2.8: Maxwell Relations

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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: $$\begin{aligned} \pabc{T}{X}{S,N}&=\pabc{y}{S}{X,N} & \pabc{T}{N}{S,X}&=\pabc{\mu}{S}{X,N} & \pabc{y}{N}{S,X}&=\pabc{\mu}{X}{S,N} \bvph\\ \pabc{T}{y}{S,N}&=-\pabc{X}{S}{y,N} & \pabc{T}{N}{S,y}&=\pabc{\mu}{S}{y,N} & \pabc{X}{N}{S,y}&=-\pabc{\mu}{y}{S,N}\bvph \\ \pabc{S}{X}{T,N}&=-\pabc{y}{T}{X,N} & \pabc{S}{N}{T,X}&=-\pabc{\mu}{T}{X,N} & \pabc{y}{N}{T,X}&=\pabc{\mu}{X}{T,N} \bvph \\ \pabc{S}{y}{T,N}&=\pabc{X}{T}{y,N} & \pabc{S}{N}{T,y}&=-\pabc{\mu}{T}{y,N} & \pabc{X}{N}{T,y}&=-\pabc{\mu}{y}{T,N} \bvph\\ \pabc{S}{X}{T,\mu}&=-\pabc{y}{T}{X,\mu} & \pabc{S}{\mu}{T,X}&=\pabc{N}{T}{X,\mu} & \pabc{y}{\mu}{T,X}&=-\pabc{N}{X}{T,\mu}\ .\bvph\end{aligned}$$