Thermodynamics

by Isaac Asimov

from Understanding Physics: Motion, Sound and Heat by Isaac Asimov.

Contents

In the previous chapter, I spoke of mixing hot and cold water and said that in the process - an intermediate temperature was reached. It is easy to see that this is achieved by the physical intermingling of the molecules of the hot water (which possess a high average kinetic energy) with the molecules of the cold water (which possess a low one). The molecules of the mixture, taken as a whole, are bound to have an average kinetic energy of an intermediate value.

Gases, too, can blunt the extremes of temperature in this fashion. Warm air masses will mingle with cold air masses (and such mingling of air masses is the fount and Origin of our weather), and the temperature of the earth's surface is kept at an intermediate value as a result. It might seem that the mixture of warm and cold on earth is not very efficient when one compares the frozen floes of the polar regions with the steaming jungles of the tropics. It could, however, be worse. Our moon is at the same average distance from the sun as the earth itself is, but unlike the earth it lacks an atmosphere. As a result, portions of its sunlit side grow hotter than even the earth's tropics do, and portions of its darkened side row colder by far than an Antarctic winter.

The transfer of heat by currents of gas or liquid is known as convection (from Latin words meaning “to carry together”).

Such actual movement of matter is not necessary for transfer of heat, however. If one end of a long metal rod is heated, the heat will eventually make itself felt at the other end of the rod. It is not to be supposed that there are currents of moving matter within the solid metal of the rod. What happens, instead, is something like this. As the end of the rod grows hot, the atoms of that portion of the metal gain kinetic energy. As long as the rod remains solid, the average position of each atom remains fixed, but each can and does vibrate about that position. As the atoms gain energy, the vibrations become more rapid, and the movements extend further from the equilibrium position. The atoms in the hottest portion of the rod, vibrating most energetically, jostle neighboring atoms, and those atoms, as a result of the impacts, vibrate more energetically themselves. In this way, kinetic energy jostles itself from atom to atom and, gradually, from one end of the rod to the other. This transfer of heat through the main body of a solid is conduction (from Latin words meaning “to lead together”).

The fact that atoms and molecules of solids vibrate with greater Amplitude as temperature rises means that each atom or molecule takes up more room. It is not surprising then that the volume of a solid, or a liquid for that matter, will increase with rising temperature and decrease with falling temperature, even though the molecules remain in virtual contact throughout the temperature range up to the boiling point.

(This is not the only factor involved in the volume change that solids and liquids undergo with temperature. There is also the matter of the nature of the molecular arrangement. The molecular arrangement for a particular substance is usually more compact in the solid state than in the liquid state, so there is generally a sudden drop in volume-and consequent rise in density-as a substance freezes. Water is exceptional in this respect. Its molecular arrangement is less compact in the solid state than in the liquid. As a result, ice is less dense than liquid water and will float in it rather than sink to the bottom.)

Both convection and conduction are explainable in mechanical terms. In both cases, there are actual impingements of energetic atoms or molecules upon less energetic atoms or molecules, and energy is therefore transferred by direct contact. Heat can, however, be transmitted without direct contact at all. A hot object encased in a vacuum will make its heat felt at a distance, even though there is no matter surrounding it to carry this heat either by convection or by conduction. The sun is separated from us by vacuum better than any we can yet make in the laboratory, and yet its heat reaches us and is evident. Such heat seems to stream out of the hot object in all directions, like the conventional rays drawn about the sun by cartoonists. The word “ray” is “radius” in Latin, and the transference of heat across a vacuum is called radiation. The detailed discussion of radiation will be left for the second volume of this book.

Interest in the laws governing the movement of heat by any or all these methods grew sharp in the first part of the nineteenth century because of the growing importance of James Watt’s steam engine, which depended in its workings on heat flow. In the steam engine, heat is transferred from burning fuel to water, converting the latter to steam. The heat of the steam then flows into the cold water bathing the condenser, and the steam, now minus its heat, is converted into water again. This heat flow that turned water to steam and back again somehow made available energy that could be converted into the kinetic energy of a piston, which, in turn, could be used to do work.

The study of the movement of heat (with particular attention, at first, to the workings of the steam engine) makes up that branch of physics called thermodynamics (from Latin words meaning “motion of heat”). Of course, all consideration of heat flow must assume, to begin with, that none of the heat will vanish into nothing or arise out of nothing. This is the law of conservation of energy, and so important is this generalization, in connection with thermodynamics in particular, that it is frequently called the first law of thermodynamics.

The first law of thermodynamics, however, merely states that the total energy content of a closed system is constant; it does not predict the manner in which the energy in such a system may shift from place to place. But even a little experience shows that some of the facts about such energy shifts seem to fall into a pattern.

For instance, suppose a closed system (that is, one that exchanges no energy with the outside world-giving off none and taking up none) consists of a quantity of ice placed in hot water. We can be quite certain that the ice will melt and the water win cool. The total energy has not changed; however, some of it has shifted from the hot water into the ice, and all the experience of mankind tells us that this shift is inevitable. Similarly, a red-hot stone will gradually cool, while the air in its neighborhood will gradually warm.

Such a flow of heat from a hot object to a cool object will continue until the temperature of different portions of the closed system are equal, and this is true whether heat is transferred by convection, conduction or radiation.

Faced with such facts about beat flow, the early workers in thermodynamics found matters most easily visualized if they thought of heat as a kind of fluid, and indeed this fluid even received a name- caloric, from a Latin word for “heat.”

The flow of heat can be pictured by uses of fluid flow as an analogy. Imagine two vessels connected by a stopcock, with the water level high on the left side and low on the right. Naturally, water pressure is higher on the left than on the right, so there is a net pressure from left to right. If the stopcock is open, water will flow from left to right and continue flowing until the levels are equal on both sides. The high level will fall; the low level will rise; and the final level on both sides will be intermediate in height. Although the total water volume of the system has not changed, there has been a change in the distribution of water within the system leading to an equalization of pressure.

By changing a few key words, we can have the previous sentence read: “Although the total heat of the system has not changed, there has been a change in the distribution of heat within the system leading to an equalization of temperature.” (Once again, we have an analogy between volume/pressure and heat/temperature.)

If we think of temperature as a kind of driving force directing the flow of heat, just as water pressure directs the flow of water, then it seems very natural, even inevitable, that heat should flow from a region of high temperature to one of low. without regard to the total heat content in each region.

Consider a gram of boiling water, for instance, and compare it with a kilogram of ice water. To freeze the kilogram of ice water, some 80,000 calories of heat must be withdrawn from it. To reduce the temperature of the gram of boiling water to the freezing point-and then freeze it-would require the withdrawal of 100 plus 80 calories; only 180 altogether. Any further cooling of the kilogram of ice obtained in the first case, as compared with the gram of ice obtained in the second, requires the withdrawal of a thousand times as much beat per Celsius degree from the former as from the latter. It is plain then that despite the difference in temperatures the total heat in the kilogram of ice water is much higher than the total heat in the gram of boiling water.

Nevertheless, if the gram of boiling water is added to the kilogram of ice water, heat flows from the boiling water into the ice water. It is not the difference in total heat content that determines the direction of heat flow. Rather, it is the difference in temperature. Again, our analogy-if in the connected vessels referred to above, the left were of narrow diameter and the right of wide diameter, water would flow from the region of smaller volume to that of greater volume. Not difference of total volume but difference of pressure would dictate the direction of water flow.

The rate at which water flowed from one portion of the system to another would depend on the size of the difference in pressure. When the stopcock is first opened, the water flows quickly, but as the difference in pressure on the two sides of the stopcock decreases, so would the rate of flow. The rate of flow becomes very small as the difference in pressure becomes small; it sinks to zero once the water “finds its level” and the differences in pressure disappear.

The flow of heat by conduction can, apparently, be pictured analogously. The rate of flow of heat from a hot region to a cold one depends in part on the difference in temperature between the two. It is conventional to calculate the quantity of heat that would flow in one second through a one-centimeter cube, where one face of the cube was 1 Celsius degree cooler than the face on the opposite side. This quantity of heat is the coefficient of conductivity, and it is measured in calories per centimeter per second per degree Celsius (cal/cm-sec-°C).

Even given a particular difference of water pressure, water flow might yet vary depending on whether it flowed through a wide orifice, a narrow orifice, a series of narrow orifices, a sponge, loosely-packed cotton, well-packed sand, and so on. The same is true for heat, and even where a given temperature difference is involved, heat will flow more rapidly through one substance than through another. In other words, the coefficient of conductivity varies from substance to substance.

Substances for which it is high are said to be good conductors of heat; those for which it is low are said to be poor conductors. In general, metals are good conductors of heat and nonmetals poor ones. The best conductor of heat is copper, with a coefficient of conductivity equal to 1.04 cal/cm-sec-°C. In comparison, water has a coefficient of conductivity of 0.0015 cal/cm-sec-°C, and some kinds of wood have coefficients as low as 0.00009 cal/cm-sec-°C.

It is for this reason that cold metal feels so much colder than cold wood. The metal and wood may be at equal temperatures, but heat leaves the hand much more quickly when it is in contact with the metal than with the wood. The temperature of the portion of the hand making contact with the substance drops much more rapidly in the first case. Analogously, it is safe to lift a kettle of boiling water by its wooden or plastic hand-grip, for the heat from the metal (which it is wiser not to touch) enters the wood or plastic slowly enough for loss by radiation to keep pace.

A system, completely surrounded by material of low heat conductivity, loses heat slowly to the outside world, or gains heat slowly, even though the temperature difference within and without is a great one. The system is made an island, so to speak, of a particular temperature in the midst of an outer sea of a different temperature. It is therefore insulated (from a Latin word for “island”), and a material of low heat conductivity is therefore a heat insulator.

Gases have low coefficients of conductivity; air, therefore, is a good heat insulator. Woolen blankets and clothes trap a layer of air in the tiny interstices between fibers; heat therefore travels from our body into the cold outer environment very slowly, and so we have a sensation of warmth that we would not otherwise have. Wool and air are not warm in themselves, but give the effect of warmness by helping us conserve our own body heat. Air alone would do equally well, if it could be relied on to remain still. The warmed air near our bodies is, however, constantly being replaced by cool air as a result of the ubiquitous air currents. Heat is carried away by convection, and a windy day feels colder than a still day at the same temperature.

All substances have coefficients of conductivity greater than zero, and there is no substance, therefore, that can qualify as a perfect insulator of heat. Suppose, though, we take the phrase “no substance” literally and surround a system with a vacuum. We would then have a better insulator than anything we could find in the realm of matter. A perfect vacuum possesses a coefficient of conductivity equal to zero, and cannot bring about heat loss through convection either. Even a vacuum is not a perfect insulator, however, for it will still serve as a pathway for the loss of heat by radiation.

Loss by radiation, however, is a slower process than loss by either conduction or convection. Consequently, some bottles are constructed with a double wall within which a vacuum is formed. Furthermore, the walls can be silvered so that any heat radiating across the vacuum, in either direction, is reflected almost entirely. In the end, passage of heat through such a vacuum flask, or “thermos bottle,” is exceedingly slow. Hot coffee placed in such a flask remains hot for an extended period of time, and cold milk remains cold.

Such devices were first constructed by the Scottish chemist James Dewar (1842-1923) in 1892. He used them to store extremely frigid substances, such as liquid oxygen, under conditions that would cut down the entry of heat from outside and thus minimize evaporation. In the laboratory, these are still called “Dewar flasks” in his honor.

The Second Law of Thermodynamics

We might therefore summarize the discussion in the preceding section by saying that it is the experience of mankind that in any closed system heat will spontaneously flow from a hot region to a cold region. It seems fair to consider this the second law of thermodynamics.

This view of heat as a kind of fluid reached its peak in the 1820's. A rigorous mathematical analysis of heat flow according to this view was advanced in 1822 by Fourier, the devisor of harmonic analysis. This view was put to further use by another French physicist, Nicolas Leonard Sadi Carnot (1796-1832).

In 1824, Carnot analyzed the workings of a steam engine in terms that we may consider analogous to those that might be applied to a waterfall. The energy of a waterfall can be made to turn a water wheel, the motion of which can then be used to run all the devices attached to the wheel. In this way energy of falling water is converted into work.

For a given volume of water, the amount of energy that can be converted to work depends on the distance through which the water drops-that is, upon the height of the pool of water at the bottom of the falls subtracted from the height of the cliff over which the water tumbles.

We could measure these two heights from any agreed-upon reference. Taking the level of the pool at the bottom of the falls as our standard, we could say that its height (h¹) was 0. Then, if the height of the cliff (h²) was 10 meters higher, its height would be +10 meters. The distance fallen by the water would be h²-h¹ -that is, 10-0, or 10 meters.

We could also let sea level be the standard. In that case, h¹ might be + 1727 meters, and h², would then be + 1737 meters; h²-h¹ would be 1737-1727, or still 10 meters. The most strictly rational zero point for height (at least on earth) would be he earth's center. In that case, the values of h¹ and h² might be 6,367,212 meters and 6,367,222 meters, respectively, and h²-h¹ would still be 10 meters. Indeed, we could let the top of the cliff be our zero point. If h², is 0, then h¹ representing the water level of the pool, ten meters lower than the cliff height, would have the value -10 meters. In that case, h²-h¹ would be 0-(-10), or still 10 meters.

I have belabored this point in order to make it perfectly clear that it is not the absolute values of h¹ and h² that count in deciding the amount of work we can extract from the energy of falling water, but only the difference between them.

Furthermore, if we continue to consider the waterfall, a clear distinction can be drawn between the total energy content of the water and the available energy content. The water drops to the bottom of the waterfall and forms p art of a quiet pool there. The pool by itself is not capable of turning a water wheel, yet it contains much potential energy. If a hole were dug, the water in that pool would drop further and some of its energy could be converted to work, provided that a water wheel was placed at the bottom of the hole. Ideally, a hole could be dug to the center of the earth, and then all the potential energy of the water (at least with respect to the earth) could be used.

However, in actual practice no hole is dug, and only the energy of the falling water of the actual waterfall is used. That energy is available. The further potential energy of the water, counting down to the center of the earth, is present but unavailable.

We can apply this sort of reasoning to the flow of heat. In the steam engine (or in any heat engine-for example, one that might use mercury vapor instead of steam) heat flows from a hot region, the steam cylinder, to a cold region, the condenser. The beat flows from the high temperature to the low temperature, as water flows from a greater height to a lesser one. It is not the value of either the high or the low temperature which dictates the amount of energy that can be converted to work. but rather the temperature difference. It is fair, then, to represent the available energy in terms of the temperature difference within the heat engine. We can express this most conveniently in terms of absolute temperature (see page 193), a concept not yet fully worked out at the time of Carnot's premature death from cholera at the age of 36. If we consider the hot region of the heat engine to be at a temperature T² and the cold region to be at T¹, then the available energy can be represented as T²-T¹.

The cold region of the steam engine still contains heat, of course. If the condenser is at a temperature of 25ºC, the water it contains (formed from the condensed steam) can, in principle, be cooled further and frozen, then cooled still further down to absolute zero; in the same way, water can be allowed to drop, -in principle, to the earth's center. The total energy of the system would be represented by the difference between the temperature of the hot region and absolute zero-that is T²-0, or simply T².

The maximum efficiency (E) of such a heat engine would be the ratio of the available energy to the total energy. If, under the conditions of the heat engine, all the energy of a system could be converted, in principle, to work, then the efficiency would be 1.0; if half the total energy could be converted into work, E would equal 0.5, and so on. Expressing available energy and total energy in terms of temperature differences, we can say then that:

T²-T¹
E=——— (Equation 15-1)
T²

Thus, suppose that steam at a temperature of 150ºC (423ºK) is condensed to water at 50ºC (323ºK). The maximum efficiency would then be (423-323)/423, or 0.236. Less than a quarter of the total heat in the steam would be available for conversion into work.

What's more, even this value is reached only if the heat engine is mechanically perfect: if there are no losses of energy through friction; none through radiation of heat to the outside world. and so on. In actual practice, heat engines are considerably less efficient than the maximum predicted by Equation 15-1. What equation 15-1 does, however, is to set a maximum beyond which even mechanical perfection cannot pass.

Equation 15-1 is derived on the assumption that heat flows only from a hot region to a cold, never vice versa. It, too, is therefore an expression of the second law of thermodynamics. The second law can therefore be viewed as setting a new kind of limitation on the utilization of energy.

The first law of thermodynamics (the law of conservation of energy) makes it plain that one cannot extract more energy from a system than the total energy present in the first place. The second law of thermodynamics maintains that it is impossible to extract more work from a system than the quantity of available energy present. and that the available energy present is invariably less than the total energy present unless a temperature of absolute zero can be attained.(1)

The second law of thermodynamics points out an important fact., In order to extract work from a heat engine, there must be a temperature difference. Suppose the hot region and the cold region were at the same temperature, both T². Equation 15-1 would then become (T²-T²)/T², or 0. There would be no available energy. (In the same way, no work could be done by a waterfall cascading down a height of 0 meters).

If this were not so, it would be conceivable that a ship traveling over the ocean could suck in water, make use of some of its energy content and then expel that water (cooler now than it was before) back into the ocean. All the ships in the world, and indeed all of man's other devices, could be run at the cost of a trifling fraction of the enormous quantity of energy in the ocean. The ocean would cool slightly in the process, and the atmosphere would warm, but the beat would flow back from air to water and all would be well.

If the second law of thermodynamics as expressed by Equation 15-1 is valid, however, this is impossible. To extract heat from the ocean, you would need a reservoir colder than the ocean and a refrigerating device to keep it colder than the ocean. The energy expended on refrigeration would be greater than the energy extracted from the ocean (assuming the refrigeration device to be mechanically imperfect, as it must be) and nothing would be gained. In fact, energy will have been lost. Virtually all “perpetual motion machines” worked up by hopeful inventors violate the second law of thermodynamics in one way or another. Patent offices will not even consider applications for such devices unless working models are supplied, and there seems little chance that a working model of such a device can ever be constructed.

Entropy

In the hands of Carnot, the second law of thermodynamics was of only limited application. He dealt only with heat engines and specifically omitted from consideration engines that worked by other means (by human or animal agency, for instance, or by the power of wind). Indeed, in Carnot's time, even the first law of thermodynamics was not yet thoroughly understood in its broadest sense.

In the 1840's, however, when Joule had demonstrated the interconversion of heat and a variety of other kinds of energy, and Helmholtz had specifically declared the law of conservation of energy to be of universal generality (see page 100), it seemed that the second law, dictating the direction of flow of heat, might also be made universally applicable. In heat engines, a temperature difference was required before energy could be converted to work, but not all work-producing devices were heat engines. It was possible to obtain work out of some systems in which there was only one level of temperature.

Thus, work can be obtained from electric batteries where no temperature differences are involved. Here, however, there are differences in electrical potential (a matter which is not discussed in this book) that represent available energy. Again, chemical reactions can be made to do work though the final products of the reaction might be at the same temperature as the original reagents. The difference in chemical potential would represent the available energy in that case.

To make the second law of thermodynamics fully general, it must be seen to apply to electrical energy, to chemical energy, indeed to all forms of energy, and not to heat alone. In whatever form energy exists, work can only be obtained if the energy is present in a state of greater intensity in one portion of the system and lesser intensity in another portion. (in the case of heat, the intensity is measured as temperature; in other forms of energy, it is measured in other ways.) It is the difference in intensity that measures the available energy. What is left of the total energy content after the available energy is subtracted is the unavailable energy.

In 1850, the German physicist Rudolf Julius Emanuel Clausius (1822-1888) saw the true generality of Carnot’s findings and announced it, specifically, as the second law of thermodynamics. (For this reason, Clausius is usually given the credit for being its discoverer.)

Now let's consider the second law again. In a heat engine, the temperature difference between the hot region and the cold region is the measure of the available energy. However, the second law states that, in a closed system heat must flow from a hot region to a cold. With time, therefore, this temperature difference must decrease, for as the heat flows in the only direction it can flow, the hot region cools down and the cold region warms up. Consequently, the available energy decreases with time. Since the total energy remains constant, the unavailable energy must increase as the available energy decreases.

Of course, we might remove the restriction of a closed system so that we can allow heat to enter the hot region from outside and keep it from cooling down. We can also pump heat out of th cold region and keep it from warming up. (This is done in actual steam engines, where burning fuel keeps the steam chamber continually hot, and running cold water keeps the condenser continually cold.) It takes energy to pump heat into the hot region and out of the cold region, however. We are increasing the total energy of the system merely to keep the available energy constant. As total energy goes up while available energy remains constant, the unavailable energy goes up, too.

In short, no matter how we argue matters in the case of a heat engine, unavailable energy increases with time. We might make this increase a very slow one, if we insulate the system well enough-to minimize heat flow from hot to cold. If we had a perfect insulator, we might even conceive of a situation in which unavailable energy did not increase.

What applies to heat engines ought also apply to all work producing devices. We might say then that the unavailable energy m any system can remain unchanged under ideal conditions, but always increases with time under actual conditions.

Clausius invented the word entropy (a word of uncertain derivation) to serve as a measure of the unavailability of energy. He showed that entropy could be expressed as beat divided by temperature. The units of entropy therefore are calories per degree Celsius. We can say then that the entropy of a system can remain unchanged under ideal conditions, but always increases with time under actual conditions. And this, too, is an expression of the second law of thermodynamics.

You must remember that the laws of thermodynamics apply to closed systems only. If we consider an open system, it is only too simple to find examples of apparent decreases in entropy.

In a refrigerator, for instance, heat is constantly being pumped from the cold objects within to the warm atmosphere outside in apparent defiance of the second law. A warm object, placed within the refrigerator, cools down; therefore, the available energy (represented by the temperature difference between the air outside and the object within the refrigerator) increases.

Where forms of energy other than heat are concerned, analogous “violations” of the second law of thermodynamics can be demonstrated. A man can walk uphill, increasing the available energy as measured by the difference in potential energy between himself and the bottom of the valley. Iron ore can be refined to pure iron and a spent storage battery can be charged-the former representing an “uphill movement” in chemical energy, the latter an “uphill movement” in electrical energy.

In every case cited, the system is not closed; energy is flowing into the system from outside. In order to make the second law of thermodynamics valid, the source of this outside energy must be included in the system so that it is “outside” no more.

Thus, material within the refrigerator does not spontaneously cool down (and remember that the original expression of the second law, speaks only of a spontaneous flow of heat). Instead, the cooling takes place only because a motor is working within the refrigerator. Although the entropy of the refrigerator's interior is decreasing, that of the motor is increasing. Furthermore, the motor's increase is greater than the interior's decrease, so the net change in entropy over the entire system-the refrigerator's interior plus its motor-is an increase.

In the same way, the entropy decrease involved in converting iron ore to, iron is smaller than the entropy increase involved in the burning coke and in the other reactions that bring about the refining of iron. The entropy increase in the electric generator supplying the electricity for the charging of the storage battery is greater than the entropy decrease of the storage battery itself as it is charged. The entropy decrease involved in a man walking uphill is less than the entropy increase involved in the reactions within his tissues which make the chemical energy of foodstuffs available for the effort involved in walking uphill.

This is true also of various large-scale, planet-wide processes that seem to involve a decrease in entropy. Examples of such entropy-decreasing phenomena are the uneven heating of the atmosphere, which gives rise to wind and weather; the lifting of uncounted tons of water miles high against the pull of gravity, which gives rise to rain and rivers; the conversion by green plants of carbon dioxide in the atmosphere to complicated organic compounds, which is the basis of the earth's never-ending food supply and of its coal and oil as well. It is because of these phenomena that the available energy on earth remains at approximately the same level through all its history; these phenomena also explain why we are in. no danger of running out of available energy in the foreseeable future.

Yet all these phenomena must not be considered in isolation for all take place at the expense of the solar energy reaching the earth. It is solar energy that unevenly heats the atmosphere, that evaporates water, and that serves as the driving force for th photosynthetic activity of green plants. In the course of its radiation of heat and light, the sun undergoes a vast increase in entropy(2) -one that is much vaster than the relatively puny decreases 0 entropy in earth-bound phenomena.

In other words, if we include within our system all the activities that affect the system, then it turns out that the net change in entropy is always an increase. When we detect an entropy decreases it is invariably the case that we are studying part of a system and not an entire one.

In actual practice we can never be sure that we are dealing with a closed system. No matter how we insulate, there are always influences from outside-energy gains and energy losses from and to the outside. All processes on the earth are affected by solar energy, and even if we consider the earth and sun together as one large system, there are gravitational and radiational influences from other planets and even other stars. Indeed, we cannot be certain that we are dealing with a truly closed system unless we take for our system nothing less than the entire universe.

In terms of the universe we can (as Clausius did) express the laws of thermodynamics with utmost generality. The first law of thermodynamics would be: The total energy of the universe is constant. The second law of thermodynamics would be: The total entropy of the universe is continually increasing.

Now suppose the universe is finite in size. It can then contain only a finite amount of energy. If the entropy of the universe (which is the measure of its unavailable energy content) is continually increasing, then eventually the unavailable energy will reach a point where it is equal to the total energy. Since the unavailable energy cannot rise beyond that point. the entropy of the universe will have reached a maximum.

In this condition of maximum entropy, no available energy remains, no processed involving energy transfer are possible, no work can be done. The universe has “run down.”

Disorder

Observations and experiments on heat in the first half of the nineteenth century assumed heat to be a fluid. From the very start of the century, however, evidence indicating that heat was not a fluid, but a form of motion, had begun to mount.

In 1798, for instance, Benjamin Thompson, Count Rumford (1753-1814), a Tory exile from the United States, was boring cannon in the service of the Elector of Bavaria. He noted that great quantities of beat were formed. Neither the cannon being bored nor the boring instrument used was at more than room temperature to begin with, and yet the heat developed by the act of boring was sufficient to bring water to a boil after a time; and the longer the boring was continued the more water could be boiled. It almost appeared as though the quantity of heat contained within the cannon and borer was infinite.

If heat were a fluid, and a form of matter, then to suppose it were formed in the act of boring raised a difficulty. Already, the French chemist Antoine Laurent Lavoisier (1743-1794) had established the law of conservation of matter, according to which matter could be neither created nor destroyed; and there was an increasing tendency among scientists to believe this generalization to be valid. If heat were being formed, then it must be something other than matter. To Rumford, the most straightforward possibility was that the motion of the boring instrument against the metal of the cannon was transformed into the motion of small parts of both borer and metal, and that it was this internal motion that was heat.

This notion was largely disregarded during the following decades. The assumption that small parts of an object might be moving invisibly seemed in 1800 to be just as difficult to accept as the assumption that matter was being created, perhaps even more difficult. A decade after Rumford's experimenting, however, the atomic theory was advanced and began to increase in popularity. By the internal movements of matter, one now meant the motions or vibrations of the atoms and molecules making it up and the assumption of such motion became continually more acceptable. In the 1840's Joule's experiments in converting work to heat extended Rumford's observations and made the victory of the atomic motion view of heat inevitable. Finally, in the 1860's, the kinetic theory of gases and the concept of heat as a form of motion on the atomic scale were established rigorously by Maxwell and Boltzmann.

This did not mean that the laws of thermodynamics, established in the first place on the basis of a fluid theory of heat turned out to be false. Not at all! The laws were based on observed phenomena, and they remained valid. What had to be change were the theories that explained why they were valid. The fluid theory of heat, to be sure, explained these phenomena very neatly,(3) but the atomic motion theory could be made to explain the fluid theory of heat could, and proved just as firm a foundation for the observation-based laws of thermodynamics.

To be sure, the view of heat as atomic motion is somewhat more difficult to picture and explain than the view of beat as a fluid. In the latter case, we can think of such familiar objects as waterfalls; in the former, the best we can do is imagine a set of perfectly elastic billiard balls bouncing about eternally in a closed chamber. One might suppose that of two theories one ought to accept the simpler, as Ockham's razor recommends. However, Ockham's razor is applied properly only when two or more theories explain all relevant facts with equal ease. This is not so in the present case.

If we confine ourselves to heat flow only, then it is easier to picture heat as a fluid than as atomic motion. However, if we are to explain the effect of heat on gas pressure and gas volume, if we are to explain specific heat, latent heat, and a host of other phenomena, it becomes very difficult to use the fluid theory. On the other hand, the atomic motion theory not only can explain heat flow but also all the other heat-involved phenomena.

Suppose, for instance, you have a hot body and a cold body in contact. The molecules in the hot body are, on the average, moving or vibrating more rapidly than the molecules in the cold body. To be sure, the molecules in both bodies possess a range of velocities, and there may be some molecules in the cold body that are moving more quickly than some molecules in the hot body, but this is an exceptional situation. When a molecule from the hot body (an “H molecule”) collides with one from the cold body (a “C molecule”) the chances are very good that -it will be the H molecule that will be moving the more quickly of the two. Another way of putting it is that if a great number of H molecules collide with a great number of C molecules, there will be a few cases where the C molecule is moving more rapidly than the H molecule with which it collides, but a vast preponderance of cases where it is the H molecule that is the more rapid of the two.

Now when two moving objects collide and rebound, the velocities of both may change in any of a large number of ways. These changes may be grouped into one of two classes. In the first class, the slower object may lose velocity in the process of collision while the faster object may gain velocity. The result would be that the slower object would finish by moving still more slowly, and the faster object would finish by moving still more quickly. In the second class, the slower object may gain velocity in the process of collision while the faster object may lose velocity. In the first class of collisions, the velocities become more extreme, in the second class more moderate.

There are many more ways in which a collision can belong to the second class than to the first. This means that over a large number of collisions in which velocity redistributes itself in a purely random manner, there will be many more collisions resulting in more moderate velocities than in more extreme velocities. Random collisions will bring about an "averaging out:, of velocities.(4)

When a hot body and a cold body are in contact, a large number of H molecules collide with a large number of C molecules; the result is that after rebounding, the H molecules are moving less quickly on the whole, and the C molecules are moving more quickly. This means that the H molecules have become cooler and the C molecules warmer. There has been a flow of beat from the H molecules to the C molecules. The temperature of the portion of the hot body in contact drops, and that of the portion of the cold body in contact rises.

Such collisions continue not only at the boundary at which the hot and cold bodies meet, but also within the substance of each. In the hot body, for instance, H molecules that have been cooled off by collisions with C molecules collide with neighboring molecules that have not been cooled off; here, too, there is a general moderation of velocities.

The result of these random collisions and random alterations of velocity throughout the entire system is that, eventually, the average velocities of the molecules in any portion of the system will be the same as in any other portion; this average will be a value that will lie between the two original extremes. (Hot and cold mix to produce lukewarm, so to speak.) Once the velocities are the same, on the average, throughout the system, collisions may continue to alter velocities, so a particular molecule may be moving quickly at one moment and slowly at another; however, the average will no longer change. The entire system having reached an intermediate equilibrium temperature, heat flow will cease.

In both the fluid theory of heat and the atomic motion theory, heat can be expected to flow spontaneously from a hot area to a cold area and this, after all, is a statement of the second law of thermodynamics. Yet there is a crucial difference between the two theories with respect to such heat flow.

In the fluid theory, the flow of heat is absolute. It is capable of going “downhill” only, and an “uphill movement is inconceivable. In the atomic motion theory, however, the flow of heat is a statistical matter and is not absolute. The random changes of velocity as a result of random collisions will result, as a matter of extremely high probability but not certainty, in the flow of heat from hot to cold. It is extremely unlikely, but not inconceivable, that in every collision, the faster molecule may gain velocity at the expense of the slower one, so heat will flow “uphill” from cold to hot.

Maxwell tried to dramatize this possibility by visualizing a scientific fantasy. Imagine two gas-filled vessels, H and C, connected by a stopcock. The H vessel is the hotter, and its molecules move the more rapidly on the average.

But it is only on the average that H molecules move more rapidly than C molecules. Some H molecules happen to move slowly, and some C molecules happen to move rapidly. Suppose that an intelligent atom-sized creature is in control of the stopcock (this creature is usually referred to as “Maxwell's Demon”). When one of the minority of slow H molecules approaches, Maxwell's Demon opens the stopcock and lets it into the C chamber. When one of the minority of fast C molecules approaches, Maxwell's Demon opens the stopcock and lets it into the H chamber. At other times, the Demon keeps the stopcock closed. In this way, there is a slow but steady drizzle of low-velocity molecules into C and an equally slow and equally steady drizzle of high-velocity molecules into H. The average velocity of the molecules in C drops, while that in H rises-and heat flows uphill from cold to hot.

The chance of such “uphill” flows of heat (or of any other form of energy) is so fantastically small in the ordinary affairs of life that it is quite safe to ignore it. However, the shift from a condition of “certainty” to a condition of “probability” is of crucial importance. As scientists probed deeper and deeper into the subatomic world during the twentieth century, statistical analysis of events and their consequences became more and more important and the improbable (but not impossible) gains a perceptible chance of taking place, while more and more of those cause/effect combinations we usually assume to be certain have been shown to be only very, very, very probable. In short, Maxwell's statistical interpretation of heat flow marks one of the first steps in the transition from the “classical physics” of the nineteenth century (with which this volume is concerned) to the “modern physics” of the twentieth century.

And how can entropy be interpreted in the light of the atomic motion view of heat? Entropy, according to the second law of thermodynamics, always increases. Well, then, what is it that always increases as a result of molecular collisions? In a manner of speaking, moderation does. If in a system, to begin with, an accumulation of heat is concentrated in one portion and there is a deficit in another, molecular collisions increase moderation and spread the heat more evenly throughout the system. In the end, when temperature equilibrium is reached, heat is spread out as evenly as possible.

Entropy can therefore be interpreted as a measure of the evenness with which energy is distributed. This can be applied to any form of energy and not merely to heat. When an electric battery discharges, its electrical energy is more and more evenly distributed over its substance and over the material involved in the electrical flow of current. In the course of a spontaneous chemical reaction, chemical energies are more evenly distributed over the molecules involved.

What's more, the evenness of energy distribution is “most even,” so to speak, when it is distributed as random motion among molecules. The conversion of any form of non-heat energy to heat represents a gain in the evenness with -which energy is distribute and is, therefore, a gain in entropy.

It is for this reason that any process involving a transfer of energy is bound to produce heat as a side-product. A body in motion will produce heat as a result of friction or air resistance, and some of its kinetic energy will be spread out over the molecules with which it has come in contact. In converting electrical energy to light or to motion, heat is also produced, as we know if we touch an electric light bulb or an electric motor.

This means, in reverse, that if heat were completely converted into some form of non-heat energy, then there would automatically be a decrease in entropy. But a decrease in entropy in a closed system is so extremely unlikely that the possibility of its occurrence under ordinary conditions can be ignored. Some heat, to be sure, can be converted into other forms of energy, but only at the expense of further increasing the entropy of the remaining heat in the system. In the steam engine, for instance, the conversion of the heat energy of the steam into the kinetic energy of the pistons is a piece of decreasing entropy that is at the expense of the (still greater) increasing entropy of the burning fuel that produces the steam.

The increasing evenness with which energy is spread out can be interpreted as increasing “disorder.” We interpret order as a quality characterized by a differentiation of the parts of a system: a separating of things into categories; a filing of cards in alphabetical order; a listing of things in terms of increasing quantities. To spread things out with perfect evenness is to disregard all these differentiations. A particular category of objects is evenly spread out among all the other categories, and that is maximum disorder.

For this reason, when we shuffle a neatly stacked deck of cards into random order, we can speak of an increase in entropy. And, in general, all spontaneous processes do indeed seem (in line with the second law of thermodynamics) to bring about an increase of disorder. Unless a special effort is made to reverse the order of things (increasing our own entropy), neat rooms will tend to become messed up, shining objects will tend to become dirty, things remembered will tend to become forgotten, and so on.

We thus find there is an odd and rather paradoxical symmetry to this book. We began with the Greek philosophers making the first systematic attempt to establish the generalizations underlying the order of the universe. They were sure that such an order, basically simple and comprehensible, existed. As a result of the continuing line of thought to which they gave rise, such generalizations were indeed discovered. And of these, the most powerful of all the generalizations yet discovered-the first two laws of thermodynamics-succeed in demonstrating that the order of the universe is, first and foremost, a perpetually increasing disorder.

Notes:

(1) It has been said that the first law of thermodynamics states, “You can’t win,” and that the second law of thermodynamics adds, “And you can’t break even either.”

(2) We might p to wonder how the sun was formed. for this formation must have Involved a vast entropy decrease in order to make it possible for the sun to continue radiating. at the expense of a continual large entropy increase, for so many billions of years. However to trace matters back beyond the sun would be more suitable in a book devoted to astronomy.

(3) In fact, it was just because it explained them so neatly that the fluid theory lasted as long as it did in the face of mounting evidence against it. It was distressing to have to give up something so convenient

(4) This does not mean that all velocities will ultimately be exactly equal If only there are enough collisions. If two objects collide at equal velocities. is becomes very probable that there will be a gain in velocity of one at the expense of the other. Too much “averaging out” becomes very unlikely. them fore. Instead, “averaging out” proceeds only to a certain point and stops. At a particular temperature the “averaging out” produces a range of velocities such as that predicted by the Maxwell-Boltzmann equations. A smaller and more limited range is extended to that point by collisions, a wider and more extended range is contracted to that point by collisions.