Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.5.
The definition of temperature in terms of entropy is
Given a formula for the entropy of a system, we can use this relation to work out its temperature. As an example, we’ll look at the Einstein solid in the low temperature case where the number of energy quanta (each of size ) is much less than the number of oscillators: . The multiplicity of such a system is approximately
The total energy of the system is so we can write this in terms of as
so the entropy is
The partial derivative in 1 implies that we’re holding fixed, so we get, using the product rule:
Since we assumed , this is equivalent to requiring , so this result is valid only for low temperatures, as we’d expect. [Note that for high temperatures, so which violates the assumption .]
Schroeder works out the energy-temperature relation for the other extreme in his section 3.1, with the result
In this case, there are enough energy quanta that every degree of freedom in all oscillators is excited and since there are two degrees of freedom per oscillator, this agrees with the equipartition theorem which says that every degree of freedom has an associated of kinetic energy.