Estimating the average time between collisions (electrons in Cu solid)

[fluidistic]

Homework Statement

The thermal conductivity of Cu at room temperature is 400 W/(mK). Use this value to determine the average time between collisions, $\tau$. Compare this value with the electrical conductivity.

Homework Equations

Not sure. I take T=300K for room temperature.
From a book and with little algebra, $\tau = \frac{3 \kappa }{C_e \overline v ^2}$. Where $C_e=nc$, the specific heat of the electrons per volume unit. $\overline v =<v^2>$.
Also, $<v^2>=\frac{3K_BT}{m_e}$ and apparently $C=\frac{3}{2}n k_B$(they do not state what n is, but I think it's the electronic density because they said that c is the specific heat per electron)

The Attempt at a Solution

Using only the given data, I do not see a clear way to solve the problem, it seems like a data is missing (for example the electron density). Am I missing a way to solve the problem?

If I had "n", I'd be done, because $\tau = \frac{2\kappa m_e }{3nk_B^2T}$ and everything is known but n.
I guess I'm missing something obvious, yet I don't see it. Thanks for any help.

 

[fluidistic]

Well I used an extra datum, namely that the distance between atoms is worth 0.2 nm. I could calculate the electron density n to be worth $1.25 \times 10 ^{29} m^{-3}$, which gave me $\tau \approx 3.40 \times 10 ^{-14}s$.
With the electrical conductivity the result is about $\tau \approx 2 \times 10 ^{-14}s$.
So they are about the same order of magnitude but not exactly the same.