Why is Cp always greater than Cv?

The relationship between Cp (heat capacity at constant pressure) and Cv (heat capacity at constant volume) is fundamentally linked to the thermodynamic processes that occur under these conditions. Cp is always greater than Cv due to the nature of the work done when a gas is heated at constant pressure.

When heat is added to a gas at constant volume, all of that heat contributes to increasing the temperature of the gas, and none is used for work. However, when heat is added to a gas at constant pressure, part of that heat goes into doing work against the external pressure as the gas expands. This expansion does not occur at constant volume, which means that additional energy is required for the gas to do this work. Therefore, the extra heat needed for this expansion at constant pressure is what results in Cp being greater than Cv.

This distinction is a key concept in thermodynamics, as it highlights how the work done by a system affects its energy changes under different conditions. Hence, it is clear why Cp is always greater than Cv; the heat required at constant pressure includes both the energy needed to increase temperature and the energy required to do work against the surrounding atmosphere.