The uniqueness of Clausius’ integrating factor ͒ Volker C. Weissa School of 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 and as expansion/compression , 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 ͑Clausius’ well-known 1/T͒, which is guaranteed to be a valid integrating factor by the second law of , multiplied by an arbitrary 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 -dependent or -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 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 . 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 . 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 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 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 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 , 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 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 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 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. Vemulapalli has suggested divid- 1 h B A ␦ ing qrev by the pressure instead of the temperature to illus- There is no heat exchange during the adiabatic processes 2 trate that there is something special about the integrating ␦ and 4, and we have q2 =q4 =0. Upon isothermal compression factor 1/T, because multiplication of qrev by 1/ P does not 3, the heat loss of the system is yield an exact differential for an ideal gas. Mathematically, q = NkT ln͑V /V ͒. ͑6͒ however, it can be shown that Pfaff differential forms in two 3 c D C variables, such as the right-hand side of Eq. ͑9͒, have infi- From the adiabat equation, which for the monatomic ideal nitely many integrating factors,5 so that 1/T appears to be gas is TV2/3=constant, we can infer that the four volumes are just one of many. In Sec. II, we will identify these more related by VB /VA =VC /VD. Using this relation, we obtain general integrating factors that, in addition to being functions ͑ ͒ q1 /Th +q3 /Tc =0, indicating that the sum integral over a of the temperature, may also depend on volume or on pres-

700 Am. J. Phys., Vol. 74, No. 8, August 2006 Volker C. Weiss 700 sure, respectively. The derivation will show that P−2/5, unlike dT b͑T,V͒ P = , ͑19͒ the factor 1/ suggested in Ref. 7, is a valid integrating dV a͑T,V͒ factor for the ideal gas that depends only on the pressure. Note that we assume 1/T to be an integrating factor by which represents the defining equation of the desired char- virtue of the second law. Although 1/T is valid for the ideal acteristics in terms of the original variables T and V. For Eq. gas, its validity cannot be extended to general thermody- ͑15b͒, the characteristics are given by: namic systems without referring to the second law.10 At- Uץ tempts to do so are flawed, but are found, for example, in P + ͩ ͪ Vץ Refs. 8 and 9. dT =− T . ͑20͒ dV CV These characteristics are nothing but the adiabats defined II. THE GENERALIZED INTEGRATING FACTORS ␦ ͑ ͒ by qrev=0; if we use this relation in Eq. 9 , we obtain Eq. A. Volume-dependent integrating factors ͑20͒. The reversible adiabats that we are considering here are also referred to as isentropes ͑lines of constant entropy͒ be- We first consider an integrating factor of the form f͑T,V͒. cause, in a reversible process, there is no change in entropy if The condition for an exact differential after multiplying both no heat is exchanged. sides of Eq. ͑9͒ by f͑T,V͒ becomes: The ordinary differential equation ͑with respect to ␩͒,

U B͑T,V͒f␩ = cf, ͑21͒ץ ץ ͔ f͑T,V͒C͓ץ V = ͫf͑T,V͒ͩ ͪ + f͑T,V͒Pͬ, ͑14͒ ץ ץ ץ V T V T is simple to solve if we choose ␩ to be a function of just one of the original variables, T or V. Either is possible, but let us which, after differentiation, can be rearranged to give the choose ␩=T to retain the empirically valid integrating factor following partial differential equation: 1/T in a transparent way. The choice ␩=V would lead to the same result. Furthermore, once we know that 1/T is a valid fץ Uץ fץ C ͩ ͪ − ͫP + ͩ ͪ ͬͩ ͪ integrating factor, it can be shown that the general integrat- ץ ץ ץ V V T V T T V ing factor f͑T,V͒ may be factorized according to f͑T,V͒ 4,5 T−1g͑T,V͒, where g is a function that we are about to= ץ ͒ ץ ץ͑ץ ץ P U/ V T CV = ͫͩ ͪ + ͩ ͪ − ͩ ͪ ͬf , ͑15a͒ determine. Because Eq. ͑21͒ is a first-order partial differen- Vץ Tץ Tץ V V T tial equation, we expect only one arbitrary function to occur ␩ ␩ in the general solution. Thus, with V =0 and T =1,wear- P rive atץ =ͩ ͪ f , ͑15b͒ Tץ V bf␩ = cf ͑22͒ 2 2 ,V͒ has or, for Eq. ͑15b͒ץTץ/U ץ͑=T͒ץVץ/U ץ͑ where Schwarz’s theorem been used because U is a state function as we know from the Pץ fץ Uץ -first law of thermodynamics. Equation ͑15b͒ is a homoge − ͫP + ͩ ͪ ͬ = ͩ ͪ f . ͑23͒ ץ ץ ץ neous first-order partial differential equation of the general V T T T V form Equation ͑23͒ simplifies to a well-known ordinary differen- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ a T,V fV + b T,V fT = c T,V f , 16 tial equation if Pץ Uץ where f is a shorthand notation for the partial derivative of x ͩ ͪ = Tͩ ͪ − P, ͑24͒ ץ ץ ͒ ͑ f x with respect to the variable . One way of solving Eq. 16 V T T V is the method of characteristics.11 The idea of this method is to identify special paths in the ͑T,V͒ plane, the characteris- which is the requirement for 1/T to be an integrating factor.8 tics, along which the partial differential equation reduces to If this condition is satisfied, we obtain an ordinary differential equation, which can then be solved fץ relatively easily. This simplification is achieved by a trans- − T = f , ͑25͒ Tץ formation of variables. We introduce ␰͑T,V͒ and ␩͑T,V͒ and ͑ ͒ use the to express Eq. 16 as −1 the solution to which we have seen already: f =c1T . This result comes as no surprise because we have assumed that af␰␰ + af␩␩ + bf␰␰ + bf␩␩ = cf. ͑17͒ V V T T Eq. ͑24͒ is satisfied, but it goes beyond the result quoted in ␰ ␰ ␩ ␩ ͑ ͒ Eq. ͑13͒ because c is actually an arbitrary function of the With A=a V +b T and B=a V +b T, Eq. 17 becomes 1 constant ␰͑T,V͒, and, therefore, a particular combination of Af␰ + Bf␩ = cf. ͑18͒ the variables T and V as defined by the characteristics, the adiabats, in Eq. ͑20͒. The integrating factor can thus be writ- The condition for Eq. ͑18͒ to reduce to an ordinary differen- ten as f =T−1g͑␰͑T,V͒͒, where g can be any differentiable tial equation with ␰ constant and ␩ as the only variable with function. ␰ ␰ respect to which derivatives are taken, is A=a V +b T =0. On For the monatomic ideal gas, the adiabats are defined by ␰͑ ͒ ␰ ␰ 2/3 2 the other hand, if T,V is constant, we have d = VdV the condition TV =constant, and we obtain the generalized ␰ −1 ͑ 2/3͒ ␩ ͑ ͒ + TdT=0, and therefore, integrating factor f =T g TV .Ifweset =V in Eq. 21 ,

701 Am. J. Phys., Vol. 74, No. 8, August 2006 Volker C. Weiss 701 2/3 ͑ 2/3͒ ͒ ͑ we would find f =V g1 TV , an equivalent result because at least, g=constant . This argument, however, will not ap- both g and g1 are arbitrary functions, so we may choose ply to the pressure-dependent integrating factors of Sec. ͑ ͒ ͑ ͒ ͒ g1 x =g x /x to obtain identical expressions. II B. For volume-dependent integrating factors, it is, never- Now that we have identified the general integrating factors theless, instructive to ignore this additional requirement for for the ideal gas, we need to determine if they hold for other the time being and to continue by defining: systems as well. For convenience, we consider the simplest dS = T−1g͑TV2/3͒␦q . ͑31͒ system that shows deviations from ideal gas behavior: The gen rev hard sphere fluid. The interaction between two particles of The application of the Legendre transformation to replace this system is infinite if the two spheres overlap, that is, if the Sgen by the variable T results in the following generalized separation of their centers, r, is less than the diameter ␴ of : Ͼ␴ the spheres; the interaction is equal to zero if r . Due to T the lack of a natural temperature scale for this system ͑nei- A = U − S . ͑32͒ gen ͑ 2/3͒ gen ther zero nor infinity is suitable for defining a temperature g TV ͒ ␴ scale , the virial coefficients Bn are determined only by . The condition that both the internal energy U and the gener- For example, the second virial coefficient is given by B2 alized Helmholtz free energy A have to be state functions ␲␴3 gen =2 /3. and thus have exact differentials yields two Maxwell re- The pressure of the hard sphere fluid will be higher than lations. Introducing the notation x=TV2/3gЈ/g, where the that of an ideal gas at the same number density ␳=N/V at 12 prime indicates differentiation with respect to the argu- any given temperature due to the excluded volume. The ment, we write internal energy is the same as for the ideal gas because the Pץ T/g͒͑ץ spheres cannot overlap and there is no interaction if they do ͩ ͪ =−ͩ ͪ , ͑33͒ ץ ץ not overlap. For low densities, the pressure can be repre- V S S V sented by a virial series:13 2xTS/͑3Vg͒ − P͔͓ץ S͑1−x͒/g͔͓ץ ϱ − ͩ ͪ = ͩ ͪ , ץ ץ ͒ ͑ ␳ ͑ ͒␳n P = kT + kT͚ Bn T . 26 V T T V n=2 ͑34͒ For the hard sphere fluid, the virial coefficients B do not n which for g=1, implying x=0, reduce to the well-known depend on temperature, and we have in standard thermodynamics. ϱ For the differential of the generalized entropy, which is ␳ ␳n ͑ ͒ ͑ ͒ ͑ ͒ P = kT + kT͚ Bn , 27 obtained by inserting Eq. 9 into Eq. 31 , n=2 Uץ U gץ g dS = ͩ ͪ dT + ͫͩ ͪ + PͬdV, ͑35͒ ץ ץ gen 3 T T V T V T U = NkT. ͑28͒ 2 to be exact, we obtain the condition Uץ P 2 x Tץ U Tץ In this case, the adiabats can be calculated explicitly from ͩ ͪ =−P + ͩ ͪ − ͩ ͪ , ͑36͒ ץ ץ ץ :Eq. ͑20͒ V T 1−x T V 3 1−x V T V ϱ 2 1 Nn−1 which, by construction, is satisfied by the monatomic ideal 2/3 ͩ ͚ ͪ ͑ ͒ TV exp − n−1 Bn = constant. 29 gas, but not necessarily by other fluids. For x=0 ͑g 3 n=2 n −1V =constant͒, Eq. ͑36͒ reduces to Eq. ͑24͒. To check the Because Eq. ͑24͒ is satisfied for the hard sphere fluid ͑just as validity of Eq. ͑36͒ for substances other than the mon- for the ideal gas, implying the validity of 1/T as an integrat- atomic ideal gas, we need the equations of state, P͑T,V͒ ing factor for both of them͒, the generalized integrating fac- and U͑T,V͒, of those fluids. The number of particles N is tor for the hard sphere fluid can be written as: assumed to remain constant. If we substitute Eqs. ͑27͒ and ϱ ͑28͒ for the hard sphere fluid and match each of the 2 1 Nn−1 f = T−1hͫTV2/3 expͩ− ͚ B ͪͬ, ͑30͒ density, keeping in mind that dBn /dT=0, we obtain: 3 n −1Vn−1 n n=2 1 B = B , ͑37͒ where h is an arbitrary differentiable function. The only way n 1−x n that the function f can be the same as for the ideal gas, f =T−1g͑TV2/3͒,isifh=g=constant due to the different ways which can be satisfied only if x=0, which implies that g in which g and h depend on T and V. The requirement of =constant. We conclude that, of all the mathematically having a universally applicable integrating factor leaves possible integrating factors, only Clausius’ choice of 1/T only T−1. works successfully for both the ideal gas and the hard To illustrate the consequences of this requirement explic- sphere fluid. itly, we define a generalized entropy based on the integrating factor of the monatomic ideal gas, f =T−1g͑TV2/3͒. At this point, the requirement that the entropy based on any integrat- B. Pressure-dependent integrating factors ing factor should be an extensive quantity ͑a of the first degree͒ rules out all nonintensive inte- In the quest for pressure-dependent integrating factors, it grating factors, that is, all but the one obtained from g=1 ͑or, is more appropriate to start from the exact differential of the

702 Am. J. Phys., Vol. 74, No. 8, August 2006 Volker C. Weiss 702 rather than the internal energy. Using the definition in the context of finding valid integrating factors without of the enthalpy, H=U+ PV, an infinitesimal amount of re- referring to the second law of thermodynamics, there is noth- versibly exchanged heat can be expressed as ing special about the integrating factor 1/T. The analogous integrating factor f͑T, P͒ for the hard ␦q = dH − VdP, rev sphere fluid is much more difficult to obtain in explicit form. It would involve an arbitrary function h of T and P in a Hץ =C dT + ͫͩ ͪ − VͬdP, ͑38͒ combination that is determined by the adiabats so as to sat- ץ P P T isfy Eq. ͑39͒, that is, h͑␰͑T, P͒͒, where ␰=constant is an -T͒ implicit representation of the adiabats. This particular comץ/Hץ͑= where the exact differential of H͑T, P͒ and C P P bination will be different from the ideal gas combination have been used. Adiabats ͑␦q =0͒ are defined by rev TP−2/5. A universally applicable integrating factor is obtained H only if we require g=h=constant, thus eliminating theץ ͩ ͪ − V pressure-dependent integrating factors and leaving only 1/T. Pץ dT =− T . ͑39͒ Thermodynamically, the properties of a pure substance are dP CP entirely determined by just two state variables if we keep the number of particles constant. In addition to T and V or T and We multiply both sides of Eq. ͑38͒ by a function f͑T, P͒ and P, a complete description of the system is also possible in ͑f␦q ͒ require the product rev to be an exact differential; the terms of P and V. If we set up a partial differential equation ͑ ͒ condition for f T, P is to determine the integrating factor f͑V, P͒ in a way that is analogous to the treatments of f͑T,V͒ and f͑T, P͒,wefind ץ ץ ͒ ͑ץ fCP H = ͭfͫͩ ͪ − Vͬͮ. ͑40͒ 2/3 ͑ 2/3 2/5͒ −2/5 ͑ 2/3 2/5͒ , f =V g V P or, equivalently, f = P g1 V P ץ ץ ץ P T P T where g and g1 are arbitrary differentiable functions for the If we rearrange the terms and use ideal gas. Both forms involve the adiabat equation for the monatomic ideal gas, so we cannot expect these integrating 2Hץ 2Hץ = , ͑41͒ factors to be valid for other systems. Imposing statement ͑2a͒ P of the second law on the partial differential equation for theץTץ TץPץ integrating factors in terms of P and V, which amounts to we obtain a partial differential equation for f͑T, P͒: insisting on 1/T being a valid solution, leads to the require- V ment thatץ fץ Hץ fץ C ͩ ͪ − ͫͩ ͪ − Vͬͩ ͪ =−ͩ ͪ f . ͑42͒ P ␣2TVץ Vץ Tץ Tץ Pץ Pץ P T T P P C − C = Tͩ ͪ ͩ ͪ = , ͑46͒ ␬ ץ ץ P V The characteristics of Eq. ͑42͒ are the adiabats given by Eq. T P T V T ͑ ͒ ͑ ͒ 39 . Each characteristic defines a path in the T, P plane, which is a well-known result derived in a rather unconven- which is represented implicitly by ␰͑T, P͒=constant. We ex- tional way. Here ␣ is the thermal-expansion coefficient and ␰͑ ͒ ␬ ploit the constancy of T, P and, for convenience and to T denotes the isothermal . ensure the validity of statement ͑2a͒ of the second law, that is, 1/T is one of the integrating factors, we set ␩=T as III. CONCLUSIONS the new variable. Equation ͑42͒ then simplifies to Based on the method of characteristics for solving partial Vץ fץ Hץ − ͫͩ ͪ − Vͬͩ ͪ =−ͩ ͪ f . ͑43͒ differential equations, we have identified integrating factors ץ ץ ץ P T T P T P that turn the inexact differential of the reversibly exchanged heat into an exact one. If the integrating factor is restricted to All thermodynamically acceptable equations of state satisfy be a function of temperature alone, Clausius’ integrating fac- V tor f =cT−1, with c being a nonzero constant, is obtained, asץ Hץ ͩ ͪ = V − Tͩ ͪ , ͑44͒ implied by the second law of thermodynamics. If we allow f Tץ Pץ T P to depend on two variables, temperature and volume or tem- which is ensured by statement ͑2a͒ of the second law of perature and pressure, then there are infinitely many more thermodynamics. Hence, we arrive, once more, at integrating factors, a fact that is known from the properties of a differential in two variables. If we assume the validity of fץ − T = f , ͑45͒ the second law, thereby ensuring that 1/T is an integrating T factor, these more general integrating factors are formed byץ −1 −1 multiplying T by an arbitrary differentiable function of the with the solution f =c2T , where c2 is now a function of T respective implicit adiabat equation ␰͑T,V͒=constant or ␰͑ ͒ and P as dictated by the adiabat equation T, P =constant. ␰͑T, P͒=constant to give f =T−1g͑␰͒. Since the functional re- −1 ͑␰͑ ͒͒ The complete solution thus reads f =T g T, P , where g lationships implied by the adiabat equations will, in general, is an arbitrary differentiable function. be different for different substances, the only way of having −1 ͑ −2/5͒ For the monatomic ideal gas, we obtain f =T g TP . a universally applicable integrating factor that is valid for all Choosing ␩= P in the process of solving the partial differen- substances is to require g to be a constant. Thus, Clausius’ tial equation ͑42͒ using the method of characteristics yields result f =1/T is recovered ͑choosing g=1 is a convention͒.It −2/5 ͑ −2/5͒ 7 f = P g1 TP , an equivalent result. If Vemulapalli had is remarkable that there is no need to refer to statement ͑2b͒ suggested division by P2/5 instead of just P, he would have of the second law of thermodynamics to eliminate all the −2/5 ͑ ͑ ͒ ͒ found that P obtained by setting g y =y or g1 =1 is a alternative integrating factors. ͑In general, it is necessary to valid integrating factor for the monatomic ideal gas and that refer to statement ͑2a͒ to ensure the existence of an integrat-

703 Am. J. Phys., Vol. 74, No. 8, August 2006 Volker C. Weiss 703 ing factor.͒ Clausius probably never considered more general must have the same efficiency, which depends only on the integrating factors because he knew about Carnot’s result between which the Carnot engine operates.2 If that the efficiency of a heat engine depends only on the tem- any two reversibly working engines had different efficien- peratures between which the engine operates. cies, we could convert heat completely into work by cou- From the equations of state alone, Clausius’ integrating pling two engines, using the more efficient one as a heat factor 1/T can be shown to work for all imperfect gases that engine and the less efficient one as a refrigerator such that have a regular virial expansion ͑“regular” means that only there is no net discharge of heat to the reservoir at the lower integer powers of the density occur in the expansion and that temperature, thereby violating Kelvin’s version of the second the virial coefficients exist͒. The question remains whether it law. The validity of the second law thus guarantees the ex- is a valid integrating factor for all substances because the istence of the integrating factor 1/T; it is therefore, as Wi- equations of state needed to evaluate Eqs. ͑24͒, ͑44͒, and dom phrased it, a “law of nature, not of .”24 ͑46͒ for an arbitrary substance are not available, in general. What ensures the general validity of 1/T as an integrating factor is the second law of thermodynamics in one of its ACKNOWLEDGMENTS standard versions according to Kelvin, Clausius, or 4,5,14,15 Carathéodory. The author is indebted to Marcel Oliver, John Wheeler, Although mathematics alone ensures the existence of inte- and Ben Widom for helpful discussions. The financial sup- grating factors for differentials in two variables, there is no port of the German Research Foundation ͑Deutsche For- guarantee of the integrability of differentials in three ͑or schungsgemeinschaft͒ is gratefully acknowledged. more͒ variables. The second law ensures that 1/T is valid even in these cases by eliminating all potential equations of a͒ state that violate the second law by failing to have 1/T as an Present address: Eduard-Zintl-Institut für Anorganische und Physikalische integrating factor for the differential of the reversibly ex- Chemie, Technische Universität Darmstadt, Petersenstrasse 20, 64287 Darmstadt, Germany; electronic address: [email protected] changed heat. In particular, Carathéodory’s work focused on darmstadt.de identifying the minimal extra-mathematical assumptions that 1 The transfers of heat and of work are two ways in which the system can would imply the second law of thermodynamics and, exchange energy with its surroundings. “Heat” denotes the transfer of thereby, ensure the existence of an integrating factor for all energy due to a ͑possibly infinitesimally small͒ temperature difference, thermodynamic systems.4,5,16–22 Carathéodory’s principle, in while “work” is energy exchanged due to a ͑possibly infinitesimally ͒ combination with the theorem that bears his name, states small pressure difference between the system and its surroundings. The system itself possesses neither heat nor work, just energy. that, “to ensure the existence of an integrating factor for a 2 See, for example, Peter Atkins and Julio de Paula, Atkins’ Physical Chem- differential in more than two variables, there have to be istry, 7th Ed. ͑Oxford University Press, Oxford, 2002͒, pp. 73–120. states in the immediate vicinity of any given equilibrium 3 The term “reversible” denotes a special way of conducting the transfer of state that cannot be reached from this state by adiabatic pro- heat or work in a given process. It ensures that, at all times, the differ- cesses.” One can then show that the integrating factor may ences in temperature and pressure between the system and its surround- be chosen to be a function of temperature alone, and refer to ings are only infinitesimal; that is, the temperature and pressure are T matched in such a way that heat and/or work are just transferred in the the ideal gas to identify it as 1/ . Without Carathéodory’s or desired direction ͑to or from the system͒, but the values of the tempera- an equivalent formulation of the second law, the existence of ture and pressure are equal for the system and for its surroundings for all an integrating factor for an arbitrary system cannot be taken practical purposes. The total entropy of the supersystem consisting of the 10,15 for granted. An exception is the special case of having a system and its surroundings remains constant in a reversible process, differential in just two variables, which is the case studied in implying that the change in entropy of the system is exactly equal, but opposite in sign, to the change in entropy of the surroundings. this paper, because mathematics alone guarantees the exis- 4 tence of integrating factors. S. M. Blinder, “Carathéodory’s formulation of the second law,” in Physi- cal : An Advanced Treatise, edited by H. Eyring, B. Henderson, There have been attempts to demonstrate that the entropy and W. Jost ͑Academic, New York, 1971͒, Vol. 1, Chap. 10. concept follows from the first law and the process of turning 5 Arnold Münster, Chemische Thermodynamik ͑VCH, Weinheim, 1969͒, the inexact differential of the reversibly exchanged heat into pp. 27–38; English translation, Classical Thermodynamics ͑Wiley- an exact one23 ͑cf. the comment in Ref. 10͒. An equivalent Interscience, New York, NY, 1970͒. 6 procedure is to infer the validity of 1/T as an integrating Stephen G. Brush, The Kind of Motion We Call Heat: A History of the ͑ factor for all substances from its validity for the ideal gas.8,9 Kinetic Theory of Gases in the 19th Century North-Holland, Amsterdam, 8 1986͒, Book 2, pp. 566–583. Djurdjević and Gutman presented what they claimed to be 7 G. Krishna Vemulapalli, “A simple method for showing entropy is a proof that 1/T is an integrating factor for all substances by function of state,” J. Chem. Educ. 63, 846–846 ͑1986͒. constructing a cyclic process in which the nonideal substance 8 Predrag Djurdjević and Ivan Gutman, “A simple method for showing is in contact with an ideal gas, but the proof seems to antici- entropy is a function of state,” J. Chem. Educ. 65, 399–399 ͑1988͒. pate the result by assuming that there is an integrating factor 9 Donald A. McQuarrie and John D. Simon, Physical Chemistry: A Mo- for the arbitrary substance without referring explicitly to the lecular Approach ͑University Science Books, Sausalito, CA, 1997͒,pp. 820 and 844–845, Problem 20–5. second law, which would then guarantee that 1/T is one such 10 John C. Wheeler, “Entropy does not follow from the first law: Critique of integrating factor. Following their line of reasoning, it would ‘Entropy and the first law of thermodynamics’,” Chem. Educ. 8, 171– be possible to show that each integrating factor for the ideal 176 ͑2003͒. gas works for any arbitrary substance as well. This conclu- 11 See, for example, E. C. Zachmanoglou and Dale W. Thoe, Introduction to sion is clearly incorrect, as we showed for the hard sphere Partial Differential Equations with Applications ͑Dover, Mineola, NY, 1986͒, pp. 133–137. fluid. 12 A different way of phrasing that 1/T is an integrating fac- This statement will be true at low and moderate densities as the first nine virial coefficients are positive; it is unknown if Bn becomes negative for tor for an arbitrary substance rests on the version of the sec- any higher value of n. See Stanislav Labík, Jiří Kolafa, and Anatol ond law of thermodynamics that states that all reversibly Malijevský, “Virial coefficients of hard spheres and hard disks up to the working Carnot engines, regardless of their working fluid, ninth,” Phys. Rev. E 71, 021105–1–8 ͑2005͒.

704 Am. J. Phys., Vol. 74, No. 8, August 2006 Volker C. Weiss 704 13 See for example, Terrell L. Hill, An Introduction to Statistical Thermo- 19 Louis A. Turner, “Simplification of Carathéodory’s treatment of thermo- dynamics ͑Dover, Mineola, NY, 1986͒, pp. 261–285. dynamics,” Am. J. Phys. 28, 781–786 ͑1960͒. 14 A. B. Pippard, The Elements of Classical Thermodynamics ͑Cambridge 20 Louis A. Turner, “Simplification of Carathéodory’s treatment of thermo- University Press, London, 1957͒, pp. 29–42. dynamics II,” Am. J. Phys. 30, 506–508 ͑1962͒. 15 Francis W. Sears and Gerhard L. Salinger, Thermodynamics, Kinetic 21 Francis W. Sears, “A simplified simplification of Carathéodory’s treat- Theory, and Statistical Thermodynamics, 3rd Ed. ͑Addison-Wesley, ment of thermodynamics,” Am. J. Phys. 31, 747–752 ͑1963͒. Reading, MA, 1975͒, pp. 138–141 and 168–172. 22 Francis W. Sears, “Modified form of Carathéodory’s second axiom,” Am. 16 H. A. Buchdahl, “On the principle of Carathéodory,” Am. J. Phys. 17, J. Phys. 34, 665–666 ͑1966͒. 41–43 ͑1949͒. 23 Panos Nikitas, “Entropy and the first law of thermodynamics,” Chem. 17 H. A. Buchdahl, “On the theorem of Carathéodory,” Am. J. Phys. 17, Educ. 7, 61–65 ͑2002͒. 44–46 ͑1949͒. 24 Ben Widom, “Thermodynamics, equilibrium,” in Encyclopedia of Ap- 18 H. A. Buchdahl, “Integrability conditions and Carathéodory’s theorem,” plied Physics, edited by George L. Trigg and Edmund H. Immergut Am. J. Phys. 22, 182–183 ͑1954͒. ͑Wiley-VCH, Weinheim, 1997͒, Vol. 21, pp. 281–310.

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