Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.8.
If we know the energy of a system as a function of temperature, we can predict its heat capacity at constant volume using
For an Einstein solid at low temperature, we have
where is the magnitude of one energy quantum. The heat capacity is therefore
We can plot versus to see the shape of the curve:
The Einstein solid has a peak in its heat capacity at a very low temperature of (differentiate with respect to and set to zero). However, as Schroeder points out, this isn’t the observed heat capacity of real solids at low temperatures.
This method of predicting heat capacity relies on being able to find , which so far we’ve been able to do only by finding an expression for the multiplicity of states in terms of the energy (and other parameters), then using that to find the entropy , then finding from the expression . In most cases, finding an expression for the multiplicity of states is not possible, so we need another method of getting which we’ll hopefully get to eventually.