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Maxwell relations from equality of mixed partial derivatives

For twice differentiable thermodynamic potentials, symmetry of mixed second derivatives yields Maxwell relations among conjugate variables.
Maxwell relations from equality of mixed partial derivatives

Statement

Let Φ(x1,…,xn)\Phi(x_1,\dots,x_n) be a thermodynamic potential expressed in its natural variables (e.g. Φ=F(T,V,N)\Phi=F(T,V,N), G(T,P,N)G(T,P,N), or Ω(T,V,μ)\Omega(T,V,\mu)). Assume Φ\Phi is C2C^2 (twice continuously differentiable) in those variables.

If the differential has the form

dΦ=∑i=1nai(x) dxi, d\Phi = \sum_{i=1}^n a_i(x)\,dx_i,

then each coefficient is a partial derivative ai=(∂Φ∂xi)xj≠ia_i = \left(\frac{\partial \Phi}{\partial x_i}\right)_{x_{j\neq i}}, and symmetry of the Hessian implies the identities

(∂ai∂xj)xk≠i,j=(∂aj∂xi)xk≠i,jfor all i≠j. \left(\frac{\partial a_i}{\partial x_j}\right)_{x_{k\neq i,j}} ={} \left(\frac{\partial a_j}{\partial x_i}\right)_{x_{k\neq i,j}} \quad\text{for all } i\neq j.

These are Maxwell relations (see ) once the aia_i are identified with conjugate variables via the standard thermodynamic definitions.

Canonical examples

  • For the F(T,V,N)F(T,V,N),

    dF=−S dT−P dV+μ dN, dF = -S\,dT - P\,dV + \mu\,dN,

    so S=−(∂F/∂T)V,NS= -(\partial F/\partial T)_{V,N} and P=−(∂F/∂V)T,NP= -(\partial F/\partial V)_{T,N}, hence

    (∂S∂V)T,N=(∂P∂T)V,N. \left(\frac{\partial S}{\partial V}\right)_{T,N} ={} \left(\frac{\partial P}{\partial T}\right)_{V,N}.

  • For the G(T,P,N)G(T,P,N),

    dG=−S dT+V dP+μ dN, dG = -S\,dT + V\,dP + \mu\,dN,

    giving

    (∂S∂P)T,N=−(∂V∂T)P,N. \left(\frac{\partial S}{\partial P}\right)_{T,N} ={} -\left(\frac{\partial V}{\partial T}\right)_{P,N}.

  • For the Ω(T,V,μ)\Omega(T,V,\mu),

    dΩ=−S dT−P dV−N dμ, d\Omega = -S\,dT - P\,dV - N\,d\mu,

    giving, for instance,

    (∂S∂V)T,μ=(∂P∂T)V,μ. \left(\frac{\partial S}{\partial V}\right)_{T,\mu} ={} \left(\frac{\partial P}{\partial T}\right)_{V,\mu}.

Key hypotheses

Key conclusions

  • Each Maxwell relation is an instance of ∂2Φ/∂xi∂xj=∂2Φ/∂xj∂xi\partial^2\Phi/\partial x_i\partial x_j = \partial^2\Phi/\partial x_j\partial x_i after substituting the derivative definitions of the conjugate variables.
  • Practically, these relations convert difficult-to-measure derivatives (e.g. derivatives of entropy) into derivatives of more accessible quantities (e.g. pressure–temperature slopes).