Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problem 5.16.

An expression similar to that relating the heat capacities can be derived to relate the isothermal and isentropic compressibilities and , defined as

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

Substituting 3 into 4 we get

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

which is the same as 21, so equation 15 checks out for an ideal gas.

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