Avogadro's Number — Question 8

Question 8

A monolayer estimate links molecular count with surface area

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. A 1.00 mg sample has assigned molar mass 200 g/mol. Every molecule spreads into one complete layer with footprint $.250\,\mathrm{nm^2}$, without overlap or uncovered gaps. Assume no material is lost.

1. Calculate the covered area and the radius if the layer is circular.

2. If the observed area is only 60.0% of that prediction and the footprint is unchanged, infer the fraction of molecules in the monolayer. Can the area alone establish where the remaining molecules went?

Question 8 - Solution

A monolayer estimate links molecular count with surface area

(a) Method and calculation

$$n=10^{-3}/200=5.00\times10^{-6}\,\mathrm{mol},\quad N=nN_A=3.01107038\times10^{18}.$$$$A=N(.250\times10^{-18})=.752767595\,\mathrm{m^2},\quad r=\sqrt{A/\pi}\approx.489503\,\mathrm m.$$

(b) Interpretation and further calculation

$$f_{\mathrm{layer}}=A_{\mathrm{observed}}/A_{\mathrm{predicted}}=.600.$$$$N_{\mathrm{layer}}=.600N,\quad N_{\mathrm{elsewhere}}=.400N.$$The area counts only molecules in the assumed single layer. Loss, dissolution or multilayer clustering cannot be distinguished without additional evidence.

Physical check

Convert square nanometres with $10^{-18}$, not $10^{-9}$.