Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 5.13 – 5.15.
We can use the Maxwell relations to derive some formulas relating to heat capacities. First, recall the thermal expansion coefficient
which is the fractional change in volume per Kelvin, assumed to be a constant pressure. For small changes, we can write this as a partial derivative:
From the Maxwell relation derived from the Gibbs energy:
The third law of thermodynamics says that as , . I’m not quite sure how we can use this to show that as since it’s not clear how entropy depends on pressure at low temperatures. However, if for any system as , then presumably it must be true no matter what the pressure is, so in that sense, is independent of pressure and then . I’m not sure that constitutes a ‘proof’ as requested in Schroeder’s problem, though.
We’re on firmer ground when we wish to derive a relation between the heat capacities (constant volume) and (constant pressure). In terms of entropy, they are
If we write then
Also, starting with we have
Inserting this into 7 and setting (constant pressure) gives
where we’ve used 5 and 6 in the last line.
From the Helmholtz energy Maxwell relation
Using 2 with its relation to the isothermal compressibility:
For an ideal gas, , so 13 becomes
which agrees with Schroeder’s equation 1.48.
From 15, we can see that provided that the term is always positive. The numerator is certainly positive, and from the definition of the isothermal compressibility
we see that it is positive provided that an increase in pressure causes a decrease in volume, which is pretty well always true for any realistic material. Since for low temperatures and as we wouldn’t expect or to change much for very low temperatures (where we’d expect most substances to be solid, or possibly liquid, such as helium), then we’d expect for low temperatures, with as temperature increases. This agrees with Schroeder’s Figure 1.14.
To put in some numbers, we can use the data from the earlier problem. For water, the values given by Schroeder are (at ):
For one mole of water, the volume is , so
for one mole of water so the difference between the heat capacities is around 1% of .
For mercury we have
One mole of mercury has a volume of so we get
The heat capacity for one mole of mercury (from the appendix to Schroeder’s book) is so the difference is around 12.8% of .
Finally, we can derive 15 by starting with and instead of and . The heat capacities are defined in terms of internal energy and enthalpy as
Writing we have
The enthalpy is defined as , so at constant pressure
Dividing through by at constant pressure gives
The Helmholtz free energy is defined as from which we can derive the relation
Using this, we have
This is the same as 11, so from here on the derivation is the same as before, and we get 15 again.