What is the compressibility of a gas of free fermions at zero temperature?

It is straightforward to show that pressure of a completely degenerate system of free fermions is given by

The compressibility is given by

Using the density of sodium (at room temperature), it is easy to show that k~10^-5 atm, which is of the order of the experimental compressibility of sodium at room temperature. Also, it is interesting to see that fermions are associated with large values of pressure, of the order of 10⁴ - 10⁵ atm, at zero temperature (due to the restrictions of the Pauli principle, of course).