They use this surface concentration to calculate the number of dye
molecules in 1 *cm*^{2} of the film:
1.3 × 10^{−7} (*mol*/*cm*^{2}) * 6.022 × 10^{23} (*molecules*/*mol*) = 7.8 × 10^{16} (*molecules*/*cm*^{2})
Knowing the approximate area of each dye molecule (1 *nm*^{2}),
they use this to calculate the ratio of the area of the dye molecules to
1 *cm*^{2}.
(There are 1 × 10^{14} *nm*^{2} in 1 *cm*^{2})
Therefore, we have:
7.8 × 10^{16} *molecules* × 1 (*nm*^{2}/*molecule*) × 1 × 10^{14} (*nm*^{2}/*cm*^{2}) = 780 *cm*^{2}.
Now, divide by 1 *cm*^{2} to obtain the roughness factor (with the
appropriate number of significant figures). This number expresses the fact that, in 1 *cm*^{2} of the film, there is approximately 780 *cm*^{2} worth of dye.