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Maxwell relations theorem

Because thermodynamic potentials are state functions (exact differentials), their mixed second derivatives agree, yielding Maxwell relations among response functions.

Maxwell relations theorem

Statement

Let $\Phi=\Phi(x_1,\dots,x_n)$ be a thermodynamic potential expressed in its natural variables, and assume $\Phi$ is twice continuously differentiable. If its differential has the form

d\Phi = \sum_{i=1}^n A_i\,dx_i,

then exactness of $d\Phi$ implies symmetry of mixed partial derivatives:

\frac{\partial A_i}{\partial x_j} = \frac{\partial A_j}{\partial x_i} \qquad (i\neq j).

These identities are called Maxwell relations.

For a simple one-component system (fixed particle number $N$ ) the standard thermodynamic potentials yield the familiar relations:

From internal energy $U(S,V)$ :
$dU = T\,dS - P\,dV, \qquad\Rightarrow\qquad \left(\frac{\partial T}{\partial V}\right)_{S} = -\left(\frac{\partial P}{\partial S}\right)_{V}.$
From Helmholtz free energy $F(T,V)$ :
$dF = -S\,dT - P\,dV, \qquad\Rightarrow\qquad \left(\frac{\partial S}{\partial V}\right)_{T} = \left(\frac{\partial P}{\partial T}\right)_{V}.$
From Gibbs free energy $G(T,P)$ :
$dG = -S\,dT + V\,dP, \qquad\Rightarrow\qquad \left(\frac{\partial S}{\partial P}\right)_{T} = -\left(\frac{\partial V}{\partial T}\right)_{P}.$

(Analogous relations hold when including particle number and chemical potential among the variables.)

Key hypotheses and conclusions

Hypotheses

The system admits an equilibrium thermodynamic description (states in thermodynamic equilibrium ).
Thermodynamic potentials are well-defined state functions and are $C^2$ in their natural variables.
The differential forms written for $dU,dF,dG,\ldots$ hold (typically derived from the first and second laws plus Legendre transforms).

Conclusions

Mixed partial derivatives of potentials coincide, producing Maxwell relations.
Maxwell relations provide experimentally useful identities that connect hard-to-measure derivatives (often involving entropy) to easier ones (often involving pressure/volume/temperature).

Cross-links to definitions

Thermodynamic objects:

internal energy , Helmholtz free energy , Gibbs free energy , grand potential .
entropy , temperature , pressure , chemical potential .

Structural/mathematical links:

Maxwell relation (definition) .
Legendre transform (thermodynamic potentials as Legendre transforms of $U$ ).
Maxwell-from-mixed-partials proposition and standard Maxwell identities corollary .

Proof idea / significance

Idea. A thermodynamic potential is a state function, so its differential is exact (see exact differential criterion ). Exactness implies equality of mixed second derivatives (Clairaut/Schwarz theorem), which directly produces the Maxwell relations once the coefficients in $dU,dF,dG,\ldots$ are identified with $T,S,P,V,\mu,N$ .

Significance. Maxwell relations are consistency conditions for equilibrium thermodynamics and a practical computational tool: they turn “entropy derivatives” into measurable response derivatives and are central in deriving identities for heat capacities, compressibilities, and susceptibilities.