Specific quantity

A specific quantity is a “per-unit-amount” version of an extensive variable . Dividing by an appropriate extensive reference makes the result independent of system size (for homogeneous systems), so specific quantities are typically intensive .

Let $X$ be an extensive variable and let $Y>0$ be an extensive reference measure of system size (commonly mass $m$, particle number $N$, or volume $V$). The specific quantity of $X$ per $Y$ is

When the system is homogeneous (no macroscopic gradients), scaling the system by a factor $\lambda$ multiplies both $X$ and $Y$ by $\lambda$, so the ratio $x$ is unchanged; this is why $x$ behaves as an intensive variable .

Physical interpretation

Specific quantities encode material properties “per unit amount,” which is often what experiments control or what constitutive laws use. For example, energy per particle compares energetic content to the number of particles present, while energy per volume compares energetic content to occupied space.

Common thermodynamic examples

• Per particle: $U/N$ (specific internal energy per particle), $S/N$ (entropy per particle).
• Per volume (densities): number density $N/V$, energy density $U/V$, entropy density $S/V$.
• Per mass (common engineering usage): $U/m$, $S/m$, and the specific volume $V/m$.

Key property: weighted averaging under composition

If a system is partitioned into two parts $A$ and $B$ with the same definition of $x=X/Y$, then

So a specific quantity is an average of subsystem values weighted by the reference amount $Y$.