Heat capacity at constant pressure

For a simple thermodynamic system in equilibrium , the heat capacity at constant pressure is a response function that quantifies how much the system’s energy-like content must change to raise the temperature while keeping the pressure fixed (and keeping composition fixed, e.g. fixed particle number $N$).

It is defined by the partial derivative

where $H$ is the enthalpy ,

with $U$ the internal energy and $V$ the volume .

Using the standard differential for $H$ for a simple compressible system, at fixed $p$ and $N$ one has $dH = T\,dS$, so an equivalent form is

in terms of the entropy .

Physical interpretation

Along a quasistatic , reversible process at fixed pressure,

Compared with heating at fixed volume (see constant-volume heat capacity ), maintaining constant pressure generally allows the system to expand as it warms. Part of the supplied heat is then “spent” on the $p\,dV$ expansion, which is why $C_P$ is typically larger than $C_V$ in stable single-phase regions.

Key relations and properties

• Extensivity: $C_P$ is typically extensive ; one often uses a specific form such as $c_P = C_P/N$ (or per unit mass).

• Gibbs free-energy curvature: In terms of the Gibbs free energy $G(T,p,N)$, one has

  $$C_P = -\,T\left(\frac{\partial^2 G}{\partial T^2}\right)_{p,N}.$$

• Difference from $C_V$: For a simple compressible system,

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

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

• Inequality (typical): In stable single-phase regions with $\kappa_T>0$, one usually has $C_P \ge C_V$, consistent with stability .

• Fluctuation formula (statistical mechanics): In the isothermal–isobaric description (fixed $T$, $p$, $N$), $C_P$ is proportional to the variance of the enthalpy as a random variable:

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

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