Now this simply gives us a constant amount of energy given by one half

times the Boltzmann constant, times the temperature.

A similar analysis can be done for the normal modes, and

those yield the same fixed amount of energy.

Hence the average energy for

a classical system is given by the sum of the degrees of freedom.

Now what is this for an ideal gas?

And what is it for a solid?

Now when we have a monatomic ideal gas, now m takes the value 0 and

n takes the value of 3n where 3n is the number of degrees of freedom for

the translational energy.

Hence all the energy is simply going to contain in the kinetic energy.

And this gives the well-known average energy for a monatomic gas of 3 pi 2 n k3.

Now on the other hand for a solid,

in addition to the kinetic energy, there's an energy contained in the vibrations.

Hence this leads to both m and n being equal to 3n.

Where n is the number of particles.

Now this leads to the average energy of a solid

that is twice as much as that of a monatomic ideal gas.