Euler equation derivation

Starting point: $S (E, V, N)$ is extensive: $S (c E, c V, c N) = c S (E, V, N)$

Differentiate both sides with respect to $c$ :

\frac{\partial S}{\partial c E} \cdot E + \frac{\partial S}{\partial c V} \cdot V + \frac{\partial S}{\partial c N} \cdot N = S (E, V, N)

Set $c = 1$ and substitute the definitions $\frac{\partial S}{\partial E} = \frac{1}{T}$ , $\frac{\partial S}{\partial V} = \frac{P}{T}$ , $\frac{\partial S}{\partial N} = - \frac{μ}{T}$ :

\frac{E}{T} + \frac{P}{T} V - \frac{μ}{T} N = S

Result:

E = T S - P V + μ N

Gibbs-Duhem — take total derivative of the Euler equation:

d E = T d S + S d T - P d V - V d P + μ d N + N d μ

Subtract the fundamental relation $d E = T d S - P d V + μ d N$ :

S d T - V d P + N d μ = 0