Solution (source code)

= Solution

One mobile <electron> per <gold> atom gives
$$
\boxed{n_e=n_{\rm Au}\simeq4.77\times10^{19}\ {\rm electrons\,m^{-2}}.}
$$
This is an areal <number density>; no layer thickness is required and it should not be converted into a volume density for the two-dimensional model.