Kinetic energy of electron in metals Will it be correct to relate temperature of metal with kinetic energy of electron in metal just like as we do to find kinetic energy of gas molecules if we know the temperature by using the following relation:
$$E=\frac32\ k\ T$$
where $E$ = kinetic energy,
$k$ =Botzmann constant, and
$T$ = absolute temperature?
I'm asking this because there is a problem in my textbook as follows:

Compute the typical de Broglie wavelength of an electron in a metal at $27$ °C and compare it with the mean separation between two electrons in a metal which is given to be about $2 \times 10^{–10}$ m.


The only way I see is as I said above, is there any problem in that procedure? (for a general case)
 A: At room temperature the electrons in metals are actually a degenerate Fermi gas and can be treated as if near absolute zero. Quoting wikipedia:

For metals, the electron gas's Fermi temperature is generally many thousands of kelvins, so in human applications they can be considered degenerate.

The total energy of Fermi gas at absolute zero is: $$E = \frac{3}{5}E_f$$ where $E_f$ is Fermi energy. 
The Wikipedia article actualy mentions the fact that this is a good model for metals: 

The three-dimensional isotropic case is known as the Fermi sphere. Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal.

To find the actual temperature dependance of energy on temperature at temperatures close but not exactly equal to absolute zero you would need to use Sommerfield expansion for internal energy. If I remember correctly, that would give you a factor of $T^2$ for the first order approximation but in most cases this is unnecessary, Fermi energy gives you the right order of magnitude. For more information on where those results come from see the links, the Wikipedia articles are fairly decent.
I think that all of this is beyond the scope of this question and it wants you to use classic Boltzman statistic to demonstrate that the nature of the problem is quantum mechanical. It says "compute" but what it probably means is "demonstrate why this kind of calculation is actually wrong".
EDIT: You mentioned that in "Note" section the book says: "This indistinguishibility has many fundamental implications which you will explore in more advanced Physics courses." I imagine Fermi gas and degeneracy is what they are talking about so they definitely do not expect you to know any of it at the moment.
A: If you use the expression you mention you are implying a classical treatment. Use the expression you mention. You should be able to the wavelength larger than the separation implying a quantum mechanical treatment for the electrons in metals. 
