20.12 : Heat Capacities of an Ideal Gas II

For a system that undergoes a thermodynamic process at a constant volume condition, the heat absorbed is used only to increase the system's internal energy and not for doing any kind of work. While for a system undergoing a thermodynamic process under a constant pressure condition, the amount of heat absorbed is used not only for increasing the internal energy (as a function of temperature) but also for doing some work. The molar heat capacity is the amount of heat required to increase the temperature of 1 mole of gas by 1 unit. Since some heat absorbed is used to do work, more heat should be added to a system under constant pressure to increase its temperature by 1 unit, the molar heat capacity for a constant pressure process is always greater than the molar heat capacity for a constant volume process. Consequently, from the derivation, which was based on the first law of thermodynamics, we get the following relationship, that is

Thus, the difference between the two molar heat capacities is a constant and is equal to the universal gas constant, R. It is also called the ideal gas constant or molar gas constant. Its value is approximately 8.314 and is expressed in the same SI unit as molar specific heat capacity J/mol⋅K. For example, for He, the measured values for C_p and C_V are 20.78 J/mol·K and 12.47 J/mol·K, respectively. Their difference is 8.31 J/mol·K which is close to R.