Extensive variable

An extensive variable is a state variable $X$ of a thermodynamic system that measures “how much” of the system is present: if you combine two weakly interacting subsystems, the total is (approximately) the sum, and if you scale the system up by a factor, $X$ scales by the same factor.

Let $X$ be a state variable defined on the thermodynamic state . We call $X$ extensive if it satisfies (to thermodynamic accuracy) the following two equivalent size-scaling properties:

1. Additivity (composition): for two macroscopic subsystems $A$ and $B$ that are non-overlapping and interact only weakly across the boundary ,
   

   $$X(A \cup B) \approx X(A) + X(B).$$

   This formalizes the additivity postulate .

2. Homogeneity of degree one (scaling): if you replicate the system by a factor $\lambda>0$ while keeping intensive control parameters fixed, then
   

   $$X(\lambda S,\lambda V,\lambda N,\dots) = \lambda\, X(S,V,N,\dots),$$

   i.e. $X$ is a homogeneous function of degree one in the extensive arguments.

Typical extensive variables include internal energy $U$, volume $V$, entropy $S$, and particle number $N$.

Physical interpretation

Extensive variables quantify the size or amount of “stuff” in the system: doubling the amount of substance (at the same macroscopic conditions) doubles $U$, $S$, $V$, and $N$ up to small boundary/interface corrections. This scaling viewpoint is most precise in the thermodynamic limit assumed by the extensivity postulate .

Key relations and consequences

• Contrast with intensive variables: Ratios of extensive variables often produce an intensive variable ; this idea is formalized by specific quantities and by densities such as number density .

• Euler relation (from extensivity): For a simple one-component system, extensivity of $U(S,V,N)$ implies the Euler relation

  $$U = TS - pV + \mu N,$$

  where $T$, $p$, and $\mu$ are the conjugate intensive variables (temperature , pressure , chemical potential ).

• Gibbs–Duhem constraint: Because extensive thermodynamics has fewer independent intensive variables than extensive ones, the intensive conjugates are not independent; this is expressed by the Gibbs–Duhem relation .