= Euler relation in thermodynamics
{c}
{title2=$E=TS-PV+\mu N$}
= Thermodynamic Euler relation
{synonym}
For a single-component equilibrium system with extensive <internal energy> $E(S,V,N)$, scaling its <entropy>, <volume> and particle number together gives $E(\lambda S,\lambda V,\lambda N)=\lambda E(S,V,N)$. The <Euler theorem for homogeneous functions> therefore gives $E=S\partial_SE+V\partial_VE+N\partial_NE$. The <first law of thermodynamics> identifies these derivatives as $T,-P,\mu$, respectively, proving $E=TS-PV+\mu N$. Dividing by volume yields the <entropy density> formula $s=(\rho+P-\mu n)/T$. This step fixes the entropy normalization rather than merely identifying its differential. Differentiating the Euler relation and subtracting $dE=T\,dS-P\,dV+\mu\,dN$ gives the <Gibbs-Duhem equation> $S\,dT-V\,dP+N\,d\mu=0$. The argument requires extensivity; it is not an assumption about arbitrary nonadditive systems.
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