Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.6.
The definition of temperature in terms of entropy is
Schroeder quotes a theorem that states that for any system with only quadratic degrees of freedom in the high temperature limit, the multiplicity is proportional to where is the energy, is the number of molecules and is the number of degrees of freedom per molecule. For an ideal gas, this is given by Schroeder’s equation 2.40:
Since for a monatomic ideal gas (3 translational degrees of freedom only), the theorem is valid here. Similarly, for an Einstein solid in the high temperature limit
and since , the number of energy quanta, is proportional to . Each oscillator in an Einstein solid is equivalent to a one-dimensional harmonic oscillator, which has two degrees of freedom. [Rather, it has two quadratic terms in its energy: for potential energy and for its kinetic energy. Each of these is interpreted as a ‘degree of freedom’.] Thus again, .
In the general case, we have
for some constant (that is, doesn’t depend on ). The entropy is therefore
and the temperature is given by
which is just what the equipartition theorem predicts ( energy per degree of freedom).
The formula is valid only for large energies, since if is small enough, eventually becomes less than 1 and the entropy 5 can become negative, which isn’t physically possible.