Intensive variable

Definition and physical interpretation

An intensive variable is a state variable whose value does not scale with the overall size of the system . Concretely, under a uniform rescaling of extensive quantities (e.g. doubling the amount of material so that $S$, $V$, and $N$ all double), intensive variables remain unchanged in the thermodynamic limit .

This contrasts with an extensive variable , which scales linearly with system size (entropy $S$, volume $V$, particle number $N$, internal energy $U$, and so on).

Intensive variables act like “generalized forces” that drive exchanges between weakly interacting subsystems. At equilibrium they equalize:

• thermal equilibrium enforces equality of temperature $T$.
• mechanical equilibrium enforces equality of pressure $p$.
• chemical equilibrium enforces equality of chemical potential $\mu$.

Conjugate derivatives from the fundamental relation

If a single-component simple compressible system admits a fundamental relation $U(S,V,N)$, the standard intensive variables arise as partial derivatives:

These relations explain why intensive variables appear as natural control parameters in thermodynamic potentials such as the Helmholtz free energy (natural in $T$), the Gibbs free energy (natural in $T$ and $p$), the enthalpy (natural in $p$), and the grand potential (natural in $T$ and $\mu$).

Homogeneity, Euler, and Gibbs–Duhem

When the system satisfies extensivity so that $U(S,V,N)$ is a homogeneous function of degree one , the Euler relation takes the form

showing $U$ as a sum of intensive–extensive products.

A key consequence is that the intensive variables are not all independent: their allowed variations are constrained by the Gibbs–Duhem relation (single component),

which encodes that changing one intensive variable generally forces changes in the others when the composition is fixed.