The following is a crude but effective method for estimating the order of
magnitude of Avogadro's number using stearic acid \(\left(\mathrm{C}_{18}
\mathrm{H}_{36} \mathrm{O}_{2}\right)\). When stearic acid is added to water,
its molecules collect at the surface and form a monolayer; that is, the layer
is only one molecule thick. The cross-sectional area of each stearic acid
molecule has been measured to be \(0.21 \mathrm{nm}^{2}\). In one experiment, it
is found that \(1.4 \times 10^{-4} \mathrm{~g}\) of stearic acid is needed to
form a monolayer over water in a dish of diameter \(20 \mathrm{~cm}\). Based on
these measurements, what is Avogadro's number? (The area of a circle of radius
\(r\) is \(\pi r^{2}\).)