Heat capacity at constant volume and pressure (C_v, C_p)

For a gas, the heat capacity depends on the conditions under which heat is added. At constant volume (C_v) all the heat raises the internal energy, hence the temperature. At constant pressure (C_p) the gas also expands and does work on its surroundings, so more heat is needed for the same temperature rise. Consequently C_p is always greater than C_v.

For an ideal gas the difference is exactly the gas constant, Mayer's relation C_p − C_v = R, because the expansion work per mole per kelvin at constant pressure is precisely R. The ratio γ = C_p / C_v (the adiabatic index, 5/3 for a monatomic gas and 7/5 for a diatomic gas at room temperature) governs adiabatic compression and the speed of sound.

The two capacities are a clean demonstration that heat can leave a system as work rather than temperature, a distinction central to the first law and to the analysis of engines and adiabatic processes.

The constant-pressure/constant-volume distinction and the relation C_p − C_v = R were established by Julius Robert von Mayer (1842) and developed in the kinetic theory of gases.