Consider a quantity of gas in a container which has a frictionless piston as shown in the diagram below. The volume occupied by the gas is changed by V.

The pressure, p, acting on the surface of the piston produces a force, F, equal to pA. During the change in volume, this force does some work.

The work done by this force is w = Fs = pAs

but As is the change in the volume occupied by the gas, V. Therefore

.....

The above result shows that if the temperature of a gas is increased at constant volume, no work is done.

However, if the temperature is increased and the gas is allowed to expand, work will be done. In this case, extra energy will have to be supplied to do this work.

For this reason, gases are said to have two principal specific (or molar) heat capacities:

It should be clear that c_p > c_v and that the difference between them is given by

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