# Energy-temperature relationship for classical, Fermi and Bose gasses

I have been trying to visualise the energy-temperature relationship for classical, fermi and bose gasses. For small T, I have that the Fermi gas avg. energy will be $$\frac{\epsilon_f}{2}$$, being half the Fermi energy. For Bosons, I have that they will all tend to zero energy and likewise for a classical ideal gas, but for Bosons to tend quicker with more occupying ground state at a higher temperature. I know that at high temperature both the Bosons and Fermions will tend towards the classical energy-temperature relationship.

For Fermions, this would be because the positive energy contribution from the Pauli exclusion will decrease, so less effective repulsion.

For Bosons, this would be because of the decrease in the negative average energy contribution from the effective attraction.

For the classical gas, I have derived the average energy-temperature relationship to $$\overline E=\frac{4kT}{\pi}$$ by plugging in the mean velocity of the Boltzmann distribution, $$\overline v=$$ $$(\frac{8kT}{\pi m})^{\frac{1}{2}}$$, into $$\frac{1}{2}m\overline v^2$$

I have found this pressure-temperature graph for the three gasses and think that it matches my expectations quite well for the energy-temperature relationships. I'm not sure of the energy-pressure relationships for the gasses at different temperatures but does this also accurately depict the energy-temperature relationships?: