Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 5.16.
These quantities measure the fractional change in volume of a substance in response to a change in pressure. To obtain the relation between them, we use a method similar to that for heat capacities and .
If we write then
Also, starting with we have
At constant temperature and we get
From the Maxwell relation from the Gibbs energy
Also, from the definition of the thermal expansion coefficient
Combining these last two equations gives
To get rid of the last partial derivative, we observe that the volume change due to a temperature change at constant pressure is
The entropy change due to an influx of heat at constant pressure at temperature is
Dividing these two relations gives
Inserting this into 10 and cancelling off a factor of gives the final result
For an ideal gas, we can use this equation to work out :
where in the third line, we’ve used Schroeder’s equation 1.48, and is the number of degrees of freedom of each gas molecule.
To check this, recall that for an isentropic (adiabatic) process in an ideal gas
where and is a constant. So