Extensive quantity

An extensive quantity is one that is proportional to the amount of material: combine two identical systems and the quantity doubles. Mass, volume, energy, particle number, and entropy are all extensive. Extensivity is precisely what forces entropy to be the logarithm of multiplicity, since multiplicities multiply when systems combine while an extensive entropy must add.

Extensive quantities contrast with intensive ones, which are independent of system size. The ratio of two extensive quantities is intensive — energy per unit volume (energy density) or entropy per particle, for example, do not change when the system is scaled. Many thermodynamic relations pair an intensive variable with the extensive variable it is conjugate to: temperature with entropy, pressure with volume, chemical potential with particle number.

The distinction is a basic organising principle of thermodynamics and underlies the structure of the fundamental thermodynamic relation dU = T dS − P dV + μ dN.

The systematic classification of thermodynamic variables into extensive and intensive types is due largely to the formalisation of thermodynamics by Gibbs in the 1870s.