Problem 171 - Chemistry

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

Molar Mass Calculation

Calculating the molar mass is crucial when we want to understand or predict the behavior of molecules in a chemical context. Molar mass is the weight of one mole (or \(6.022 \times 10^{23}\) entities) of a substance. It is usually expressed in grams per mole (g/mol). To find the molar mass of stearic acid, \(\mathrm{C}_{18} \mathrm{H}_{36} \mathrm{O}_{2}\), we need to consider the contribution of each type of atom in the molecule:

\[ \text{Molar mass} = (18 \times \text{molar mass of C}) + (36 \times \text{molar mass of H}) + (2 \times \text{molar mass of O}) \]

- **Carbon (C):** 12.01 g/mol
- **Hydrogen (H):** 1.01 g/mol
- **Oxygen (O):** 16.00 g/mol

Putting this together:
\[ \text{Molar mass} = (18 \times 12.01) + (36 \times 1.01) + (2 \times 16.00) = 284.48 \text{ g/mol} \]

This calculation helps us convert the mass of a substance in grams into moles, which is necessary to explore further experiments, reactions, or compositions.

Molecular Monolayer

In chemistry, a monolayer refers to a single, closely packed layer of molecules. When molecules like stearic acid are spread over a surface, they can form a monolayer that is one molecule thick. This unique formation can be used to compute the number of molecules in a given area, which in this exercise, is crucial for estimating Avogadro's number.

To find out how many molecules are present, you must first determine the number of moles from the given mass using the molar mass obtained earlier. Here:

- **Mass of stearic acid used:** \(1.4 \times 10^{-4} \text{ g}\)
- **Molar Mass:** \(284.48 \text{ g/mol}\)

Calculate the number of moles:
\[ \text{Number of moles} = \frac{0.00014 \text{ g}}{284.48 \text{ g/mol}} = 4.92 \times 10^{-7} \text{ mol}\]

To convert moles into molecules, multiply by Avogadro's number (\(6.022 \times 10^{23} \text{ molecules/mol}\)):
\[ \text{Number of molecules} = 4.92 \times 10^{-7} \text{ mol} \times 6.022 \times 10^{23} \text{ mol}^{-1} = 2.96 \times 10^{17} \text{ molecules}\]

Surface Area Conversion

The concept of surface area conversion is crucial when dealing with measurements in different units. In this exercise, we encounter the need to convert the area of a dish from meters squared to nanometers squared for consistency with the cross-sectional area of the molecule.

The dish, being circular, has its area calculated using the formula:
\[ \text{Area} = \pi \times (\text{radius})^2 \]

- **Radius of the dish:** 10 cm = 0.1 m
- **Pi (\(\pi\)):** approximately 3.14159

Calculate the area in square meters:
\[ \text{Area (m}^2\text{)} = 3.14159 \times (0.1)^2 = 0.031416 \text{ m}^2\]

Next, convert this area into nanometers squared, knowing that \(1 \text{ m}^2 = 10^{18} \text{ nm}^2\):
\[ \text{Area (nm}^2\text{)} = 0.031416 \times 10^{18} = 3.14 \times 10^{16} \text{ nm}^2\]

This conversion allows us to directly assess the number of molecules that cover a given area, facilitating our estimation of Avogadro's number in terms of the molecular monolayer.