Maxwell Relations

MAXWELLRELATIONS

In Section 3.6 we observed that quantities such as the isothermal compressibility, the coefficient of thermal expansion, and the molar heat capacities describe properties of physical interest. Each of these is essentiallya derivative (0X/0Y)2.w... rfl which the variablesare either extensive or intensive thermodynamic parameters. With a wide range of extensive and intensive parameters from which to choose, in general systems,the number of such possible derivativesis immense.But there are relations among such derivatives, so that a relatively small number of them can be considered as independent; all others can be expressedin terms of these few. Needlessto say such relationships enormously simplify thermodynamic analyses.Nevertheless the relationships need not be mem- orized. There is a simple, straightforward procedure for producing the appropriate relationships as needed in the course of a thermodynamic calculation. That procedure is the subject of this chapter. As an illustration of the existence of such relationshios we recall equations 3.70 to 3.77 a2u a2u (7.1) ASAV AVAS

taP\ | aT\ -tx),.*,.*,: (7.2) I avJ,.,,.r,.

This relation is the prototype of a whole class of similar equalities known as the Maxwell relations. These relations arise from the equality of the mixed partial derivatives of the fundamental relation expressedin any of the various possible alternative representations. 18r 182 Maxwell Reltttions

Given a particular thermodynamicpotential, expressed in terms of its (/;1) natural variables,theie are t(t + 7)/2 separate-pairsof mixed rela- ,r"o.rd derivatives.Thus eachpotential yields t(t + 1)/2 Maxwell tions. For a single-componentsimple system the internal energyis a function -(t:2), mixed of three va;iables and the three[: (2'3)/2] pairs of : -.. second derivatives are AzU/aSAV A2U/AV AS, A/U/AS-9N arUt aX aS,and 02(17 AV AN : A2U/ i-NTV,The.complete set of Maxwell given the following i"tuiiot -i*r foi a single-"o*pon"nt simple systemis in iirti"!, which [t. nrri column siates the potential from which the t"tutl6tt derives,the secondcolumn statesthe pair of independentvaria- bles with respectto which the mixed partial derivativesare taken,and the last column itut"r the Maxwell relationsthemselves. A mnemonicdiagram i. U" describedin Section7.2 provides a mental device for recalling relationsof this form. In Section7.3 we presenta procedurefor utilizing theserelations in the solutionof thermodynamicproblems.

I ar\ -t^),| aP\ (73) U S,V \*) ,": , gI) - + p'dN s,N f (j4) (i 4) dll: TdS PdV \ dN/s.u \ ds/r.,r, -fI aP\ | ap\ (7'5) V,N *), ": \f;n)',N I aP\ (1.6) U[Tl= P T,V (#),"\ n),.r

dF:-SdT-PdV+P'dN T,N Ct 9 o- I aP\ v,N -\a*),., tl ll tt UlPl= H s,P (#),":(#),, (.s) Ie C( dH: TdS+ VdP+ pdN S,N (#)",(7.10) (' (#),": P P,N (K),, fl o ult'j S,V (#),.:-(#), (1.12) fr r n dUlt'l: TdS - PdV - Ndt' S,P (#),.:-(#),, eL3) It (r V,p. (#),, (714) (#)".: PI A ThermodynumicMnemonic Diagram tB3 I av\ Pl = 6 r,P -(#),"": ('t.Ls) {/lT, \ ar), r as\ I op\ dG: -SdT + vdP + pdN T,N t-*l (1.16) 0N ) r.r AV\ ( ap\ P,N (7.1'/) aN ),.r: \n),r ulr,p) T,V (#).. (7.18) AN\ dU[T,p'l: -SdT - Pdv T,P (+) _l (1.re) \ olL lr.v 0T J v.u "Ndp

(aP\ ,A/\ V,p | -l (720) \ op),, fr),, ar\ u[P, p] s,P (1'2t) aP) ,.* #) , r AZ\ dU[P,pl: TdS + VdP + Ndp S,p (122) dlr^l J s.p #) , r AV\ P, y" (i23) oll^l I s.p #),.-

7-2 A THERMODYNAMIC MNEMONIC DIAGRAM

A number of the most useful Maxwell relations can be remembered conveniently in terms of a simple mnemonic diagram.r This diagram, given in Fig. 7.7, consistsof a squarewith arrows pointing upward along the two diagonals. The sides are labeled with the four common thermodynamic potentials, F, G, H, and U, in alphabetical order clockwise around the diagram, the Helmholtz potential F at the top. The two corners at the left are labeled with the extensive parameters V and S, and the two corners at the right are labeled with the intensive parameters T and P. (" Valid Facts and Theoretical Understanding Generate Solutions to Hard Problems" suggeststhe sequenceof the labels.) Each of the four thermodynamic potentials appearing on the square is flanked by its natural independent variables.Thus U is a natural function of V and S; ,F is a natural function of V and Z; and G is a natural function of ?n and P. Each of the potentials also depends on the mole numbers, which are not indicated explicitly on the diagram.

tThis diagram was presentedby ProfessorMax Born in 1929 in a lecture heard by ProfessorTisza. It appeared in the literature in a paper by F. O. Koenig, J. Chem. Phys 1,29 (1935), and 56,4556 (1972). See also L. T. Klauder, Am. Journ. Phys.36,556 (1968),and a number of other variants oresented bv a successionof authors in this iournal. : :: ::,: ' :l:: . ,--. ..,.-,- :-,.-----:: . ,.,:. - -.*"::j-.,:

t84 Maxwell Relations

FIGURE 7.1 H P The thermodynamic square.

In the differential expressionfor each of the potentials, in terms of the differentials of its natural (flanking) variables, the associated algebraic sign is indicated by the diagonal arrow. An arrow pointing away from a natural variable implies a positive coemcient, whereas an arrow pointing toward a natural variable implies a negative coefficient. This scheme becomes evident by inspection of the diagram and of each of the following equations: dU: TdS - PdV +lp.odNo (1 24) k dF: -SdT - PdV +LpodNo (1.25) k

dG: *SdT+VdP+LprdNo \7.26) k dH : Tds + vdP +lpodNo (7.21) k Finally the Maxwell relations can be read from the diagram. We then deal only with the corners of the diagram. The labeling of the four corners of the square can easily be seen to be suggestiveof the relationship

(9I\ : 1.o'stantN1, Nr,...) (t.zs) \ ds/p H\iJP ls v i---l i---',r iz i i \l s L---rP sL-__rP By mentally rotating the square on its side, we find, by exactly the same construction /as\ l av\ t-l -l*l (constantNr,1{r,...) (7.29) \0Plr \ or I p si----l r---tV tl t1 r\, 1 t/ | | :l PL--_JT Pt---'t T 185

Therninus sign in this equationis to be inferredfrom the unsymmetrical of the arrowsin this case.The two remainingrotations of the ^ir."r.nt Maxwell relations i;;;r" give the two additional taP\ i as\ \ar),:\m), (constantNr, Nr,...) (7.30) and tar\ taP\ -\ (constantN,, Nr,...) (7.31) \av),: as/"

These are the four most useful Maxwell relations in the conventional of thermodynamics. applications'fhe mnemonic diagram can be adapted to pairs of variables other than S and V. If we are interested in Legendre transformations dealing with S and {, the diagram takes the form shown in Fig. 7.2a. The arrow connecling N, and F; has been reversedin relation to that which previ- ously connected V and P to ake into account the fact that p,,is analogous 'l "7 to - P. Equations .4, 7.7 , .I3, and 7.I9 can be read directly from this diagram. Other diagrams can be constructed in a similar fashion, as indicated in the seneral casein Fie.'l .2b.

F=UITI T uIP2] D " I

IITD D 1 UlT, ttjl v\LI,t2J

u. D s ulu;l 'l UIPJ (a) (b) FIGURE 7.2

PROBLEMS

7.2-1. ln the immediate vicinity of the state ?"6,u0the volume of a particular system of 1 mole is observedto vary according to the relationship u: Do+ a(T - 4) + b(P - Po) Calculate the transfer of heat dQ to the systemif the molar volume is changed by a small increment du : u - uo at constant temperature ]n0.

Answer" /,s\ do:r(H),a,:4n.0,:- f o, , ;,-.,-.':+i-t--ii;. . ;-:::'':r: ,t:.- -.:., := .:;!;iE

t86 Maxwell Relations

of a particular state, a 7.2-2. For a particular system of 1 mole, in the vicinity by a heat flux change of pressure dP at-constant?n is observedto be accompanied expansion of this de l- ,q ai. Wnat is the value of the coefficient of thermal system,in the samestate? 7.2-3. Show that the relation I *T implies that c" is independentof the pressure | 0c,\ \ *),:o

1-3 A PROCEDURE FOR THE REDUCTION OF DERIVATTVES IN SINGLE.COMPONENT SYSTEMS

In the practical applications of thermodynamics the experimental situa- tion to be analyzediiequently dictates a partial derivative to be evaluated. For instance, we may be con"ettted with the analysis of the temperature change that is required to maintain the volume of a single-component temperature systeir constant ii the pressure is increased slightly. This change is evidentlY o,: (#). .0, \ t .J/)

and consequently we are interested in an evaluation of the derivative (ar/ann,".e number of similar problems will be consideredin Section th.ey i.q.'A g"ti";,uf feature of the derivaiives that arise in this way is that generally are likEly to involve constant mole numbers and that they involve both intensive and extensive parameters' Of all such deriuatiues, in^ only three can be independent,and any g-iuenderiuatiue can be expressed set ts t"r^, of an arbitrarily chosenset of three basic deriuatiues.This conventionally-it" chosenaS cp, c' and Kr- choice of cp, a, rcr'is an implicit transformation to the Gibbs ^nd are reDresentation, fof the three iecond derivatives in this representation |ig/AT2. 0zg/0T0P, and Tzg/dP2; thesederivatives are equal,tt:p.tj- tne tiiJy, rc - rl7f . ua, and -uKr. For constantmole numbersthese are only indepetid"nt secondderivatives. and intensiue itt qoi deriuatiues(inuoluing both extensiue ry,rory,'i.::) vtnrctt. bi written in terms of secondderiuatiues of the Gibbs,potential,oJ can set we haue now seen that cr, a, and rc, constitite a complite independent mole numbers). (at constant ls procedure to be followed in this " reduction of derivatives" The bY straightforward in principle; the entropy S need only be replaced A Procedureforthe Reduction of Deriuatiues in Single-Componen! Sj'stems 187

-AG/AT and V must be replacedby 0G/0P, therebyexpressing the original derivativein termsof secondderivatives of G with respectto ?r ana P.In practicethis procedurecan becomesomewhat involved' It is essentialthat the studentof thermodynamicsbecome thoroughly oroficientin the " reductionof derivatives."To that purposewe presenta procedure.based upon the "mnemonicsquare" and organizedin a stepby itep recipe that accomplishesthe reduction of any given derivative. Students are urged to do enough exercisesof this type so that the becomesautomatic. 'orocedure Consider a partial derivativeinvolving constantmole numbers.It is desiredto expressthis derivativein termsof cp, a, and rc..We first recall the following identitieswhich are to be employedin the mathematical manipulations(see Appendix A).

tax\ i7 ?1) \av), :'l(#). and tax\ tax\ lt avt \av),:\a*),1\a*), \7.34) I ax\ | az\ lt azt -lav/ (7.3s \ar),: rf\axl, )

The following steps are then to be taken in order: 1. If the derivative contains any potentials, bring them one by one to the numerator and elirninate by the thermodynamic square (equations 7.24 to 7.27).

Example Reducethe derivative(0P/0U)o.*. toPt ltaut l-t (by \au)".": |.\ 7.33) '.P)".r1 :[.(#)""-,(#)""]-' (by 7.2a) -4ffi,/(#)" ".'(#).,1(#),.]-' (by7.3s) - s(aT/aP) + v -s(aT/aP)v.N+ vl-' -T s,N +P -l - s?r/as) P.N - s\ar/ av) P.N l \by 7 26)

i. I ]88 Maxwell Relations

The remaining expressiondoes not contain any potentials but may involve a number of derivatives. Choose these one by one and treat each according to the following procedure. 2. If the derivative contains the chemical potential, bring it to the numerator and eliminate by means of the Gibbs-Duhem relation, dp : -sdT * udP.

Example Reduce(0p./0V)r.*. ( | 0u\ -'(| AT\ AP\ \av)''': M)''"*'\ av)''* 3. If the derivative contains the entropy, bring it to the numerator. If one of the four Maxwell relations of the thermodynamic square now eliminates the entropy, invoke it. If the Maxwell relations do not eliminate the entropy- put a 0T under dS (employ equation 7.34 with w -- T). The numerator will then be expressible as one of the specific heats (either c, or cr).

Example Considerthe derivative(07/0P)r." appearingin the exampleof step1:

taT\ as\ asl -\aP),.*l\n),.*/ lt (by I ar/".": 7.3s) tav\ ltr \n),.*lT', \by 7 .29)

Example Consider the derivative (AS/ AV) p,N. The Maxwell relation would give (AS/AV)p,N: (AP/AT )"," (equation 1.28), which would not eliminate the entroDv. We therefore do not invoke the Maxwell relation but write as (as/ar)P,N (N/T)c, / \ _ fty 13a) \ av) ,.*: ( avTn;o (av/ aT)P.N The derivative now contains neither any potential nor the entropy. It consequently contains only V, P, ?n(and N). 4. Bring the volume to the numerator. The remaining derivative will be expressible in terms of a and rr.

Example Given(AT/AP)v.N

(#)"":-(#)..:ffi),.:+ (by 7.35) 189

5. The originally given derivative has now been expressedin terms of the four quantities cu, cpt dr and rr. The specificheat at constantvolume is eliminated bY the equation

co: cp - Tua2/rc, (7.36)

This useful relation, which should be committed to memory, was alluded to in equation 3.75. The reader should be able to derive it as an exercise 7.3'2). (see' Problem This method of reduction of derivativescan be applied to multicompo- nent systemsas well as to single-comPonentsystems, provided that the chemical potentials F; do not appear in the derivative (for the Gibbs-Duhem relatiori, which eliminates the chemical potential for single-componentsystems, merely introduces the chemical potentials of other componentsin multicomponentsystems).

PROBLEMS

73-1. Thermodynamicistssometimes refer to the "first IdS equation"and the "secondZ/S equation"; T dS : Nc,dT + (ra/ xr) dV ( N constant)

TdS:NcrdT-TVadP ( N constant)

Derive theseequations. 7.3-2. Show that the secondequation in the precedingproblem leads directly to the relation 4#),:,,-r,"(#). and so validatesequation 7.36. 73-3. Calculate (0V/0V)r.*interms of the standard quantities cp, d, K7, T, and P.

Answer: (#),, (ra- 7)/x,

7.3-4. Reduce the derivative(0u/0s)r. 7.3-5. Reduce the derivative(0s/0fl". 7.3-6. Reduce the derivative(0s/0 flr. 7.3-7. Reduce the derivative(0s/0u) 1,.