The Equipartition Theorem

I'll be deriving this with a little help from the partition function and a particle in a box.
First let's find out what the partition function is for this box of volume:





The wave function for this configuration is of the form:

To find the eigenstates of the Free-Particle Hamiltonian, that is the allowed energy levels of this configuration, we simply use the Free-particle wave equation:

The partition function is given by:

Where the spacing between modes is relatively small compared to tau, this can be written in integral form and calculated:

Now we define the quantum concentration, the concentration associated with one atom in a cube of side equal to the thermal average de Broglie wavelength:

Where n_c is just the concentration of particles. As ideal gas is defined as a gas of noninteracting atoms in the classical regime, that is Z1 >> 1.

Now that we have the partition function, we can go about calculating our thermal average energy similarly to how we did in a previous post.

It is useful now to define the Helmholtz free energy function, which is just how much energy is free to use from a closed system at constant temperature and volume; how obtainable energy is with regards to entropy:







Where sigma is the entropy of the system, which is just the logarithm of the number of states accessible to the system.

From differential relations of F and the thermodynamic identities relating entropy, thermal average energy, pressure, and volume it can be shown that:











Using logarithmic rules and our latest formula for Z in terms of tau, we can note that Log(Z) has only one term involving tau, namely: -3/2log(1/tau). We have thus found the equipartition theorem, the energy per atom of an ideal gas:

Sources: Thermal Physics 2nd edition by Charles Kittel and Herbert Kroemer.