2.8: Maxwell Relations
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1$#1_$
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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=TdS−pdV+μdN ,
Relations deriving from F(T,V,N)
The energy F(T,V,N) is a state function, with dF=−SdT−pdV+μdN ,
Relations deriving from H(S,p,N)
The enthalpy H(S,p,N) satisfies dH=TdS+Vdp+μdN ,
Relations deriving from G(T,p,N)
The Gibbs free energy G(T,p,N) satisfies dG=−SdT+Vdp+μdN ,
Relations deriving from Ω(T,V,μ)
The grand potential Ω(T,V,μ) satisfied dΩ=−SdT−pdV−Ndμ ,
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+pTdV−μTdN .
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+μdN∑iF=E−TSdF=−SdT+ydX+μdN∑iH=E−yXdH=TdS−Xdy+μdN∑iG=E−TS−yXdG=−SdT−Xdy+μdN∑iΩ=E−TS−μNdΩ=−SdT+ydX−Ndμ .