JANUARY, 1948
HEAT AND ENTROPY HAROLD H. STEINOUR Portland Cement Association, Chicago, Illinois
A FEW YEARS AGO, Brpinsted (3) criticized traditional thermodynamics for lack of clarity in its concepts of heat and work. In particular, he contended that "heat produced or consumed within the boundary of the system itself" was not adequately defined, that only heat communicated was consistently handled. To this charge, MacDougall (11) replied that "it is not allowed to speak of the quantity of heat in a system unless some additional properly defined significance is given to the term 'heat' " in thermodynamics. Thelatter view appears to be a widely accepted one. Various writers make explicit denial of the presence of heat within a body. For example, Luder (10) states in a recent article, " . . .there is no heat as such present in any thermodynamic system." The validity of such statements followssimply from the fa& that the authors defineheat not merely as energy that can be transferred by reason of a temperature difference, but as energy undergoing such transfer. However, the view that thermal energy should be considered heat only when it is passing through the boundary of a system is a di5cult one to maintain consistently. It is contrary to the concept of heat as commonly taught in general physics, and it is highly artificial since thermal energy leaving the boundary of a system must be closely related to that which remains behind and accounts for the temperature of the system. It is not surprising, therefore, that the restriction of the heat concept to transfer of thermal energy is frequently violated even by those who attempt to observe it. MacDougall admits that most chemists and physicists, when expounding thermodynamics, are inclined to use "misleading" expressions such as the remark that "work done against frictional forces is converted into heat." Since the mechanical equivalent of heat can be determined by work of this kind, it is easy to see why such a "misleadmg" expression comes to be used. The question that arises, then, is why the limitation of concept should be attempted. Probably there is value in using the symbol q only for thermal energy in process of transfer, but this need not prevent one from assuming that a body can and does contain energy of essentially the same kind. Apparently, the belief pre-
vails that in thermodynamics all mention of heat, however the term is used, is best reduced to a minimum in the interest of a clearer and more precise presentation (7). I n agreement with this view, it is said that the concept of heat resident in a body seems to have no utility for thermodynamics (11). Some authors apparently take the position that no simple basis exists for consistently relating energy changes and heat content.' For example, it is said: "One can pass heat into a body but it is no longer heat when it gets in. Take by way of illustration the passage of heat into a body that is expanding a t constant temperature. The heat has gone to do external work, and if it is converted, it is no longer heat. Heat enters a system but immediately becomes something that is not heat. In general, there is an increase in the internal energy of the system and also some external work is done" (9). On the other hand, the concept of a heat eontent has apparently been acceptable to some of the most prominent workers in thermodynamics, even if they did not make much use of it. Clausius ($), Maxwell (IS), and Sackur (16)all wrote of heat as a part of the energy content of bodies or systems. Lewis and Randall (8), though they advocate minimum reference to heat, use the term broadly, as when they say, "the energy of chaotic motion which we call heat." Some authors define heat content very definitely, as will appear later. The view that wi1l be developed in the present paper is that thermodynamics already deals with a sharply defined energy quantity that amounts, in all essentials, to a heat content. Thus, the writer takes the position that to limit heat to thermal energy in transfer is not the most logical course, and is inadvisable because it creates inconsistency with the rest of physics without adequate reason. Moreover, in his opinion, it makes entropy more difficult to understand because it limits artificially the extent to which this factor can be related to heat. A major object of the paper is to show that entropy, when viewed in terms of the broader conception of heat, can become a much more tangible concept. No originality is claimed for these views. 'Throughout this paper, "heat content" is used to denote atorage of heat, never as a name for enthalpy, H .
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JOURNAL OF CHEMICAL EDUCATION
TS REGARDED AS HEAT CONTENT
these being temperature, T , and entropy, S , respectively. Dod6 considers that definition of entropy as the capacity factor of the "heat energy" is more rigorous than definition in terms of the integral of dq/T. Also, like the present writer, he considem the former concept to be the more concrete or tangible one. Identification of heat with TS evidently requires revision of some of the statements that have been made regarding heat in connection with kinetic theory, but this should cause no difficulty; acceptance that heat is TS interferes in no way with the kinetic interpretations of heat transmission and energy partition. Evidently, , both the potential and kinetic energies of molecules can contribute to T S , but in classical thermodynamics the whole of TS functions simply as energy of thermal position and is potential energy in that sense. In the further discussion, TS will sometimes be referred to as potential energy in cases where the context makes the meaning clear. In what follows, the adequacy of T S as heat content E = A + TS, (1) will be investigated by considering how TS stands in reone of which, A, is energy which, in so far as it can he lation to heat-in-transfer, q, and whether its functionr~leased,is able to do work a t temperature T . The ing is ever out of character with the role of a heat conquantity A can, therefore, be regarded as the work con- tent. tent a t temperature T , and it is, in fact, often called ISOTHERMAL CHANGES work content. Thus, T S appears to be cast for the role With respect to changes a t constant temperature it of heat content, though in the past the term "heat seems rather obvious that TS does meet the requirecontent" has not been applied to it but, instead, has
If a system does have a heat content the rest of its energy should evidently constitute a work content. That is, since in matters of energy transfer the energy is identified either as heat or work, it is natural to suppose that if part of the internal energy exists as heat the rest is a work content. On this basis, energy received or released by a system as heat or work should presumably cause an equivalent change in the heat or work content, except as this might be obviated by rational internal transformations. Any quantity assumed to function as a heat content should apparently be judged primarily by its ability to fit into such a scheme. In addition, one would expect to find the heat content related to the temperature of the system in a manner consistent mith the fact that temperature difference is the cause of heat transfer. With these considerations in mind it is instructive to examine the familiar equation2 which divides internal energy, E, into two parts,
opposing general recognition of the appropriateness of the term vith respect to TS. However, if internal energy is to be divided simply into work and heat contents, the latter should apparently be the energy, T S , that is isothermally unavailable for work, the isothermal qualification being necessary to a precise distinction. These views on the significance of equation (1) may be compared with the following statement by Sackur (16) : ''The energy U of a body is the sum of two parts, viz., the heat energy and the so-called volume energy, the increment of which is the work done in an isothermal compression." It is evident that in this statement TS is called the heat energy of the body, i . e., its heat content. Mach (12) appears also to have regarded heat as TS. Trevor (19, 20) shows clearly that he once thought of heat as T S , though he later (21) conformed to the convention of defining heat simply in terms of heat transfer. Berthelot (2) and Leaf (5) call T S "thermal energy." Dod6 (4) calls it "heat energy," and, in one instance, "heat." Babor and Thiessen (1) state, significantly, that TS "is actually heat or some energy form that can only he manifested as heat." In conformity with ideas presented in earlier discussions by Zeuner (25), Mach (12), and others, Trevor (19) and later writers point out that, like other forms of energy, TS is divisible into an intensity and a capacity factor,
q, =
TAS
(2)
When, in such processes, heat leaves or enters a body, that body loses or gains an equal increment of TS. Moreover, by reason of the conservation of entropy, S , in reversible processes, that is the only way that the T S energy of the body can change isothermally under reversible conditions. In an irreversible isothermal process, any heat transferred still represents an equivalent transfer of T S , and vice versa. Moreover, an irreversible process, like a reversible one, leaves a body with a TS value that is determined solely by the final state of the body, as should be expected if T S is heat content. After an irreversible isothermal process there is, in system plus surroundmgs, more TS energy than before; but all the excess arises, through creation of new entropy, from a loss of work or work content. Once produced, this new TS energy can be transferred isothermally only as heat. In all respects, therefore, the functioning of T S a t constant temperature is precisely what should be expected if T S is heat content. It remains, however, to show more directly the fallacy of the argument, one example of which has already been quoted, that heat which enters and causes isothermal expansion of a body cannot remain entirely as heat because some of it is immediately "converted" to work. I n other versions, this argument is based on the reversa Symbals used in this paper conform to the usage of Lewis ible expansion of a perfect gas because the heat aband Randall (8)except in quotations and with respect to omssorbed is then exactly equal to the work done. The arsional subscripts the significance of which is evident from the gument is a non sequitur; the fact that absorption of context.
JANUARY, 1948
heat enables work to be done does not show that the heat has been transformed into the work. Owing to the equivalence of heat transfer and TS transfer, it is reasonable to assume that the heat absorbed is, throughout, the quantity T AS,, where AS, is the entropy transferred. The heat and TAS, undergo transfer a t the same time and in equal amounts, yet the heat accounts for the whole of the energy transfer through the given boundary. If this is a correct statement, as it seems to be, then there is really no escape from the conclusion that the heat is TAS,. The only way to avoid the couclusion is to assume that the entropy does not undergo transfer in the ordinary sense, i. e., as a persistent entity, but that entropy is destroyed within the system that the heat leaves and is immediately reproduced within the system that the heat enters. However, such a viewpoint is unnecessarily complicated, for the change in location of the entropy is not discontinuous. Moreover, a system undergoing reversible isothermal expansion loses in work content, A, by an amount exactly equal to the work done. This is always true, irrespective of whether the expanding body is a perfect gas. Surely, it is reasonable to consider that the energy lost as work content is the energy t.hat becomes work. CHANGES INVOLVING CHANGE IN TEMPERATURE
With respect to processes involving changes in temperature, the validity of regarding TS as heat content is indicated by consideration of the ideal heat engine. The net effect of a Carnot cycle, besides the work done, is the transfer of a quantity of entropy, S,, from the high to the low temperature level. Since S, must always have associated with it thermal energy equal to TS, it takes from the high temperature reservoir a t Tz thermal energy equal to T2S, and delivers a t the low temperature, TI, the energy TIS,. Thus, the heat that is converted to work is given directly by (TI - TI)&. In the reversed Carnot cycle, the work that is converted to heat is also given by (Tz - T1)S,. In both cases the heat transformed or produced equates directly to the change in value of TS though temperature change is responsible for the whole of the change. Furthermore, in any reversible process, whether cyclical or not, any change that gives rise to a quantity SdT must have a similar explanation except that a change in work content, A, can take the place of work itself. In any event no irrationality develops. If the process is irreversible, the only difference is the production of some TS from energy that was work or work content, the entropy being newly created. This increase in TS is logically regarded as heat content since it is energy that can thereafter be transferred isothermally only as heat. Moreover, if the same change in state were brought about reversibly, the energy needed for the increase in TS that in the irreversible process results from degradation of work or work content would actually have to be supplied as heat, q. Let there be no misunderstanding regarding the statement made above concerning "the heat that is
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converted to work" during the Carnot cycle. In conformity with what was said before, none of the heat should be regarded as undergoing conversion to work during the isothermal stage a t Tz. Though the net work is actually done during this isothermal expansion, it is done at the expense of a part of the work content, A, which is restored during the ensuing adiabatic expansion when S, gives up the energy (Tz - T1)S,. It is sometimes argued that the Foncept of a heat content is untenable because the heat absorbed in a change of state is not independent of the path. However, since TS undergoes well-defined interchanges with other energy which make its over-all change independent of the path, all such arguments fail when TS is regarded as heat content. If heat and work are interconvertible, there is no reason why heat content and work content should not be also. Actually, according to the ideas being presented, it is proper to consider that the heatto-work transformation always occurs hy conversion of heat content to work or work content (See above). HEAT IN TRANSFER
As already mentioned, the view that TS is heat content requires that y, or heat-in-transfer, be regarded also as an amount of TS, for a heat transfer can bemade without the heat ever leaving material systems. By the same reasoning, the opposing view, that there is no such thiig as a heat content, should not permit of a TS formulation for q. This fact emphasizes the extreme difficulty of maintaining the no-heatrcontent view, for temperature is generally regarded as the thermal level, potential, or intensity factor of heat, irrespective of whether the term "heat" is used broadly or is restricted to transfer of thermal energy. Indeed, the acceptance in thermodynamics of the fact that heat will not undergo uncompensated transfer except down a temperature gradient amounts to a recognition of temperature as the intensity factor for heat. However, if temperature is the potential or intensity factor of heat, the capacity factor can be only the entropy. To a given potential factor must correspond just one fundamental kind of capacity factor (also called "charge function" (6)). In electrical theory, for example, we do not expect nor do we find any fundamental capacity factor other than electrical charge, whether it is measured in coulombs or some other unit. True, there is a quantity called electrical capacity which has a unit of its own, but it is simply the electrical charge required to raise the potential of some object by unit amount-not a term for some special charge function. This is mentioned because heat capacity, cal. per degree, is given by some authors as the capacity factor of heat (along with temperature as the intensity factor). That this viewpoint is faulty should be evident from the fact that heat capacity is insufficiently general in application; it is applicable only when a temperature change is involved and not in processes like fusion and evaporation. Moreover, even as applied to a temperature change, heat capacity is merely the ability of a particular body to receive heat from an external source. Hence, it
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JOURNAL OF CHEMICAL EDUCATION
think of entropy as moving "just as we think of heat as moving." Indeed, it is perhaps best to thiik of the entropy as moving and delivering the heat. In this connection it is worth noting that we do not o r d i i i l y think of hydraulic energy as flowing; instead, we think of the water as flowing and delivering the energy. Zeuner (23) and Trevor (19) called entropy heatweight because, as already indicated, the thermal energy associated with a unit of entropy varies with temENTROPY perature in a way wholly analogous to the variation in As already shown, heat-in-transfer must be regarded the potential energy of a weight with change in altitude as TS energy if entropy is thought of as being trans- (the gravitational field being assumed uniform). Melferred from one system to another instead of being de- lor (14) has reported upon the heat-weight concept, apstroyed and reproduced. Conversely, identification of parently sympathetically. The term "heat-weight" emphasizes an apt analogy, heat with TS energy leads naturally to the thought t,hat entropy does undergo transfer, like mass and electric hut does not seem well suited to formal usage. Accharge. If entropy is regarded as randomness or dis- cordingly, the writer prefers to call entropy simply the order, this mobile quality should not seem strange. carrier of the thermal energy. Each entropy unit has Anything done t o reduce the randomness a t one place associated with it an amount of thermal energy, or heat, can simply he regarded as moving the disorder else- precisely equal numerically to the absolute temperawhere, often with production of addit,ional disorder ture. This thermal energy can be released for work only hut never with any decrease. This mobility is a prop- when the temperature lowers, and then only in quantity erty to he valued since it enables entropy to he re- equal numerically to the temperature change. Howgarded as indestructible. Thus, it increases the tangi- ever, if the energy lost by entropy units during a temperature drop is lost through conduction, it remains as bility of the entropy concept. Indeed, it seems strange that entropy transfer is not heat and associates itself with newly created entropy generally recognized as the common-sense view. En- units, each of which receives its full energy quota equal tropy (of system plus surroundings) remains constant in in magnitude to the existing absolute temperature. reversible processes and can only increase in irreversible When temperature rises, the energy associated with ones. This increase represents a creation of new en- each entropy unit must increase by an amount equal tropy under well-understood conditions which imply in to the temperature increase. This additional energy no way the concurrent destruction of any preexisting cannot be supplied as heat; work or consumption of entropy. Thus, the entropy existing a t any one time work content is necessary to raise the potential of any can be regarded as something that is thereafter con- form of energy (cf. Trevor (18, 19) and Br@nsted(3)). served, i. e., that always remains the same number of However, after the additional energy is acquired by the units. Furthermore, entropy can be traced along con- entropy unit, it is to he regarded as heat, just as it is tinuous paths in its redistributions. Other entities, at the end of the reversed Carnot cycle. such as gravitational mass or electrical charge, that HEAT CAPACITY exhibit such properties are not thought of as being d e It is instructive to consider the concept of heat capacstroyed and re-created but as moving about. There appears to be every reason to regard an entropy unit ity in terms of these views. Consider first the molar in the same way. When we do so, we find that we have heat capacity a t constant volume. A definite quantity something strikingly analogous-an indestructible en- of heat. is absorbed by a mole of a given substance for tity that moves under a potential difference, and that each degree of temperature rise. The absorption of always has associated with it potential energy propor- heat causes the temperature rise, but the heat itself is tional to its potential level, or temperature. The fact simply taken on as the energy associated with the addithat additional entropy is created in irreversible proces- tional entropy which the substance acquires as it rises ses is a complication, of course, but this is true no ma& in temperature. The energy supplied to each entropy ter what mental construct is adopted. As a matter of unit to provide for the increase in temperature must fact, the creation of entropy during heat conduction is come a t the expense of work content, A. These facts no dierent from the creation of entropy during free are expressed by the two well-known equations flow of electricity. The difference in effect arises simply because the entropy produced during heat conduction is identical in nature with the entity (entropy) whose loss in potential produced it, whereas in the electrical case and this is not so. Thus, heat remains undiminished in amount whereas electrical energy does not (12). ,. The view that entropy is transferred from one body to It is interesting to speculate on the signifiosnoe of this equaanother is by no means new. As early as 1904 Swin- tion at high temperatures, where it is indicated that A will hurne (17) argued that it is both logical and useful to become zero. cannot he a fundamental capacity factor, or charge function, in the same sense as is entropy, mass, or coulombs. Recognition of temperature as the potential factor of heat illumines the fact, already developed, that heat is not converted to work isothermally even in a noncyclical process. Potential energy can be released for work only when a drop in potential is involved.
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JANUARY, 1948
19
heat from these other forms of potential energy. For example, the form of the second law enunciated by Clausius, namely, that "heat cannot, of itself, pass from a colder to a hotter body" states no more than we should expect of energy that has temperature as its potential factor. It is only when we consider irreversible processes, and find that in them entropy is created, that the However, the free energy, F, appears in equation (6) really distinctive nature of heat becomes evident. As instead of work content, A . This is simply because the Dod6 (4)emphasizes, that which gives heat a unique work of expansion is done a t the expense of a part of position relative to gravitational and electrical energy A; hence, to obtain the reduction in A that supplies is the fact that under appropriate circumstances, work the additional energy to the entropy units, it is neces- can be transformed into heat without change in the sary to subtract the work of expansion. The change in thermal potential, i. e., by creation of entropy. Hence, those forms of the sec6nd law that indicate F at constant pressure gives precisely this difference. The foregoing discussion shows that heat capacity is what happens in an irreversible process are the only simply capacity for receiving heat from the outside ones that show, directly, the peculiar content of the and is not the amount by which the heat content is in- heat concept. This is not to say, however, that statecreased per degree (cf. Trevor (SO)). On the other ments of the second law like that by Clausius are inhand, heat utilized for isothermal change of phase does adequate, for one can utilize them in conjunction with represent the whole change in the heat content since a reversible cycle to define entropy as a fundamental no SdT is involved. An interesting sidelight is that so- quantity and can then proceed to irreversible processes called sensible heat, though capable of measurement by and show that in them entropy does increase. temperature rise, is, in one sense, just as latent as heat EASILY APPLIED CONCEPTS taken on during change of phase, since the increase in Adoption of the viewpoint that TS is heat and that the energy associated with each entropy unit, which registers as temperature rise, is supplied by the work the entropy, S , is the carrier of the heat involves no new content of the system. The absorptiod of heat can be derivations and no new mathematical treatment such regarded, however, as the cause of the readjustment of as that to which Br$nsted (3) was led by his dissatisinternal energy. This explanation of what happens to faction with traditional thermodynamics. Instead, the the internal energy of a system when heat is absorbed concepts represent imply a way of interpreting the should not seem strange if one accepts the view that classical quantities and equations. The gist of the viewheat is energy of thermal potential, for it is simply a point that has been presented is simply that it is logical matter of working out the implications of that concept. to regard all TS energy as heat, and that to do so is The result is that equations like (3) to (6) acquire a advantageous for the better understanding of heat, ensignificance beyond that of the bare mathematical rela- tropy, and various thermodynamic relationships. In developing the discussion, i t was convenient to tions. use the expression "heat content" when referring to the TS energy within a system. However, in view of the Carnot's principle that the amount of work that can wide misuse of "heat content" as a designation for be produced from a given amount of heat is the same enthalpy, H, general use of the same expression for T S for all reversible engines working between the same is not recommended. The single word "heat" is better. temperatures follows naturally from the conception of Of course, "isothermally unavailable energy" is now heat here presented. The number of entropy units the only designation which when used alone can be retaken into the engine a t the higher temperature, Ta, lied upon to mean TS to everyone. It is, in reality, a must all be released a t the lower temperature, T I , if definition, and to anyone who adopts the point of view the working substance is to return to its initial condi- that has been presented in this paper it will mean heat. tion. Since each entropy unit has associated with it LITERATURE CITED originally heat equal to T2,and must leave with heat (1) BABOE;J.A., AND G. W. T ~ E S S E N"HOW , to Solve Probequal to T I ,the efficiency of any ideal engine that takes lems in Physical Chemistry," Thomas Y. Crawell Co., on heat only a t T2 and discharges heat only a t T I is New York, 1944, p. 159: ohviouslv (Tz - T -I,,) / T D.. Bull. w e . chim. 141. 35. 257 (1924). (2) BERTHELOT. - ~ . Only the properties of heat that are involved in ?4/ chim. diat (paris). 31.'7-28 11944). ~ &B. , -M.:~drn.'seruice~ ' ~, reversible processes are brought out by this reasoning. (5) LEAF,'B.,>. Chem. ~ h y s . 12,89-98' , (1944j. They are properties that are shown also by other forms (6) LEWIS,G. N., AND M . RANDALL, "Thermodynamics," 1st of potential energy, when the charge function is coned., McGraw-H111Book Co., New York, 1923. ibid., p. 53. (7) LEWIS,G. N., AND M . RANDALL, served-a fact pointed out long ago by Mach (IS). (8) LEWIS,G. N., AND M. RANDALL, ibid., p. 126. Hence, the various ways of stating the second law of (9) LEWIS,W. McC., "A System of Physical Chemistry," thermodynamics that merely provide such an assumpLonemans. Green and Co.. London. 1916. Vol. 11. D. 53. tion as will establish Carnot's principle do not them- (10) LUDE; W. F., J. CHEM.ED&., 23, 5i-9, il0-5 (1946). selves disclose the peculiar property that distinguishes (11) M ~ c D o n a ~ m F.,H., J. Phys. Chem., 44,715-15 (1940). Similar equations apply to the absorption of heat a t constant pressure, namely:
20 (12) MACH,ERNST,"Die Principien der Warmelehre," Johan Ambrosius Barth, Leipzig, 1896, pp. 3-6. (13) MKXWELL, C., "Theory of Heat," D. Appleton and Co., New York, 1872, p. 6. (14) MELWR,J. W., "A Comprehensive Treatise on Inorganic and Theoretical Chemistry," Longmans, Green and Co., London, 1922 (1941 impression), Vol. I, p. 724. (15) SACKWE,0.(Translation by G. E. Gibson), "Thermochemistry and Thermodynamics," Maemillan and Co., London, 1917, p.19. (16) SACK^, O., ibid., p. 428. (17) SWINBURNE, J., "Entropy," Archibald Constable and Co., Westminster, 1904, p. 94.
JOURNAL OF CHEMICAL EDUCATION (18) TREVOR,J. E.,J. Phus. Chem., 3, 3 3 W 8 (1899). J. E., ibid., 4,514-28 (1900). (19) TREVOR, (20) TREVOR,J. E., ibid., 4, 529-32 (1900).
J. E., ''The General Theory of Thermodynamics," (21) TREVOR, Ginn and Co., Boston, 1927. (22) TUNELL,G., J. P h p C h m . , 36,1746 (1932). (23) ZEUNER,G., "Grundziige der meohnischen Wdrmetheorie," Leipeig, 1865; seen only as Zeuner's "Teohnicd Thermodynamics," translated from 5th German ed. by J. F. KLEIN,D. Van Nostrand Co., New York, 1907, Vol. I, pp. 44-5, 55-7.
