Maxwell-Boltzmann distribution and total energy per unit volume

Question

We know that

$$n(E) ~=~ \frac {2 \pi (N/V)}{(\pi k_B T)^{3/2}} E^{1/2} e^{-E/(k_B T)} dE,$$

where $V$ is total volume.

If then, how do we derive total energy per unit volume from this equation?

Answer 1

Integrate n(E)*E over all possible energies and divide the result by the total volume, this gives the average energy per unit volume.

Ali

Answer 2

You should first wonder what is the total energy per unit volume. Your formula gives n(E): the density of particles whose energy is between E and E+dE. There is n(E)*V such particles, each one carris the energy E. Their combined energy is thus: n(E)*V*E. Integrate over E to get the total energy of your system. At last divide by V to get the average energy by unit volume.