Standard derivative identities from Maxwell relations

Let

• $U$ be internal energy ,
• $F$ be Helmholtz free energy ,
• $G$ be Gibbs free energy ,
• $\Omega$ be the grand potential .

Let $S$, $T$, $P$, and $\mu$ denote entropy , temperature , pressure , and chemical potential , respectively.

Assume equilibrium and sufficient smoothness so that mixed partial derivatives commute, as in Maxwell's relations theorem .

Statement

For a simple compressible system with particle number $N$, the potentials in their natural variables satisfy

Therefore the conjugate variables are recovered by partial differentiation, for example:

• from $F(T,V,N)$,

  $$S = -\left(\frac{\partial F}{\partial T}\right)_{V,N},\qquad P = -\left(\frac{\partial F}{\partial V}\right)_{T,N},\qquad \mu = \left(\frac{\partial F}{\partial N}\right)_{T,V};$$

• from $G(T,P,N)$,

  $$S = -\left(\frac{\partial G}{\partial T}\right)_{P,N},\qquad V = \left(\frac{\partial G}{\partial P}\right)_{T,N},\qquad \mu = \left(\frac{\partial G}{\partial N}\right)_{T,P};$$

• from $\Omega(T,V,\mu)$,

  $$S = -\left(\frac{\partial \Omega}{\partial T}\right)_{V,\mu},\qquad P = -\left(\frac{\partial \Omega}{\partial V}\right)_{T,\mu},\qquad N = -\left(\frac{\partial \Omega}{\partial \mu}\right)_{T,V}.$$

Commuting mixed partials yields the standard Maxwell relations, e.g. from $F(T,V,N)$,

Key hypotheses

• The equilibrium state admits differentiable thermodynamic potentials.
• Each potential is expressed in its natural independent variables (obtained via a Legendre transform from $U$).
• The relevant second derivatives exist and mixed partial derivatives commute.

Key conclusions

• $S$, $P$, $V$, $N$, and $\mu$ are partial derivatives of a single potential in natural variables.
• Maxwell relations provide measurable identities between derivatives (often turning difficult entropy derivatives into accessible pressure/volume derivatives).

Proof idea / significance

Differentiate the fundamental identity for $U(S,V,N)$ and apply the defining Legendre transforms to obtain $F$, $G$, and $\Omega$. The displayed differential forms follow by direct differentiation. Maxwell relations then come from commuting mixed partial derivatives (see Maxwell relations from mixed partials ). These identities are the primary computational bridge between different experimentally accessible response functions.