This should make sense. A system wants to minimize its total energy, and the fermions would pack into the lowest energy states first. If the temperature is low, there is little thermal excitation to promote any fermions to orbitals with higher energy.
We can solve for the Fermi energy by setting the total number of particles below that energy equal to the total number of particles in the system. We obtain:
We use the term "ground state" to refer to the state in which no fermions are excited to higher energy states beyond the Fermi energy. We can calculate the energy of the ground state by summing up the energies of the orbitals below the Fermi energy, to obtain:
We can go through and calculate all of the other relevant quantities just as we did for the ideal gas.
The Fermi gas appears throughout any study of physics. Electrons form a Fermi gas. The electrons in a metal, the "sea of electrons", act as a Fermi gas. In astrophysics, white dwarf stars are prevented from collapsing upon themselves by the pressure of the Fermi gas and the resistance it gives to having its orbitals pushed together.
A Bose gas is a gas consisting of bosons. We will treat the topic briefly as above with the Fermi gas.