Euler Relation in Thermodynamics

Definition and physical interpretation

Consider a single-phase thermodynamic system whose equilibrium states admit a fundamental relation in the energy representation of the form $U=U(S,V,N)$, with $(S,V,N)$ being extensive variables .

If $U(S,V,N)$ is homogeneous of degree one (i.e. extensive, as assumed in the thermodynamic limit with negligible boundary contributions), then Euler’s theorem implies the Euler relation

Here $T$, $P$, and $\mu$ are the conjugate intensive variables : the temperature $T$, pressure $P$, and chemical potential $\mu$, while $S$ is the thermodynamic entropy , $V$ the volume , and $N$ the particle number .

Physically, the relation says that for an extensive system the internal energy is a sum of “intensive $\times$ extensive” contributions. The sign of the mechanical term matches the chosen pressure–volume work sign convention (and the chemical term matches the chemical work convention ).

An equivalent form in the entropy representation $S=S(U,V,N)$ is

Key consequences

• Combining the Euler relation with the definitions of common state functions gives useful identities, e.g. using the definitions of enthalpy , Helmholtz free energy , and Gibbs free energy :

  $$H=U+PV=TS+\mu N,\qquad F=U-TS=-PV+\mu N,\qquad G=U-TS+PV=\mu N.$$

• Differentiating the Euler relation and comparing with the exact differential from the first law ,

  $$dU = T\,dS - P\,dV + \mu\,dN,$$

  yields the Gibbs–Duhem relation , which constrains the intensive variables.