Volume

Definition and physical interpretation

The volume $V$ of a thermodynamic system is an extensive variable that measures the system’s spatial size and, in macroscopic equilibrium thermodynamics, serves as a state coordinate for “simple compressible” systems. Operationally, $V$ is determined by the system’s boundary (e.g., the container’s geometry for a fluid) and is well-defined when the macroscopic state is stationary and reproducible.

Volume is thermodynamically conjugate to pressure : changing $V$ against $p$ is the prototypical mechanical mode of energy exchange.

Role in the fundamental differential and work

For a simple compressible system in equilibrium , the fundamental differential in the energy representation $U=U(S,V,N)$ is

The appearance of $-p\,dV$ identifies $V$ as the coordinate whose quasistatic change produces pressure–volume work under the standard sign convention .

Because work is path-dependent , knowing only the initial and final volumes does not generally determine the mechanical work; it does so only under specified process conditions (e.g., quasistatic with a known $p(V)$ relation).

Extensivity, additivity, and densities

For macroscopic matter with short-range interactions, volume is typically additive across weakly coupled subsystems (see additivity ) and scales linearly under “copying” of the system (see the extensivity postulate ). These properties underlie the use of densities such as

• number density $n = N/V$,
• energy density $u = U/V$,
• entropy density $s = S/V$.

When long-range forces or strong surface effects matter, $V$ can remain a useful geometric variable, but extensivity/additivity may fail or require careful thermodynamic-limit conventions (see thermodynamic limit ).

Relations to response functions and equations of state

Material behavior is encoded in an equation of state relating $V$ to other state variables, e.g. $p=p(T,V,N)$ or $V=V(T,p,N)$. Two standard response coefficients that quantify how volume reacts to changes in intensive controls are:

• isothermal compressibility $\kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T,N},$
• thermal expansion coefficient $\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p,N}.$

Stability and equilibrium constraints (see stability ) restrict these derivatives; for example, $\kappa_T\ge 0$ is required for mechanical stability.