They use this surface concentration to calculate the number of dye molecules in 1 cm² of the film:

1.3 × 10⁻⁷ (mol/cm²) * 6.022 × 10²³ (molecules/mol) = 7.8 × 10¹⁶ (molecules/cm²)

Knowing the approximate area of each dye molecule (1 nm²), they use this to calculate the ratio of the area of the dye molecules to 1 cm².

(There are 1 × 10¹⁴ nm² in 1 cm²)

Therefore, we have:

7.8 × 10¹⁶ molecules × 1 (nm²/molecule) × 1 × 10¹⁴ (nm²/cm²) = 780 cm².

Now, divide by 1 cm² to obtain the roughness factor (with the appropriate number of significant figures). This number expresses the fact that, in 1 cm² of the film, there is approximately 780 cm² worth of dye.