The difference between the molar heat capacities at constant pressure ($C_p$) and constant volume ($C_v$) for an ideal gas is a constant value given by the gas constant $R$. This relationship is expressed mathematically as:

Here, $R$ is the universal gas constant, with a value of approximately 8.314 J/(mol·K).

The reason behind this relationship lies in the energy needed to perform work against the external pressure when the gas is heated at constant pressure, which does not happen in the case of heating at constant volume. When an ideal gas is heated at constant pressure, not only does the internal energy of the gas increase (which is also the case at constant volume) but some of the energy is also used to do work on the surroundings by expanding, which contributes to the larger value of $C_p$ compared to $C_v$. This difference is equal to the amount of work done per unit increase in temperature, which is quantitatively equivalent to the gas constant $R$.

Therefore, the expression indicates that the difference in heat capacity (which essentially is the difference in the ability to store thermal energy under different conditions) is fundamentally linked to the work done by the gas during expansion at constant pressure. The equation $C_p - C_v = R$ holds true for an ideal gas because it assumes no interactions between the molecules and that all processes are reversible.