MAXWELLRELATIONS 7-I THE MAXWELL RELATIONS 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 thermody- namic 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 G 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.