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The Uniqueness of Clausius' Integrating Factor

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The Uniqueness of Clausius' Integrating Factor The uniqueness of Clausius’ integrating factor ͒ Volker C. Weissa School of Engineering and Science, International University Bremen, P. O. Box 750561, 28725 Bremen, Germany ͑Received 23 August 2005; accepted 3 March 2006͒ For a closed system that contains an arbitrary pure substance, which can exchange energy as heat and as expansion/compression work, but no particles, with its surroundings, the inexact differential of the reversibly exchanged heat is a differential in two variables. This inexact differential can be turned into an exact one by an integrating factor that, in general, depends on both variables. We identify the general form of the integrating factor as the reciprocal temperature ͑Clausius’ well-known 1/T͒, which is guaranteed to be a valid integrating factor by the second law of thermodynamics, multiplied by an arbitrary function of the implicit adiabat equation ␰͑T,V͒ =constant or ␰͑T, P͒=constant. In general, we cannot expect that two different equations of state ͑corresponding to two different substances͒ predict identical equations for the adiabats. The requirement of having a universal integrating factor thus eliminates the volume-dependent or pressure-dependent integrating factors and leaves only a function of temperature alone: Clausius’ integrating factor 1/T. The existence of other integrating factors is rarely mentioned in textbooks; instead, the integrating factor 1/T is usually taken for granted relying on the second law or, occasionally, one finds it “derived” incorrectly from the first law of thermodynamics alone. © 2006 American Association of Physics Teachers. ͓DOI: 10.1119/1.2190685͔ I. INTRODUCTION roundings, the infinitesimal amount of work can be ex- ␦ pressed as wrev=−PdV, where P denotes the pressure and V A central theme in the history of thermodynamics is the the volume of the system. Thus, we obtain quest for state functions that describe the changes of all mea- dU = ␦q + ␦w = ␦q − PdV. ͑2͒ surable equilibrium quantities in terms of a set of thermody- rev rev rev namic state variables, that is, a set of variables that uniquely ␦ ␦ Because wrev is given by purely mechanical variables, qrev determine a thermodynamic state. Equilibrium thermody- is the only quantity left to provide us with the sought-for namics is based on two laws, each of which identifies such a second state function. As state functions have exact differen- state function. The principle of energy conservation implies tials, an integrating factor is needed at this point. An inte- that any change of the internal energy U of a closed system grating factor f is a function that accomplishes the goal of is caused by a transfer of heat q into or out of that system or ␦ turning an inexact differential, such as qrev, into an exact by work w being done on or by the system.1 In differential ␦ one. This new exact differential f qrev=dSgen, where the sub- form, the first law of thermodynamics thus reads script “gen” stands for “generalized” in a sense that will become clear later, is now associated with a state function dU = ␦q + ␦w. ͑1͒ Sgen. The second law of thermodynamics can be divided into two important parts, which are summarized by the following The differential of the state function U is exact, while the statements:4 differentials of heat and work are not ͑see below͒; the latter fact is indicated by using the symbol ␦ instead of d for these • ͑2a͒ The reciprocal of the absolute temperature 1/T is an differentials. Quantities that have exact differentials are state integrating factor for the differential of the reversibly ex- ␦ ͑ functions, which means that the difference of the values of changed heat qrev for all thermodynamic systems Car- such a quantity at two different state points depends only on not’s theorem͒, which implies the existence of a new state the position of these points, but not on the path that is taken function, the entropy S ͑with the exact differential dS −1␦ ͒ to get from the initial point to the final one. In contrast, for =T qrev . quantities with inexact differentials, the difference depends • ͑2b͒ In any irreversible adiabatic process, the entropy on the initial and final state points and the choice of the path increases—as implied by Clausius’ inequality dSജ␦q/T, connecting them.2 where the equality holds only for reversible processes. The introduction of the state function U separates possible processes ͑those that conserve energy͒ from impossible pro- In the following, we will illustrate that 1/T is an integrat- cesses ͑those that violate the principle of energy conserva- ing factor for the ideal gas and then ask if, for this fluid, there ͒ ␦ tion . This separation is not sufficient because not all pro- are other integrating factors for qrev. It can be shown that, if cesses that can happen will happen; some of them are ͑much͒ there is one integrating factor, there are infinitely many more likely to occur than others. Hence, a second state func- more,5 although most textbooks omit this point. We will tion is needed to separate the likely ͑spontaneously occur- identify these alternative integrating factors, but we will not ͒ ͑ ring processes from the unlikely ones. If we focus on revers- be concerned about whether the state functions Sgen gener- ible processes3 and assume that the system can only alized entropies͒ based on these alternative integrating fac- exchange heat and expansion/compression work with its sur- tors really separate probable from improbable processes, that 699 Am. J. Phys. 74 ͑8͒, August 2006 http://aapt.org/ajp © 2006 American Association of Physics Teachers 699 closed cycle is zero, as required for a path-independent change. A more formal way of deriving an integrating factor that does not require using the Carnot cycle considers the first law directly.7–9 From Eq. ͑2͒, the differential of the revers- ibly exchanged heat is obtained as ␦ ͑ ͒ qrev = dU + PdV. 7 We regard U as a function of T and V and write its exact differential as Uץ Uץ dU = ͩ ͪ dT + ͩ ͪ dV. ͑8͒ ץ ץ T V V T Substituting Eq. ͑8͒ into Eq. ͑7͒, we obtain Uץ Fig. 1. Representation of the Carnot cycle in the pressure-volume plane. An ͑ ͒ ␦q = C dT + ͫͩ ͪ + PͬdV, ͑9͒ ץ isothermal expansion 1 at temperature Th takes the system from state A to rev V state B; the system then expands adiabatically ͑2͒ to point C and the tem- V T ͑ ͒ -T͒ has been used. The difץ/Uץ͑= perature drops to Tc. At this temperature, a compression 3 to state point D where the definition C occurs isothermally. An adiabatic compression ͑4͒ completes the cycle and V V ferential in Eq. ͑9͒ is exact if ͑and only if͒: takes the system back to the starting point A at the higher temperature Th. Uץ Pͬ + ͪ ͩͫץ Vץ Cץ V = T , ͑10͒ Tץ Vץ is, whether an analogous law of increasing entropy Sgen holds for them. Instead, we will concentrate on the question of the ␦ which is not the case for the monatomic ideal gas because uniqueness of the integrating factor that turns qrev into an exact differential. Nk/V0. Based on Carnot’s observation that the efficiency of a re- If we now multiply Eq. ͑9͒ by a function f͑T͒, such that Uץ -versibly working heat engine depends only on the tempera f͑T͒ͩ ͪ + f͑T͒Pͬͫץ tures of the heat source and the heat sink, Clausius assumed Vץ ͔ f͑T͒C͓ץ that the integrating factor is a function of temperature T only V T ͑ ͒ ␦ = 11 Tץ Vץ and gave 1/T as the solution to the problem of turning qrev into an exact differential.6 Many textbooks2 use the Carnot ␦ cycle to illustrate that ␦q /T is an exact differential for an is satisfied, qrev is turned into an exact differential by the rev ͑ ͒ ͑ ͒ ideal gas: Starting from point A in Fig. 1, the working sub- integrating factor f T . For the monatomic ideal gas, Eq. 11 stance, a monatomic ideal gas, expands isothermally along leads to an ordinary differential equation for f͑T͒: Pץ path 1 at temperature Th to state point B. From there, an adiabatic expansion takes it to point C at a lower temperature 0=fЈ͑T͒P + f͑T͒ͩ ͪ , ͑12͒ Tץ Tc at which the system is, subsequently, compressed isother- V mally to point D. An adiabatic compression returns the sys- where the prime denotes the derivative of f with respect to ͒ ץ ץ͑ .tem to starting point A its argument. Because P/ T V = P/T for the ideal gas, the The equations of state for the pressure and the internal solution of the differential equation ͑12͒, which can be writ- energy of a monatomic ideal gas are ten as 0=TfЈ͑T͒+ f͑T͒,is NkT ͑ ͒ −1 ͑ ͒ P = , ͑3͒ f T = c0T , 13 V where c0 is an arbitrary constant. No other function of T ␦ 3 alone can turn qrev into an exact differential; the integrating U = NkT, ͑4͒ factor 1/T is therefore unique among all f͑T͒. ͑The value of 2 the constant c0 is irrelevant as long as c0 0, so we may ͒ where N is the number of particles and k is Boltzmann’s choose c0 =1 for convenience. The new state function was ␦ constant. During the isothermal expansion 1, the heat trans- termed entropy by Clausius, its differential is dS= qrev/T. ͑ ͒ −1 ferred to the system is The question remains as to whether f T =c0T is the only ␦ 7 q = NkT ln͑V /V ͒. ͑5͒ integrating factor for qrev.
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