Heat capacity at constant volume

For a closed, simple thermodynamic system in thermodynamic equilibrium , the heat capacity at constant volume is the response function that quantifies how the internal energy changes with the temperature when the volume is fixed (and the composition is fixed, e.g. fixed particle number $N$ for a single-component system).

The heat capacity at constant volume is

Because $U$ and $T$ are state functions (for equilibrium states), $C_V$ is a property of the equilibrium state .

A useful equivalent form follows from the fundamental energy differential for a simple compressible system, which implies that at fixed $V$ and $N$ one has $dU = T\,dS$. Hence

where $S$ is the thermodynamic entropy .

Physical interpretation

Along a quasistatic , reversible process in which $V$ and $N$ are held fixed, the heat increment satisfies

This is why $C_V$ is often described as “the heat required per unit temperature rise at constant volume”: the container is effectively rigid, so there is no $p\,dV$ expansion and the energy input shows up entirely as a change in internal energy.

Key relations and properties

• Extensivity: $C_V$ is typically an extensive quantity. One often uses a specific or per-particle form such as $c_V = C_V/N$ (or per unit mass).

• Free-energy curvature: In terms of the Helmholtz free energy $F(T,V,N)$, one has

  $$C_V = -\,T\left(\frac{\partial^2 F}{\partial T^2}\right)_{V,N}.$$

• Stability (typical sign): For stable single-phase equilibrium states, one expects $C_V \ge 0$; this is tied to thermodynamic stability and can be viewed as a consequence of entropy concavity (or equivalently energy convexity in appropriate variables).

• Difference from constant-pressure heat capacity: The difference between $C_V$ and the constant-pressure heat capacity is

  $$C_P - C_V = \frac{T\,V\,\alpha^2}{\kappa_T},$$

  where $\alpha$ is the thermal expansion coefficient and $\kappa_T$ is the isothermal compressibility .

• Fluctuation formula (statistical mechanics): In the canonical description at fixed $T$, $V$, and $N$, $C_V$ can be written in terms of the variance of the energy as a random variable:

  $$C_V = \frac{1}{k_B T^2}\left(\langle E^2\rangle - \langle E\rangle^2\right),$$

  where $k_B$ is the Boltzmann constant and $\langle\cdot\rangle$ denotes the ensemble expectation .