Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A9 / b

Using $M_{\mathrm{Au}}\simeq 196.97\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ and the Avogadro constant, the areal atom count isOne atom occupies each square cell of area $a^2$, so the inferred square lattice spacing isThis uses the given deposited mass and the assumed single square layer; it does not substitute a bulk gold lattice parameter for the measured areal count.

One mobile electron per gold atom givesThis 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.

Substitution in part (a) givesandThe conversion uses $1\mathrm{eV}=1.60218\times 10^{-19}\mathrm{J}$. These are the predictions of the stipulated two-dimensional free electron model, with the vacuum electron mass and one carrier per inferred square cell.