How do you calculate the density of free electron for alluminium and silicon? How do you calculate the density of free electron for alluminium and silicon?
For gold I have used the formula: $N=\dfrac{n\,\rho \,N_a}{A}$ where $n$ is the number of electron valence, $\rho$ is the density, $N_a$ is the Avogadro number and $A$ is the atomic number. In the case of gold, this formula gives me $n=5.9\,\times10^{22}$ electrons per cube centimeter, which is correct, but I get $n=1.8\,\times 10^{22}$ electrons per cube centimeter in the case if aluminium, which is wrong because the tabulated value is $6\,\times 10^{22}$ electrons per cube centimerter
 A: One cannot answer your question using classical hhysics rather one has to use quantum mechanical ideas.
Here is my attempt to explain in simple and not always rigorous terms, why aluminium has one electron per atom available for the electric conduction process rather than three.
It is certainly true that aluminium has three valence electrons which are designated $\rm 3s^23p^1$ meaning that there are two electrons in the $\rm 3s$ shell and one electron in the $\rm 3p$ shell of an isolated aluminium atom.  
In the solid state those three electrons from each of the aluminium atoms become delocalised - they are no longer linked to one particular aluminium nucleus.
Those three electrons from each of the aluminium nuclei "exist" within the whole of the metal.  
Whist electrons in an isolated atom exist in energy levels, in the solid state those three electron which have be liberated from particular aluminium nuclei now exist in energy bands - a continuum of energy levels. 
As electrons are fermions they obey the Pauli exclusion principle and so no two electrons can exist in the same state.
So the electrons fill up the energy levels in a band from the bottom (lowest energy state) upwards.  
To illustrate what happens next assume that for aluminium the bands formed by the $\rm 3s$ and $\rm 3p$ electrons do not overlap.
The $\rm 3s$ band (lower energy) is populated first and filled with electrons.
The $\rm 3p$ band is populated next but only partially filled with electrons.
The maximum energy (ones largest in energ) in the $\rm 3p$ band is called the Fermi energy.
You might think that all of these electrons are able to contribute the the electrical (and heat) conduction process but that is not so.
For one of these electrons to contribute to the electrical conduction process it must gain momentum and hence energy.
Having gained energy it must be able to be promoted to a higher energy level and that is where a problem will arise for electrons in the $\rm 3s$ band as they have no vacant energy levels above them except in the $\rm 3p$ band and they do not have enough energy to make that jump.
So all of those $\rm 3s$ cannot contribute to the electrical conduction process.
However the electrons in the $\rm 3p$ band, to which nucleus has contributed one electron, do have empty energy above them a so can contribute to the electrical conduction process.
So although aluminium has three valence electrons only one of them can contribute to the electrical conduction process.
For silicon (four valence electrons) the highest populated energy band, called the valence band, is full and so those electrons cannot gain energy and stay within the band - they cannot contribute to the electrical conduction process.
However above the valence band there is another band called the conduction band which is not populated with electrons, it is empty.
This is what might be imagined if the electrons had no thermal energy - the temperature was exceedingly low.
The energy difference between the top of the valence band and the bottom of the conduction band is called the band gap.  
At higher temperatures if electrons in the valence band have enough thermal energy they can jump across from the valence band into the conduction band.
The larger the gap the harder it is for thermal electrons to jump across.
Now you have a situation where the valence band is no longer full so electrons can gain energy and move into a higher energy within the valence band and electrons in the conduction band can also gain energy.
So there are now electrons which can gain energy and contribute to the electrical conduction process. 
For silicon the band gap is quite large so not that many electrons have enough energy to jump across the gap so there are not that many electrons which can contribute to the electrical conduction process - silicon is not a very good conductor but not as bad as some materials and is called a semiconductor.
If silicon is heated it becomes a better of conductor of electricity because more electron can jump across the band gap.
The density of free electrons in silicon is thus very temperature dependent.  
We call materials which have large band gaps insulators, they do not have many electrons which are able to contribute to the electrical conduction process.
