Relations Between Heat Capacities - Ideal Gas

Ideal Gas

This is a derivation to obtain an expression for for an ideal gas.

An ideal gas has the equation of state:

where

P = pressure

V = volume

n = number of moles

R = universal gas constant

T = temperature

The ideal gas equation of state can be arranged to give:

or

The following partial derivatives are obtained from the above equation of state:

The following simple expressions are obtained for thermal expansion coefficient :

and for isothermal compressibility :

One can now calculate for ideal gases from the previously-obtained general formula:

Substituting from the ideal gas equation gives finally:

where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows:

This result would be consistent if the specific difference were derived directly from the general expression for .