2.8: Maxwell Relations

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\)
\( \newcommand\Vrho

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\)
\( \newcommand\Vvarrho

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\)
\( \newcommand\Vsigma

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\)
\( \newcommand\Vvarsigma

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\)
\( \newcommand\Vtau

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\)
\( \newcommand\Vupsilon

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\)
\( \newcommand\Vphi

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\)
\( \newcommand\Vvarphi

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\)
\( \newcommand\Vchi

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\)
\( \newcommand\Vpsi

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\)
\( \newcommand\Vomega

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\)
\( \newcommand\VGamma

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\)
\( \newcommand\VDelta

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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
\( \newcommand\BE{\mib E}\)
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\( \def\tket#1{|  #1 \rangle}\)
\( \def\tbra#1{\langle #1|}\)
\( \def\tbraket#1#2{\langle #1  |   #2 \rangle}\)
\( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
\( \def\sket#1{|  \, #1 \,  \rangle}\)
\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
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\( \newcommand\pz{\partial}\)
\( \newcommand\pzb{\bar{\partial}}\)
\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
\( \newcommand\nd{^{\vphantom{\dagger}}}\)
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\( \newcommand\oc{\omega_\Rc}\)

\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
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\( \newcommand\Rep{\textsf{Re}\,}\)
\( \newcommand\Imp{\textsf{Im}\,}\)
\( \newcommand\ncdot{\!\cdot\!}\)
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\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
\( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
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\( \newcommand\subA{\ns_\ssr{A}}\)
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\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
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\( \newcommand\rhohat{\hat\rho}\)
\( \newcommand\vrhhat{\hat\vrh}\)
\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
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\( \newcommand\vet{\tilde\ve}\)
\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
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\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
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\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
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\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
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\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\)
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
\( \newcommand\ndap{\nd_{\alpha'}}\)
\( \newcommand\labar

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\)
\( \newcommand\msa{m\ns_\ssr{A}}\)
\( \newcommand\msb{m\ns_\ssr{B}}\)
\( \newcommand\mss{m\ns_\Rs}\)
\( \newcommand\HBx{\hat\Bx}\)
\( \newcommand\HBy{\hat\By}\)
\( \newcommand\HBz{\hat\Bz}\)
\( \newcommand\thm{\theta\ns_m}\)
\( \newcommand\thp{\theta\ns_\phi}\)
\( \newcommand\mtil{\widetilde m}\)
\( \newcommand\phitil{\widetilde\phi}\)
\( \newcommand\delf{\delta\! f}\)
\( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
\( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
\( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
\( \newcommand\vbar{\bar v}\)
\( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
\( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
\( \newcommand\gla{g\nd_{\RLambda\nd}}\)
\( \newcommand\TA{T\ns_\ssr{A}}\)
\( \newcommand\TB{T\ns_\ssr{B}}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \newcommand\NS{N\ns_{\textsf S}}\)

Maxwell relations are conditions equating certain derivatives of state variables which follow from the exactness of the differentials of the various state functions.

Relations deriving from \(E(S,V,N)\)

The energy \(E(S,V,N)\) is a state function, with \[dE=T\,dS-p\,dV+\mu\,dN\ ,\] and therefore \[T=\pabc{E}{S}{V,N} \qquad,\qquad -p=\pabc{E}{V}{S,N} \qquad,\qquad \mu=\pabc{E}{N}{S,V}\ .\] Taking the mixed second derivatives, we find \[\begin{aligned} \qabc{E}{S}{V}&=\pabc{T}{V}{S,N}=-\pabc{p}{S}{V,N}\\ \qabc{E}{S}{N}&=\pabc{T}{N}{S,V}=\pabc{\mu}{S}{V,N}\bvph\\ \qabc{E}{V}{N}&=-\pabc{p}{N}{S,V}=\pabc{\mu}{V}{S,N}\ .\bvph\end{aligned}\]

Relations deriving from \(F(T,V,N)\)

The energy \(F(T,V,N)\) is a state function, with \[dF=-S\,dT-p\,dV+\mu\,dN\ ,\] and therefore \[-S=\pabc{F}{T}{V,N} \qquad,\qquad -p=\pabc{F}{V}{T,N} \qquad,\qquad \mu=\pabc{F}{N}{T,V}\ .\] Taking the mixed second derivatives, we find \[\begin{aligned} \qabc{F}{T}{V}&=-\pabc{S}{V}{T,N}=-\pabc{p}{T}{V,N}\\ \qabc{F}{T}{N}&=-\pabc{S}{N}{T,V}=\pabc{\mu}{T}{V,N}\bvph\\ \qabc{F}{V}{N}&=-\pabc{p}{N}{T,V}=\pabc{\mu}{V}{T,N}\ .\bvph\end{aligned}\]

Relations deriving from \(\CH(S,p,N)\)

The enthalpy \(\CH(S,p,N)\) satisfies \[d\CH=T\,dS + V dp + \mu\,dN\ ,\] which says \(\CH=\CH(S,p,N)\), with \[T=\pabc{\CH}{S}{p,N} \qquad,\qquad V=\pabc{\CH}{p}{S,N} \qquad,\qquad \mu=\pabc{\CH}{N}{S,p}\ .\] Taking the mixed second derivatives, we find \[\begin{aligned} \qabc{\,\CH}{S}{p}&=\pabc{T}{p}{S,N}=\pabc{V}{S}{p,N}\\ \qabc{\,\CH}{S}{N}&=\pabc{T}{N}{S,p}=\pabc{\mu}{S}{p,N}\bvph\\ \qabc{\,\CH}{p}{N}&=\pabc{V}{N}{S,p}=\pabc{\mu}{p}{S,N}\ .\bvph\end{aligned}\]

Relations deriving from \(G(T,p,N)\)

The Gibbs free energy \(G(T,p,N)\) satisfies \[dG=-S\,dT + V dp + \mu\,dN\ ,\] therefore \(G=G(T,p,N)\), with \[-S=\pabc{G}{T}{p,N} \qquad,\qquad V=\pabc{G}{p}{T,N} \qquad,\qquad \mu=\pabc{G}{N}{T,p}\ .\] Taking the mixed second derivatives, we find \[\begin{aligned} \qabc{\,G}{T}{p}&=-\pabc{S}{p}{T,N}=\pabc{V}{T}{p,N}\\ \qabc{\,G}{T}{N}&=-\pabc{S}{N}{T,p}=\pabc{\mu}{T}{p,N}\bvph\\ \qabc{\,G}{p}{N}&=\pabc{V}{N}{T,p}=\pabc{\mu}{p}{T,N}\ .\bvph\end{aligned}\]

Relations deriving from \(\Omega(T,V,\mu)\)

The grand potential \(\Omega(T,V,\mu)\) satisfied \[d\Omega=-S\,dT -p\,dV - N\,d\mu\ ,\] hence \[-S=\pabc{\Omega}{T}{V,\mu} \qquad,\qquad -p=\pabc{\Omega}{V}{T,\mu} \qquad,\qquad -N=\pabc{\Omega}{\mu}{T,V}\ .\] Taking the mixed second derivatives, we find \[\begin{aligned} \qabc{\Omega}{T}{V}&=-\pabc{S}{V}{T,\mu}=-\pabc{p}{T}{V,\mu}\\ \qabc{\Omega}{T}{\mu}&=-\pabc{S}{\mu}{T,V}=-\pabc{N}{T}{V,\mu}\bvph\\ \qabc{\Omega}{V}{\mu}&=-\pabc{p}{\mu}{T,V}=-\pabc{N}{V}{T,\mu}\ .\bvph\end{aligned}\]

Relations deriving from \(S(E,V,N)\)

We can also derive Maxwell relations based on the entropy \(S(E,V,N)\) itself. For example, we have \[dS={1\over T}\,dE + {p\over T}\,dV - {\mu\over T}\,dN\ .\] Therefore \(S=S(E,V,N)\) and \[\qabc{S}{E}{V}=\pabc{(T^{-1})}{V}{E,N}=\pabc{(pT^{-1})}{E}{V,N}\ ,\] et cetera.

Generalized thermodynamic potentials

We have up until now assumed a generalized force-displacement pair \((y,X)=(-p,V)\). But the above results also generalize to magnetic systems, where \((y,X)=(H,M)\). In general, we have \[\begin{aligned} \hbox{\tt THIS}&\hbox{\tt\ SPACE AVAILABLE}& dE&=T\,dS + y\,dX + \mu\,dN \vph\\ F&=E-TS & dF &= -S\,dT+ y\,dX + \mu\,dN \vph\\ \CH&=E-yX & d\CH &= T\,dS - X\,dy + \mu\,dN \vph\\ G &= E - TS - yX & dG &= -S\,dT -X\,dy + \mu\,dN \vph\\ \Omega &= E - TS - \mu N & d\Omega &= -S\,dT+y\,dX -N\,d\mu\ .\end{aligned}\] Generalizing \((-p,V)\to (y,X)\), we also obtain, mutatis mutandis, the following Maxwell relations: \[\begin{aligned} \pabc{T}{X}{S,N}&=\pabc{y}{S}{X,N} & \pabc{T}{N}{S,X}&=\pabc{\mu}{S}{X,N} & \pabc{y}{N}{S,X}&=\pabc{\mu}{X}{S,N} \bvph\\ \pabc{T}{y}{S,N}&=-\pabc{X}{S}{y,N} & \pabc{T}{N}{S,y}&=\pabc{\mu}{S}{y,N} & \pabc{X}{N}{S,y}&=-\pabc{\mu}{y}{S,N}\bvph \\ \pabc{S}{X}{T,N}&=-\pabc{y}{T}{X,N} & \pabc{S}{N}{T,X}&=-\pabc{\mu}{T}{X,N} & \pabc{y}{N}{T,X}&=\pabc{\mu}{X}{T,N} \bvph \\ \pabc{S}{y}{T,N}&=\pabc{X}{T}{y,N} & \pabc{S}{N}{T,y}&=-\pabc{\mu}{T}{y,N} & \pabc{X}{N}{T,y}&=-\pabc{\mu}{y}{T,N} \bvph\\ \pabc{S}{X}{T,\mu}&=-\pabc{y}{T}{X,\mu} & \pabc{S}{\mu}{T,X}&=\pabc{N}{T}{X,\mu} & \pabc{y}{\mu}{T,X}&=-\pabc{N}{X}{T,\mu}\ .\bvph\end{aligned}\]