Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.9.
The entropy is related to temperature by
Using the chain rule, and keeping everything at constant and , we can measure the change in entropy due to a change in temperature as
where is the heat capacity at constant volume:
If we know as a function of temperature, we can therefore find the change in entropy for a finite change in temperature by integration:
The total entropy in a system at temperature could theoretically be found by setting in the integral
In theory, at absolute zero, any system should be in its (presumably) unique lowest energy state so the multiplicity of the zero state is 1, meaning that , and this integral does in fact give the actual entropy in a system at temperature . It’s also obvious that for this integral to be finite (and positive) as at a rate such that the integral doesn’t diverge at its lower limit. Thus we must have where as . Either of these conditions is a statement of the third law of thermodynamics, which basically says that at absolute zero, the entropy of any system is zero.
In practice, as a substance is cooled, its molecular configuration can get frozen into one of several possible ground states, so that there is a residual entropy even when .
Example Carbon monoxide molecules are linear and in the solid form, they can line up in two orientations: OC and CO. Thus at absolute zero, the collection of molecules can be considered as a frozen-in matrix of molecules oriented randomly, so for a sample of molecules, there are possible structures. For a mole, the residual entropy is therefore