Equation is the definition of specific heat given in terms of the fermi-dirac distribution function $f$ and the density of states equation $g$

${C}_{e}({T}_{e})={\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\frac{\mathrm{\partial}f(\epsilon ,\mu ,{T}_{e})}{\mathrm{\partial}{T}_{e}}g(\epsilon )\epsilon \text{}d\epsilon ,$

The first consideration that I make is that the fermi level is a function of temperature, so I use the following conservation equation to solve for the fermi level

${N}_{e}={\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(\epsilon ,\mu ({T}_{e}),{T}_{e})g(\epsilon )d\epsilon .$

where ${N}_{e}$ is the number of free electrons. For example in Copper it would be 1 electron/atom.

My question is around how to determine the number of free electrons in a metal as a function of temperature. All the analysis that I have seen just assumes a static amount of free electrons for a material, but we know as $T\to \mathrm{\infty}$ the number of free electrons goes towards the atomic number of the metal.

My question is how can I solve for ${N}_{e}(T)$ if I know ${N}_{e}({T}_{0})$?