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Chapter 3: Problem 171

The following is a crude but effective method for estimating the order of magnitude of Avogadro's number using stearic acid \(\left(\mathrm{C}_{18} \mathrm{H}_{36} \mathrm{O}_{2}\right)\) shown here. 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 crosssectional 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 A vogadro's number?

Short Answer

Expert verified
Avogadro's number as estimated from this monolayer of stearic acid is \(9.42 \times 10^{23}\) molecules/mol.

Step by step solution

01

Calculate the molar mass of stearic acid

Firstly, we need to calculate the molar mass of stearic acid \( \mathrm{C}_{18} \mathrm{H}_{36} \mathrm{O}_{2}\). Molar mass of carbon (C) is 12.01 g/mol, hydrogen (H) is 1.01 g/mol, and oxygen (O) is 16.00 g/mol. Hence, molar mass of stearic acid = number of moles of C * molar mass of C + number of moles of H * molar mass of H + number of moles of O * molar mass of O = 18 * 12.01 g/mol + 36 * 1.01 g/mol + 2 * 16.00 g/mol = 284.48 g/mol.
02

Estimate the number of stearic acid molecules in the monolayer

Next, we need to find out how many moles of stearic acid are in the 0.00014 g of stearic acid. Remember that number of moles = mass in g / molar mass. So, number of moles of stearic acid = 0.00014 g / 284.48 g/mol = 4.92 * 10^-7 mol. To find the number of molecules, we multiply it by Avogadro's number (6.022 * 10^23). Hence, number of stearic acid molecules = moles * Avogadro's number = 4.92 * 10^-7 mol * 6.022 * 10^23 mol^-1 = 2.96 * 10^17 molecules.
03

Calculate the area of the monolayer

Then, calculate the area of the dish. The dish is a circle with diameter 20 cm, so its radius is 10 cm = 0.1 m. Hence, Area = Pi * (radius)^2 = 3.14159 * (0.1 m)^2 = 0.031416 m^2. Then, convert this to nm^2 (1 m^2 = 10^18 nm^2), we get 3.14 * 10^16 nm^2.
04

Estimate Avogadro's number

Finally, estimate Avogadro's number by dividing the number of molecules by the area of the monolayer in nm^2: Avogadro's number = number of molecules / area = 2.96 * 10^17 molecules / 3.14 * 10^16 nm^2 = 9.42 * 10^23 molecules/mol.

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Most popular questions from this chapter

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