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 $\Phi(x_1,\dots,x_n)$ be a thermodynamic potential expressed in its natural variables (e.g. $\Phi=F(T,V,N)$ , $G(T,P,N)$ , or $\Omega(T,V,\mu)$ ). Assume $\Phi$ is $C^2$ (twice continuously differentiable) in those variables.

If the differential has the form

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

then each coefficient is a partial derivative $a_i = \left(\frac{\partial \Phi}{\partial x_i}\right)_{x_{j\neq i}}$ , and symmetry of the Hessian implies the identities

\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 Maxwell relations ) once the $a_i$ are identified with conjugate variables via the standard thermodynamic definitions.

Canonical examples

For the Helmholtz free energy $F(T,V,N)$ ,
$dF = -S\,dT - P\,dV + \mu\,dN,$
so $S= -(\partial F/\partial T)_{V,N}$ and $P= -(\partial F/\partial V)_{T,N}$ , hence
$\left(\frac{\partial S}{\partial V}\right)_{T,N} ={} \left(\frac{\partial P}{\partial T}\right)_{V,N}.$
For the Gibbs free energy $G(T,P,N)$ ,
$dG = -S\,dT + V\,dP + \mu\,dN,$
giving
$\left(\frac{\partial S}{\partial P}\right)_{T,N} ={} -\left(\frac{\partial V}{\partial T}\right)_{P,N}.$
For the grand potential $\Omega(T,V,\mu)$ ,
$d\Omega = -S\,dT - P\,dV - N\,d\mu,$
giving, for instance,
$\left(\frac{\partial S}{\partial V}\right)_{T,\mu} ={} \left(\frac{\partial P}{\partial T}\right)_{V,\mu}.$

Key hypotheses

The system is described by a state function $\Phi$ in equilibrium (see thermodynamic equilibrium ).
$\Phi$ is $C^2$ in its natural variables, so mixed partial derivatives commute.
Conjugate variables are defined by partial derivatives of the potential (e.g. temperature , pressure , chemical potential ).

Key conclusions

Each Maxwell relation is an instance of $\partial^2\Phi/\partial x_i\partial x_j = \partial^2\Phi/\partial x_j\partial x_i$ 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).

Cross-links to definitions

Proof idea / significance

Since $\Phi$ is a state function, $a_i=\partial\Phi/\partial x_i$ . If $\Phi$ is $C^2$ , then mixed derivatives commute:

\frac{\partial}{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right) ={} \frac{\partial}{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right).

Identifying $a_i$ with (signed) conjugate variables via the differential (e.g. $-S$ is the coefficient of $dT$ in $dF$ ) yields the standard Maxwell relations.

Significance: Maxwell relations are consistency conditions expressing the integrability of thermodynamics, and they underpin many experimentally useful identities.