their strength. However, most pure metals are actually quite soft. In order to obtain a tough hard metal, something else has to be added. For instance, the earliest useful metal was not copper but bronze, which is copper alloyed with tin. Similarly, iron is never used in the pure state but as some form of steel.
The softness of a pure metal results from a lack of perfection in the crystal frame work formed by its atoms (see diagram). Even when the most rigorous conditions are employed, it is impossible to grow any material in perfect crystalline form. There will always be some atoms in the wrong place or missing from their proper place. When solidification occurs fairly rapidly, as when a molten metal is cooled in a mold, even more defects occur. Under bending or shearing stresses, such defects can move and allow the metal to change shape easily. When the foreign atoms of an alloying element are present, they usually have a different size from those of the host and cannot easily fit into the crystal lattice. They therefore tend to site themselves where the lattice is irregular, i.e. where the defects are. The effect of this is to prevent the defects from moving, and so to increase the rigidity of the metal.
* ELECTROMAGNETISM
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* MINING, MINERALS AND METALS
* IRON AND STEEL
* HUMAN PREHISTORY
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Metals (page 3)
ftsTitle
Metal corrosion. A flake of badly corroded car bodywork, magnified and shown in false color: the rust at the bottom is covered by three coats of original paint and one coat of a later respray. Most metals are prone to corrosion, or surface oxidation; the problem is particularly acute in the case of iron, because it forms porous oxides that break away and allow corrosion to continue.
Metals (3 of 4)
Tarnishing and corrosion
Nearly all metals are prone to surface oxidation, i.e. the surface of the metal reacts with oxygen or other components of the atmosphere. The major exceptions are the coinage metals and those of the platinum group (see Periodic Table), and even these react with sulfur compounds in industrially polluted atmospheres and turn black. All other metals should, in principle, react with moisture and the oxygen in air, yet some corrode badly and others appear to be inert. In fact, they all oxidize, but in many cases a thin layer of oxide adheres firmly to the metal surface and prevents further reaction. This is the case with aluminum and titanium. On the other hand, iron forms porous oxides that readily break away, allowing corrosion to continue. Stainless steels are produced by alloying iron with chromium and sometimes also with nickel, which form a protective oxide on the surface; the thickness of this layer is so small that the surface still appears shiny and metallic.
An alternative way of preventing corrosion is essentially electrochemical. Corrosion can be prevented by connecting a metal object to a piece of more reactive metal and completing the circuit through the earth. The more reactive metal be comes the anode and gradually `dissolves' or degrades. This method is sometimes used for metal tanks that have to stand outdoors: a block of magnesium buried and connected by a wire to the tank slowly oxidizes - it acts as a `sacrificial' anode. This is also the reason why galvanization works. Contrary to what would be expected, the layer of zinc on the iron needs to be somewhat porous so that water can bring the two metals into electrochemical contact. It is then the zinc that reacts instead of the iron.
* ELECTROMAGNETISM
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* MINING, MINERALS AND METALS
* IRON AND STEEL
* HUMAN PREHISTORY
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p050-4
ftsTitleOverride
Metals (page 4)
ftsTitle
Metal and the Periodic Table.
Metals (4 of 4)
THE REACTIVITY SERIES
The widely varying reactivity of metals can be related to their positions in the Periodic Table. The s-block metals are highly reactive, while the transition metals typically become less reactive from left to right across the table, with the noble and coinage metals least re active of all.
When metals react, they usually lose electrons to form positively charged ions (cations). This charge (also known as the oxidation state) is again related to the position of a metal in the Periodic Table. In the s-block, the charge equals the group number, +1 or +2 (e.g. K+, Mg2+). In the transition metals, variability of oxidation state is the rule: for instance, iron may lose 2 or 3 electrons (Fe2+, Fe3+), and this fact is indicated in the names and formulae of its compounds - iron(II) chloride (FeCl2) and iron(III) chloride (FeCl3).
The reactivity of a metal can thus be explained in terms of its readiness to lose electrons to form cations: potassium (K) readily loses an electron to form a K+ ion, while gold (Au) is highly unreactive and dissolves only in aqua regia, a fiercely oxidizing mixture of hydrochloric and nitric acids. Metals can be placed in order of reactivity, in a sequence known as the reactivity series ; for some of the more important metals, the series runs as follows (in order of decreasing reactivity): K Na Ca Mg Al Zn Fe Pb Cu Hg Ag Au Pt
A metal can be displaced from a solution of one of its salts simply by addition of a metal higher (earlier) in the series. For in stance, if zinc metal (Zn) is added to a solution of copper(II) sulfate (CuSO4), the zinc becomes coated by copper metal and the blue color of the solution fades, as the colored copper ions in solution are displaced by zinc ions: Zn(s) + Cu2+(aq) + SO42-(aq) Zn2+(aq) + SO42-(aq) + Cu(s)
The reactivity series also indicates the affinity of a metal for oxygen. As such, it explains the differing susceptibility of metals to corrosion (surface oxidation) and underlies the extraction of metals from their oxides. The more reactive a metal is, the higher the temperature required to reduce its oxide by carbon. In practice, the most reactive metals cannot be economically reduced by carbon, and are therefore obtained by electrolysis or by displacement by an even more reactive metal.
The reactivity series can also be seen as an electrochemical series. When two different metals are dipped into an electrolyte solution, a voltage forms between them and the metal higher in the series becomes the anode (positive electrode). The distance between the two metals in the series reflects the size of the voltage produced. Electrochemical reactions of this kind underlie electroplating and the operation of electrolytic cells and batteries. Frequently some hydrogen is also produced: such a reaction often occurs in domestic central heating systems, where copper pipes and iron radiators are both in contact with hot water; the `air' that accumulates in the system is actually hydrogen.
* ELECTROMAGNETISM
* ELECTRICITY IN ACTION
* ATOMS AND SUBATOMIC PARTICLES
* ELEMENTS AND THE PERIODIC TABLE
* CHEMICAL BONDS
* CHEMICAL REACTIONS
* MINING, MINERALS AND METALS
* IRON AND STEEL
* HUMAN PREHISTORY
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p052-1
ftsTitleOverride
Natural Compounds (page 1)
ftsTitle
Many organic compounds, such as alanine and limonene, are built up asymmetrically around a central carbon atom and can exist in two mirror-image forms, known as enantiomers. In the case of amino acids such as alanine, one enantiomer predominates greatly over the other, the latter having a small role in nature. This apparently superficial difference in form can have a startling effect on the properties of the compound concerned. A trivial but striking example of this is provided by the enantiomers of limonene: one smells strongly of lemons, the other of oranges.
Natural Compounds (1 of 5)
The molecular basis for life processes, which have evolved with such remarkable elegance around carbon as the key element, is beginning to be understood, thanks to the combined triumphs of biological, chemical and physical scientists during the last hundred years.
Although the chemist can now make synthetically almost any chemical compound that nature produces, the challenge re mains to achieve this objective routinely with the efficiency and precision that characterizes the chemistry of living systems.
There is something very special about the chemistry of carbon that has singled it out as the atomic building block from which all naturally occurring compounds in living systems are constructed. The subject that deals with this important area of science, nestling between biology and physics, has become so vast and significant that it has earned recognition as a separate field of scientific investigation. As it was originally thought that such carbon-based compounds could be obtained only from natural sources, this field of study became known as organic chemistry.
* THE BEGINNINGS OF LIFE
* GENETICS AND INHERITANCE
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p052-2
ftsTitleOverride
Natural Compounds (page 2)
ftsTitle
Structural formulae. Chiral carbon atoms are conventionally indicated by an asterisk (*). Covalent bonds located in the plane of the paper are represented by lines (
), while bonds orientated (tetrahedrally) above and below this plane are displayed as wedges( ) and dashes (| | |) respectively.
In simplified structural formulae (such as those given for limonene), the junctions and termini of lines, wedges and dashes represent carbon atoms.
Natural Compounds (2 of 5)
The unique carbon atom
Carbon's unique feature is the readiness with which it forms bonds both with other carbon atoms and with the atoms of other elements. Having four electrons in its outer shell, a carbon atom requires four more electrons to attain a stable noble-gas configuration. It therefore forms four covalent bonds with other atoms, each of which donates a single electron to each bond. In this way the electronic requirements are satisfied, and a three-dimensional `tetracovalent' environment is built up around the car-bon atom.
Carbon bonds are found both in pure forms of carbon (graphite, diamond and newly discovered buckminsterfullerene) and in association with other atoms in a vast array of compounds. Compounds consisting of just carbon and hydrogen - hydrocarbons - are extremely important, notably as the principal constituents of fossil fuels. In addition, carbon readily bonds with many other atoms, including oxygen, nitrogen, sulfur, phosphorus and the halogens, such as chlorine and bromine. Often the covalent bonds between carbon and other atoms are stable enough for us to handle the resulting compounds at room temperature; yet these compounds are not so strongly bonded that they cannot be manipulated by means of well-known chemical reactions.
* THE BEGINNINGS OF LIFE
* GENETICS AND INHERITANCE
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p052-3
ftsTitleOverride
Natural Compounds (page 3)
ftsTitle
A selection of important functional groups. An 'R' indicates a site where another functional group or an atom may be attached.
Alkenes are hydrocarbons that contain one or more carbon double bonds. Alkenes with just one double bond form a series including ethene (ethylene; C2H4), propene and butene.
Alcohols. Examples in clude methanol (CH3OH) and ethanol (C2H5OH).
Ketones. Examples include propanone (acetone; CH3COH3) and MVK.
Aldehydes. An important example is methanal (formaldehyde; HCOH), used in the production of formalin (a disinfectant) and of synthetic resins.
Carboxylic acids. As well as occurring in organic acids, such as acetic (ethanoic) acid (CH3CO2H; vinegar), this group occurs in all amino acids, including alanine.
Amines, together with the carboxylic-acid group, occur in all amino acids.
Amides. The most important type of amide bond is that formed in protein synthesis, when the carboxylic-acid group of one amino acid condenses with the amine group of another to give a peptide bond.
Thiols. This group is characterized by a strong disagreeable odor. An example is ethanethiol.
Aromatic compounds. This six-membered ring containing three double bonds is highly stable and is thus a very common characteristic of organic compounds.
Natural Compounds (3 of 5)
Functional groups and reactivity
Carbon combines with itself and other atoms to produce open-chain (acyclic) and ring (cyclic) skeletons, into which are built highly characteristic arrangements of atoms, known as functional groups. The diverse but predictable chemical behavior of the different functional groups is a consequence of their ability either to attract or to repel electrons compared with the rest of the carbon skeleton. The overall effect of the resulting charge distribution is to create a molecule in which some regions are slightly negatively charged (nucleophilic), and others slightly positively charged (electrophilic).
Most organic reactions involve the electrophilic and nucleophilic centers of different molecules coming together as a prelude to the formation of new covalent bonds. An appreciation of how particular compounds behave towards others and of the various mechanisms by which such reactions occur forms the basis of classical organic synthesis. This allows chemists to build up large molecules, containing many different functional groups and with a great diversity of chemical properties, in a controlled and predictable manner.
* THE BEGINNINGS OF LIFE
* GENETICS AND INHERITANCE
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p052-4
ftsTitleOverride
Natural Compounds (page 4)
ftsTitle
The retrosynthetic analysis of limonene. Limonene is 'disconnected' in two steps (yellow arrows) to the readily available starting materials 2-methyl-butadiene (2MB) and methylvinylketone (MVK). In step A, the CH2 group is replaced by an oxygen atom; in step B, the ring is disconnected to 2MB and MVK. The large green arrows indicate the actual synthetic reactions required to transform the two starting materials into limonene; the small green arrows indicate how the six-membered ring is formed in the Diels-Alder reaction.
Natural Compounds (4 of 5)
Chirality and the tetrahedral carbon atom
Alanine is one of the 20 naturally occurring amino acids from which proteins are synthesized in living organisms. Amino acids are characterized by their possession of two functional groups - a carboxylic-acid group (CO2H) and an amine group (NH2). Different amino acids, often with very different properties, are distinguished by the identity of a third group - a methyl group (CH3) in the case of alanine.
A more detailed examination of alanine reveals another feature of paramount importance to the modern chemist. The four groups bonded to the central carbon atom are arranged in such a way as to define a tetrahedron in three dimensions. This spatial arrangement (or configuration) can exist in two different forms, one the non-superimposable mirror image of the other. They differ as our right hand does to our left, so the central carbon atom is said to be chiral (from the Greek for `hand'), or asymmetric. The two different forms are known as enantiomers.
The physical consequences of this apparently minor difference can be quite startling. Limonene is a liquid hydrocarbon with one chiral carbon atom, and occurs as two enantiomers. While one enantiomer smells strongly of lemons, the other smells strongly of oranges. The different spatial arrangements of the groups and consequently the different overall shapes of the two molecules cause them to interact differently with molecular sensors in our nose, so each initiates a different message that is then sent to our brain.
Molecular recognition of this kind, based upon chirality, is prevalent in the chemistry of the molecules of life. Nucleic acids (DNA, RNA), polysaccharides (large natural sugar molecules) and proteins, especially enzymes, all discriminate between enantiomers in their respective modes of action. The enzymes are nature's catalysts, providing a very efficient environment in which molecules can come together and react. Like other proteins, they are built up from chains of amino acids, joined together in numerous different combinations. The chains twist and coil, so causing the functional groups of different amino acids to come together or `converge', thereby creating specific regions known as active sites. It is at the active site that particular molecules may be held briefly while reactions are performed on them, before being released as new molecules. However, an enzyme is generally very selective about which molecules it will accept; often, only one of a pair of enantiomers will be accepted, the other being the wrong shape to fit comfortably into the active site. In this way, life itself depends vitally upon chirality.
Aspects of design
If a desired compound does not occur naturally, it must be made, or synthesized, by modifying a molecule that already exists. Such a chemical synthesis may involve a number of different steps, and even relatively simple molecules could, in principle, be synthesized in many different ways from many different starting materials. Using their knowledge of chemical reactions, chemists examine several possible routes to a molecule before setting out upon its synthesis. Retrosynthetic analysis is a design method in which the desired product is broken down theoretically, or `disconnected', into smaller and smaller fragments until a convenient starting material is reached. It relies upon a knowledge of how different functional groups can be manipulated to build up the desired molecule gradually.
The art of synthetic design has progressed rapidly over the last 30 years. Chemists have learned to handle and manipulate new families of compounds, and discovered new synthetic transformations that may operate under milder conditions than existing ones or at a much faster rate. This has been coupled with great advances in the methods used to purify and analyze molecules; such methods have allowed the structures of molecules to be probed more deeply, thereby revealing how they react together and how a particular molecule may interact with its surroundings. In principle, the expertise already exists to synthesize any molecule, however complex: the only constraint is time. As we learn more and more about the chemicals that exist all around us - and within us - we put ourselves in an increasingly strong position to tackle the many intricate scientific and environmental issues facing mankind as the new century approaches.
* THE BEGINNINGS OF LIFE
* GENETICS AND INHERITANCE
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p052-5
ftsTitleOverride
Natural Compounds (page 5)
ftsTitle
Natural Compounds (5 of 5)
The phosphorus atoms (light blue), each surrounded by four oxygen atoms (red), give a strong sense of the right-handed twist of the sugar-phosphate backbone of this form of DNA (B-DNA). The dark-blue spheres, representing nitrogen atoms, form part of the nucleotide bases that link the two strands of the molecule. Green represents carbon, and white hydrogen.
Nucleic acids - DNA and RNA - are present in all living cells, with the exception of red blood cells. The highly variable sequences of bases allow in formation relating to the characteristics of the individual cell to be encoded in a molecular fashion. This information controls both the inherited characteristics of the next generation and the life processes of the organism itself.
* THE BEGINNINGS OF LIFE
* GENETICS AND INHERITANCE
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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ftsTitleOverride
Man-made Products (page 1)
ftsTitle
The liquid-crystal phase occurs between the solid (ordered) and liquid (disordered) phases. The intermediate phase may take a number of slightly different forms, of which 'smectic' and 'nematic' are examples. The characteristic orientations of the molecules of a liquid-crystalline substance are due to the long, rod-like shape of the molecules, which allows a weak, long-range order to develop between each molecule and its neighbors. A typical example is 4-n-hexyl-4-cyanobiphenyl.
Man-made Products (1 of 5)
Our chemical knowledge of natural compounds will ultimately be dwarfed by that relating to man-made, or unnatural, products. The reason for this is that chemistry is not only the science that deals with molecules, but also an exercise in design and engineering at the molecular level. In so far as chemistry creates its own world, its practice in an abstract form has infinite possibilities. Synthetic zeolites, which have important applications in the petrochemical industry, and ceramics, which have been found to behave as high-temperature superconductors, are typical examples of man- made products that are totally inorganic in their atomic composition. Likewise, liquid crystals - although based on organic compounds - are for the most part wholly synthetic compounds. Man-made products are also beginning to show promise in other areas, notably as artificial enzymes, i.e. tailor-made biological catalysts.
Liquid crystals
Discovered in 1888 by the Austrian botanist Friedrich Reinitzer, liquid crystals have been heralded as the fourth state of matter, occurring at the interface between the solid and the liquid phases. About 5% of crystalline compounds do not simply melt when heated: they form turbid (cloudy) liquids, which may also exhibit marked color changes as the temperature is raised, before becoming normal liquids. The process is reversible: upon cooling, the liquid passes back through the so-called `liquid-crystalline' state before solidifying again. Generally, it is rod-shaped molecules, such as 4-n-hexyl-4'-cyano biphenyl, that exhibit liquid-crystal behavior.
Because of the relatively weak forces between the molecules of a liquid crystal, the interactions between the molecules and hence their relative orientations can be changed not only by temperature and pressure but also by electric and magnetic fields. The effect of temperature on the color of liquid crystals has led to their use in the detection of tumors that are `warmer' than surrounding healthy tissue. Their most familiar application is in the liquid-crystal displays (LCDs) used in watches and calculators, where their optical properties are controlled by applying electric fields to change the orientation of the molecules in the liquid crystal.
* ELECTRICITY IN ACTION
* NATURAL COMPOUNDS
* CHEMICALS IN EVERYDAY LIFE
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p054-2
ftsTitleOverride
Man-made Products (page 2)
ftsTitle
A computer-graphics representation of ZSM-5, with a methane molecule in the central channel. ZSM-5 is just one of a range of synthetic zeolites that have been tailor-made to accomplish specific tasks. The role of zeolites as molecular sieves and size-selective catalysts is due to their distinctive crystalline structure; the tiny channels and cavities are formed from tetra hedrons based on aluminum (red) and silicon (yellow).
Man-made Products (2 of 5)
Synthetic zeolites
Natural zeolites are highly porous crystalline minerals, consisting mainly of silicon, aluminum and oxygen. They are built up of three-dimensional networks of silicate (SiO4) and aluminate (AlO4) tetra- hedrons, arranged in such a way that they form tiny submicroscopic channels. In the natural state, these channels may hold water molecules; however, if the water is driven off by heat, other molecules of appropriate size can be absorbed.
Because the size of the channels deter mines the size of the molecules that can enter, zeolites are said to be size-selective. This property allows zeolites to be used as molecular sieves, which separate mixtures of compounds purely on the basis of their different sizes. For example, the zeolite chabazite contains channels of which the diameter is just 0.39 manometers - or 0.00000000039 meters; these allow straight-chain hydrocarbons to pass through, while bulkier branched-chain hydrocarbons are retained. Zeolites can also be used as size-selective catalysts, since only molecules of a particular size can enter and undergo chemical reactions.
By altering the ratios of aluminum and silicon, along with conditions under which the zeolites are formed, chemists can fashion synthetic zeolites with different channel sizes and shapes. In this way a new generation of zeolites is being specifically designed to accomplish particular tasks. For instance, the synthetic zeolite ZSM-5 is used to convert toluene (C6H5CH3) into benzene (C6H6) and paraxylene (CH3C6H4CH3), an intermediate in the production of polyester fibers. The catalytic activity of synthetic zeolites is increasingly being exploited in the production of petroleum (gasoline) to break down large straight-chain hydrocarbon molecules into smaller branched ones that lead to smoother performance in internal-combustion engines.
* ELECTRICITY IN ACTION
* NATURAL COMPOUNDS
* CHEMICALS IN EVERYDAY LIFE
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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ftsTitleOverride
Man-made Products (page 3)
ftsTitle
The '1-2-3' compound (YBa2Cu3O7) was the first substance shown to become superconducting at liquid-nitrogen temperatures, just below 100 K (
F). The lattice structure of superconductors causes moving electrons to join together in pairs; because the pairs are in the lowest possible energy state, they cannot lose energy and so electrical resistance is reduced to zero.
Man-made Products (3 of 5)
High-temperature superconductors
The amazing ability of a material to transmit an electrical current without showing any electrical resistance is known as superconductivity . In 1911 the Dutch physicist Heike Kamerlingh Onnes (1853-1926) discovered that the resistance of mercury falls to zero in liquid helium, which boils at 4.2 K(-268.8 deg C / -452 deg F). Progress was subsequently made in increasing the transition temperature to the superconducting state, notably by using certain alloys of niobium and titanium, but the major breakthrough came in 1986, when it was announced that a ceramic metal oxide of lanthanum (La), barium (Ba) and copper (Cu) loses its resistance at 30 K (-243 deg C/ -405 deg F). The frantic research activity that followed this announcement resulted in an oxide in which yttrium (Y) replaces lanthanum, with the formula YBa2Cu3O7, which was demonstrated to be superconducting at liquid-nitrogen temperatures, just below 100 K (-173 deg C / -279 deg F). This material is often referred to as `1-2-3' because of the ratios of the metals involved. Further advances have since in creased the transition temperature to around 125 K (-148 deg C / -234 deg F).
* ELECTRICITY IN ACTION
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Man-made Products (page 4)
ftsTitle
The Meissner effect: a rotating magnet is seen levitating above a nitrogen-cooled specimen of the superconducting material '1-2-3'. A major technological revolution is expected to follow the discovery of materials that become superconducting at less extreme temperatures. Superconducting magnets cooled by liquid helium have already been used in prototypes of magnetically levitated trains.
Man-made Products (4 of 5)
Another remarkable feature of superconducting materials is that within a bulk specimen there is no magnetic field. One important consequence of this is the so-called Meissner effect (named after one of its two discoverers), an awe-inspiring phenomenon in which a rotating magnet levitates above a superconductor. Why does this happen? Simply because, when a magnet approaches a superconductor, a supercurrent is induced in the superconductor, which generates a magnetic field of the same polarity as that of the magnet and therefore repels it. Thus a superconductor - which can be viewed as a kind of magnetic mirror - responds electromagnetically to the magnet rotating above it and keeps it levitated.
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ftsTitleOverride
Man-made Products (page 5)
ftsTitle
The crystal structure of 1-2-3, as displayed in a computer-graphics image. Copper atoms occur at the center of layers of copper oxide pyramids (green) and planes (dark blue); oxygen atoms are red, yttrium atoms light blue, and barium atoms yellow.
Man-made Products (5 of 5)
The discovery of a practical room-temperature superconductor would undoubtedly bring about a major technological revolution. For instance, the transmission of electricity could be achieved effortlessly without energy loss, while the efficiency of computer chips could be greatly enhanced without risk of their burning themselves up. Yet even now in the early 1990s, magnetic resonance imaging is already established in medical practice, using superconducting magnets composed of niobium-titanium alloys and operating at liquid-helium temperatures. Such magnets, again cooled by liquid helium, have also been used in experimental magnetically levitated (maglev) trains.
* ELECTRICITY IN ACTION
* NATURAL COMPOUNDS
* CHEMICALS IN EVERYDAY LIFE
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p056-1
ftsTitleOverride
Chemicals in Everyday Life (page 1)
ftsTitle
A selection of synthetic polymers. The advent of the synthetic fiber nylon in the 1930s caused a major revolution in the fashion industry. Technological demands for light, high strength materials were the driving force behind the production of the latest synthetic polymers, such as Kevlar and Victrex PEEK. The square brackets indicate that the illustrated structure (the monomer unit) recurs again and again to form long polymer chains.
Chemicals in Everyday Life (1 of 4)
We live with chemicals all around us - and we are ourselves a highly complex assembly of chemical compounds. Chemicals are to be found where we live, where we work and as we travel. They influence the way we grow our food, control our environment, and protect our health; they can brighten our lives and enhance the quality of our everyday experiences.
Few people realize the extent to which chemistry supports and pervades many aspects of modern life. Man-made fibers such as nylon and polyester have revolutionized the fashion industry, while computers rely on materials developed through chemistry for their operation. However, most of the chemicals we come into contact with are mixtures, or formulations, rather than pure chemical compounds. In some formulations, the properties of certain chemicals are enhanced by the presence of others. In other products, such as soap powders, particular chemicals may be present simply in order to provide a more convenient medium in which the active chemicals can operate.
Synthetic polymers
One of the most far-reaching influences chemistry has had on our lives in the 20th century has been the introduction of increasingly advanced materials based on synthetic polymers . 316-17). These are materials composed of repeating small molecules, or monomers, joined together to make long chains. The way in which these chains interact with one another depends on the monomers from which they are composed, and can differ greatly. This accounts for the wide range of properties that polymers can possess, including flexibility, strength and heat resistance. Demand for improved materials, both for technological and household application, is still increasing, and - although already widespread - there is still a lot more polymer-based technology to come into our lives.
Along with the emergence of excellent new dyes and paints, the fashion and display industries have been revolutionized by the advent of polyester fibers such as Dacron, and polyamide fibers such as Nylon 6 and Nylon 66. These early synthetic polymers were all composed of moderately flexible chains. More recently, stiff-chain polymers - often incorporating aromatic rings - have found many applications. Because of their low flammability, high thermal stability and great tensile strength, compounds such as Kevlar and Victrex PEEK are becoming very widespread; they may be found in an enormous range of products, from electrically heated hair-styling brushes to bullet-proof vests.
Not only has Kevlar a tensile strength higher than that of steel but it also has a strength-to-weight ratio that is six times better. Composite materials containing Kevlar have been used to build the tail sections of jumbo jets; it is transparent to radar, it eliminates corrosion, and its lightness lowers fuel costs. The outstanding mechanical and electrical properties of Victrex PEEK, particularly when employed in fiber-reinforced composites with glass, carbon or Kevlar, are such that it can be used to replace metal alloys to great advantage in many engineering situations. The day of the all-plastic motor car is just around the corner.
* WHAT IS CHEMISTRY?
* NATURAL COMPOUNDS
* MAN-MADE PRODUCTS
* MEDICAL TECHNOLOGY
* THREATS TO THE ENVIRONMENT
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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Chemicals in Everyday Life (page 2)
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Chemicals in Everyday Life (2 of 4)
Agrochemicals
At the end of the 20th century, the challenge of feeding the world's growing population can only be met with the aid of chemicals produced by the fertilizer industry. In addition to sunlight, oxygen, carbon dioxide and water, plants require macro-nutrients - as well as trace elements - from the soil. By far the most important of these are nitrogen (N), phosphorus (P) and potassium (K). The fertilizer industry provides compounds in which these macro-nutrients are in a form that can be readily used by plants, principally as water-soluble salts: nitrogen as nitrates (NO3-) of sodium and potassium, and as ammonium sulfate ((NH4)2SO4); phosphorus as phosphates (PO43-) and potassium as potash (K2O). Commercially produced fertilizers are currently responsible for approximately 50% of Europe's agricultural output.
Pesticides have been developed in order to protect crops from predators such as insects (insecticides) and from fungi (fungicides), or simply to prevent them from being overrun by weeds (herbicides). Pesticides have two general modes of action by which they achieve these ends: non-systemically active pesticides do not actually penetrate the plant surface - for this reason they are known as contact killers; systemically active pesticides are taken up by the plant's vascular system and inhibit some life-sustaining process. The latter is more desirable, since the chemical is thereby located within the plant itself and is less prone to being washed away into the surrounding water table, but it is a much more difficult goal to achieve. It is only in the last 50 years that effective systemically active pesticides have proved successful. An example is Glyphosate, a widely used herbicide that kills weeds by inhibiting the biological synthesis of a family of vital amino acids:
* WHAT IS CHEMISTRY?
* NATURAL COMPOUNDS
* MAN-MADE PRODUCTS
* MEDICAL TECHNOLOGY
* THREATS TO THE ENVIRONMENT
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p056-3
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Chemicals in Everyday Life (page 3)
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Ampicillin is one of a family of closely related antibiotics
the penicillins. Because of the shape of these drugs, the enzyme in the invading bacterium responsible for cell-wall construction mistakes them for constituents of the cell wall. Once this has occurred, the
enzyme cannot continue its work, the wall protecting the cell is left unfinished and the cell can be destroyed by the body's immune system.
} Chemicals in Everyday Life (3 of 4)
Chemistry in medicine
The enormous influence of pharmaceutical products upon modern medical practice is an illustration of the problem-solving capacity of chemistry. Some of the greatest advances in medicine in the last 150 years have come about as a result of advances in chemical thinking and a growing appreciation of the chemical processes occurring in our bodies. The development of chemistry allowed people not only to isolate the active components in herbal remedies, but to start to synthesize them - and, more importantly, to synthesize closely related analogs that often display even greater activity.
The first advance was the development of anaesthetic gases, used to render patients insensitive during surgical operations. The first widely used an aesthetics were chloroform (CHCl3) and ether (CH3CH2OCH2CH3); however, chloroform was discovered to have undesirable side-effects and was replaced by the gas nitrous oxide (N2O) and fluorocarbons. Antiseptics, such as phenol (C6H5OH), were also introduced to surgery in the 19th century, to prevent bacterial infections. A related category of compounds - disinfectants - also kill bacteria and microorganisms, but because of their strength they are only used on non-living objects. Common examples of disinfectants include sodium hypochlorite (NaOCl) and hydrogen peroxide (H2O2). Some of these compounds have survived, but most have been replaced by more effective and less toxic alternatives.
One of the most important classes of drugs to be discovered are the antibiotics. These are drugs that generally work by aiding the body's own defense system, helping the white blood cells to stop bacterial cells from multiplying. The first antibiotic was discovered by chance in 1928 by Alexander Fleming. However, it was not until after World War II that the active ingredient - penicillin - could be identified and produced in large enough quantities for clinical trials; its success was then immediate. Many different types of penicillin exist, but they all share the same basic structure as ampicillin.
Certain medical problems, particularly those associated with heart disease and ulcers, have been aggravated by the stresses imposed by modern life styles. Combating such medical conditions has become a major concern of pharmaceutical companies, and a number of spectacularly successful drugs have recently been produced. SU
d. SU
* WHAT IS CHEMISTRY?
* NATURAL COMPOUNDS
* MAN-MADE PRODUCTS
* MEDICAL TECHNOLOGY
* THREATS TO THE ENVIRONMENT
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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Chemicals in Everyday Life (page 4)
ftsTitle
Chemicals in Everyday Life (4 of 4)
BETA-BLOCKERS AND ULCER DRUGS
In 1958 the Scottish pharmacologist James Black (1924- ), convinced that the challenge in pharmacology was to reduce the randomness in the game of `molecular roulette' played by drug companies at the time, applied his unconventional thinking to coronary heart disease. He initiated a search for a compound that would combat the undesired effect of increasing heart rate and muscle tension brought about by adrenaline, the hormone released under panic to prepare the body for `fight or flight'. The molecules of such a compound, he argued, would have to bind to the so-called beta-adrenergic receptors in the heart and thus block the action of adrenaline molecules, without exhibiting their undesired physiological activity. This led to the synthesis of propanolol (Inderal), which was launched in 1964 and was the first of the beta-blockers. Many more have subsequently appeared on the market, and today atenolol (Tenormin) is the most widely prescribed drug for the management of hypertension and coronary heart disease.
In 1964 James Black applied a similar line of reasoning in search of a drug to treat ulcers. Histamine in the body is capable of interacting with two receptor sites - at the H1 receptor during allergic responses and at the H2 receptor, causing increased gastric-acid secretion, which in turn leads to ulceration of the stomach and/or the duodenum. The need was to find a drug that would compete successfully for the H2 receptor without blocking the H1 receptor, at which the older anti-histamines were known to act. After 12 years and the synthesis of 12 000 compounds at a cost of millions of dollars, the anTAGonist ciMETidine or `TAGaMET' was introduced into clinical practice to wide acclaim. By preventing gastric-acid secretion, it proved to be spectacular in healing peptic ulcers, and surgical operations for duodenal ulcers became less and less necessary. Subsequently a team of researchers discovered that a simple modification of cimetidine led to the ulcer drug ranitidine (Zantac), which has fewer side-effects than Tagamet. Not only have these drugs proved to be a blessing for thousands of people with ulcers, but they have also benefited national economies by reducing drastically the numbers of patients requiring hospitalization.
The ulcer drugs Zantac and Tagamet are currently two of the three most important pharmaceutical products, with annual sales throughout the world running at $1500 million and $1000 million respectively. The third, also with annual sales of $1000 million, is Tenormin - one of the descendants of James Black's first beta-blocker.
* WHAT IS CHEMISTRY?
* NATURAL COMPOUNDS
* MAN-MADE PRODUCTS
* MEDICAL TECHNOLOGY
* THREATS TO THE ENVIRONMENT
* OIL AND GAS
* RUBBER AND PLASTICS
* CHEMICALS AND BIO TECHNOLOGY
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p058-1
ftsTitleOverride
The Scientific Method (page 1)
ftsTitle
Sir Karl Popper. According to Popper, knowledge is better advanced by scientists attempting to disprove theories, rather than trying to prove them.
The Scientific Method (1 of 3)
The spectacular successes of the natural sciences from the 17th century onwards have prompted a search for `the scientific method'. Until the 20th century this was seen as the search for a general set of instructions or recipe for getting scientific results. But nowadays it has become an attempt to describe the general aims of science. Scientific method is now thought of as whatever in practice serves to promote those aims.
Asking what all the various subjects popularly called `sciences' have in common would yield only platitudes like `Don't jump to conclusions in the absence of firm evidence.' So attention has focused on clear paradigms - or models - of science such as physics. Physical theories are paradigms of comprehensiveness because they explain physical processes that vary in scale from the subatomic (about 10 to the power of -15 m or less) to the astronomical (billions of billions of kilometers), and vary in time from about 10 to the power of -24 seconds to billions of years. Furthermore, physical theories yield paradigms of accurate prediction. For example, NASA was able to calculate the speed, direction and time of launch needed to send the Voyager 2 spacecraft, without any later course correction, on a journey of thousands of millions of miles passing close to each of the outer planets in turn.
By studying paradigms of science we may hope to answer still controversial questions such as `Is Freudian psychiatry really scientific?' or `Can social or historical processes be explained scientifically, as Karl Marx claimed?'
* THE HISTORY OF ASTRONOMY
* THE HISTORY OF SCIENCE
* THE HISTORY OF MEDICINE
* PHILOSOPHY
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p058-2
ftsTitleOverride
The Scientific Method (page 2)
ftsTitle
The medieval world view is broken. This 16th-century German woodcut is a symbolic celebration of the attempts of Renaissance astronomers to penetrate appearances and find the mechanisms underlying the universe.
The Scientific Method (2 of 3)
Physics as a paradigm of science
Galileo and Descartes were the first to insist clearly that science should employ only precise mathematical concepts in its theories. The application of such a theory to physical reality will then be a calculation in applied mathematics. This is the feature that makes physical theories so comprehensive and such precise predictors.
Newton, for example, needed only four laws to explain the orbits of the moon and planets to the accuracy afforded by available measuring instruments. But those same laws also explain the rate at which a body falls, the motion of a pendulum, and even a simplified version of the relation between the temperature, pressure and volume of a gas. Set out in full the calculations would be very long, because the very same small set of laws are being used to explain such diverse phenomena. Such mathematical calculations are logical, deductive inferences . Deductive inferences using non- mathematical concepts, on the other hand, cannot in practice be sustained for long without losing their credibility. Hence the central role of mathematics in physics. Theories employing non-mathematical concepts could not achieve such comprehensiveness and precision.
However, not all successful sciences match the paradigm of physics. Darwin's theory of evolution and Pasteur's germ theory of disease are examples of theories using non-mathematical concepts. But biologists seek to use quantitative mathematical conceptions wherever they can.
* THE HISTORY OF ASTRONOMY
* THE HISTORY OF SCIENCE
* THE HISTORY OF MEDICINE
* PHILOSOPHY
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p058-3
ftsTitleOverride
The Scientific Method (page 3)
ftsTitle
An experimental biochemical laboratory. Many of the observed phenomena upon which scientists seek to build theories are in the form of specially designed experiments. Such experiments are often used to test a hypothesis - a suggested explanation for a collection of known facts. If the experiments confirm the hypothesis, the hypothesis may be elevated to the more certain status of a theory. However, such a theory is never - in the strictest sense - a logical conclusion from the observed facts.
The Scientific Method (3 of 3)
The ideal of science, and the practice
Full comprehensiveness is an ideal yet to be achieved. The two current leading theories, quantum mechanics (which explains atomic processes) and the general theory of relativity (which explains astronomical processes) are mutually inconsistent, although both are firmly accepted by all physicists. But comprehensiveness remains an ideal of physics, because physicists recognize this inconsistency as a problem requiring resolution.
Surprisingly, a theory can be accepted as true even though it is known to make some false predictions. For example, it was well known in the 19th century that Newton's laws could not be squared with the precise orbit of the planet Mercury. However, because Newton's laws were so successful elsewhere, 19th-century physicists regarded Mercury's orbit as an unexplained anomaly that did not shake their belief in Newton's laws. Only after those laws had been superseded by the theory of relativity was the orbit of Mercury regarded as one of the facts that refuted Newton.
DEDUCTIVE AND INDUCTIVE INFERENCES
Here is a simple deductive inference.
Given the premises All rabbits are mammals.
All mammals have kidneys. we deduce the conclusion All rabbits have kidneys. The defining characteristic of a valid deductive argument is that it is impossible for all the premises to be true and the conclusion false, because the information contained in the conclusion is already stored in the premises, taken collectively. Mathematical calculations are deductive inferences.
Here is a simple inductive inference.
Given the premise All observed ravens are black. we induce the conclusion All ravens, whether observed or not, are black. The defining characteristic of an inductive argument is that the information contained in the conclusion goes beyond the information contained in the premises. Hence it is possible for inductive arguments to let us down - for their premises to be true but their conclusions false.
Indeed, inductive inferences have been known to let us down. Once upon a time we were in a position to assert All observed swans are white. and hence to induce the conclusion All swans are white. But then black swans were discovered in Australia.
Deductive inferences, no matter how long, have been codified and the rules for their validity worked out. Inductive inferences have resisted codification, and their validity is controversial.
Explanations of particular experimental results by theories use deductive inferences. Justification of a theory by experimental results uses inductive inferences, although of a kind more complex than the simple example given.
* THE HISTORY OF ASTRONOMY
* THE HISTORY OF SCIENCE
* THE HISTORY OF MEDICINE
* PHILOSOPHY
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The History of Science (page 1)
ftsTitle
An experiment with a bird in a steampump (1768) by Joseph Wright. Although great advances in science were made during the Scientific Revolution of the 17th century, much science was still dressed up as entertainment to obtain an audience, and the practical application of much of the new knowledge was limited.
The History of Science (1 of 2)
If we consider science as the systematic investigation of reality by observation, experimentation and induction, then among early civilizations science did not exist. Certainly discoveries were made, but they were piecemeal. Myth and religion dominated as modes for explaining the world.
This began to change with the speculations of the early Greek philosophers, who excluded supernatural causes. By the 3rd century bc Greek science was highly sophisticated and producing theoretical models that have shaped the development of science ever since.
With the fall of Greece to the Roman Empire, science fell from grace. Few important advances were made outside medicine, and the work done was firmly within the Greek traditions and conceptual frameworks.
For several centuries from the fall of Rome in the 5th century ad, science was practically unknown in Western Europe. Islamic culture alone preserved Greek knowledge and later transmitted it back to the West. Between the 13th and 15th centuries some advances were made in the fields of mechanics and optics, while men like Roger Bacon insisted on the importance of personal experience and observation.
The 16th century marked the coming of the so-called 'Scientific Revolution', a period of scientific progress beginning with Copernicus and culminating with Newton. Not only did science break new conceptual ground but it gained enormously in prestige as a result. Science and its trappings became highly fashionable from the later 17th century, and also attracted a great deal of royal and state patronage. The founding of the Acadumie des Sciences by Louis XIV in France and the Royal Society by Charles II in Britain were landmarks in this trend.
In the course of the 19th century science became professionalized, with clear-cut career structures and hierarchies emerging, centered on universities, government departments and commercial organizations. This trend continued into the 20th century, which has seen science become highly specialized and dependent on technological advances. These have not been lacking.
Modern science is immense and extremely complex. It is virtually impossible to have an informed overview of what science as a whole is up to. This has made many people regard it with some suspicion. Nevertheless Western civilization is fully committed to a belief in the value of scientific progress as a force for the good of humanity. While some of the world's greatest dangers and horrors have their roots in scientific endeavor, there is some hope that science will also eventually provide viable solutions to them.
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* PHYSICS
* CHEMISTRY
* THE SCIENTIFIC METHOD
* MATHEMATICS
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* THE HISTORY OF MEDICINE
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The History of Science (page 2)
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Marie Curie (1867-1934), the Polish scientist who, with her French husband Pierre (1859-1906) and Henri Becquerel (1852-1908), received the 1903 Nobel Prize for Physics for the discovery of radioactivity. She also received the 1911 Prize for Chemistry for her discovery of the elements radium and polonium.
/U/The History of Science (2 of 2)
MAJOR DEVELOPMENTS IN SCIENCE
3500-3000 BC
The Sumerians develop metallurgy and the use of a lunar calendar.
3000-2500 BC
Multiplication tables are invented and mathematics used for calculating areas. In Egypt a solar calendar is used.
2500-2000 BC
A superior lunar calendar is used in Babylon. Units of time such as the minute and hour are introduced.
2000-1500 BC
Babylonians use maths to plot planetary positions. The stellar constellations are identified. Simple taxonomies for classifying animals are used.
1500-1000 BC
Mathematics continues to develop. Chemicals are used to make paints and cosmetics.
1000-500 BC
Early Greek philosophers conceive rational theories of the universe. Those of Thales of Miletus (640-560 bc) and Anaximander (611-547 bc) are notable. Anaximander introduced the concept of infinity into cosmology and believed that life had evolved from the sea. The notion that the world is a sphere is attributed to Pythagoras (c. 580-c. 500 bc), who also formulated basic laws of geometry.
500-400 BC
The concept of elementary matter was introduced by Empedocles of Agrigentum (c. 490-430 bc), who believed that there are four elements, namely earth, water, fire and air. Democritus (c. 460-c. 370 bc) and Leucippus (c. 500-450 bc) conceived of matter as consisting of minute invisible particles called atoms.
400-300 BC
The first fully comprehensive cosmology to give a rational account of all physical phenomena was devised by Aristotle (384-322 bc; see also p. 486). He divided the universe into two distinct regions. Below the sphere of the Moon was the realm of the four elements and of change and decay. Above was the realm of a fifth element, the ether, changeless and divine. Each element had its natural place and motion, the ether moving in circles around the Earth, and carrying the stars with it. Aristotle's cosmology and physics ruled until the time of Galileo and Newton. Aristotle did the first systematic work on comparative biology.
300-200 BC
Archimedes of Syracuse (287-212 bc) pioneered the sciences of mechanics and hydrostatics invented the lever and the Archimedian screw for raising water, and made many contributions to mathematics. Observational astronomy reached its peak with Aristarchus of Samos (c. 310-250 bc), who realized that the Earth rotates on its own axis and orbits the Sun.
200-100 BC
The most accurate ancient star catalogue was constructed by Hipparchus of Nicaea (c. 190-120 bc), who also discovered the precession of the equinoxes.
100 BC-AD 100
Little original science was done in these centuries, although Greek astronomy was perfected by Ptolemy (Claudius Ptolemaeus, ad 100-170), in whose system the Earth was the center of the universe, so rejecting the theory of Aristarchus. The earliest known alchemical text appeared; alchemy was a mystical forefather of chemistry that sought to transmute base metals into gold and produce the elixir of eternal life.
AD 200-1200
Much of classical learning disappeared from Europe during the so-called 'Dark Ages', but was preserved by Islamic scholars such as Avicenna (Ibn Sinna, ad 980-1037) and Averros (ibn-Rushd, 1126-98). From c. 1100 it was transmitted back when Christian scholars such as Gerard of Cremona (1145-87) translated Arabic texts into Latin and began to assimilate ancient knowledge.
1200-1300
Albertus Magnus (Count von Bollstadt, c. 1193-1280), a German scholastic philosopher, patron saint of scientists and teacher of Aquinas, worked to reconcile Aristotelian science and philosophy with Christian doctrine. The English friar Roger Bacon (c. 1214-92) became a great advocate of experimentation. He did important work in optics and was the first European to describe the manufacture of gunpowder. He was also a great speculator, proposing flying machines and mechanically powered ships and carriages. The man considered by Bacon to be the greatest experimental scientist of his day was the French crusader Petrus Peregrinus (active 13th century), who described in detail the use of the magnetic compass in navigation.
1300-1400
The English philosopher William of Ockham (c. 1285-1349) propounded the principle (known as Ockham's razor) that 'entities are not to be multiplied beyond necessity'. This principle, that the simplest explanation is the best, was adopted by many later scientists, including the French bishop Nicole d'Oresme (c. 1325-82), who worked on cosmology and motion. In the latter field, Oresme confirmed the theories of the Merton School at Oxford, and both Oresme and the Mertonians worked on the mathematization of science.
1400-1500
There was little of scientific note in this century, although at the end of the century Leonardo da Vinci (1452-1519) began his studies of all kinds of natural phenomena. The discovery of the New World in 1492 contradicted the geographical teachings of Ptolemy, so helping to free science from its psychological dependence on ancient authorities.
1500-1550
Nicholas Copernicus (1473-1543), the Polish astronomer, revived the heliocentric theory, placing the Sun at the center of the universe. Because this theory threatened the Church's cosmology, Copernicus only circulated it among a few friends. Chemistry was to some extent freed from its alchemical bonds by Paracelsus (real name Theophrastus Bombastus von Hohenheim, 1493-1541).
1550-1600
The study of terrestrial magnetism was developed by English physician William Gilbert (1540-1603), who introduced the concept of magnetic poles. Tycho Brahe (1546-1601) produced a very accurate star catalogue, and his assistant Johann Kepler (1571-1630) demonstrated that planetary orbits round the Sun are elliptical. The English statesman and philosopher Sir Francis Bacon (1561-1626) revived the use of induction in scientific method.
1600-1650
The modern science of mechanics (statics) was founded by Galileo Galilei (1564-1642). Galileo formulated laws of motion that conflicted with ancient physics, and tended to support the heliocentric hypothesis. The French philosopher and mathematician Renu Descartes (1596-1650; see also pp. 62, 418 and 486) proposed a radically mechanistic model of the universe that rendered God virtually redundant. He invented coordinate geometry.
1650-1700
The controversy between ancient and modern cosmologies and physics was resolved in the work of Englishman Sir Isaac Newton (1643-1727). He formulated the law of universal gravitation and three laws of motion and made important contributions to optics and calculus. Chemistry continued to be separated from its alchemical roots by the work of men such as the English scientists Robert Boyle (1627-1716) and Robert Hooke (1635-1703), who studied the chemistry of gases and the nature of respiration and combustion.
1700-1750
Combustion was also tackled by the German chemist Georg Stahl (1660-1743), who suggested a hypothetical substance called phlogiston as the causal agent of combustion.
1750-1800
The Swedish botanist Carl Linnaeus (1707-78) introduced his binomial system of biological classification. The phlogiston theory was rendered obsolete with the discovery of oxygen by the English chemist and radical Joseph Priestley (1733-1804), who also invented soda water. However, it was left to the Frenchman Antoine Lavoisier (1743-94) to name oxygen and demonstrate its role in combustion. Lavoisier also formulated the important law of conservation of matter and recognized that air and water are chemical compounds. In geology the Scotsman James Hutton (1726-97) introduced the notion that the Earth is millions of years old, denying catastrophes such as Noah's Flood. The Frenchman Charles Augustin Coulomb (1736-1806) first identified the electric force. In Italy Count Alessandro Volta (1745-1827) made important experiments with electricity, while the Frenchman Andru Ampere (1775-1836) did pioneering work on electricity and magnetism. The concept of 'biology' was established by the Frenchman Jean-Baptiste Lamarck (1744-1829), who also set out a theory of evolution.
1800-1850
The conceptual groundwork of modern chemistry was laid by the Englishman John Dalton (1766-1844) when he revived atomic theory and applied it to gases. The Englishman Michael Faraday (1791-1867) and the American Joseph Henry (1797-1878) separately discovered electromagnetic induction, the basis of electricity generation. Study of the nature of heat was furthered by American-born physicist Benjamin Thompson (Count Rumford, 1753-1814), who suggested that it was a form of motion rather than a substance. The English amateur scientist James Joule (1818-89) did important work on thermodynamics, discovering the principle of the mechanical equivalent of heat, and helping to develop the principle of the conservation of energy.
1850-1900
Thermodynamics was furthered by the Scottish physicist William Thomson (Lord Kelvin, 1824-1907). The Russian chemist Dmitri Mendeleyev (1834-1907) compiled the first periodic table of chemical elements. The English naturalist Charles Darwin (1809-92) revolutionized biology with his theory of evolution by natural selection. The study of genetics was furthered by the Austrian monk Gregor Mendel (1822-84), who demonstrated that inheritance involves dominant and recessive characteristics. The Scottish physicist James Clerk Maxwell (1831-79) established the concept of the electromagnetic force, and in 1887 the existence of electromagnetic waves was demonstrated experimentally by the German physicist Heinrich Rudolf Hertz (1857-94). At the end of the century another German physicist, Wilhelm Rontgen (1845-1923), discovered X-rays, a fundamental research tool in physics and a vital diagnostic tool in medicine. Ernest Rutherford (1871-1937), an English physicist, used X-rays to investigate gases, and discovered alpha, beta and gamma rays.
1900-PRESENT
Mendel's work was developed by the American geneticist Thomas Hunt Morgan (1866-1945), the discoverer of chromosomes. After the Canadian bacteriologist Oswald Avery (1877-1955) had demonstrated that DNA is responsible for inheritance, the Anglo-American team of Francis Crick (1916- ), James Watson (1928- ) and Maurice Wilkins (1916- ) were able in 1953 to unravel its structure. Following the cracking of the genetic code by Wilkins and others in the 1960s, Paul Berg (1926- ) and colleagues developed the use of restriction enzymes to cut DNA and allow the insertion of new genes. This originated the now widespread technology of genetic engineering. The major biological activity of the century, the human genome project, is expected, by the year 2000, to map the entire genetic structure of human DNA and enormously advance understanding of human physiology and disease.
In astronomy, Erwin Hubble (1889-1953) proved that the universe was rapidly expanding. The first space telescope, launched into permanent orbit in April 1990, was named after him.
The structure of atoms was investigated by Ernest Rutherford, who discovered the atomic nucleus. The German physicist Albert Einstein (1879-1955) radically revised classical physics with his theories of special and general relativity. The German Max Planck (1858-1947) formulated quantum theory, which was applied to Rutherford's atom by the Dane Niels Bohr (1885-1962), thereby effecting another major revision of classical physics. Understanding of the structure of the atom and the tremendous forces locked into it led to the development of nuclear power and nuclear weapons. Erwin Schrodinger (1887-1961) and Werner Heisenberg (1901-76) fundamentally advanced ideas on the nature of the atom by developing quantum mechanics. Deepening research into subatomic particles led Murray Gell-Mann (1929- ), Henry Kendall (1926- ) and others to the concept of quarks, from which other subatomic particles are made.
Modern science is dominated by expensive technology and extreme specialization. In physics, subatomic particles continue to be investigated, and are thought to hold the key to understanding the origin and ultimate nature of the universe.
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* ASTRONOMY
* PHYSICS
* CHEMISTRY
* THE SCIENTIFIC METHOD
* MATHEMATICS
* EARTH SCIENCES
* LIFE SCIENCES
* THE HISTORY OF MEDICINE
* TECHNOLOGY
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p062-1
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Mathematics and its Applications (page 1)
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Mathematics and its Applications (1 of 4)
Many people think of mathematics in terms of rules to be learned in order to manipulate symbols or study numbers or shapes in the abstract for their own sake. Mathematical theory does develop in the abstract; it need have no dependence on anything outside itself. The truth of the theory is measured by logic rather than experiment. However, one of its most valuable uses is in describing or modeling processes in the real world, and thus there is constant interaction between pure mathematics and applied mathematics.
Mathematics may be considered as the very general study of the structure of systems. Since the study is unrelated to the physical world, rigorous formal proofs are sought, rather than experimental verifications. Theory is presented in terms of a small number of given truths (known as axioms) from which the entire theory can be inferred.
Thus, the aims are for generality in approach and rigor in proof, aims that explain the traditional concern of mathematicians for the unification of seemingly different branches of mathematics. As an example, Descartes showed that geometrical figures could be described in terms of algebra, enabling geometric proofs to be established in terms of arithmetic, so that both generality and rigor were advanced.
Applied mathematics and modeling
There is no sharp boundary between the study of mathematical systems in the abstract (the field of pure mathematics) and the study of such systems to make inferences about certain physical systems that are described by the mathematical theory (the field of applied mathematics). In principle, any branch of mathematics may turn out to describe some physical, economic, biological, medical, or other system. Modeling a physical system consists of seeking a formal mathematical theory that conforms with the properties of the physical system. Often, as for example in computer simulations of space travel, the mathematical theories are very large and complex, but sometimes the model can be quite simple. Sometimes, known mathematics can describe and predict the behavior of the system; at other times, the modeling can give rise to completely new branches of mathematics.
Applied mathematics encompasses many specialized fields in which the relation ships between the experimental findings and the mathematical theories are well established. Although the subject can include the application of statistical theory to such areas as sociology, the term is usually restricted to the application of the methods of advanced calculus, linear algebra and other branches of advanced mathematics to physical and technological processes.
Triangulation, geometry and trigonometry
A simple example of a mathematical model is the representation of a portion of the Earth's surface by a set of interlocking triangles, from the measurement of which maps may be constructed. The triangulation model uses the rules of geometry and trigonometry to derive angles and distances that cannot be measured directly.
Geometry establishes that two triangles each have angles of the same sizes if, and only if, corresponding pairs of sides are in the same proportions.
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* THE HISTORY OF SCIENCE
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Mathematics and its Applications (page 2)
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Mathematics and its Applications (2 of 4)
Here D, E and F are the center-points of sides AB, BC and CA respectively. So, DE is half the length of AC, EF is half the length of AB, and FD is half the length of BC. Thus, the shaded triangle, DEF, is similar to the large triangle, and the angles at D, E and F are, respectively, equal to those at C, A and B. Furthermore, the triangles ADF, FEC, DBE and EFD are all congruent, i.e. identical in shape and size, and are thus all similar to triangle ABC.
A right-angled triangle is a triangle where one of the angles is 90 deg. Pythagoras' theorem states that, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides. So, in the triangle shown, below, AC' = AB' + BC'. Trigonometry relies on the recognition that in a right-angled triangle the ratio of the lengths of pairs of sides depends only on the sizes of the two acute angles (i.e. angles less than 90 deg) of the triangle.
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Mathematics and its Applications (page 3)
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Mathematics and its Applications (3 of 4)
These ratios are given names. For example, the sine of an angle is the ratio of the side opposite the given angle to the hypotenuse. The Greek letters (theta) and (phi) are usually used to denote the angles; thus in the triangle shown we say that the sine of , usually written sin , is BC/AC. Similarly, since the cosine (cos) of the angle is the ratio of the side adjacent to the given angle to the hypotenuse, cos is AB/AC. The third basic ratio is the tangent (tan), which is the ratio of the opposite to the adjacent side, BC/AB in the example; it is easy to see that tan must always equal sin / cos . Pythagoras' theorem can be used to establish some very useful values for sin, cos and tan.
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Mathematics and its Applications (page 4)
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&Mathematics and its Applications (4 of 4)
In triangle DEF, DE = EF = 1, so the angles at D and F are equal, that is they are each 45 deg (the internal angles of a triangle add up to 180 deg ). Using Pythagoras' theorem, DF' = 1' + 1' = 2, so DF = 2. We can therefore conclude:
1
sin 45 deg = ---
1
cos 45 deg = ---
tan 45 deg = 1
In triangle GHK, GH = HK = KG = 2, so the angles at G, H and K are equal, that is they are each 60 deg . Using Pythagoras' theorem, KL' + 1' = 2', so KL = 3. We therefore have:
sin 60 deg = 1/2
3 = cos 30 deg
cos 60 deg = 1/2 = sin 30 deg
tan 60 deg =
tan 30 deg = 1 /
SOME EMINENT MATHEMATICIANS
Pythagoras (c. 582-500 BC), Greek philosopher. Born in Samos, he founded a religious community at Croton in southern Italy. The Pythagorean brotherhood saw mystical significance in the idea of number. Popularly remembered today for Pythagoras' theorem.
Euclid (c. 3rd century BC), Greek mathematician. Euclid devised the first axiomatic treatment of geometry and studied irrational numbers. Until recent times, most elementary geometry textbooks were little more than versions of Euclid's great book.
Archimedes (c. 287-212 BC), Greek mathematician, philosopher and engineer, born in Syracuse, Sicily. His extensions of the work of Euclid especially concerned the surface and volume of the sphere and the study of other solid shapes. His methods anticipated the fundamentals of integral calculus.
Descartes, Renu (1596-1650), French philosopher, mathematician and military scientist. Descartes sought an axiomatic treatment of all knowledge, and is known for his doctrine that all knowledge can be derived from the one certainty: Cogito ergo sum (`I think therefore I am'). One of his major mathematical contributions was the development of analytical geometry, whereby geometrical figures can be described in algebraic terms.
Newton, Sir Isaac (1643-1727), English mathematician, astronomer and physicist. Newton came to be recognized as the most influential scientist of all time. He developed differential calculus and his treatments of gravity and motion form the basis of much applied mathematics.
Euler, Leonhard (1707-83), Swiss-born mathematician, who worked mainly in Berlin and St Petersburg. He was particularly famed for being able to perform complex calculations in his head, and so was able to go on working after he went blind. He worked in almost all branches of mathematics and made particular contributions to analytical geometry, trigonometry and calculus, and thus to the unification of mathematics. Euler was responsible for much of modern mathematical notation.
Gauss, Carl Friedrich (1777-1855), German mathematician. He developed the theory of complex numbers. He was director of the astronomical observatory at Gottingen and conducted a survey, based on trigonometric techniques, of the kingdom of Hanover. He published works in many fields, including the application of mathematics to electrostatics and electrodynamics.
Cauchy, Baron Augustin-Louis (1789-1857), French mathematician and physicist. He developed the modern treatment of calculus and also the theory of functions , as well as introducing rigor to much of mathematics. As an engineer he contributed to Napoleon's preparations to invade Britain, and he twice gave up academic posts to serve the exiled Charles X.
Boole, George (1815-64), English mathematician. Despite being largely self-taught, Boole became Professor of Mathematics at Cork University. He laid the foundations of Boolean algebra, which was fundamental to the development of the digital electronic computer.
Cantor, Georg (1845-1918), Russian-born mathematician who spent most of his life in Germany. His most important work was on finite and infinite sets. He was greatly interested in theology and philosophy.
Klein, Christian Felix (1849-1925), German mathematician. Klein introduced a program for the classification of geometry in terms of group theory. His interest in topology (the study of geometric figures that are subjected to deformations) produced the first description of a Klein bottle - which has a continuous one-sided surface.
Hilbert, David (1862-1943), German mathematician. In 1901, Hilbert listed 23 major unsolved problems in mathematics, many of which still remain unsolved. His work contributed to the rigor and unity of modern mathematics and to the development of the theory of computability.
Russell, Lord Bertrand (1872-1970), English philosopher and mathematician. Russell did much of the basic work on mathematical logic and the foundations of mathematics. He found the paradox now named after him in the theory of sets proposed by the German logician Gottlob Frege (1848-1925), and went on to develop the whole of arithmetic in terms of pure logic. He was jailed for his pacifist activities in World War 1. In 1950, he was awarded the Nobel Prize for Literature.
CHAOS THEORY
From its beginnings, science has been a quest for orderly laws that govern nature. And with each advance it has seemed that some element of disorder has been conquered. Complex dynamical systems, in particular, could be understood and quantified when the calculus was invented ( pp. 70-71). But scientists have long recognized that many natural phenomena - the movement of clouds, turbulence in streams or in the rising smoke from a cigarette ( box, p. 23), the movement of a leaf in the wind, the patterns of brain waves, disease epidemics or traffic jams - are so inherently disordered and chaotic as to seem to defy any attempt to find governing laws.
As early as 1903, however, the French mathematician Jules Henri Poincaru (1854-1912) - famous for his work on topology - recognized that there are circumstances in which tiny inaccuracies in initial conditions can be multiplied so as to lead to huge differences in the outcome. Poincaru's work was largely forgotten until in 1961 the American meteorologist and mathematician Edward N. Lorenz, working with a crude early computer, set out to produce a mathematical model of how the atmosphere behaves. In the course of this work Lorenz accidentally hit upon the first mathematical system in which small changes in the initial conditions led to overwhelming differences in the outcome. Lorenz showed that this phenomenon made long-range weather prediction almost impossible. His work and the analogies that developed from it attracted the attention of scientists in other fields and led to the development of a new branch of mathematics - chaos theory. One of the most striking of these analogies is known as the `butterfly effect' - the idea that the air perturbation caused by the movement of a butterfly wing in China can cause a storm a month later in New York.
By the 1970s some scientists and mathematicians, and even some economists, were beginning to investigate disorder and instability. Physiologists were considering patterns of chaos in the action of the heart-patterns that could lead to sudden cardiac arrest; electronic engineers were investigating the sometimes chaotic behavior of oscillators; ecologists were examining the seemingly random way in which wildlife populations changed; chemists were studying unexpected fluctuations in chemical reactions; and economists were wondering whether some order might be found in random stock-market price fluctuations.
The first indication for an underlying pattern in chaos was found by the American physicist Mitchell Feigenbaum. In 1976 Feigenbaum noticed that when an ordered system starts to break down into chaos, it often does so in accordance with a consistent pattern in which the rate of occurrence of some event suddenly doubles over and over again. This is exactly what happens in fractal geometry - in which any part of a figure is a reduced copy of a larger part. Feigenbaum also discovered that at a certain constant number of doublings, the structure acquires a kind of stability. This numerical constant, called Feigenbaum's number, can be applied to a wide range of chaotic systems.
To understand what mathematicians mean by chaos it is best to consider a simple example. Iteration is the mathematical process in which the result of a calculation is applied as the starting point for a repeat of the same process and so on. One might, for instance, take a number and halve it, then take the result and halve that, and so on repeatedly. The set of numbers that result is called the orbit of the number. Starting with, say, 16, the orbit would be 8, 4, 2, 1, 1/2, 1/4, 1/8, 1/16... Again, one might perform an iterative process on any number (x) between 0 and 1, the process being `multiply the product of x and 1
x by 3'. This gives a readily predictable orbit. Surprisingly, iteration for numbers between 0 and 1 using the process `multiply the product of x and 1
x by 4' produces a chaotic orbit for some numbers and a predictable one for others. Closely related starting values give orbital numbers that are widely different. In other words, the system is sometimes highly sensitive to its starting values, sometimes not. This is characteristic of what mathematicians mean by chaos. Chaos theory attempts to describe how such systems change from predictable to wholly disordered.
Today there is much debate as to whether chaos theory, so far as it goes at present, actually does adequately describe seemingly disordered dynamical systems in nature - whether it really is, as some have claimed, a new mathematical tool of the same order of importance as calculus, or even that it is a discipline to rank in importance with relativity and quantum mechanics. The controversy rages on but the level of interest and the volume of research continue to rise. Major developments, one way or the other, are to be expected soon.
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Number Systems and Algebra (page 1)
ftsTitle
The Chinese abacus usually has two beads representing 5s on each wire above the cross bar, and five beads representing 1s on each wire below the bar. The beads are moved towards the bar. Two numbers are shown here, 8654 on the left and 93 on the right.
Number Systems and Algebra (1 of 2)
The natural numbers or whole numbers are those we use in counting. We learn these at an early age, perhaps pairing them with our fingers or else learning to chant their names in order: 'one, two, three, four, . . . '. These are both important features of our number system - that these numbers can be used to count sets of objects, and that they form a naturally ordered progression that has a first member, the number 1, but no last member: no matter how big a number you come up with, I can always reply with a bigger one - simply by adding 1.
However, even quite simple arithmetic, as we shall see, cannot be carried out wholly within the natural numbers. Ordinarily we take the principles that govern such systems for granted, yet merely to be able to subtract and divide, for example, requires other, more complex, number systems, such as fractions and negative numbers.
Natural numbers and arithmetic
If I have 3 sheep and you give me 4 more, I can count that I now have 7 sheep, or I can use the operation of addition to get the same answer: 3 + 4 = 7. If I promise to give 5 children 4 sweets each, again I can count out 20 sweets altogether, or I can use the operation of multiplication: 5 x 4 = 20. Here, we have examples of another principle of natural numbers: any addition or multiplication of natural numbers gives another natural number. Such a system is said to be closed under these operations. (A closed system is one where an operation on two of its elements produces another element of that system.) If I had 3 sheep and when you gave me your sheep I had 7, I can use the operation of subtraction to find how many sheep you gave to me: 7 - 3 = 4. If I distribute 20 sweets equally to 5 children, I can use the operation of division to find how many I gave to each: 20
5 = 4. Subtraction is the inverse operation of addition; division is the inverse operation of multiplication. However, the natural numbers are not closed under the operations of subtraction and division, as we shall see later.
In simple algebra, we generalize arithmetic by using letters to stand for unknown numbers whose value is to be discovered, or to stand for numbers in general. Usually letters from the beginning of the alphabet are used in the latter way - for example, to express a general truth about numbers, such as a + b = b + a. The letters at the end of the alphabet are generally used to represent unknown numbers. For example, the information about the sheep can be expressed by the equation, 3 + x = 7, where x represents the unknown number of sheep you gave to me. Since the two sides of this equation are equal, they remain equal if we treat them both the same way. If we then subtract 3 from each side we get x = 7 - 3, that is x = 4. We have solved the equation.
Subtraction and the integers
The set of natural numbers is not closed under the operation of subtraction; for example, 3 - 7 does not give a natural number as an answer. We need a system of numbers that is closed under subtraction. The smallest set of numbers that is closed under subtraction is the set of integers, i.e. the set ..., -3, -2, -1, 0, 1, 2, 3, ..... Here, the positive integers can be identified with the natural numbers; zero (0) is defined as the result of subtracting any integer from itself; and the negative integers are the result of subtracting the corresponding positive integers from zero (e.g. -3 = 0 - 3).
Now, every subtraction has an answer within the number system of integers, that is, the integers are closed under subtraction.
Division and the rational numbers
The integers, however, are still not closed under the operation of division. We can construct a system that is by defining the result of any division, a
b to be the pair of integers, a and b, written in a notation that clearly distinguishes which divides which. Thus, we write a
b as the ratio or fraction, a/b, and we have the system of rational numbers.
It is important to note that rational numbers are not identical with their symbols. The same rational number may be represented by many different fractions (in fact, an infinite number of them). For example, 24/8 is the same rational number as 12/4 or 6/2. We adopt the convention of representing them, where possible, by the unique fraction in which there is no common factor that can be canceled out (thus, 14/21 becomes 2/3, where the factor, 7, has been canceled out). It should also be noted that decimals are rational numbers, since, for example, 0.5 = 5/10 = 1/2, and 1.61 = 161/100.
We do have a problem, however: the rationals cannot be closed under division, because of the integer 0. We cannot give value to a/0 for any rational number a. This problem, however, cannot be avoided: we have to be content with the fact that the rationals, excluding the integer 0, are closed under division.
Roots and irrational numbers
69, which we read as, '6 to the power 9' means 6 multiplied by itself 9 times ( 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6). Generally, ab, which we read as, 'a to the power b', means a multiplied by itself b times. These are closed operations for the systems of numbers we have so far considered. However, none of these systems guarantees the possibility of the inverse operation, the extraction of roots. If b = an, (where n represents an integer), then a is the nth root of b, written a = n
b. For example, since 3 x 3 = 9, the second or square root of 9 (written 2
9 or more usually
9) equals 3. To give another example, since 2 x 2 x 2 = 8, the third or cube root of 8 (written 3
8) is 2. But none of the systems we have considered is closed under this operation. For example,
3, and
5 cannot be expressed as fractions or as terminating decimals; they are examples of what are called irrational numbers. They have exact meaning - for example, by Pythagoras' theorem,
2 is the length of the hypotenuse of a right-angled triangle whose other sides are each length 1;
5 is the length of the hypotenuse of a right-angled triangle whose other sides have lengths 1 and 2, etc. Obviously, we need to add their rationals to our number systems to ensure closure under these calculations.
All the systems we have discussed, the natural numbers, the integers, the rational numbers and the irrationals form together the system of real numbers.
Imaginary and complex numbers
However, now we have admitted the extraction of roots, we have opened up a new gap in our number system: we have not, as yet, defined the square root of a negative number. At first sight, we may wonder why this omission should be of any great importance, but without the development of a system to include such numbers, many valuable applications to engineering and physics would not be possible. Surprisingly, we need only extend the number system by one new number. Since all negative numbers are positive multiples of -1 (for example, -6 is 6 x -1, so that -6 = 6 x -1) we are concerned only with the square root of -1. The square root of -1 is denoted by the letter i, so we have i to the power of 2 = -1.
Real multiples of i, such as 3i, 2.7i, 2i/3, i
2, etc., are called imaginary numbers. The sum of a real number and an imaginary number, such as 5 + 3i, is a complex number. It can be shown that every complex number can be expressed uniquely as the sum of its real and imaginary parts.
The rules for using complex numbers are the same as those for real numbers. It can be shown, for example, that (a + ib) (a - ib) = a to the power of 2 + b to the power of 2.
The terms in brackets are thus the factors of a to the power of 2 + b to the power of 2. In fact it turns out that in the complex number system any algebraic expression with integer powers has exactly the same number of factors as the highest power in the expression. This result is so important that it is called the fundamental theorem of algebra.
* SETS AND PARADOXES
* CORRESPONDENCE, COUNTING AND INFINITY
* COMPUTERS
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Number Systems and Algebra (page 2)
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Number Systems and Algebra (2 of 2)
LOGARITHMS
Since a to the power of 3 = a x a x a, and a to the power of 2 = a x a, then a to the power of 2 x a to the power of 3 = a x a x a x a x a = a to the power of 5. This is an instance of the general rule for the multiplication of powers of the same base: ax x ay = ax + y.
From this it is easy to see that a0 = 1, whence also a-x = 1/ax , and the corresponding rule for division is ax / ay = ax - y Similar considerations enable us to give a meaning to ax even where x is not an integer; for example, since
x x
x = x = x1,
x must be x1/2.
The logarithm of a number to a given base is simply the power of that base that is equal to the given number. Tables of common logarithms, which use base 10, were used in the days before pocket calculators to assist with complicated multiplications and divisions. For example, it is obviously quite difficult to multiply 135.763 by 4386.734, but it is much easier to add their logarithms, which can be found in a table. Since 135.763 is 10 to the power 2.1327 we find that the logarithm of 135.763 is 2.1327; similarly, since 4386.734 is 10 to the power 3.642l, we find that the logarithm of 4386.734 is 3.6421. We then add these logarithms to find the logarithm of the product of the given numbers. Thus the logarithm of the product is 5.7748, which we can look up in a table of antilogarithms to find the answer 595400 (since 10 to the power 5.7748 is 595400). (NB: This is only approximate because the tables are only made up to four figures; the precise answer is 595556.168042.) A slide rule is a mechanical device that applies this principle. You can add two numbers using two ordinary rulers (where the numbers are equally spaced) by placing the zero of one scale against one of the numbers and reading their sum off the other ruler opposite the second number.
A slide rule has a scale that shows numbers spaced according to their logarithms, so that the same method has the effect of adding the logarithms, so that the number read off as the answer is the product of the two given numbers: Here, if we place 1 on the lower slide against 2.5 on the top slide, then the number on the top slide opposite any number on the bottom slide is the result of multiplying it by 2.5; for example, 2.5 x 3 = 7.5 as shown.
OTHER NUMBER NOTATIONS
The usual notation for numbers is a decimal place-value system. This means that there are ten distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and that the position of each digit determines what it contributes to the value of the number. Each position gives a value 10 times as great as the position to its right, so, for example, 7234 can be written as 4 units (4 x 10 to the power of 0) on the right, plus 3 tens ( 3 x 10 to the power of 1) plus 2 hundreds (2 x 10 to the power of 2) plus 7 thousands (7 x 10 to the power of 3). We say that 10 is the base of the decimal place-value system.
We can easily construct systems with other bases to suit particular needs. The binary system uses only the digits 0 and 1; so it has base 2. This is used in the representation of numbers within computers, since the two numerals correspond to the on and off positions of an electronic switch. In the binary system we count as follows: 1, 10 (= 2 + 0), 11 (= 2 + 1), 100 (= 4 + 0 + 0), 101 (= 4 + 0 + 1), 110 (4 + 2 + 0), 111 (4 + 2 + 1), 1000 (= 8 + 0+ 0 + 0), 1001 (= 8 + 0 + 0 + 1), etc.
Sometimes, especially in computing, it is convenient to use octal arithmetic (with base 8) or hexadecimal arithmetic (base 16). In base 16, the letters A to F are used as well as the numerals 0 to 9. Obviously it is necessary to know which base is being used, so the base is indicated by a subscript, for example, 3110 = 1F16 = 378 = 111112.
There are many other ways in which numbers systems can vary. Sometimes one can see vestiges of other systems in the numerical terms of a language: in French one counts up to 100 in a mixture of base 10 and base 20; for example, quatre-vingt-dix (four times twenty plus ten) equals 90. Even English retains vestiges of base 12, with the words 'eleven' and 'twelve'. The traditional Chinese abacus uses a mixture of base 5 and base 10.
The system of Roman numerals is not a place-value system: the letters have fixed values and are ordered from the largest to the smallest. For example, MDCLXVI = 1000 + 500 + 100 + 50 + 10 + 5 + 1 = 1666. When, however, a letter representing a smaller value precedes a larger, it is subtracted; thus CM = 900, and IX = 9. This makes calculations very difficult, and it has been suggested that the superiority of Eastern mathematics over that of early medieval Europe was a result of the system of Roman numerals.
PRIME NUMBERS
A prime number is a natural number that has no proper factors - that is, which cannot be divided by any natural numbers other than itself and 1. We can find the primes by taking a sequence of numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 . . .
and first deleting all the numbers divisible by 2 (excluding 2 itself, which is only divisible by itself and 1), then all those divisible by 3, then (since anything divisible by 4 has already been deleted) all those divisible by 5, and so on.
All non-prime natural numbers must by definition be divisible by other numbers apart from themselves and 1; these other numbers can in turn be repeatedly divided until one is left with a series of prime factors. Hence, all non-prime numbers can be expressed as the product of a series of primes - in fact, for each number, the expression is unique.
The prime numbers have been studied since the days of the ancient Greeks, who knew, for example, that there is no largest prime. Their proof is quite easy to understand: Suppose there is a largest prime, so that all the prime numbers can be listed in order of size. Now consider the number we obtain if we multiply all these primes together, and add 1; call this number N. Clearly N cannot be divided by any of the list of primes without leaving a remainder of 1. But since these are (we are assuming) all the primes, any other number is non-prime and so has prime factors. Therefore it cannot divide N unless its prime factors divide N - but no primes can divide N. Thus N must itself be prime. But it is a bigger prime than what we supposed was the biggest prime, so that supposition has led us to a contradiction and must be false. The largest known prime number (August 1989) is 391582 x 2 to the power of 216193 - 1, which is a number of 65087 digits.
On the other hand it is not known whether or not there are infinitely many twin primes. These are pairs of successive odd numbers that are both prime, like 5 and 7, 11 and 13, or 29 and 31. Another famous conjecture about prime numbers is that of Christian Goldbach (1690-1764), who postulated that every even number is the sum of two prime numbers. It is not known whether this is true or false.
Prime numbers have recently become of great interest to cyptographers: certain codes are based on the result of multiplying two very large primes together, and because even the fastest possible computer would take years to factorize this product, the resulting code is virtually unbreakable.
LAWS OF ARITHMETIC
Commutative law for addition: a + b = b + a
for multiplication: a x b = b x a
Associative law for addition: (a + b) + c = a + (b + c)
for multiplication: (a x b) x c = a x (b x c)
Distributive law for multiplication
over addition: a x (b + c) = (a x b) + (a x c)
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Sets and Paradoxes (page 1)
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Family pets fed and looked after by Sue.
Sets and Paradoxes (1 of 7)
Sets can be considered simply as any collections of objects. However, in the early 20th century, when attempts were made to formalize the properties of sets, contradictions were discovered that have affected mathematical thinking ever since.
A set can be specified either by stipulating some property for an object as a condition of membership of the set, or by listing the members of the set in any order.
Sets are usually indicated by the use of curly brackets , known as braces.Thus, suppose we are considering a family that has a cat, a rabbit, a horse, a dog, a mouse and a piranha: we could represent the pets fed and looked after by Sue as cat, rabbit, horse. In that case cat, rabbit, horse = x: x is a family pet looked after by Sue. Sets are often pictured by drawing a circle around representations of their members, thus:
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Sets and Paradoxes (page 2)
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Sets and Paradoxes (2 of 7)
Union and intersection
We can use circles to show the relationship between two (or more) sets. Let us suppose that Sue has a brother, Tim, who feeds and looks after the dog and the mouse; he also helps Sue look after the horse. If S is the set of pets looked after by Sue and T is the set of pets looked after by Tim, we can show their responsibilities like this: It is easy to see that the set of all the family pets looked after by the children is cat, rabbit, horse, dog, mouse. This is called the union of the two sets, and is written S
T (we say, 'S union T ').
The two sets have a member in common, the horse. The set of members that belong to both of two given sets is known as their intersection; here, it is the set whose only member is the horse. This is written S T = horse. This is a set, even though it has only one member (as far as the family's pets are concerned) and we write 'horse
horse', where the symbol '
' means 'is a member of '.
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Sets and Paradoxes (page 3)
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Sets and Paradoxes (3 of 7)
Subsets
Formally, a set is a subset of another set if all the members of the first set are members of the other set, that is, one set is contained within another. Thus, among the family's pets, horse is a subset of cat, rabbit, horse, dog, mouse. If the bigger set is A and the subset is H, then we write H
A, to mean that H is a subset of A, or A
H, to mean that set A contains set H. These could be shown:
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Sets and Paradoxes (page 4)
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Sets and Paradoxes (4 of 7)
The universal set and complements
Often, we need to know which objects are not members of a given set. For example, we might wish to know which pets are not looked after by Sue. If we think of this as the set of all pets not looked after by Sue, this would then include any other pets in the world - which is obviously not what we intended. We are concerned only with the pets looked after by Sue and Tim and their family. A universal set contains all the objects being discussed - in this case, all the pets looked after by the whole family. Those pets not looked after by Sue, within the universal set, form what is known as the relative complement of the set of her pets. Where S is the set of pets looked after by Sue, the complement set is written C (S) or S
. Let us suppose that Sue and Tim's parents look after the only other family pet, a piranha, and that the children are banned from the piranha. The circle for Sue's pets is as before. The rectangle, identified U, is the universal set of the family's pets. We can see that the complement of S is S
= dog, mouse, piranha.
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Sets and Paradoxes (page 5)
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Sets and Paradoxes (5 of 7)
Empty and disjoint sets
The empty set (sometimes called the null set) has no members. It is written {} or f. If we consider the family's pets, we could write, for example, {parents' pets}
{Sue's pets} = f,
meaning the set of pets looked after by both Sue and her parents is the empty set - it has no members.
Disjoint sets have no members in common. As the diagram below shows, the set of pets looked after by Sue does not intersect with the set of pets looked after by her parents, P; S and P have no members in common and their intersection is the empty set. That is, S
P = f.
Sets and logic
There is a direct link with logic that becomes apparent if we write the formal definitions of union, intersection and complement in set notation:
T = {x: x is a member of S or a member of T }
T = {x: x is a member of S and a member of T }
S' = { x: x is not a member of S }
These words, 'and', 'or' and 'not', represent what logicians call truth-functions. That means that when they are attached to sentences to form more complex sentences, the truth or falsehood of the latter depends only on that of the former. For example, 'John is in London and Mary is in Paris' can only be true if both 'John is in London' and 'Mary is in Paris' are separately true; and in that case, 'John is not in London' must be false. These relations can be shown by the following tables:
P - P P Q P & Q P v Q
T F T T T T
F T T F F T
F T F T
F F F F
Here, P and Q stand for any sentences whatever, '-P' is 'not-P', 'P & Q' is read 'P and Q', and 'P v Q' represents 'P or Q'. The tables show every possible combination of values, and can be used to work out tables for more complex formulas. When 'T' represents a positive signal and 'F' the absence of one (sometimes written '1' or '0' respectively), these tables show the outputs from the electronic logic gates of the same names out of which computers are built.
Paradoxes
Although developments from the simple concept of sets - such as have been outlined - seem to work well enough for practical purposes, various paradoxes were discovered when axioms for the theory of sets were sought. The German Gottlob Frege (1848-1925) and the English man Bertrand Russell (1872-1970) were independently interested in showing that all of mathematics could be reduced to pure logic, and looked to set theory as a link. In 1908, just as Frege was publishing a major work on the subject, Russell discovered, and communicated to Frege, that his axioms generated an important contradiction; this has become known as Russell's paradox.
The simplest way of explaining Russell's paradox is by a particular example (a more general account is given in the box). Let us consider a doctor who serves a community. This doctor treats only those in the community who do not treat themselves. Now, if the doctor treats himself, he cannot be included in the set of those who do not treat themselves. If he does not treat himself, then he is included in the set of those he does treat. Either way, there is a contradiction; but there are only two possibilities and we cannot make sense of either of them. There has to be something wrong with the definition itself from which we were able to derive the contradiction. This, and other paradoxes, proved a great blow to mathematical logicians and new philosophies such as intuitionism grew up partly as a result.
RUSSELL'S PARADOX
Obviously, sets in general can be members of other sets; for example, 1,2 is a member of 0,1, 1,2, 2,3,
and that set is a member of the set of three-membered sets, which in turn is a member of the set of large sets (since there are many three-membered sets). More particularly, some sets are members of themselves, like the set of large sets, while others are not members of themselves, like the set of small sets (since there are many small sets).
Let us then consider the set W of all sets that are not members of themselves: is it a member of itself or not? An element of a set must have the property that defines the set, so if W is a member of the set of sets that are not members of themselves, then it can't be a member of itself - but that just means that it can't be a member of W. On the other hand, if W is not a member of itself that just is the property that defines W, so that W must be a member of W - that is, it is a member of itself.
There are only two possibilities: if you think of any entity you like and any set you like, either the thing is in the set or it is not; there is no third possibility. Thus in particular, either W is a member of itself or it is not; yet whichever supposition we make leads straight to the contradiction. This is a deeper problem than the paradox in the text; there we said that although the description of the doctor's contract looks inoffensive, analysis shows it is really self-contradictory. Here there is no such description to reject; W was constructed out of pure logic. So the contradiction can only lie in one place - at the heart of logic itself.
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Sets and Paradoxes (page 6)
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Sets and Paradoxes (6 of 7)
VENN DIAGRAMS AND VALIDITY
Venn diagrams are very useful when testing the validity of arguments. It is, however, always important that all possible combinations of the sets are represented in the diagram. If, for example, we are interested in three distinct but possibly intersecting sets, we can use three circles, making sure we include areas for the intersections of each pair and of all three. For more than three sets, the diagrams can become quite complex. The use of colors or shading can be helpful.
To check that an argument is valid, we have to ensure that the information given in the conclusion to an argument is already contained in the premises. For example, suppose we wish to show whether the following is a valid argument - or not.
Premise 1: All ice skaters have a good sense of balance.
Premise 2: Some pianists are ice skaters.
Conclusion: So some pianists have a good sense of balance.
We have three sets to show on our Venn diagram: ice skaters (S), pianists (P) and those with a good sense of balance (B). Now, premise 1 is equivalent to saying that all members of S are also members of B. That is, the part of set S that is outside B must be empty (i.e. have no members). On the diagram this area is colored red.
Premise 2 tells us that at least one pianist is an ice skater, that is, the intersection of sets S and P has at least one member. Part of this area (colored red) is already known to be empty, so any members must be in the area colored green. We mark this area with a tick.
The conclusion that some pianists have a good sense of balance means that the intersection of B and P is not empty. This area is surrounded by a heavy line in the diagram. We see that the area already contains a tick, meaning it is not empty, so the conclusion must be true when the premises are; that is the argument is valid.
In contrast, we can consider an invalid argument:
Premise 1: All ice skaters have a good sense of balance.
Premise 2: Some pianists are ice skaters.
Conclusion: So all pianists have a good sense of balance.
Premises 1 and 2 are the same as before, so we can consider the same Venn diagram. However, the conclusion asserts that set P is empty except for its intersection with set B (the heavily outlined area). The premises do not tell us whether this is true or false; the conclusion goes beyond the premises and the argument is thus shown to be invalid.
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p066-7
ftsTitleOverride
Sets and Paradoxes (page 7)
ftsTitle
Sets and Paradoxes (7 of 7)
VENN DIAGRAMS AND NUMERICAL DATA
Venn diagrams can be used to assist with calculations that involve a subdivided population about which only fragmentary numerical data are available. The story of 'The Thirty-Sixth Man' provides an entertaining example . . .
Vennseta is a spymaster. To his horror, he discovers that three hostile powers, Xylia, Yoravia and Zenobia, have cracked the cover of some of his agents. Unfortunately for Vennseta, the information he has is only fragmentary, but he has to find out which of his agents he can still trust.
Of his 36 operatives, he knows that 21 have been cracked by Xylia, only 4 of whom have been cracked by no other power. He knows that 5 have been uncovered by Zenobia alone and that the cover of 12 has been broken by Xylia and Zenobia. Of those who are known to Yoravia, 9 are also known to Xylia and 7 are also known to Zenobia. His information tells him that 18 are entirely unsuspected by Yoravia.
Wondering whether he has enough data, Vennseta draws one of those diagrams for which he is famed in the world of espionage. E is the set of all his espionage operatives. The circles he labels X, Y and Z, to represent the sets of agents cracked by, respectively, Xylia (X), Yoravia (Y) and Zenobia (Z). He then writes in the numbers to represent the data he has.
He still has one piece of information that he cannot fit into his diagram, but he is quite sure it will be useful later.
The spymaster decides he can now determine how many of his agents are known to all three powers, by considering the subdivisions of X. He adds the 4 who were cracked only by X, the 9 also cracked by Y and the 12 also cracked by Z. He gets a total of 25 agents cracked. But he sees that only 21 have been cracked by X, so there are, seemingly, 4 spies too many. He realizes what this means: 4 of the 9 + 12 must have been cracked by both Y and Z, as well as X. He decides to draw another diagram, putting this information into it.
From his second diagram, he sees that 12 - 4 = 8 are not known to Y ; 9 - 4 = 5 are not known to Z ; and 7 - 4 = 3 are not known to X. The only subsets he has to determine are those agents known only to Y and those agents not known to any of the three hostile powers. He marks these sets by question marks.
Vennseta puzzles a while, and then reaches for his one unused piece of information: 18 agents are unsuspected by Y. He quickly adds up the agents cracked by X and Z but out side Y: 5 + 8 + 4 = 17. This leaves one agent who is safe from all three powers.
The spymaster is elated; not only have his diagrams, once again, solved his problem for him, but out there in hostile regions there is an agent on whom he can rely, absolutely. He idly scribbles in a 6 as the missing number in Y; he writes a large 1 in the rectangle outside the circles. Suddenly he stiffens and turns pale: how on earth can he find out which of his agents he can still trust?
* THE SCIENTIFIC METHOD
* CORRESPONDENCE, COUNTING AND INFINITY
* COMPUTERS
* LOGIC
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p068-1
ftsTitleOverride
Correspondence, Counting and Infinity (page 1)
ftsTitle
Correspondence, Counting and Infinity (1 of 6)
The simplest way of counting a set of objects is to point at them one by one, making sure that none have been missed or repeated, and saying 'one', 'two', 'three', etc., as each object is indicated. These are, in order, the names of the natural or counting numbers. The number of objects in the set is just the last number counted out in this process. This fact can be used to define the size of a finite set (i.e. one that can be completely counted off) in a way that our intuition tells us is acceptable. However, when we seek to apply this intuitively satisfactory account to infinite sets, we discover that our intuitions break down, and we are forced to the puzzling conclusion that there must be different sizes of infinity.
Mappings
Sometimes it is useful to compare two sets of objects without actually counting them. For example, a shepherd might check that he has equal numbers of black and white sheep, without using numbers. Even today, cricket umpires drop pebbles or coins into their pockets, corresponding to the number of balls that have been bowled in an over; they do not count them in numbers.
The first set (in the examples, the white sheep, or the umpire's pebbles) is called the domain. The other set (the black sheep, or the balls bowled) is called the codomain. A mapping associates members of the domain with members of the codomain.
We shall now move to a different domain, that of friends at a party. For the codomain, we shall consider what snacks they choose to eat. We can show the relation in a mapping diagram, where the mapping arrows all lead from the domain to the codomain.
* NUMBER SYSTEMS AND ALGEBRA
* SETS AND PARADOXES
* FUNCTIONS, GRAPHS AND CHANGE
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Correspondence, Counting and Infinity (page 2)
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Correspondence, Counting and Infinity (2 of 6)
Here, we have a many-to-one relation, since more than one party-goer (member of the domain) chooses the same snack. The codomain, that is the set of snacks available, is crisps, nuts, cheese, fruit. Although fruit was available, none of Al, Beth, Charles, Dirk, Eve choose it. The set crisps, nuts, cheese is called the range.
As the party continues, some of the friends get bored with the same food. The diagram looks like this: Many friends choose one snack; at least one friend chooses many snacks; this shows a many-to-many relation.
* NUMBER SYSTEMS AND ALGEBRA
* SETS AND PARADOXES
* FUNCTIONS, GRAPHS AND CHANGE
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Correspondence, Counting and Infinity (page 3)
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A one-to-one mapping
Correspondence, Counting and Infinity (3 of 6)
As the hours pass, Beth, Dirk and Eve go home from the party, leaving Al and Charles still nibbling. We see that at least one of the friends (Al) prefers many (crisps and cheese) snacks. The diagram now shows a one-to-many relation.
* NUMBER SYSTEMS AND ALGEBRA
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Correspondence, Counting and Infinity (page 4)
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Correspondence, Counting and Infinity (4 of 6)
Correspondence and number
There is one other kind of relation - one-to-one correspondence. Let us suppose that Beth, Dirk and Eve decide to have a bedtime drink when they get home. Their preferences are shown in the next diagram, which illustrates a one-to-one mapping.
The earlier examples of the shepherd pairing his black and white sheep, and of the cricket umpire pairing pebbles and balls, are also one-to-one relations. One-to-one correspondence forms the foundation of arithmetic. When sets are paired in this way, they are said to be equivalent, and two such equivalent sets must have the same number of members. It is possible to prove formally that every set whatsoever is in some family of equivalent sets, and that no set can be in more than one family. Thus, one-to-one correspondence separates out all sets into families of sets that have the same number of members. Two equivalent sets are said to have the same cardinality, and this serves to base arithmetic in set theory. So all sets with, for example, seven members, have the same cardinality.
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* FUNCTIONS, GRAPHS AND CHANGE
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Correspondence, Counting and Infinity (page 5)
ftsTitle
Correspondence, Counting and Infinity (5 of 6)
Infinite sets
The sequence of sets, 1, 1,2, 1,2,3, etc., has the property that every finite set (i.e. a set with a finite number of members) corresponds to one of them. Thus every set corresponds to a unique set of natural numbers starting with 1 and in their natural order, and the size of the set is given by the largest number in that unique set.
However, a set that does not have a largest member cannot be equivalent to any set that does, and this leads to the idea of an infinite set as one that cannot be counted. In particular, the set of all the natural numbers cannot be counted, so it might seem possible to define infinity in terms of equivalence to the set of all the natural numbers.
All we need to do to show that two sets are equivalent is to show that their members can be put in one-to-one correspondence. But this immediately gives rise to a puzzle. It is easy to show that the mappings that associate each natural number with its double (and so, each even natural number with its half) are one to one.
Thus, we have shown there are exactly as many even natural numbers as there are natural numbers. However, since every natural number is either even or odd and there are as many even numbers as odd, there must be twice as many natural numbers as even natural numbers.
This is only a paradox if we expect that a subset of a set must be smaller than the set itself; but since our idea of an infinite set means that it is possible to list members of the set forever, we can see why adding extra members at the end does not increase its size. Our intuition that a set must have more members than any of its subsets is false for infinite sets. In fact, the usual definition of an infinite set is that it can be put into one-to-one correspondence with some subset of itself.
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Correspondence, Counting and Infinity (page 6)
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Correspondence, Counting and Infinity (6 of 6)
Different infinities
One remarkable application of this definition of an infinite set shows that there are no more rational numbers than there are natural numbers. Again, we look for a one-to-one correspondence between the two sets. There are many possible mappings, but the simplest is by first counting the fractions whose numerator and denominator add up to 1, then those for which the total is 2, then those totaling 3, and so on. Within each group, the fractions are put in order of size and any fraction that equals a previously listed fraction is omitted. This results in the rationals being listed in a unique order: Obviously, we can calculate both which natural numbers any rational corresponds to, and which rational corresponds to any given natural number, and this is sufficient to prove that there are no more rationals than there are natural numbers: It would be wrong, however, to conclude from all this that all infinite sets are equivalent. In fact, it is possible to prove that real numbers (the rationals and irrationals together) cannot be counted off in the way that we counted off natural numbers and rationals. This gave rise, at the end of the 19th century, to new studies of what was called transfinite arithmetic (transfinite means 'extending beyond the finite'), and attempts were made to prove that the cardinality of the reals is the next transfinite cardinal after that of the integers. In 1963 it was finally shown that this conjecture can neither be proved nor disproved; this is a 'gap' in mathematics itself.
Intuitionism and infinity
The various paradoxes and discoveries about infinity caused much attention to be paid in the late 19th century to the philosophical and logical foundations of mathematics.
One school of thought, called intuitionism, blamed the contradictions that be came apparent on the assumption that it is possible to complete infinite processes. Intuitionists resolved to ban infinity from mathematics, with the result that statements have to be allowed to be neither true nor false.
As an example, let us consider . Suppose we define N to be 0 if the sequence 0123456789 occurs in the decimal part of , and to be 1 if this sequence never occurs. A classical mathematician will be prepared to assert that 'N is either 0 or 1' is true, even though, at the time, he or she has no way of knowing it is. An intuitionist claims that, until we are able to prove one result or the other, the statement cannot be said to be either true or false.
Arithmetic, however, requires that there is no largest integer. The intuitionist resolves this by looking at things in a different way from the classicist. For example, the classicist might think of a distance of 2 units as actually made up of infinitely many steps of length 1 unit, then 1/2 unit, then 1/4 unit, then 1/8 unit, etc. The intuitionist, however, would interpret the result that the infinite series of 1 + 1/2 + 1/4 + 1/8 . . . + 1/2n + . . . has a sum (in fact 2) to mean that the total of finitely many terms in order can be made as close as we like to 2 by taking sufficiently many - but still finitely many - terms of the series.
HILBERT'S INFINITE HOTEL AND TRISTRAM SHANDY
The German mathematician David Hilbert, who was probably the most eminent mathematician of his generation, dramatized the paradoxical property of infinite sets by an exercise of the imagination.
Imagine a hotel with an infinite number of rooms; then it can be full and yet able to accommodate more guests. The manager simply moves the guest in room 1 into room 2, the guest in room 2 into room 3, and so on. Each guest is then in a room with a number higher than before, so that room 1 remains vacant for a late arrival.
Unfortunately, not one latecomer, but an infinite busload of them arrive. Instead of the moves as before, the manager puts the guest in room 1 into room 2, the guest from room 2 into room 4, the guest from room 3 into room 6, and so on. The infinite number of rooms with odd numbers are now vacant for the infinite number of latecomers.
The novelist Laurence Sterne constructed a similar paradox in his novel Tristram Shandy. The hero is trying to write his autobiography. Since, after two years, he has only described the first two days of his life, he concludes that his efforts are doomed to failure.
However, Bertrand Russell remarked that this would not follow if Tristram Shandy were immortal. Then the infinite set of the days of his life would be the same size as the 365-times larger set of days needed to describe the 'smaller' set.
CANTOR'S DIAGONAL THEOREM
The German mathematician Georg Cantor was responsible for the development of the whole theory of cardinality outlined in the text.
He proved that every set has a power set (set of all its subsets) that is strictly bigger than the given set; that is, the power set cannot be put in one-to-one correspondence with the given set - even in the case of an infinite set. The proof is easiest to understand in terms of the real numbers, by assuming that such a correspondence is possible and then showing that this assumption cannot be true since it leads to a contradiction.
We consider the real numbers between 0 and 1, expressed as decimals (0.47936421 ... is an example), so that each number has an infinite number of digits after the decimal point. Where decimals terminate, we continue the number with zeros. Suppose real numbers can be listed in order, that is put into one-to-one correspondence with the natural numbers. We could then write all the real numbers in this form: 0. a1 a2 a3 a4 ...
0. b1 b2 b3 b4 ...
0. c1 c2 c3 c4 ...
and so on.
Now, we try to construct a new number. We make the first digit after the decimal point differ from a1, the second digit differ from b2, the third digit differ from c3, and so on. Thus, we have constructed a new real number between 0 and 1, but we have constructed it so that it differs from every member of the list of real numbers that we began by supposing was complete: we have derived our contradiction. It follows, therefore, that such a list of the real numbers is not possible.
However, the construction required by Cantor's proof requires the completion of an infinite number of steps, and intuitionists therefore claim that the new number is ill-defined; they therefore reject 'Cantor's paradise' of different infinities.
infinities.
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Functions, Graphs and Change
(page 1)
ftsTitle
Functions, Graphs and Change (1 of 6)
In studying events in real life, we are often concerned with continuous change: a boulder rolling downhill picks up speed; a balloon expands as air is blown into it; our reaction-time slows as we grow older. Processes like these can be represented by a function. A function can be represented by a curve, so allowing us to picture how a process changes and develops.
Calculus is the branch of mathematics that studies continuous change in terms of the mathematical properties of the functions that represent it, and these results can also be interpreted in geometric terms relating to the graph of the function. Calculus was developed independently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Because their presentation involved paradoxical references to infinitesimals (infinitely small quantities), many scientists rejected their 'infidel mathematics', but at the same time there was a considerable dispute about who should have the credit for its discovery.
Functions and mappings
Suppose we go out for a cycle run, and keep up a speed of 15 km/h. Then our distance from home is determined by how long we have been traveling. For example, after half an hour we will have traveled 7.5 km; after an hour 15 km; after 2 hours 30 km, and so on. We can express this relationship by saying that the distance we traveled is a function of the time we have been traveling. Here the two quantities, time and distance, might be represented by the variables t and d, and the mathematical relationship between them would then be written d = 15t. What that means is that for any number of units of time, t, we can work out the number of units of distance traveled, d, by multiplying t by 15.
A function is just a mapping, but the term function is preferred when the domain and codomain are quantities that can be represented by sets of numbers, and especially real number. In general, the notation for a function is y = f(x), which indicates that the value of y depends upon the value of x; in that case, y is called the dependent variable, and x is called the independent variable. The variables are thought of as running through a range of values - for example, if our journey takes a total of 3 hours, the range of t is the interval (0,3), and the range of d is the interval (0,45).
Cartesian coordinates
The real numbers can be represented geometrically by a line (an axis) marked off from the origin (0) using some numerical scale. Any point in a plane, a two-dimensional area, can similarly be represented by the pair of numbers that correspond to its respective distances from two such axes, as shown below; these numbers are the coordinates of the point. Thus the coordinates of the point P in the diagram below are (1,2):
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p070-2
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Functions, Graphs and Change
(page 2)
ftsTitle
Functions, Graphs and Change (2 of 6)
Here the independent and the dependent variables of a function are represented by two lines at right angles (the x-axis and the y-axis) that cross at the origin. The curve representing the function is then the line that passes through the points whose coordinates satisfy the function. For example, the curve of the function y= x to the power of 2 is the set of pairs, (x, y), of real numbers for which y is the square of x; thus, for example, (2,4), (-1,1), (-2,4), (2,2), etc., are all in the graph of the function. The curve corresponding to this function is shown above.
This system of coordinates is named after the French philosopher and mathematician Renu Descartes (or des Cartes, whence the adjective 'Cartesian'.
Graphs and curves
Because a function associates elements of one set with those of another, it defines the set of all pairs of elements, (x,y), in which x is a value of the independent variable and y is the value of the function for the argument x. Another way of expressing this is that any point that satisfies the function y = f(x) can be represented by the point (x, f(x)). Since a function must be a many-one relation, every such pair has a different first element, so the pairs can be listed in a unique order. The function can then be thought of as moving through the values of the dependent variable as the value of the independent variable increases. This is what is represented by a graph in the Cartesian coordinate system: if we now draw a line joining the points (x, f(x)) as x increases, this line passes through all and only the points whose coordinates satisfy the function. Such a line is usually called a graph, although mathematicians prefer to use that term for the set of values of the variables, and call the diagram a curve. Since this way of representing change and dependency is equivalent to the function itself, curves provide us with a way of visualizing processes of change.
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p070-3
ftsTitleOverride
Functions, Graphs and Change
(page 3)
ftsTitle
Functions, Graphs and Change (3 of 6)
Average rates of change
Let us now consider a different example: suppose I throw a ball straight up in the air; it is slowed down by gravity until it stops and falls back to the ground, falling faster all the time. This is an example of a functional relationship between time (t) and height (h). If this relationship is described, say, by the equation h = 20t - 5t to the power of 2 then its graph looks like this: Velocity is defined to be the rate of change of displacement (here height), that is, displacement per unit time. Thus in the form
increase of height
velocity = ------------------
time taken
the well known formula for velocity permits the average velocity over any time interval to be calculated. In the figure, the average velocity between A and B is therefore the change of height, CB, divided by the change of time, AC. Where the dependent variable (in this case height) is represented by the vertical axis, and the independent variable (time) by the horizontal axis, this average is equivalent to the gradient or slope of the line joining the points on the curve that correspond to the ends of the interval, as indicated by the bold line above. However, in this case the average clearly conceals more than it reveals, since we know that between t = A and t = C, the ball actually changed direction, so that its upward velocity changed from positive to negative.
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ftsTitleOverride
Functions, Graphs and Change
(page 4)
ftsTitle
Functions, Graphs and Change (4 of 6)
Instantaneous rates of change
Similarly if you want to know how fast the Orient Express was traveling as it flashed through the Simplon Tunnel, you obviously get a very poor answer if you divide the whole distance from London to Venice by the total time taken to cover it. You get a better approximation if you measure the distance and time between Paris and Milan, and a better approximation still if you time the train between the stations at either end of the tunnel. This suggests that if we had accurate enough clocks and measuring tapes we would be able to get closer and closer to a precise answer by timing the train over increasingly shorter distances. Although this still never tells us the speed at any one instant, it suggests that the instantaneous speed is the limit that this sequence of averages tends to as the length of the interval gets smaller.
Let us now return to the example of the ball, and apply this reasoning formally: here the vertical height could be expressed in terms of elapsed time as h = 20t - 5t to the power 2; for example, the height after 0.5 second is 8.75 m, and after 1 second is 15 m. Now consider an arbitrary time t after the ball is thrown and take an interval of duration d on either side of it: We can see that the average velocity over this distance is the difference between the values of the distance function at the arguments t + d and at t - d, divided by the difference between these arguments. This is the slope of the chord PQ, that is
[20(t+d)-5(t+d)2]-[20(t-d)-5(t-d)2]
-----------------------------------
(t+d)-(t-d)
which simplifies to: (40d - 20td)/2d = 20 - 10t. Since this value is independent of d, this remains the average velocity round t no matter how small the interval d becomes, so that we can infer that the instantaneous velocity at time t from the starting point is actually equal to 20 - 10t.
This, of course, is another function of t, so that the rate of change of a function is another function of the same independent variable. Where the original function was y = f(x), the new function, known as its derivative, is written f'(x) or dy/dx (read as 'dy by dx'), where dy and dx represent small increments in y and x respectively; here, for example, the derivative of f(t) = 20t - 5t to the power of 2 is f'(t) = 20 - 10t. Similarly, in the expression y = x to the power of 2, f(x) = x to the power of 2, and the derivative of f(x) = x2 is
(x+d)2 - (x-d)2 4xd
--------------------- = ----- = 2x
(x+d)-(x-d) 2d
* MOTION AND FORCE
* MATHEMATICS AND ITS APPLICATIONS
* SETS AND PARADOXES
* CORRESPONDENCE, COUNTING AND INFINITY
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Functions, Graphs and Change
(page 5)
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Functions, Graphs and Change (5 of 6)
Derivatives and differentiation
The process of finding the derivative of a function is called differentiation, and this branch of mathematics is known as the differential calculus. This process can also be interpreted geometrically: as d be comes smaller, the points P and Q come closer together until finally they coincide, at which point the slope has the value we have calculated, and represents the tangent to the curve.
However, it is not necessary always to work out a derivative either as we have done, or by drawing the graph. Instead, certain general principles apply; for example, as we have seen, the derivative of x to the power of 2 with respect to x is 2x, and we can generalize this to state that the derivative of axn with respect to x is anxn-1.
The derivative of a function can itself be differentiated; for example, acceleration is the rate of change of velocity, and we can repeat the line of reasoning in the previous example to find the derivative of the velocity function with respect to time. This is the second derivative of the distance function.
In the previous example, the average acceleration over the interval (t - d, t + d) is [20-10(t+d)]-[20-10(t-d)]
(t+d)-(t-d)
which simplifies to the constant -10. It is because acceleration is a second derivative, that is the rate of change of a rate of change, that it is measured in units of distance per second per second. In this case the value -10 approximates to the downward force of gravity, which slows the ball down and returns it to the ground.
Totals and integrals
So far we have been considering how much we can find out about speed if we are given functional information about distance in terms of time. Now consider the converse problem: if we know a train's velocity as a function of time, how can we calculate the total distance it has traveled? Clearly if we knew the average speed, the total distance would be the result of multiplying the overall average speed by the total duration of the journey; and if the journey was undertaken in stages for which we knew separate average speeds and durations, the total distance traveled would be the sum of the distances calculated by this method for each separate stage. However, this formula requires the journey to be broken down into separable stages whose average speeds are known; where we only know the instantaneous speed at any moment expressed as a function of time, it is of no help at all. On the other hand, we can conjecture that we can approximate to the right answer by breaking the whole journey down into more and more, shorter and shorter stages.
Let us now go back to the previous example: the above figure shows the function derived for the velocity of the ball. The complete duration is divided into equal intervals, and in each we take the speed at the midpoint as an approximation to the average. But this really represents the stepped graph (shaded), rather than the true continuous function. However, if we double the number of intervals and halve the duration of each, we get a better approximation, and carrying on in the same way gives a sequence of better and better approximations whose limit can be thought of as the sum of the areas of infinitely many infinitely thin slices of the area under the graph. The integral calculus is the study of such processes, and it enables us to calculate infinite sums that can be expressed in terms of continuous functions.
* MOTION AND FORCE
* MATHEMATICS AND ITS APPLICATIONS
* SETS AND PARADOXES
* CORRESPONDENCE, COUNTING AND INFINITY
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Functions, Graphs and Change
(page 6)
ftsTitle
Functions, Graphs and Change (6 of 6)
The sum of the value of the function y = f(x) between arbitrary x = a and x = b (the shaded area on the figure) is written
f (x) dx, and is equal to F (b) - F (a), where F(x) is the indefinite integral of f(x); this is another continuous function of x, written
f(x) dx. For example, in the ball example, the indefinite integral,
(20 - 10t) dt, of y = 20 - 10t, is 20t - (10/2)t2 + c (where c can be any constant). If we denote this new function Int(x), then the definite integral between a and b is Int(b) - Int(a), that is
(20b - 5b2 + c) - (20a - 5a2 + c)
= 5(a2 - b2) + 20(b - a).
In fact it turns out that this process of integration is the inverse of differentiation and it is therefore sometimes called antidifferentiation. This means that the indefinite integral of the derivative of a given function, and the derivative of its indefinite integral, are both equal to the given function - a result so important that it is called the fundamental theorem of calculus.
TABLE OF
DERIVATIVES AND
INTEGRALS
Function Derivative Integral
x 1 x to the power of 2/2
x to the power of 2 2x x to the power of 3/3
xn nxn-1 xn+1/(n+1)
sin x cos x -cos x
cos x -sin x sin x
x -sin x sin x
x -sin x sin x
x -sin x sin x
x -sin x sin x
* MOTION AND FORCE
* MATHEMATICS AND ITS APPLICATIONS
* SETS AND PARADOXES
* CORRESPONDENCE, COUNTING AND INFINITY
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p072-1
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Probability: Chance and Choice (page 1)
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Card games offer very many more combinations of possibilities than tossing coins or throwing dice. Some of the odds are staggering: for example, the odds against dealing 13 cards of one suit are 158 753 389 899 to 1, while the odds against a named player receiving a 'perfect hand' consisting of all 13 spades are 653 013 559 599 to 1. The odds against each of four players receiving a complete suit (a 'perfect hand') are in excess of 2 x 1027 to 1.
Probability: Chance and Choice (1 of 2)
Not all actions and happenings have completely predictable results. Often we know that there is only a limited range of possible outcomes, but we do not know with certainty which of these to expect.
Probability theory enables us to describe with mathematical rigor the chance of an action or happening having a particular outcome. We may not, even then, make the right choice, but it will at least be a justifiable choice.
Probability and frequency
When we toss a coin or throw a die, we cannot predict which of the sides will land facing upwards - this, after all, is the point of tossing coins and throwing dice. Assuming we accept the fairness of the coin and the way it is tossed, we know it is just as likely to come up heads as tails, and there is no other possible outcome. Similarly, with a fair die, it is just as likely to fall with any of its numbers - from 1 to 6 - upwards, and there are no other possible outcomes. We describe these examples by saying that all the possible outcomes are equiprobable, and that the a priori probability (i.e. the theoretical probability) of a coin coming up heads is 1 in 2 or 1/2, and that of throwing a 6 on a single die is 1 in 6 or 1/6.
On the other hand, empirical probability (often called a posteriori probability) is based on observation and experiment. Here, the probability of a particular outcome is calculated from the proportion of times it has been observed to have happened before under the same conditions - its relative frequency. Thus, if you tossed a coin 10 times and the coin came up heads 3 times, the empirical probability that one of these throws came up heads is 3/10.
The probability scale
When an outcome is certain, it occurs every time: 1 in 1, 2 in 2, etc. Expressing this as a fraction, we say the probability is 1/1, that is, one. When an outcome is impossible, it occurs no times in any number of tests, so we say the probability is zero. For example, when throwing a die, the probability of throwing a number greater than 6 is zero, and the probability of throwing a number between 1 and 6 is one.
Probabilities between certainty and impossibility are expressed as fractions. So if, for example, we know that the 6 sides of a die are equiprobable, and the probability of throwing any of them is 1, the probability of each must be 1/6. Furthermore, if we consider only two possible outcomes, an odd number or an even, the probability of each must be 1/2. The fact that there are three odd outcomes each with probability 1/6, and 1/6 + 1/6 + 1/6 = 1/2, demonstrates, very simply, the addition rule: we can add up the individual probabilities of the different possible outcomes in a particular trial to get the combined probability.
Particularly in gaming and gambling, we see odds used as a scale for measuring chance. 'Odds' - more formally known as likelihood ratio - means the proportion of favorable to unfavorable possibilities, and is a different way of expressing probability. As we have seen, the probability of throwing, say, a 4, with a die is 1/6. Therefore, the probability of not throwing a 4 is 5/6. The 'odds' are thus expressed as 1 to 5 on throwing a 4 (or 5 to 1 against throwing a 4).
The law of large numbers
Suppose we toss a coin 10 times and the outcome is only 3 heads. The probability of a head is 1/2, so why do we not get 5 heads? We try a total of 100 tosses of the coin and the outcome is now, say, 40 heads, the last 6 being all heads. A gambler might back the chance that the 101st toss would fall tails, because, previously, there had been more tails than heads. Another gambler might back heads, because there seemed to be a 'run of heads', which would conform with the so-called 'law of averages'.
However, we know that the probability of a head or a tail at any one toss is 1/2 and a coin cannot remember - it cannot be influenced by what has gone before. Both gamblers are relying on empirical probability where theoretical probability is what matters - both are therefore betting on hope.
There is no 'law of averages'. Experimental and theoretical probability are connected only by the law of large numbers, which states that as the number of trials increases, the observed empirical probability comes closer and closer to the theoretical value. Thus in this example it means only that in the very long run, the relative frequency settles down towards 1/2.
*MATHEMATICS AND ITS APPLICATIONS
*NUMBER SYSTEMS AND ALGEBRA
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p072-2
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Probability: Chance and Choice (page 2)
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A roulette wheel. On a normal roulette wheel there are 37 compartments numbered from 0 to 36. The probability of, say, 0 winning is 1/37; the odds against 0 winning are 36 to 1. The casino will not usually offer odds better than 35 to 1, so in the long run the casino will make a profit.
Probability: Chance and Choice (2 of 2)
Rules of probability
If, we wish to find the combined probability of two independent trials, we use the multiplication rule. When we throw a pair of dice, we can consider these as two independent trials - because how one die falls does not affect the other. No matter what the first die shows, the second can show any of its six faces - and there are six ways the first can fall. So there are 6 x 6 = 36 possible outcomes for the ordered pair. Since these are equiprobable, the probability of any particular outcome, say a 1 with the first die and a 4 with the second, is 1/36, which is 1/6 x 1/6; that is, we multiply the individual probabilities to get the probability of a particular ordered outcome using the two dice. Since there are six ways of throwing a double, the probability of throwing the same number with two dice is 6/36 = 1/6, whence the probability of throwing different numbers is 1 - 1/6 = 5/6.
When a third die is thrown, there are now only four available faces differing from those of the first two dice, so that the probability of this showing a third different number is 4/6. Hence, the probability of throwing three different numbers with three dice is 5/6 x 4/6. Throwing six dice, the probability of an outcome of six different numbers is 5/6 x 4/6 x 3/6 x 2/6 x 1/6, which equals 5/324, or approximately 0.015. Thus we can expect the outcome to occur only once or twice in every hundred trials.
However, if we wish to specify the order beforehand (say, 1, 2, 3, 4, 5, 6 or 6, 4, 2, 5, 3, 1, for example), the probability is 1/6 for the first throw multiplied by 1/6 for the second throw, and so on. With the six dice the probability is therefore (1/6) to the power of 6 which equals 1/46656, or approximately 0.000021. Thus, we can expect such an outcome only about twice in every 100 000 trials.
It is very important that we correctly define a problem before applying the addition or multiplication rule. Many problems in fact require both. Suppose, for example, we wish to achieve a total of 8 with two dice. This could be from 6 and 2, 5 and 3, or 4 and 4; but there are two other possibilities: 2 and 6, and 5 and 3. That is, there are two ways of throwing a pair of distinct numbers, so that the probability that one die will show a 6 and the other a 2 is 2/36. Similarly, for a 5 and a 3. But there is only one way of throwing a double, so that the probability of a double 4 is only 1/36. The probability of an outcome of a total of 8 is the sum of these probabilities, that is, 5/36.
Decision-making
Very often we are required to make decisions based on only a little knowledge of the likely circumstances. As examples: a doctor may have to choose between different treatments for a patient with relatively little experimental evidence as to which is more successful; or the directors of a business may have to choose between different advertising strategies on the basis of rival claims about the effectiveness of the different media. In such cases, the decision-makers need ways of measuring the competing strategies against one another. One way of doing this involves calculating the expected value.
A simple way to explain this is by considering a hockey or soccer league table, where 2 points are awarded for a win, 1 for a draw, and 0 for a loss. Suppose a particular team in the league decides, at the beginning of the season, that - based on all the available evidence - its probability of winning any match is 1/4 and of drawing is 1/3. Then its probability of losing is 1 - 1/4 - 1/3 = 5/12. Over a run of 12 matches the team would expect to win 3, draw 4, and lose 5. The points it would expect to gain in 12 average matches would be (3 x 2) + (4 x 1) + (5 x 0) = 10 points. Thus the average number of points they can expect in each match is 10/12. This is the expected value.
By calculating an expected value for each course of action available to us, we are able to choose the one that has the best likely outcome. On the basis of only partial knowledge, we are enabled to make a rational choice, even though it may not be the one with the highest probability.
SHARING A BIRTHDAY
Suppose we are looking for a pair of people who have the same birthday. What is the smallest number of people, chosen at random, for which there is a better than evens chance of two sharing a birthday? Since, allowing for leap years, there are 366 possible birthdays, many people guess that 183 is the answer. In fact, the answer is 23. The probability of the second person we ask not matching the birthday of the first, is 365/366. The probability of the third person not matching either of the other two is 364/366. So the chance of three people not sharing the same birthday is 365/366 x 364/366. For n people, the chances of them all having different birthdays is, thus, 365/366 x 364/366 x 363/366 x .... to n - 1 terms. We need to find how many terms of this sequence we need to multiply together before their product becomes less than 1/2 - that is, before there is a less than evens chance that this number of people do not include a single pair with the same birthday.
If we work it out, we find that the probability of 22 people having different birthdays is 0.5252, and for 23 people, it is 0.494. Therefore, 23 is the smallest number of people for which there is a better than evens chance of at least one shared birthday. On the other hand, we need 367 to be certain that two of them have the same birthday!
RATIONAL CHOICE
When my car breaks down, my garage in forms me that the cause is either the gearbox or the final drive, with odds of 3 to 2 on the gearbox being the problem. The cost of repairing the gearbox will be $200 and the final drive $150, both costs including dismantling and reassembly charges. However, if the component they examine first is undamaged, the cost of stripping it down and reassembling is $60 for the gearbox and $30 for the final drive, in addition to the cost of the repair. Where should they begin?
Suppose they examine the gearbox first. In the long run, on 3 occasions out of 5 the fault will actually be there, and the cost will be $200. Otherwise, and hence with a probability of 2/5, the cost will be $60 for inspecting the gearbox and $150 for repairing the final drive - a total of $210. Thus, the expected cost of the strategy of examining the gearbox first is (
3/5 x $200) + (2/5 x $210) = $204.
Suppose, however, they decide to examine the final drive first. In the long run, there will be no fault on 3 occasions out of 5, incurring a cost of $30 for inspecting the drive plus $200 for repairing the gearbox. On the 2 out of 5 occasions that the fault is actually in the final drive, the cost will only be $150. Thus, the expected cost for this strategy is
(3/5 x $230) + (2/5 x $150) = $198.
It is thus better if the garage examines the final drive first, even though the fault is more likely to be in the gearbox. This is because this strategy has the lower expected cost. Of course, this will be of little consolation to me if the fault is in the gearbox and I have to pay $230.
The garage, on the other hand, might decide to make a standard fixed charge of $230 for the job. It would still be better for the garage to examine the drive first, since - over a run of repairs - they would make an average gain of $32, whereas if they investigated the more probable cause first, their average gain would be only $26.
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