The Relationship between Cp and CV for an Ideal Gas

The Relationship between Cp and CV for an Ideal Gas

The heat capacity of a system depends on the conditions under which the heating occurs. Two special cases are particularly important:

• Heat capacity at constant volume, $C_V$: the heat required to raise the temperature by one degree while keeping the volume fixed.
• Heat capacity at constant pressure, $C_p$: the heat required to raise the temperature by one degree while keeping the pressure fixed.

At constant volume, $q_V = C_V\Delta T = \Delta U$ (since no work is done). At constant pressure, $q_p = C_p\Delta T = \Delta H$.

For an ideal gas, we can derive a simple relationship between $C_p$ and $C_V$. Consider one mole of an ideal gas. From the definition of enthalpy:

For an ideal gas, $pV = RT$ (for one mole), so $\Delta(pV) = \Delta(RT) = R\Delta T$ (since $R$ is constant). Therefore:

Now substitute $\Delta H = C_p\Delta T$ and $\Delta U = C_V\Delta T$:

Dividing through by $\Delta T$ (which is non-zero):

This is a fundamental result for ideal gases. The molar heat capacity at constant pressure is always larger than that at constant volume by exactly the gas constant $R$ (approximately $8.314\ \text{J mol}^{-1}\text{K}^{-1}$).