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Maxwell Relations MAXWELLRELATIONS 7.I THE MAXWELL RELATIONS In Section 3.6 we observedthat quantities such as the isothermal compressibility, the coefficientof thermal expansion,and the molar heat capacities describe properties of physical interest. Each of these is essentiallya derivativeQx/0Y)r.r..,. in which the variablesare either extensiveor intensive thermodynaini'cparameters. with a wide range of extensive and intensive parametersfrom which to choose, in general systems,the numberof suchpossible derivatives is immense.But thire are azu azu ASAV AVAS (7.1) -tI aP\ | AT\ as),.*,.*,,...: \av)s.N,.&, (7.2) 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 expressed in any of the various possible alternative representations. 181 182 Maxwell Relations Given a particular thermodynamicpotential, expressedin terms of its (l + 1) natural variables,there are t(t + I)/2 separatepairs of mixed second derivatives.Thus each potential yields t(t + l)/2 Maxwell rela- tions. For a single-componentsimple systemthe internal energyis a function : of three variables(t 2), and the three [: (2 . 3)/2] -arUTaSpurs of mixed second derivatives are A2U/ASAV : AzU/AV AS, AN : A2U / AN AS, and 02(J / AVA N : A2U / AN 0V. Thecompleteser of Maxwell relations for a single-componentsimple systemis given in the following listing, in which the first column states the potential from which the ar\ U S,V I -|r| aP\ \ M)',*: ,s/,,' (7.3) dU: TdS- PdV + p"dN ,S,N (!t\ (4\ (7.4) \ dNls.r, \ ds/r,r,o V,N _/ii\ (!L\ (7.5) \0Nls.v \0vls.x UITI: F T,V ('*),,.: (!t\ (7.6) \0Tlv.p dF: -SdT - PdV + p,dN T,N -\/ as\ a*) ,.,: (!"*),. (7.7) V,N -(#)..: (7.8) ( '!or),. U[Pl=H s,P (7s) (#)"":(!"^),,. dH: TdS + VdP+ pdN s,N (#)"":(^e)", (7.10) P,N (K),,:(#)"N (711) u[t'] s,v -(#), (712) (f),r: , dUlpl: TdS - PdV - Ndp s,p (#),.:-(#),, e 13) v,lt (#),.:(#),, (..t4) A Thermodynamic Mnemonic Diagram 183 -\/ as\ UlT,Pl=Q T,P n), * dG: -SdT + VdP+ p.dN T,N -(r"a),,":(#),. (ir6) tav\ P,N (ir't) \a*),,: (#),. / as\ ulr, pl T,V \an),, (7.18) dUlT,pl: -SdT - PdV T,1t' (H),, -Ndt, V,IL (#)., I ar\ | av\ ulP,pl ,S,P (1.2r) \a./".,: \as/"." dUlP,tll: TdS + VdP + Ndp" S,p (#),,:-(#)", (7zz) P,p (fr),,: -(#)"- (723) 7.2 A THERMODYNAMIC MNEMONIC DIAGRAM A number of the most useful Maxwell relationscan be remembered convenientlyin terms of a simple mnemonicdiagram.I This diagram, given in Fig. 7.L, consistsof a squarewith arrowspointing upward along the two diagonals.The sidesare labeledwith the four common thermody- namic potentials, F, G, H, and U, in alphabeticalorder clockwisearound the diagram, the Helmholtz potential F at the top. The two cornersat the left are labeled with the extensiveparameters V and S, and the two corners at the right are labeled with the intensive parametersT and P. ("Valid Factsand TheoreticalUnderstanding Generate Solutions to Hard Problems"suggests the sequenceof the labels.) Each of the four thermodynamicpotentials appearingon the squareis flanked by its natural independentvariables. Thus {/ is a natural function of V and S; F is a natural function of V and,T; and G is a natural function of 7 and P. Each of the potentials also dependson the mole numbers,which are not indicated explicitly on the diagram. 1This diagram was presented by Professor Max Born in 1929 in a lecture heard by Professor Tisza. It appeared in the literature in a paper by F. O. Koenig, J. Chem. Phys. 3, 29 (1935), and 56, 4556 (1972). See also L. T. Klauder, Am. Journ. Phys 36,556 (1968), and a number of other variants presented by a succession of authors in this journal. t84 Maxwell Relations v U FIGURE 7 T P The thermodynamicsquare. In the differential expressionfor eachof the potentials,in terms of the differentials of its natural (flanking) variables, the associatedalgebraic sign is indicated by the diagonal arrow. An arrow pointing away from a natural variable implies a positive coefficient,whereas an arrow pointing toward a natural variable implies a negative coefficient. This scheme becomesevident by inspectionof the diagramand of eachof the following equations: dU: TdS- PdV +lp,odNo (7.24) k dF: -SdT- PdV+EpodNo (7.2s) k dG: -SdT+VdP+LpodNo (7.26) k dH: TdS+ VdP+EpodNo (7.27) k Finally the Maxwell relations can be read from the diagram. We then deal only with the cornersof the diagram.The labeling of the four corners of the squarecan easilybe seento be suggestiveof the relationship I av\ | aT\ : (constantN"N""') (7.2s) (iSl, t#J" V r----t r----rT tttl iz i i \i s L___rP sL___rP By mentally rotating the squareon its side, we find, by exactly the same construction / as\ taY\ -\7't,, (constant N1, N2,. .) (7.2e) \aP)r: s;---,' i---jv iz i i\i PL___rT Pr____tT Problems r85 The minus sign in this equationis to be inferred from the unsymmetrical placementof the arrows in this case.The two remaining rotations of the squaregive the two additional Maxwell relations (#)"(#).(constantNr,lfr,...) (7.30) and (#),-(K). (constantNr, Nr,...) (7.31) These are the four most useful Maxwell relations in the conventional applications of thermodynamics. The mnemonic diagramcan be adaptedto pairs of variablesother than S and V. lf we are interestedin Legendretransformations dealing with ,S and N, the diagram takes the form shown in Fig. 7.2a. The arrow conneciing N, and F7 has been reversedin relation to that which previ- ously conneciedv and P to ake into accountthe fact that p,, is analogous to - P. Equations7 .4, 7.7, 7.73, and 7.19 canbe read Oiie6tlyfrom"this diagram. other diagrams can be constructed in a similar iashion, as indicatedin the generalcase in Fig.7.2b. Nj F = UITI ulP2'J xr D UlT,tt jl u[Pr, P2] x2 Pr s Ulull Itj UIP] (a) (b) FIGURE 7 2 PROBLEMS 1.2-1. ln the immediate vicinity of the state ft, uo the volume of a particular system of 1 mole is observed to vary according to the relationship u: uo+ a(T - fo) + b(p - po) calculate the transfer of heat dQ to the system if the molar volume is changed by : - a small increment du u uo at constant temperature %. Answer: : / ,.s \ : - do r(#),0, 4#),dv: f on 186 Maxwell Relations 7.2-2. For a particular system of 1 mole, in the vicinity of a particular state, a change of pressure dP at corstant T is observed to be accompanied by a heat flux : 'expansion dQ A dP. what is the value of the coefficient of thermal of this system, in the same state? 7.2-3. Show that the relation 1 d:j implies thal c, is independentof the pressure | 0c"\ l-:-l :0 \OPJ, 7-3 A PROCEDURE FOR THE REDUCTION OF DERIVATIVES IN SINGLE.COMPONENT SYSTEMS dT: (#).*0, (7.32) and consequently we are interested in an evaluation of the derivative (aT/ aP) ,.r. A number of similar problems will be consideredin Section AII first deriuatiues(inuoluing both extensiueand intensiueparamercrs) can be written in terms of secondderiuatiues of the Gibbs potential, of whicit we haue now seen that,cp, a, and Kr constitutea complite independentset (at constantmole numbersl. The procedure to be followed in this "reduction of derivatives" is straightforward in principle; the entropy s need only be replaced by A Procedurefor the Reduction of Deriuatiues in Single-Component Systems 187 -AG/AT and V must be replacedby 0G/0P, thereby expressingthe original derivative in terms of secondderivatives of G with respectto r and P. In practice this procedurecan becomesomewhat involved. It is essentialthat the student of thermodynamicsbecome thoroughly proficient in the "reduction of derivatives."To that purposewe presenta procedure,based upon the "mnemonic square" and organizedin a stepby step recipe that accomplishesthe reduction of any given derivative. Students are urged to do enough exercisesof this type so that the procedurebecomes automatic. consider a partial derivative involving constant mole numbers. It is desiredto expressthis derivativein termsof cp, c, and rcr.we first recall the following identities which are to be employed in the mathematical manipulations(see Appendix A). tax\ \av),:'l(#). (7.33) I ax\ (ax\l(ar\ \av),:\awlzl\aw)z (7.34) lax\ \av),:-ffi).1(K), (7.3s) The following stepsare then to be taken in order: 1. If the derivativecontains any potentials,bring them one by one to the numerator and eliminate by the thermodynamicsquare (equations 7.24 to 7.2t). Example Reducethe derivative (0P/0U)o.*. (#)".:tffi),"]-' (bv7.33) :[.(#)".-,(#),"]-' (by7.2a) : t-.(#),.l(#)" ".,(#)..1(#). "]-' (by7.3s) _[_r-s(aT/ap)s,N+v * r-s(ar/ap)r.*+ vl-l -s(ar/as)P.N -s(ar/av)P.N t l (by 7.26) 188 Maxwell Relatiow The remaining expressiondoes not contain any potentials but may involve a number of derivatives.Choose these one by one and treat eacir according to the following procedure. 2. If the derivative contains the chemical potential, bring it to the numerator and eliminate by meansof the Gibbs-Duhem relation, dp : -sdT * udP. Example Reduce(0p./0V)r.*. (#),,.:-"(#), .*,(#), . Example Considerthe derivative (07/0P)r," appearingin theexample of stepl: (#)"":-(#)..1(#),. (by7.3s) : (#),.f+,, (by7.2e) Example consider the derivative Qs/av) p,N. The Maxwell relation would give (AS/AV),,N: (AP/ID*," (equation7.28), which would not eliminatethe entropy. we thereforedo not invoke the Maxwell relation but write (as/ar)p.N (N/r)c, f !q) _ -_ (by7 sa) \ ovI p.w (av/aDp.N evTal r^ The derivative now contains neither any potential nor the entropy.
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