# Heat Capacities of an Ideal Gas II

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Heat Capacities of an Ideal Gas II

### Nächstes Video20.13: Heat Capacities of an Ideal Gas III

Consider two identical flasks filled with an ideal gas at different temperatures that can undergo different thermodynamic processes.

Suppose, in a constant-volume process, the first flask's gas temperature is increased to equate to the temperature of the second flask.

Then, its internal energy will be the same as the heat change and can be written in terms of molar heat capacity.

In a constant-pressure process, if the temperature is increased by the same amount, then the system's internal energy changes with some work being done.

As the internal energy is a state function, the internal energy change of the system is the same under both processes.

Using the ideal gas equation in differential form, the work done can be written in terms of temperature change.

By canceling the common factors on both sides, the relation between the molar heat capacities for both processes is obtained.

Here, R is the universal gas constant, and hence, the molar heat capacity of an ideal gas under a constant-pressure process is always greater than that under a constant-volume process.

## 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 Cp and CV 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.