Heat capacity tells us how much heat energy we need to add to raise the temperature of a substance by one degree. There are two important types based on what we keep constant during heating. Isochoric heat capacity, denoted C V, measures heat needed when volume stays constant, like in a rigid container. Isobaric heat capacity, denoted C P, measures heat needed when pressure stays constant, like with a movable piston.
In a constant volume process, we heat a gas inside a rigid container that cannot expand or contract. Since the volume stays fixed, all the heat energy we add goes directly into increasing the internal energy of the gas molecules. The gas cannot do any work on its surroundings because there is no volume change. This makes heating at constant volume very efficient for raising temperature.
In a constant pressure process, the gas is free to expand as we heat it. The pressure stays constant, like atmospheric pressure pushing on a movable piston. When we add heat, some energy increases the internal energy of the gas, but some energy also goes into doing expansion work against the constant pressure. This means we need more heat to achieve the same temperature rise compared to constant volume heating. Therefore, C P is always greater than C V.
The key relationship is that C P is always greater than C V for any substance. This happens because at constant pressure, the added heat must do two things: increase the internal energy of the substance and also do work against the constant pressure during expansion. For an ideal gas, the difference between C P and C V equals the gas constant R. The ratio of C P to C V is called gamma, which is an important parameter in thermodynamics.