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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=TdSpdV+μdN ,

and therefore T=(ES)V,N,p=(EV)S,N,μ=(EN)S,V .
Taking the mixed second derivatives, we find 2ESV=(TV)S,N=(pS)V,N2ESN=(TN)S,V=(μS)V,NNN2EVN=(pN)S,V=(μV)S,N .NN

Relations deriving from F(T,V,N)

The energy F(T,V,N) is a state function, with dF=SdTpdV+μdN ,

and therefore S=(FT)V,N,p=(FV)T,N,μ=(FN)T,V .
Taking the mixed second derivatives, we find 2FTV=(SV)T,N=(pT)V,N2FTN=(SN)T,V=(μT)V,NNN2FVN=(pN)T,V=(μV)T,N .NN

Relations deriving from H(S,p,N)

The enthalpy H(S,p,N) satisfies dH=TdS+Vdp+μdN ,

which says H=H(S,p,N), with T=(HS)p,N,V=(Hp)S,N,μ=(HN)S,p .
Taking the mixed second derivatives, we find 2HSp=(Tp)S,N=(VS)p,N2HSN=(TN)S,p=(μS)p,NNN2HpN=(VN)S,p=(μp)S,N .NN

Relations deriving from G(T,p,N)

The Gibbs free energy G(T,p,N) satisfies dG=SdT+Vdp+μdN ,

therefore G=G(T,p,N), with S=(GT)p,N,V=(Gp)T,N,μ=(GN)T,p .
Taking the mixed second derivatives, we find 2GTp=(Sp)T,N=(VT)p,N2GTN=(SN)T,p=(μT)p,NNN2GpN=(VN)T,p=(μp)T,N .NN

Relations deriving from Ω(T,V,μ)

The grand potential Ω(T,V,μ) satisfied dΩ=SdTpdVNdμ ,

hence S=(ΩT)V,μ,p=(ΩV)T,μ,N=(Ωμ)T,V .
Taking the mixed second derivatives, we find 2ΩTV=(SV)T,μ=(pT)V,μ2ΩTμ=(Sμ)T,V=(NT)V,μNN2ΩVμ=(pμ)T,V=(NV)T,μ .NN

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=1TdE+pTdVμTdN .

Therefore S=S(E,V,N) and 2SEV=((T1)V)E,N=((pT1)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 \tt THIS\tt\ SPACE AVAILABLEdE=TdS+ydX+μdNiF=ETSdF=SdT+ydX+μdNiH=EyXdH=TdSXdy+μdNiG=ETSyXdG=SdTXdy+μdNiΩ=ETSμNdΩ=SdT+ydXNdμ .

Generalizing (p,V)(y,X), we also obtain, mutatis mutandis, the following Maxwell relations: (TX)S,N=(yS)X,N(TN)S,X=(μS)X,N(yN)S,X=(μX)S,NNN(Ty)S,N=(XS)y,N(TN)S,y=(μS)y,N(XN)S,y=(μy)S,NNN(SX)T,N=(yT)X,N(SN)T,X=(μT)X,N(yN)T,X=(μX)T,NNN(Sy)T,N=(XT)y,N(SN)T,y=(μT)y,N(XN)T,y=(μy)T,NNN(SX)T,μ=(yT)X,μ(Sμ)T,X=(NT)X,μ(yμ)T,X=(NX)T,μ .NN


This page titled 2.8: Maxwell Relations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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