the Fermi level () at higher temperatures as: where is the temperature and we define as the Fermi temperature ( is Boltzmann constant). The perturbative approach is justified as the Fermi temperature is usually of about 105 K for a metal, hence at room temperature or lower the Fermi energy and the chemical potential are practically equivalent. Compressibility of metals and degeneracy pressure The total energy per unit volume (at ) can also be calculated by integrating over the phase space of the system, we obtain which does not depend on temperature. Compare with the energy per electron of an ideal gas: , which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by where is the volume and is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the electron degeneracy pressure and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or bulk modulus of the metal This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account. Additional predictions Heat capacity One open problem in solid-state physics before the arrival of the free electron model was related to the low heat capacity of metals. Even when the Drude model was a good approximation for the Lorenz number of the Wiedemann–Franz law, the classical argument is based on the idea that the volumetric heat capacity of an ideal gas is . If this was the case, the heat capacity of a metal could be much higher due to this electronic contribution. Nevertheless, such a large heat capacity was never measured, raising suspicions about the argument. By using Sommerfeld's expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an electron gas, given by: , where the prefactor to is considerably smaller than the 3/2 found