Euler equation and Gibbs-Duhem

Two results that follow directly from extensivity of entropy. See Euler equation derivation.

Euler equation

E = T S - P V + μ N

Derived by differentiating $S (c E, c V, c N) = c S (E, V, N)$ with respect to $c$ and setting $c = 1$ .

Gibbs-Duhem equation

S d T - V d P + N d μ = 0

Derived by taking $d (E = T S - P V + μ N)$ and subtracting $d E = T d S - P d V + μ d N$ .

Implies $T, P, μ$ are not independent — only two can be varied freely for a one-component system. Breaks down for systems with long-range interactions (gravity, black holes where $S \propto A$ not $V$ ).

Partial derivative identities from Gibbs-Duhem:

{(\frac{d P}{d T})}_{μ} = \frac{S}{V}, {(\frac{d T}{d μ})}_{P} = - \frac{N}{S}