Question

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

Short answer

Avogadro's number is approximately \(6.01 \times 10^{23}\).

Step-by-step solution

Calculate the Area of the Dish

First, calculate the area of the dish. The dish is a circle with a diameter of \(20\, \text{cm}\), so the radius \(r\) is half of the diameter, which is \(10\, \text{cm}\). The area \(A\) of the dish is given by the formula for the area of a circle: \[ A = \pi r^2 = \pi (10)^2 = 100\pi \text{ cm}^2. \]

Convert Units to Square Nanometers

Since the cross-sectional area of a stearic acid molecule is given in \(\text{nm}^2\), we need to convert the area of the dish from \(\text{cm}^2\) to \(\text{nm}^2\). 1 cm = \(10^7\, \text{nm}\), therefore \[100\pi \text{ cm}^2 = 100\pi \times (10^7 \text{ nm})^2 = 100\pi \times 10^{14} \text{ nm}^2.\]

Calculate the Total Number of Molecules Needed

To find the number of stearic acid molecules needed to form a monolayer on the dish, divide the total area of the dish by the area occupied by one molecule:\[ N = \frac{100\pi \times 10^{14} \text{ nm}^2}{0.21 \text{ nm}^2} \approx \frac{314 \times 10^{14}}{0.21} \approx 1.493 \times 10^{17}.\]

Calculate the Number of Molecules per Gram

Next, calculate the number of molecules per gram of stearic acid. The molar mass of stearic acid is \(284 \text{ g/mol}\). Using Avogadro's number \(N_A\), we know \[ 1 \text{ mole} = 284 \text{ g} = N_A \text{ molecules}.\] So, \[ N_A = \frac{N \times 284\, \text{g/mol}}{1.4 \times 10^{-4} \text{ g}}.\] Then \[ N_A = \frac{1.493 \times 10^{17} \times 284}{1.4 \times 10^{-4}} \approx 6.01 \times 10^{23}.\]

Estimate Avogadro's Number

Finally, based on our calculations, we estimate Avogadro's number to be approximately \[ N_A \approx 6.01 \times 10^{23}.\] This is in good agreement with the known value of Avogadro's number.

Key concepts

Stearic Acid Monolayer

When stearic acid molecules are added to water, they have an efficient way of arranging themselves. They form what is known as a monolayer. This means the molecules spread out to make just a single layer over the water's surface. It's like a thin film made of just one layer of molecules. Every molecule tries to occupy as much area as possible but in only one molecule thick layer.
Such monolayers are particularly interesting because they allow us to make estimates regarding molecular sizes and numbers. In this experiment, the total mass of the stearic acid needed to cover the dish indicates how many molecules are involved when the acid forms a monolayer. This forms the basis of estimating Avogadro's number.

Molecular Cross-section Area

The molecular cross-section area becomes essential in understanding how molecules occupy space. For stearic acid, this area has been measured at 0.21 nm² per molecule. This value signifies the footprint of each molecule on the surface in the monolayer.
Imagine you are laying down tiles on a floor. Each tile would be equivalent to the molecular cross-section area of the stearic acid. To find out how many tiles (molecules) you need to cover a surface (the dish), you can use the cross-section area to determine this number. In this specific scenario, you can calculate the number of molecules covering the given area by dividing the total area by the cross-section area of one molecule.

Unit Conversion

Unit conversion is a vital mathematical tool in scientific calculations. Here, it's crucial since the area was measured in cm² but must be converted to nm² to match the molecular cross-section units.

• 1 cm = 10⁷ nm
• 1 cm² = (10⁷ nm)² = 10¹⁴ nm²

For the stearic acid experiment, the area of the dish was first calculated in cm² using the formula for the area of a circle. The conversion is done by multiplying the area in cm² by 10¹⁴ to get nm². Mastering such conversions ensures accurate cross-comparisons and calculations in science.

Molar Mass of Stearic Acid

The molar mass of a substance gives us the mass of one mole, which is a fundamental unit in chemistry connecting atoms to measurable quantities. For stearic acid, the molar mass is 284 g/mol. Knowing the molar mass allows us to find the number of moles in a given amount of stearic acid.
To find Avogadro's number using the monolayer, you compute how many molecules are in the given mass of stearic acid. The molar mass forms a bridge, telling us how grams relate to the count of molecules through Avogadro's number. This calculation uses proportions, relating the mass of stearic acid used to the number of moles and further to the number of individual molecules, eventually giving an estimate of Avogadro's constant.