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 \(\left.\pi r^{2} .\right)\)

Step by step solution

01

Calculate the area of the dish

First, convert the diameter of the dish (20cm) into meters (0.2m) to make units consistent. Then, calculate the area using the formula \( \pi r^{2} \), where \( r \) is the radius of the dish which is half of the diameter, so \( r = 0.2 m / 2 = 0.1 m \). That results in an area of \( \pi (0.1 m)^{2} = 0.0314 m^{2} \).

02

Calculate the number of stearic acid molecules

The number of molecules in the monocle can be calculated by diving the total area of the dish by the cross-sectional area of a single molecule. Given that the cross-sectional area of a stearic acid molecule is \(0.21 nm^{2} = 0.21 \times 10^{-18} m^{2} \) (converting from nm² to m² for consistency), the total number of molecules are \( 0.0314 m^{2} / 0.21 \times 10^{-18} m^{2} = 1.5 \times 10^{20} \) stearic acid molecules.

03

Use the mass of stearic acid to estimate Avogadro's number

The total mass of stearic acid used is \(1.4 \times 10^{-4} g = 1.4 \times 10^{-7} kg\) (converting from grams to kilograms). Hence, the mass of one stearic acid molecule is \( 1.4 \times 10^{-7} kg / 1.5 \times 10^{20} = 9.3 \times 10^{-28} kg \). Avogadro's number, usually denoted by \(N_{A}\), is the number of particles in one mole. We have the molar mass of stearic acid (\(C_{18}H_{36}O_{2}\)) as 296.5 g/mol = 0.2965 kg/mol. Therefore, Avogadro's number can be estimated as \( 0.2965 kg/mol / 9.3 \times 10^{-28} kg = 3.19 \times 10^{23} \) particles/mol.

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