For a perfect gas, the relationship between heat capacities is:

For a perfect gas, the relationship between the heat capacities at constant pressure (C_p) and constant volume (C_V) is defined by the equation C_p − C_V = nR. This equation arises from the first law of thermodynamics and the definitions of heat capacities.

C_p represents the heat capacity at constant pressure, which quantifies the amount of heat required to raise the temperature of the gas by one degree Celsius while maintaining constant pressure. Conversely, C_V signifies the heat capacity at constant volume, indicating the heat required to raise the temperature of the gas by one degree Celsius while the volume remains constant.

The term nR in the equation accounts for the number of moles of the gas (n) and the ideal gas constant (R), representing the additional heat required to account for the work done when the pressure of the gas remains constant during expansion. Thus, when transitioning from a constant volume process to a constant pressure process, the difference in heat capacities reflects this work done on or by the gas.

The relationship is foundational in thermodynamics and is crucial for various applications, including understanding the behavior of gases under different thermodynamic processes.