With Clausius from Energy to Entropy Mkimo B a r b Departamento de Fisica. Facultad de Ciencias Exactas y Naturals, Pabelion 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
"Nature. ..seems delighted with transmutations." Isaac Newton, Optics, Book Ill,query 30. Prelude Science. wrote Jose Orteea v Gasset over 60 vears aco ( I ) . is "whatever effort we maketdunderstand somkhing." his is of particular importance when teaching thermodvnamics a t &introductor~level. Its formal aspects are not difficult, hut the concepts involved are subtle and require much thinking hefore they can be understood. Entropy is a typical example of how understanding is replaced by arithmetic efficiency, as shown in the answer a professor gave to an inquiring student: "Don't worry what i t is, just learn how to calculate its changes." A more reasonable attitude can he reached looking with a little patience, in the book hv Rudolf Clausius: The Mechanical Theory of Heat (2).lcwas first published in German in 1864, based on the Davers he wrotein the 1850's. The presentation is beautifull; ciear and deals with all the subjeets that are referred to as classical thermodynamics. There are few fiaures and the algebra is minimal. keading it, one has the impression of being in front of Clausius, watching him think aloud. Clausius, not considering himself the founder of thermodynamics, recalls how William Thomson (3) brought to his attention the original ideas published by Sadi Carnot in 1824 (4) and their description in graphs and equations by Henri Clapeyron 10years later (5).He recalls also that later Robert
Mayer proposed in his "Observations on the Forces by Inanimate Matter" (1842 ( 6 ) )an eauivalence between heat and s spent several work. With this mkerial at h a n d ~ l a u s i u then years thinking as he wrote, using clear mental images. T o reach the concept of entropy, following as closely as possible the route taken by Clausius, who first visualized it, the concept of energy appears to he good starting point. 1i this respect it is interesting to remark that in a recent paper G. M. &row proposes a v& attractive restatement of thermodynamics based precisely on the energy concept (7).
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Energy a s Heal and Work Contained In Bodies The first step is Robert Mayer's principle or law of equivalence: the effect equals the cause. Later this became known as the firstlaw of thermodynamics, and Clausius called it the principle (or law) of equivalence between heat and work. Furthermore, since energy can neither he destroyed nor created, hut only transformed, Clausius asserted: The energy of the universe remains constant. T o understand this statement, we can imagine a system of any kind, small or large, and place ourselves as observers inside or outside of it. We will then he able to see the flow of energy in any of its forms (heat or work) through its houndaries, if they are well defined.,
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This point of view, which can be considered classical, allows us to define a system as large as the universe. The svstem's boundaries. whatever thev . mav. be.. turn i t into a ciosed system where k y change or passage from one state to another implies the flow of a certain amount of heat in one direction and another, of work of the same magnitude, in the opposite one. Since, as stated hv Clausius, the enerw of the universe has a cons& value, ihanges ofany kindare only possible through the interconversion of one form of energy into another: heat into work (mechanical, chemical, electrical, or magnetic) and vice versa. But for this to happen energy cannot be distributed uniformly in the universe, and since changes do in fact occur constahtly in the world around us this mieht well be true. ~lnderstandingthe relationship of energy to heat and work, Clausius called enerw "the heat and work content of bodies". Therefore i t can -de said that there is energy in a body whenever the body is capable of giving (or receiving) heat or providing (or absorbing) work. This is then the case of all fuels, foods, geological enclosures a t high temperatures and pressures, winds, chemical reactions, the sun, etc. However, the mere existence of energy in the universe does not imply necessarily the possibility of its transformation. Transformations and Thelr Direction The occurrence of transformations indicates that energy mav be transferred or flow from one place of the universe to andther; therefore, energy distribution throughout cannot he uniform. Calculations regarding this flow have the quantitative limitations stated by the first law. It is possible to calculate how much work c& be drawn from the amount of heat released by a burning fuel, or the amount of heat needed to melt a certain weight of ice, and soon. Conversely, it can also be calculated how much work should be done on a vapor to condense i t into a liquid or how much heat must he withdrawn from a certain weight of water to turn i t into ice. The above are nothine more than schematic descri~tions of heat engines, the evaporation of water from rivers, lakes, and oceans. the meltineof ice. the fall of rain. snow. and hail. and any coiceivable chemic&reaction. But iaturai transfor: mations exist in pairs, one being the inverse of the other (i.e., evaporation of water and fall of rain). Both are equally valid under the first law, which does not establish any difference regarding feasibility, nor any preference in direction. However, by simply observing nature it will be realized that transormations do occur spontaneously only under certain conditions. T o establish these or, in other words, to determine the direction of a certain spontaneous transformation, an experiment discussed by the French physicist Gay Lussac (8) a t the beginning of the 19th century is helpful. Instead of using vessels and valves, as he did, let us consider a laree rectaneular room well inside a buildine. with no windows a i d far away from any external wall. 'l'h&oom can then be taken to be effectively isolated from the ourside world. Many projection rooms comply with these requirements, because when their doors are closed, it is impossible to know what happens inside or vice versa. Let us imagine further that the room is divided into two equal cubic parts by a vertical wall that is perfectly rigid to resist any pressure, absolutely impervious to anything, and also collapsible a t will. As a result we can make i t disappear whenever we wish. One of the twin rooms is now considered t o be com~letelvevacuated. while the other is left a t anv pressure (the actual value isirrelevant), and there is a the;. mometer in each. If the total enclosure is effectively isolated from any external observer, i t is impossible to know if anything happens in the interior, unless the thermometers have some sortof external connection. If there is such a connection a t the start of the experiment, the readings on both thermometers will be identical. The thermometer in the
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evacuated room is in thermal eauilibrium with the walls through radiation, but the wall temperature is also the temperature of the eas in the other room: therefore. both thermometer readings correspond to the'thermal A t e of the walls and the eas. We can nowimagine that the dividing wall collapses suddenly and disappears completely, so that the gas in the nonevacuated room is free to expand (actually diffuse) into the other. Gay Lussac predicted that any attempt to follow this process through temperature readings would fail because no change should he observed in the thermometers. Under these circumstances nothing would seem to have happened within the enclosure. No work was done, heat did not flow to or from the outside, so that the internal energy of the gas cannot have changed. In other words the gas has the same enerev and is still able to do work or deliver heat. As Clausius ~ & d say, "there has been no change in the heat or work content of the gas." The total energy of the room and its surroundings is the same, but its distribution is different. During the experiment the capacity to do work that the system had a t the beginning by virtue of its configuration has been lost. In Clausius's words. the svstem has become impoverished of work content. Can the lost capacity be recovered? Yes! If the gas is compressed hack t o the initial conditions. Can this happen spontaneously? At this point we do not know. We do know, however, that this could be done from the outside if the necessary means are provided to compress the gas from the total volume into the orieinal one. In the proposed e x p k m e n t the enclosure (the twin rooms) was considered so perfectly isolated from the outside that we have no way of knowing what happened. There has been no flow of information. So, to return to the initial stage the isolation must be broken and the outside world be made to provide the necessary work for the compression. This is precisely what we do when a refrigerator or an air-conditionine device is operated. Heat is withdrawn from a cold bodv an2 through external work is released to a hot one. ~ i k k pushing a load uphill. This somewhat abstract discussion can be easily applied to a very common phenomenon, namely the evaporation, condensation and even freezine of water. A certain amount of water (from a lake for inscnce) can be converted into vapor throueh heat from a suitable source (the sun). but it can also freeze to ice when in contact with'a large mass of polar (cold) . . air. Therefore. it is evident that both transformations seem to be equallyspontaneous in the natural world aroundus. However, in both cases there is aflow of heat from a hot source or body to a cold one: from the sun to the water (evaporations) and from the water t o the cold air (freezing). But this is nothing more than the content of the second law, stated as: Heat flows naturally (without external intervention) from a hot body to a cold~one. In the case of water both cold (polar air) and hot (sun) sources exist in nature; however, observation of reality indicates that hot sources or bodies are a t the origin of phenomena..s i m.. ~ l because v cold sources a m e a r alwavs as a result of a previous action. A sudden fall in-the thermal state of atmospheric air, which can lead to an intense downpour, is the result of the very complex ascending movement of large masses of air. Its displacement is due ultimately to heat received from the sun. Consequently, although the energy of the universe may be constant (first law), its distribution is such that flow or spontaneous transformation is possible only in the direction set by the second law. But to find a statement containing this requirement, the concept of transformation has to be examined further. Uneven Distribution of Energy In the phenomena described so far, transformations appear whenever a system is allowed to evolve freely from an
initial to a final state. The changes are spontaneous and unless particular care is taken intermediate steps are impossible to identify. Only the initial and final stages are c l e d y known. This happens, for instance, if we take a mass of hot water and dump it into another one of cold water. We end up with a third mass of warm water. The total energy of the initial set of the two masses (considered as one system) will not have changed, upon mixing, but its distribution will have changed indeed. The capacity to do work, present a t the beginning by virtue of the configuration (two separate masses a t different temperatures) has been lost. Now, to recover it, the mass of warm water has to he divided into two parts, and heat must be carried from one t o the other until the original situation is reached. This will not occur spontaneously (second law) and can only be reached through external assistance. Some portion of the rest of the universe must be in a position to provide the needed flow of energy. Any machine that can be onerated with electricitv for instance takes advantage of the uneven distribution of energy that can then be transferred or flow. In this case. there is a flow of energy, through the cables, from the powerplant (surroundings) t o the system (machine). The water cycle is again a very good example. The sun nrovides enerw to transform ice into water. and this water into steam, b i t i t also provides the needed energy to create cold masses through expansion of air that act as bodies or sources to invert the process, converting water back into ice. Clausius called the capacitv to transform (or to flow) "the transformation value of heat". However, the correct denomination should he the transformation value of energy, because i t is energy that transforms (or flows), while heat is only one of its forms. Therefore, as a result of all the transformations occurring constantly in nature, the energy of the universe becomes successively impoverished in this transformation value (caoacitv or content) and eets continuouslv degraded as a res&t'of itsconstant flow. w i e n all the energy of the universe has become uniformly, distributed and can no longer flow, some kind of minimum will be reached. During Clausius's time and for years thereafter, a name was given to this final instant: the heat death of the universe. The problem now is how to assign - meaningful numbers to this transformation value. Invertible and/or Reversible If we return t o spontaneous or natural transformations we will realize that a number of them can oroceed in two directions. The case of water serves again as a good example, because the passages from ice to steam and from steam back t o ice are equally possible. The transformation is invertible. This property is also found in a very simple experiment that can be carried out with the hand inflator that children sometimes use to inflate bicycle tires. If the exit valve is held firmly between the fingers, i t is easy t o feel that it gets hot as air is heing pumped through it. The initial and final states of the air can be readily identified through their pressure, volumes and temperatures, but unless appropriate care is taken. the intermediate staees are unknown. However, if instead oTallowing the air to escape into the tire, the valve is plugged up and the inflator carefully isolated (with a layer of asbestos or glass wool or whatever) the air will remain hot when compressed. If now the piston is released the gas will expand A d , of course, cool do&. This can be repeated a t will, and the successive heatings and coolings are ohviously ruled by the first law. If isolation is perfect the initial and final states will be equal. The transformation is again perfectly invertible. If we reason in terms of energy, before itscompression the air had no capacity todo any work aeainst the atmosoheric nressure. but after comnression it iideed does, as c& be seen when the handle is reieased and shoots back. Under these circumstances the work exchaneed (received or done) by the inflator is related exactly & a
certain amount of heat, so that the energy of the system (the inflator) before compression is exactly the same after expansion. Essentially similar are the well-known experiments with a rubher band. When i t is stretched, work is done on it and it warms up. Conversely, upon relaxing, i t does work and therefore cools down. Both operations, as with the tire pump, can be carried out in succession, and the accompanying heating and cooling is easily felt by placing the band just touching any sensitive portion of the facial skin (upper lip, cheek). This stretching and relaxing between two fixed lengths (well-defined initial and final states) can be carried out in any desired manner and are both invertible. Again, as with the tire pump, the actualnumber of routes is indefinite. Among these there is at least one, hut there may be more, that can be followed step by step, continuously, so that the value of the energy is known for each and every one of the steps and can he represented by continuous functions. By choosing such a route or path we may know constantly not oulv where we are but also where we have iust been and where we are going to be next. Every point o n a route of this kind is usually called an eauilihrium state. It means that the route can be ;overed in e=actly the same way in both directions. If we call this trip a transformation, it will be not only invertible hut also reversible (9). Compensation and Noncompensation
Looking again a t both our inflator and rubber hand exneriments. we will realize that. althoueh nerfectlv invertible. the proces$es will be reversible if andonfy if the; are carried out so that their progress can he followed step by step from the outside. Actions in the inside and the outside must be coupled and are, in Clausius's words, mutually compensated. This is one of his great conceptual contributions. He called reuersible processes compensated and irreversible processes noncompensated (or uncompensated). Since a system's development can be followed step by step from the outside world only in reversible rever$bility and compensation are intimately related. At this point i t is important to realize that the rate of any reversible process is a consequence of the need for exact compensation along a reversible path and that there is no loss of any kind. This is not the case with irreversible processes where there is no compensation. The capacity to transform is lost. If we now look aeain a t the free-exnansion exneriment. we may recall that in& initial state th;! gas confiked in onk of the rooms has a certain canacitv to transform. If. instead of allowing i t to diffuse freely intdthe evacuated section, some external connection is provided, useful work can be obtained. However, if this is not done, the capacity to transform is lost, and to recover it the gas would have to be compressed back by means of work done on it from the outside. This amount of work represents the transformation value the gas had hefore heing allowed to expand freely (or
-..Now,iftherestorationofthecapacity ...--, to transform isdone riiffi,.~~ .
reversibly, through a compensated process, the correspunding transformation value can be calculated exactly. Dlsgregation The above concept of transformation value was reached starting from the notion of work. It is imoortant now to see if a simicar conclusion can be reached starting from that of heat. T o do so i t is useful to take the opening sentence of Sadi Carnot's famous paper (4),"Everyone knows that heat can he the cause of movement and that it also has a large motive force.This is avery old concept, so i t is only natural that Clausius should have considered that heat would act upon the "constitutive parts of bodies". He did not talk about molecules. He considered that heat was able to overVolume 66
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come the cohesion among particles, even t o the point of leading to a phase change (melting or boiling). But in this case, as in many others, Clausius wanted to reach a formal expression (mathematical, he called it) and proposed t o represent the degree of subdivision in a body through . what he called "disgregation". He defined the action of heat on bodies saying that "the heat tends to increase its disgregation." But he went further to consider that any state of a body (or system in our modern laneuase) could be characterized bv a "disereeation value". ~ h i r e g r kany , heat exchanged alters the & r e g a t i o n and will represent a "change in the disgregation value". So, any substance has different disgregation values depending on whether it is in the solid, liquid, or vapor phase. Clausius pointed out that the forces needed to overcome the actions of heat increase with the temperature of the heated bodies (or systems in our language). Consequently he proposed that "The mechanical work that heat may perform during a change of disgregation in a body is proportional to the absolute temperature a t which the change occurs." Note that Clausius does not talk of the work oerformed bv heat but refers instead to work that heat may perform. In doing so he actuallv defines a c a ~ a c i t vand sueeests the idea of oossibility. I t :an also be seen that the work in question;^ the maximum obtainable as a result (or a function) of the thermal state (absolute temperature) of the body. At this point we must recall that, since any process or transformation implies the exchange of heat and work, a transformation value can be assigned to it. But, since the exchanee of heat or work is a s s o c k d with a chanee in the disgregkion value, both concepts can be linked togeiher and considered to be equivalent. Entropy Summing up, i t can be said that both transformation and disgregation values represent one single - propertv, - . or a auality or content" of &y system. Therefore, a ceriain amount of enerw, under defined conditions, will possess a definite transfo%ation ualue or, what is equivalknt, can suffer or undergo a certain change in disgregation ualue. Clausius wanted to devise a suitable denomination for this concept, so that i t could be easily recognized and handled. T o do so he took advantage of a proposal he had already made when he called energy "work and heat content". Therefore. he chose the greek word rowa6 . . (transformation) for the new concept and coined the word entropy. Theentropv -.of anv svstem will then be a measure both of its transformation content or ualue or its disgregation ualue, and-the entropy of a system can be said to represent its capacity, or lack of it, to transform. Clausius saw that the entropy concept is a t the very heart of the second law and sought a way to express this law as a function of entropy and its relation to the universe, in a manner similar t o t h a t he used in the case of energy and the first law, when he stated that the energy of the universe is constant. Now, for reversible-path processes, whatever happens to
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the system is exactly compensated by something equal happening in the rest of the universe. Therefore, if a value is assigned t o the change in entropy for the system, there should be another one, equal and of opposite sign, assigned to the universe. Adding up both would give zero, and the entropy of the universe should remain constant. However, natural processes follow irreversible paths. Consequently they are not compensated. The entropy value must then change, and it could increase or decrease toward a limiting value. Since by convention Clausius assigned a positive value to entrow chanees in the universe. there should be an uooer limit it. ~ s ' h arrived e a t this cbnc~usionstarting fromihe second law, he had found his answer and proposed the following statement: The entropy of the uniuirse increases toward a maximum ualue. Conclusion The energy of the universe has the capacity t o flow as a result of its inhomogeneous distribution and entropy is a measure of this capacity.Entropy, therefore, is a measure of the capacity energy may haue to transform or flow from one place of the uniuerse to another. Epilogue . From Clausius's statements of the first and second law it would be possible to conclude that the universe could be a closed system, but modern cosmological theories indicate that this may not be necessarily true. Furthermore, the very existence of living beings could be a violation of the classical view of entropy because i t is not accessible to the concept of closed systems, heat engines operating between two temperatures and equilibrium reactions. However, classical thermodynamics was a true starting point for a new vision of reality as Einstein (10)acknowledged by saying "It is a theory of principles, the only one of universal content that, within the applicability of its basic concepts, will never be overthrown." Acknowledgment Careful discussion of the original manuscript by H. C. von Baever (Collcee of William and Marv. Williamsburr. ". VA). is graiefuiy ackLowledged. Detailed and valuable suggestions of the referee are very sincerely appreciated. Literature Clted 1. Ortoga y Geanot, J. El Tm", d. Nuestro Tiompa; Espesa Calpe Argentina: Buenas Aires. 1950: Chapter 2.p 26: (TheModern Theme: Harper: New Yark, 196L:English translation). 2. Clau~ius.R. Thhorie MOehoniove de lo Chalaur: Lanoir.. Ewene. Ed.:. Paris.. 1868:. ~ ~ e n edition eh revised by th; author. 3. Thomnon, W. Tram. ROY.SOC.Edinburgh 1849.16.Ml-547. 4. Carnot, S. Rhflerions aur lo Puirronce Motrice du Feu; Paris 18%. Reprinted by Lihrairie Scientifrqueet Technique, Blanchsrd:Paris. 1953. 5. Clapeyon, H. Taylm. Scientific Memoirs; Yo1 I, pp U7-376 (2837). 6. Mayer, R. Liebigs A n ~ I e n1842. XLII. 233-210. 7. Barror,G. M.J. Chem.Educ. 1988.65.123-125. 8. Gay Lusrae. L. J. Arcud, Mem. de Phys. 1807.1.180-204. 9.Csmpball.J. A. J.Cham. Educ. 1980.57.345. 10. Kloin, M. J. Science 1967,157,509-516. Quoted from Einstein, A."Autobiogrsphleai Notes''. In Schlipp. P. A. Albsn Einstein. Philosopher Scionfis(: 1919.
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