sign in sign up
<<
Home , Entropy, Equation of state, Maxwell relations, Thermodynamic square

MAXWELLRELATIONS

7.I THE

In Section 3.6 we observedthat quantities such as the isothermal , the coefficientof ,and the molar 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 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 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 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 . 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 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 is changed by : - a small increment du u uo at constant %.

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 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 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 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 , 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. It consequentlycontains only V, p, T (and N). 4. Bring the volumeto the numerator.The remainingderivative will be expressiblein terms of a and rc..

Example Given @T/AP)v,N

(#)..:-(#)..l(#),*:T (by7.3s) r89

- 5. The o{elnallv given derivativehas now been expressedin terms of the four quantities c,, cpt dt and rr. The specificheat at constantvolume is diminated by the equation

- cr: cp Tuaz/te, (7.36)

This useful relation, which should be committed to memory,was alluded to in equation 3.75.The readershould be able to derive it as an exercise (seeProblem 7.3-2).

PROBLEMS

73-1. Thermodynamicistssometimes refer to the "first ?d,s equation,'and the "secondTd^S equation";

TdS : Nc,dT + (fa7rcr) aV (/f constant)

TdS:NcrdT-TVq.dP (N constant) Derive theseequations. 7.3-2. show that the secondequation in the precedingproblem leads directly to the relation ,(#) - ,,"(#) u:", " and so validatesequation 7.36. 7.3-3. calcularc (aH/av)2iryinterms of the standardquantities cp, d, K7, T, and P.

Answer: (#),.: (ru- t)/r,

7.3-4. Reducethe derivative(0u/0s)r. 7.3-5. Reducethe derivative(0s/0fl,. 7.3-6. Reducethe derivative(0s/ 0flr. 7.3-7. Reducethe derivative(0s/0u)6. 190 Maxwell Relatiorc

7.4 SOME SIMPLE APPLICATIONS

In this section we indicate severalrepresentative applications of the manipulationsdescribed in Sectioni.3.In eachcase to 6e consideredwe

cp, l, and rc, are assumedknown and if the changesin parametersare small.

Adiabatic Compression consider a single-componentsystem of somedefinite quantity of (characterizedbythe mole numberN) enclosedwithin bn adiibatic wall.

we consider in particular the changein temperature.First, we assume 4he fundamental equationto be known. By diffeientiation, we can find the

- Ar: r(s,4,N) Z(S,P,,N) (7.37) If the fundamental equation is not known, but c", a, and K.r are given, and if the pressure change is small, we have dT: (#),*0, (7.38) By the method of Section 7.3, we then obtain TUA dT: dP (7.3e) cP Some Simple Applications 191

The change in chemical potential can be found similarly. Thus, for a small pressure change

dp:(#) (7.40) " .o' :(,-#)* (7.4r)

I sothermal Compression we now consider a systemmaintained at constant temperatureand mole number and quasi-staticallycompressed from an initial pressurep, to a final pressurePr. we may be interestedin the prediction of the changesin the values of U, S, V, and,,r. By appropriate elimination of variables among the fundamental equation and the equations of state, any such parameter can be expressedin terms of T, p, and N, and the changein that parametercan then be computeddirectly. For small changesin pressurewe flnd

,':(#),.0, (7.42)

: -aVdP (7.43) also dU: $Y).*0, (7.44) : (-TaV + pVK) dp (t.+s) and similar equationsexist for the other parameters. one may inquire about the total quantity of heat that must be extracted from the system by the heat reservoir in order to keep the system at constant temperatureduring the isothermal compression.First, assume that the fundamentalequation is known. Then

LQ : rAS : TS(r, P,,N) - TS(7,P,, N) (7.46) 192 Maxwell Relations whereS(U,V, N) is reexpressedas a functionof Z, p, and N in standard fashion. If the fundamental equationis not known we consideran infinitesimal isothermal compression,for which we have,from equation7.43

dQ: -TaVdP (7.47)

Finally, supposethat the pressurechange is large,but that the fundamen- tal equation is not known (so that the 7.46is not available).Then, if c and v areknownas functionsof z and P, we integrateequation7.47 at constant temperature -r LQ: t"ovar (1.48)

This solution must be equivalentto that given in equation 7.46.

Free Expansion

septum separating the sections is suddenly fractured the sponta- neously expandsto the volume of the whole container.We seel to predict the changein the temperatureand in the various other parameterj of the system. The total internal €nergyof the systemremains constant during the free expansion. Neither heat nor are transferred to the systembv anv external agency. If the temperatureis expressedin terms of (J, V, and N, we find

Tr- T,: T((r,Vr,N)- T(U,V,,N) (7.4e)

If the volume change is small

dT= i?u,*d, (7.50) &)* (7.51) Some Simple Applications 193

This process,unlike the two previouslytreated, is essentiallyirreversible and is not quasi-static(Problem 4.2-3).

Example In pr4cticethe processesof interestrarely are so neatlydefined as thosejust considered.No single thermodynamicparameter is apt to be constant in the

processeswill occur readily to the reader,but the generalmethodology is well representedby the following particular example.

Solution we first note that rhe tabulatedfunctions co(T,p), a(7, p), rcr(T,p), and a(7, P) are redundant. The first three functioirs imply the last, as has already been shown in the exampleof Section3.9. Turning to the statedproblem, the equation of the path in the T-v plane is T:A+ BV; A:(71V2- T2V)l(V2- Vr); B:(Tr- Tt)l(Vz- V) Furthermore, the pressureis known at each point on the path, for the known function u(7, P) can be inverted to expressp as a function of r and u. and thenceof u alone P : P(T,V): P(A + BV,V) The work done in the processis then w: + BV,v) tv,[',r1,e dv

This integral must be performednumerically, but generallyit is well within the capabilitiesof even a modestprogrammable hand calculator. The heat input is calculatedby considerings as a function of r and,v. d^s:/Pq\ sr *(ls\'\i/v),dv \orlv-- : N laP\ ir,dr +\fr ),0, :(+ - n"'\ar+Lav \ I rcrl Kr But on the path, dT : B dV, so that as:(Na2-BVa2* \ .f K.r t)* r94 Maxwell Relations

Thus the heat input is -(,t O: lqlu nc, + BV)(BVa-r)aTtcrldv

Again the factors in ttre integral must be evaluatedat the appropriatevalues of p and 7 correspondingto the point v on the path, and the integral over z must then be carried out numerically. It is often convenientto approximatethe given data by polynomial expressions in the region of interest; numerouspackaged computer programs for such "fits" are available. Then the inte$als can be evaluatedeither numerically or. analyti- cally.

Example In the P-u plane of a particular substance,two states,A and D, aredefined by

Pt : 705Pa ut : 2 x 10-2 rrt/mole

: Po lDa Pa uo: l0-r m3/mole and it is also ascertainedthat T,r : 350.9K. If 1 mole of this substanceis initially in the stateA, and if a thermal reservoirat temperature150 K is available,how much work can be deliveredto a reversiblework sourcein a processthat leaves the systemin the stateD? The following data are available.The adiabatsof the systemare of the form

Pu2: constant (fors: constant)

Measurementsof co and a are known only at the pressureof 10s pa.

Co: Bu2/3 (forP: tO5Pa);

B : 708/3: 464.2J/mzK

a: 3/T (for P : 105Pa) and no measurementsof r, are available. The reader is strongly urged to analyzethis problem independentlybefore reading the following solution. Solution In order to assessthe maximumwork that can be deliveredin a reversibleprocess A --+D it is necessaryonly to know z, - un and so - s,q. The adiabatthat passesthrough the stateD is describedbypuz: 102pa . m6; it intersectsthe isobar P : lOs Pa at a point C for which Pc : 105Pa uc: 10-3/2m3 : 3.16x 10-2m3 Some Simple Applications 195

As a two-stepquasi-static process joining A and D we choosethe isobaricprocess -, A c followed by the c -- D. By considering thele two processesin turn we seekto evaluatefirst a. - un and Jc - J,r and then uo - uc and s, - s., yieldingfinally uo - u1 and s, - sr. We first considerthe isobaricprocess A -- C. tc- \ /1 \ du:Tds-- Pdu: - P)du: t/3r_ l# (ir, r^)^ we cannot integrate this directly for we do not yet know (u) along the isobar. To calculate I(u) we write lar\ 7 r l-l (forP: Pn) \0uJp ud. 3u '(;): i"(;) and T : 350.9x(s0u)1/3 (on p : 10spa isobar) Returningnow to the calculationof u, - un : - a, lia x 350.9x(50)1/3 tot] du = rls du or uc - ut: 105x (u, - u): l.t6 x 103J We now require the differencauo - ur. Along the adiabat we have - -rc'["'+:fi2[ujL up- uc: - r;t] : -2.t6x 103J f'oPdr:ur, tr. n" Finally, then, we have the requiredenergy difference uo- ut: -103J We now - turn our attentionto the entropydifference s o s,q: Jc _ sr. Along the isobarlC : ^ (#) #a, : !nu-,r,a, and "^: - - so st rc - s,t: iBlryt - uf]: 6.t I/K Knowing au and, A.r for the process, we turn to the problem of delivering maximum work. The increase in entropy of the system permits us to extract from the thermal reservoir. (-Q,."): 4",Ar:150x 6.1:916J The total energy that can then be delivered to the reversible work source is (-Lu) * (-Q,..), or

work delivered : I.92 x 103 J tF.

196 Maxwell Relations

PROBLEMS

7.4-1. ln the analysisof a -Thomsonexperiment we may be given the initial and final molar volumesof the gas, rather than the initial and final . Expressthe derivative(07/0u)^ in termsof co, e, and rr. 7.4-2. T\e adiabatic bulk modulusis definedbv Fs: -,(#)":-v(#),.

Expressthis quantity in terms o-f,?,- and r, (do not eliminate cr). what is ",,.a, the relation of your result to the identity K,/Kr: cu/co (re,callprobiem 3.9-5)? 7.4-3. Evaluatethe changein temperaturein an inflnitesimal free expansionof a simple (equation 7.51). Does this result alsohold if the changein volume is comparableto the initial volume?can you give a more generalargument for a simple ideal gas,not basedon equation 7.51? 7.4-4. Show that equation7.46can be written as Q: Urfp,pl - UJp,pl so that uIP, trl can be interpretedas a "potential for heat at constant T and,N.,, 7.4'5. A 1% decreasein volume of a systemis carried out adiabatically.Find the change in the chemical potential in terms of. co, c, and r, (and the state functions P, T, u, u, s, etc). 7.4-6. Two moles of an imperfect gas occupy a volume of 1 liter and are at a temperatureof 100 K and a pressureof 2 Mpa. The gas is allowed to expand freely into an additional volurne,initially evacuated,of 10 cm3.Find the change in . At the initial conditionsc,:0.8 J/mole. K, Kr = 3 x 10-6Pa-', and c : 0.002K-r.

Answer: I p -(r"- puo)I AH : l;----z---^1 lAu: 15J - Tua') | (co"r l 7.4-7.Show that (0c,/0u)r: TGzp/072), andevaluate this quantityfor a systemobeying the van der Waalsequation of state. 7.4-8. Show that (*),: -ruf,'.(#),1

Evaluate this quantity for a systemobeying the equationof state "(,*#): * Problems r97 7.4'9. one moleof problem thesystem of 7.4-gis expandedisothermally from an Po pr. to a final pressure calculatethe heat flux to the systemin t1t1t_ry:*rerrusprocess.

Answer: O: -RZh( 2Aef- p,)/r, i) 7.4-10. A system obeys the van der waals . one more of this systemis expandedisothermally at temperaturez from an initial volume u0 to final a volume ur. Find the heat transfer to the systemin this "*pu"rion.

co: 26.20+ t7.49x 10-3r _ 3.223x t0_6Tz

where co is in J/mole and T is in .

Answer: Pt=15x105Pa

7-4'13- calculate the changein the molar internal energyin a throttling process in which the pressure changeis dp, expressingthe reiilt in terms of standard parameters. Assuming !.4:14- that a gasundergoes a free expansionand that the remperature is found to changeby dr, calcuhtJthe differencedp betweenthe initial pressure. and final 198 Maxwell Relations

7.4-16. Assuming the expansionof the ideal van der waals of problem 7.4-15 to be carried out quasi-staticallyand adiabatically, again find the final temperature|. Evaluateyour result with the numericaldata specifiedin problem 7.4-15. 7.4'17. It is observedthat an adiabaticdecrease in molar volume of 17oproduces a particular changein the chemicalpotential p. what percentagechange in molar volume, carried out isothermally,produces the samechange in p? 7.4-18. A cylinder is fitted with a piston, and the cylinder contains helium gas. The sidesof the cylinder are adiabatic,impermeable, and rigid, but the bottom of the cylinder is thermally conductive,permeable to helium, and rigid. Through this permeablewall the systemis in contact with a reservoirof constant r and pr"" (the chemical potential of He). calculate the compressibility of the system in properties FG/V)(dV/dP)l termsof the of helium (co,'Discuss u, d., K7t etc.) and thereby demonstrate that this compressibility diverges. the physical reasonfor this divergence. 7.4'19. The cylinder in Problem 7.4-18 is initially filled with ft mole of Ne. Assumeboth He and Ne to be monatomicideal .The bottom of the cylinder is again permeableto He, but not to Ne. Calculatethe pressurein the cylinder and the compressibility(-[/V)(dv/dp) as functionsof Z, V, and p,s.. Hint: RecallProblems 5.3-1, 5.3-L0, and 6.2-3. 7.4'20. A system is composedof 1 mole of a particular substance.In the p-u plane two states(l and B) lie on the locus Pu2 : constant,so that pnu) : peuzB. The following properties of the system have been measuredalong this locus: co: Cu,, a: D/u, and rr: Eu, whereC, D, and E are constants.Calculate the temperatureT" in termsof Ta,P,a,u,4,us, &rrd the constantsC, D, and E.

Answer: TB: T,q* (ur - u,q)/D+ 2EpAalD-rln(ur/u")

7.4-21. A systemis composedof 1 mole of a particular substance.Two thermody- namic states, designatedas A and B, lie on the locus pu : constant. The following propertiesof the systemhave been measured along this locus; co: Cu, a : D/u2, and r": Eu, where C, D, and, E are constants.Calcullte the differencein molar (u"- u) in terms of Tn,pa,u1,uB,Lrrd the con- stants C, D, and E. 7.4-22. The constant-volumeheat capacityof a particular simple systemis c,: AT3 (l : constant) In addition the equationof stateis known to be of the form (u-uo)r:B(r) where B(Z) is an unspecifiedfunction of T. Evaluatethe permissiblefunctional form of B(1). Generali zations : M agnet i c Sys tems r99

In terms of the undeterminedconstants appearing in your functional represen- tation of B(T), evaluated, cD, and rc. as functions of Z and u. Hint: Examine the derivative'|zs / 0T 0u.

Answer: cp : AT3 + (73/ DT * .E),where D and.E areconstants.

7.4-23. A system is expandedalong a straight line in the p-u plane, from the

C..KT (foru:uo) ;:AP c_ #: B, (forP: Pr)

Answer: o : +A(pf- p:) + lB(uf - r3)+ iQo- p)(u:_ uo)

7.4-u. A nonideal gas undergoesa throttling process(i.e., a Joule-Thomson expansion)from an initial pressurePo to a final pressurepr. The initial tempera- ture is ro and the initial molar volume is uo.calculate the hnal temperatureI if it is given that : *r: 4 alongthe T Toisotherm(l > 0) D" c : do along the T : To isotherm and ,| alongthe P : Pt isobar "o: what is the condition on 7i in order that the temperaturebe lowered by the expansion?

7-5 GENERALIZATIONS: MAGNETIC SYSTEMS

U : U(S,V,I,N) (7.s2) Legendre transformationswith respectto s, v, and N simply retain the magneticmoment I as a parameter.Thus the enthalpy is a function of s, Maxwell Relations

P, I, and N. H = U[p] : Il + pV: FI(,S,p,I,N) (7.53) An analogoustransformation can be made with respectto the magnetic coordinate UlB.l: IJ- B"I (7.s4) and -this potential is a function of s, v, 8", and N. The condition of equilildum for a systemat constantexternal field is that this potential be mrmmum. v-ariousother potentialsresult from multiple Legendretransformations, as depicted in the mnemonic squaresof Fig. 7.3. Maxwell relations and the relationships-betweenpotentials can belead from thesesquares in a completely straightforwardfashion.

UIB"J B"

utP,8"1

UI.P]

UTT,BJ (#)., B" (#)., UF] UF,P,B)

P UTT,P]

UTN (H).. T (#),, uIr,BJ

UTB) B" FIGURE 7.3 Probletns 20t

The "magnetic enthalpy" IIIP,B"l= U + pV - B"I is an interesting and useful potential. It is minimum for systems maintained at constanl pressure and constant external field. Furthermore, as in equation 6.29 for the enthalpy, dUIP,B"l: TdS: dQ at constant p, B;, and N. Thus the magnetic enthalpy uIP, B"l acts as a "potential for hiat" for systems maintained at constant pressure and magnetic field.

Example A particular material obeys the fundamental equation of the "paramagnetic model" (equation3.66), with To : 200 K and l3l2R : 10 Tesla2trVm2J. Two moles of this material are maintained at constantpressure in an external field of B, = 0.2 Tesla(or 2000gauss), and the systemis heatedfrom an initiar tempera- ture of 5 K to a final temperatureof 10K. what is the heatinput to the system? Solution

e: N[^o'- t *':^G)] -- 218.314x 5 + 10 x 0.04 x 0.11J : 83.22J (Note that the magnetic contribution, arising from the second term, is small ompared to the nonmagneticfirst-term contribution; in reality the nonmagnetic ontribution to the heat capacityof real falls rapidly at low remperarures end would be comparablysmall. Recall Problem3.9-6.)

PROBLEMS

751. calculate the "magneticGibbs potential" uIT,B"l for the paramagnetic model of equation 3.66. Conoborate that the derivative of this Dotential with rcspectto B. at constant 7 has its proper value. ?52. Repeat Problem7.5-1 for the systemwith the fundamentalequation given in Problem 3.8-2.

Answer: - utr,B"l:l*f;,t: lNnrh(k,n2e)

7.$'3. calculate (01/07)" for the paramagneticmodel of equation 3.66. Also calculate@s/aB)r what is the relationshipbetween these derivatives, as read from the mnemonicsquare? I

202 Maxwell Relations

7.5-4. Show that c,"_C,:#(#),.

and Ca" _ Xr Cr Xs

where C"" and C, are heat capacities and Xr ffid Xs are susceptibilities: Xr= Fo(01/08")r.