THE HEIGHT OF UP
ASIMOV ON PHYSICS
by: Isaac Asimov
Most of us would consider the surface of the sun to be pretty hot. Its temperature, as judged by the type of radiation it emits, is about 60000 K. (with “K.” standing for the Kelvin scale of temperature). However, Homo sapiens, with his own hot little hands, can do better than that. He has put together nuclear fission bombs which can easily reach temperatures well beyond 100,0000 K.
To be sure, though, nature isn’t through. The sun’s corona has an estimated temperature of about 1,000,0000 degrees K., and the center of the sun is estimated to have a temperature of about 20,000,000 degrees K. Ah, but man can top that, too. The hydrogen bomb develops temperatures of about 100,000,0000 degrees K.
And yet nature still beats us, since it is estimated that the central regions of the very hottest stars (the sun itself is only a middling warm one) may reach as high as
2,000,000,0000 degrees K. Now two billion degrees is a tidy amount of heat even when compared to a muggy day in New York or Tampa, but the questions arise: How long can this go on? Is there any limit to how hot a thing can be? Or to put it another way, How hot is hot?
That sounds like asking, How high is up? and I wouldn’t do such a thing except that our twentieth century has seen the height of upness scrupulously defined in some respects. For instance, in the good old days of Newtonian physics there was no recognized limit to velocity. The question, How fast is fast? had no answer.
Then along came Einstein, he advanced the notion, now generally accepted, that the maximum possible velocity is that of light, which is equal to 186,274 miles per second, or, in the metric system, 299,776 kilometers per second. That is the fastness of fast. So why not consider the hotness of hot?
One of the reasons I would like to do just that is to take up the question of the various temperature scales and their interconversion for the general edification of the readers. The subject now under discussion affords an excellent opportunity for just that. For instance, why did I specify the Kelvin scale of temperature in giving the figures above? Would there have hen a difference if I had used Fahrenheit? How much and why? Well, let’s see.
The measurement of temperature is a modern notion, not more than 350 years old. In order to measure temperature, one must first grasp the idea that there are easily observed physical characteristics which vary more or less uniformly with change in the subjective feeling of “hotness” and “coldness.” Once such a characteristic is observed and reduced to quantitative measurement, we can exchange a subjective, “Boy, it’s getting hotter,” to an objective, “The thermometer has gone up another three degrees.”
One applicable physical characteristic, which must have been casually observed by countless people, is the fact that substances expand when warmed and contract when cooled. The first of all those countless people, however, who tried to make use of this fact to measure temperature was the Italian physicist Galileo Galilei. In 1603 he inverted a tube of heated air into a bowl of water. As the air cooled to room temperature, it contracted and drew the water up into the tube. Now Galileo was ready. The water level kept on changing as room temperature changed, being pushed down when it warmed and expanded the trapped air, and being pulled up when it coaled and contracted the trapped air. Galileo had a thermometer (which, in Greek, means “heat measure”). The only trouble was that the basin of water was open to the air and air pressure kept changing. That also showed the water level up and down, independently of temperature, and queered the results.
By 1654, the Grand Duke of Tuscany, Ferdinand II, evolved a thermometer that was independent of air pressure. It contained a liquid sealed into a tube, and the contraction and expansion of the liquid itself was used as an indication of temperature change. The volume change in liquids is much smaller than in gases, but by using a sizable reservoir of liquid which was filled so that further expansion could only take place up a very narrow tube, the rise and fall within that tube, for even tiny volume changes, was considerable. This was the first reasonably accurate thermometer, and was also one of the few occasions on which the nobility contributed to scientific advance.
With the development of a desire for precision, there slowly arose the notion that, instead of just watching the liquid rise and fall in the tube, one ought to mark off the tube at periodic intervals so that an actual quantitative measure could be made. In 1701, Isaac Newton suggested that the thermometer be thrust into melting ice and that the liquid level so obtained be marked as 0, while the level attained at body temperature be marked off as 12, and the interval divided into twelve equal parts. The use of a twelve-degree scale for this temperature range was logical. The English had a special fondness for the duodecimal system (and need I say that Newton was English?). There are twelve inches to the foot, twelve ounces to the Troy pound, twelve shillings to the pound, twelve units to a dozen and twelve dozen to a gross. Why not twelve degrees to a temperature range? To try to divide the range into a multiple of twelve degrees-say into twenty-four. or thirty-six degrees-would carry the accuracy beyond that which the instrument was then capable of.
But then, in 1714, a German physicist named Gabriel Daniel Fahrenheit made a major step forward. The liquid that had been used in the early thermometers was either water or alcohol. Water, however, froze and became useless at temperatures that were not very cold, while alcohol boiled and became useless at temperatures that were not very hot. What Fahrenheit did was to substitute mercury. Mercury stayed liquid well below the freezing point of water and well above the boiling point of alcohol. Furthermore, mercury expanded and contracted more uniformly with temperature than did either water or alcohol. Using mercury, Fahrenheit constructed the best thermometers the world had yet seen. With his mercury thermometer, Fahrenheit was now ready to use Newton’s suggestion; but in doing so, he made a number of modifications. He didn’t use the freezing point of water for his zero (perhaps because winter temperatures below that point were common enough in Germany and Fahrenheit wanted to avoid the complication of negative temperatures). Instead, he set zero at the very lowest temperature he could get in his laboratory, and that he attained by mixing salt and melting ice.
Then he set human body temperature at 12, following Newton, but that didn’t last either. Fahrenheit’s thermometer was so good that a division into twelve degrees was unnecessarily coarse. Fahrenheit could do eight times as well, so he set body temperature at 96. On this scale, the freezing point of water stood at a little under 32, and the boiling point at a little under 212. It must have struck him as fortunate that the difference between the two should be about 180 degrees, since 180 was a number that could be divided evenly by a large variety of integers including 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60 and 90. Therefore, keeping the zero point as was, Fahrenheit set the freezing point of water at exactly 32 and the boiling point at exactly 212. That made body temperature come out (on the average) at 98.60, which was an uneven value, but this was a minor point. Thus was born the Fahrenheit scale, which we, in the United States, use for ordinary purposes to this day. We speak of “degrees Fahrenheit” and symbolize it as “degrees F.” so that the normal body temperature is written 98.6 degrees F.
In 1742, however, the Swedish astronomer Anders Celsius, working with a mercury thermometer, made use of a different scale. He worked downward, setting the boiling point of water equal to zero and the freezing point at 100. The next year this was reversed because of what seems a natural tendency to let numbers increase with increasing heat and not with increasing cold. Because of the hundredfold division of the temperature range in which water was liquid, this is called the Centigrade scale from Latin words meaning “hundred steps.” It is still common to speak of measurements on this scale as “degrees Centigrade,” symbolized as “degrees C.” However, a couple of years back, it was decided, at an international conference, to call this scale after the inventor, following the Fahrenheit precedent. ‘Officially, then, one should speak of the “Celsius scale” and of “degrees Celsius.” The symbol remains “degrees C.”
The Celsius scale won out in most of the civilized world. Scientists, particularly, found it convenient to bound the liquid range of water by 0 degrees at the freezing end and 100 degrees at the boiling end. Most chemical experiments are conducted in water, and a great many physical experiments, involving heat, make use of water. The liquid range of water is therefore the working range, and as scientists were getting used to forcing measurements into line with the decimal system (soon they were to adopt the metric system which is decimal throughout), 0 and 100 were just right. To divide the range between 0 and 10 would have made the divisions too coarse, and division between 0 and 1000 would have been too fine. But the boundaries of 0 and 100 were just right.
However, the English had adopted the Fahrenheit scale. They stuck with it and passed it on to the colonies which, after becoming the United States of America, stuck with it also. Of course, part of the English loyalty was the result of their traditional traditionalism, but there was a sensible reason, too. The Fahrenheit scale is peculiarly adapted to meteorology. The extremes of 0 and 100 on the Fahrenheit scale are reasonable extremes of the air temperature in western Europe. To experience temperatures in the shade of less than 0 degrees F. or more than 100 degrees F. would be unusual indeed. The same temperature range is covered on the Celsius scale by the limits -18 degrees C. and 38 degrees C. These are not only uneven figures but include the inconvenience of negative- values as well.
So now the Fahrenheit scale is used in English-speaking countries and the Celsius scale everywhere else (including those English-speaking countries that are usually not considered “Anglo-Saxon”). What’s more, scientists everywhere, even in England and the United States, use the Celsius scale.
If an American is going to get his weather data thrown at him in degrees Fahrenheit and his scientific information in degrees Celsius, it would be nice if he could convert one into the other at will. There are tables and graphs that will do it for him, but one doesn’t always carry a little table or graph on one’s person. Fortunately, a little arithmetic is all that is really required. In the first place, the temperature range of liquid water is covered by 180 equal Fahrenheit degrees and also by 100 equal Celsius degrees. From this, we can say at once that 9 Fahrenheit degrees equal 5 Celsius degrees. As a first approximation, we can then say that a number of Celsius degrees multiplied by 9/5 will give the equivalent number of Fahrenheit degrees. (After all, 5 Celsius degrees multiplied by 9/5 does indeed give 9 Fahrenheit degrees.)
Now how does this work out in practice? Suppose we are speaking of a temperature of 2O degrees C., meaning by that a temperature that is 20 Celsius degrees above the freezing point of water. If we multiply 20 by 5/9 we get 36, which is the number of Fahrenheit degrees covering the same range; the range, that is, above the freezing point of water. But the freezing point of water on the Fahrenheit scale is 32 degrees. To say that a temperature is 36 Fahrenheit degrees above the freezing point of water is the same as saying it is 36 plus 32 or 68 Fahrenheit degrees above the Fahrenheit zero; and it is degrees above zero that is signified by the Fahrenheit reading. What we have proved by all this is that 20 degrees C. Is the same as 68 degrees F. and vice versa.
This may sound appalling, but you don’t have to go through the reasoning each time. All that we have done can be represented in the following equation, where F represents the Fahrenheit reading and C the Celsius reading:
F = 9/5 C + 32 (Equation 15)
To get an equation that will help you convert a Fahrenheit reading into Celsius with a minimum of thought, it is only necessary to solve Equation 1 for C, and that will give you:
C = 9/5 (F — 32) (Equation 16)
To give an example of the use of these equations, suppose, for instance, that you know that the boiling point of ethyl alcohol is 78.5 C. at atmospheric pressure and wish to know what the boiling point is on the Fahrenheit scale. You need only substitute 78.5 for C in Equation 15. A little arithmetic and you find your answer to be 173.3 degrees F. And if you happen to know that normal body temperature is 98.6 degrees F. and want to know the equivalent in Celsius, it is only necessary to substitute 98.6 for F in Equation 16. A little arithmetic again, and the answer is 37.0 degrees C.
But we are not through. In 1787, the French chemist Jacques Alexandre César Charles discovered that when a gas heated, its volume expanded at a regular rate, and that when it was cooled, its volume contracted at the same rate. This rate was 1/ 273 of its volume at 0 degrees C. for each Celsius degree change in temperature.The expansion of the gas with heat raises no problems, but the contraction gives rise to a curious thought. Suppose a gas has the volume of 273 cubic centimeters at 0 degrees C. and it is cooled. At - 1 degree C. it has lost 1/ 273 of its original volume, which comes to 1 cubic centimeter, so that only 272 cubic centimeters are left. At -2 degrees C. it has lost another 1/ 273 of its original volume and is down to 271 cubic centimeters. The perceptive reader will see that if this loss of 1 cubic centimeter per degree continues, then at -273 degrees C., the gas will have shrunk to zero volume and will have disappeared from the face of the earth.
Undoubtedly, Charles and those after him realized this, but didn’t worry. Gases on cooling do not, in actual fact, follow Charles’s law (as this discovery is now called) exactly. The amount of decrease slowly falls off and before the -273 degrees C., point is reached, all gases (as was guessed then and as is known now) turn to liquids, anyway; and Charles’s law does not apply to liquids. Of course, a “perfect gas” may be defined as one for which Charles’s law works perfectly. A perfect gas would indeed contract steadily and evenly, would never turn to liquid, and would disappear at -273 degrees. However, since a perfect gas is only a chemist’s abstraction and can have no real existence, why worry?
Slowly, through the first half of the nineteenth century, however, gases came to be looked upon as composed of discrete particles called molecules, all of which were in rapid and random motion. The various particles therefore possessed kinetic energy (i.e. “energy of motion”), and temperature came to be looked upon as a measure of the kinetic energy of the molecules of a substance under given conditions. Temperature and kinetic energy rise and fall together. Two substances are at the same temperature when the molecules of each have the same kinetic energy. In fact, it is the equality of kinetic energy which our human senses (and our nonhuman thermometers) register as “being of equal temperature.”
The individual molecules in a sample of gas do not all possess the same energies, by any means, at any given temperature. There is a large range of energies which are produced by the effect of random collisions that happen to give some molecules large temporary supplies of energy, leaving others with correspondingly little. Over a period of time and distributed among all the molecules present, however, there is an “average kinetic energy” for every temperature, and this is the same for molecules of all substances.
In 1860, the Scottish mathematician Clerk Maxwell worked out equations which expressed the energy distribution of gas molecules at any temperature and gave means of calculating the average kinetic energy. Shortly after, a British scientist named William Thomson (who had just been raised to the ranks of the nobility with the title of Baron Kelvin) suggested that the kinetic energy of molecules be used to establish a temperature scale. At 0 degrees C. the average kinetic energy per molecule of any substance is some particular value. For each Celsius degree that the temperature is lowered, the molecules lose 1/ 273 of their kinetic energy. (This is like Charles’s law, but whereas the decrease of gas volume is not perfectly regular, the decrease in molecular energies-of which the decrease in volume is only an unavoidable and imperfect consequence - is perfectly regular.) This means that at -273 degrees C., or, more exactly, at -273.16 degrees C., the molecules have zero kinetic energy. The substance-any substance-can be cooled no further, since negative kinetic energy is inconceivable.
The temperature of -273.16 degrees C. can therefore be considered an “absolute zero.” If a new scale is now invented in which absolute zero is set equal to 0 degrees and the size of the degree is set equal to that of the ordinary Celsius degree, then any Celsius reading could be converted to a corresponding reading on the new scale by the addition of 273.16 (The new scale is referred to as the absolute scale or, more appropriately in view of the convention that names scales after the inventors, the Kelvin scale, and degrees on this scale can be symbolized as either “degrees A.” or ‘degrees K.”) Thus, the freezing point of water is 273.1 6 degrees K. and the boiling point of water is 373.16 degrees K.
In general: K = C + 273.16 (Equation 17)
C = K - 273.16 (Equation 18)
You might wonder why anyone would need the Kelvin scale. What difference does it make just to add 273.16 to every Celsius reading? What have we gained? Well, a great many physical and chemical properties of matter vary with temperature. To take a simple case, there is the volume of a perfect gas (which is dealt with by Charles’s law). The volume of such a gas, at constant pressure, varies with temperature. It would be convenient if we could say that the variation was direct; that is, if doubling the temperature meant doubling the volume.
If, however, we use the Celsius scale, we cannot say this. If we double the temper-ature from, say, 20 degrees C. to 40 degrees C., the volume of the perfect gas does not double. It increases by merely one-eleventh of its original volume. If we use the Kelvin scale, on the other hand, a doubling of temperature does indeed mean a doubling of volume. Raising the temperature from 20 degrees K. to 40 degrees K., then to 80 degrees K., then to 160 degrees 0 K., and so on, will double the volume each time. In short, the Kelvin scale allows us to describe more conveniently the manner in which the universe behaves as temperature is varied-more conveniently than the Celsius scale, or any scale with a zero point anywhere but at absolute zero, can.
Another point I can make here is that in cooling any substance, the physicist is withdrawing kinetic energy from its molecules. Any device ever invented to do this only succeeds in withdrawing a fraction of the kinetic energy present, however little the amount present may be. Less and less energy is left as the withdrawal step is repeated over and over, but the amount left is never zero. For this reason, scientists have not reached absolute zero and do not expect to, although they have done wonders and reached a temperature of 0.00001 degree K. At any rate, here is another limit established, and the question: How cold is cold? is answered.
But the limit of cold is a kind of “depth of down” as far as temperature is concerned, and I’m after the “height of up,” the question of whether there is a limit to hotness and, if so, where it might be. Let’s take another look at the kinetic energy of molecules. Elementary physics tells us that the kinetic energy (E) of a moving particle is equal to ½mv2, where “m” represents the mass of a particle and “v” its velocity. If we solve the equation E = ½mv2 for “v”, we get:
Now then, if in Equation 20 the temperature (T) is given in degrees Kelvin, and the mass (in) of the particle is given in atomic units, then the average velocity (v) of the particles will come out in kilometers per second. (If the numerical constant were changed from 0.158 to 0.098, the answer would come out in miles per second.)
For instance, consider a sample of helium gas. It is composed of individual helium atoms, each with a mass of 4, in atomic units. Suppose the temperature of the sample is the freezing point of water (273 degrees K.). We can therefore substitute 273 for ‘T’ and 4 for “in” in Equation 20. Working out the arithmetic, we find that the average velocity of helium atoms at the freezing point of water is 1.31 kilo- meters per second (0.81 miles per second). This will work out for other values of “T” and “in.” The velocity of oxygen molecules (with a mass of 32) at room temperature (300 degrees K.) works out as 0.158V 300/32 or 0.48 kilometers per second. The velocity of carbon dioxide molecules (with a mass of 44) at the boiling point of water (373 degrees K.) is 0.46 kilometers per second, and so on.
Equation 20 tells us that at any given temperature, the lighter the particle the faster it moves. It also tells us that at absolute zero (where T = 0) the velocity of any atom or molecule, whatever its mass, is zero. This is another way of looking at the absoluteness of absolute zero. It is the point of absolute (well, almost absolute) atomic or molecular rest.
ASIMOV ON PHYSICS
Copyright @1976 , by Isaac Asimov
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ISAAC ASIMOV, Hugo Award-winning author of science fiction, is also America’s foremost writer of science fact for the layman. Dr. Asimov has written more than two hundred (200+ ) books on every subject from neutrinos and quasars to the Bible
He has also authored hundreds of articles for publications ranging from Esquire to pamphlets of the Atomic Energy Commission. Among Dr. Asimov’s classics of science reportage are FROM EARTH TO HEAVEN, LIFE AND TIME, THE NEUTRINO, THE PLANET THAT WASN’T; QUASAR, QUASAR, BURNING BRIGHT, THE UNIVERSE FROM FLAT EARTH TO QUASAR and VIEW FROM A HEIGHT (all available in Avon Discus editions).
His other books include ASIMOV ON SCIENCE FICTION (Avon Discus) and ASIMOV’S GUIDE TO THE BIBLE (two-volume Avon edition).
Isaac Asimov was born in Russia, but came to this country with his parents at the age of three and grew up in Brooklyn. In 1948 he received his Ph.D. in Chemistry from Columbia University, then joined the faculty of Boston University, where he is Associate Professor of Biochemistry of the Boston University School of Medicine.
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