Derivation of Calculation for the Ratio of Specific Heats of Gases

For ideal gases the heat capacity is:

The relationship between the two heat capacities is:

The first law of thermodynamics is defined as:

If there is no heat exchange between a system and its surroundings then it is an adiabatic process. This results in:

For ideal gases the following derivation can be applied:

Using this derivation and the equations above:

            Therefore, by a manipulation of equations one can easily see that the ratio of the specific heats at constant pressure and constant volume (γ) can be obtained from plotting the natural log of pressure against the natural log of volume of a substance. Gamma (γ) will be equal to the slope of that graph.