Question

Some substances that are only very slightly soluble in water will spread over the surface of water to produce a film that is called a monolayer because it is only one molecule thick. A practical use of this phenomenon is to cover ponds to reduce the loss of water by evaporation. Stearic acid forms a monolayer on water. The molecules are arranged upright and in contact with one another, rather like pencils tightly packed and standing upright in a coffee mug. The model below represents an individual stearic acid molecule in the monolayer. (a) How many square meters of water surface would be covered by a monolayer made from \(10.0 \mathrm{g}\) of stearic acid? [Hint: What is the formula of stearic acid?] (b) If stearic acid has a density of \(0.85 \mathrm{g} / \mathrm{cm}^{3}\), estimate the length (in nanometers) of a stearic acid molecule. [Hint: What is the thickness of the monolayer described in part a?] (c) A very dilute solution of oleic acid in liquid pentane is prepared in the following way: $$\begin{aligned} &1.00 \mathrm{mL} \text { oleic acid }+9.00 \mathrm{mL} \text { pentane } \rightarrow \text { solution }(1)\\\ &1.00 \mathrm{mL} \text { solution }(1)+9.00 \mathrm{mL} \text { pentane } \rightarrow \text { solution }(2)\\\ &1.00 \mathrm{mL} \text { solution }(2)+9.00 \mathrm{mL} \text { pentane } \rightarrow \text { solution }(3)\\\ &1.00 \mathrm{mL} \text { solution }(3)+9.00 \mathrm{mL} \text { pentane } \rightarrow \text { solution }(4) \end{aligned}$$ A 0.10 mL sample of solution (4) is spread in a monolayer on water. The area covered by the monolayer is \(85 \mathrm{cm}^{2} .\) Assume that oleic acid molecules are arranged in the same way as described for stearic acid, and that the cross-sectional area of the molecule is \(4.6 \times 10^{-15} \mathrm{cm}^{2}\). The density of oleic acid is \(0.895 \mathrm{g} / \mathrm{mL} .\) Use these data to obtain an approximate value of Avogadro's number.

Short answer

The 10.0g sample of stearic acid will cover approximately 97 m^2. The length of a stearic acid molecule is about 1.21 nm. Avogadro's number can be approximated as 5.81*10^{23}/mol.

Step-by-step solution

Calculate the number of moles

The first step is to calculate the number of moles of stearic acid in the 10.0g and the 1.00mL of oleic acid. The molar mass of stearic acid is \(284.5 g/mol\), and the molar mass of oleic acid is \(282.5 g/mol\). To calculate the number of moles, we divide the given mass by the molar mass. Thus, there are \(10.0 g / 284.5 g/mol = 0.035 mol\) of stearic acid, and \(0.895 g / 282.5 g/mol = 0.00317 mol\) of oleic acid.

Calculate the area covered by the monolayer

Next, calculate how many molecules are in the monolayer. Avogadro's number tells us how many molecules are in one mole. Therefore, there are \(0.035mol * 6.022*10^{23} / mol = 2.1077*10^{22}\) stearic acid molecules. Since stearic acid molecules are standing upright, their cross sectional area is the area of the water surface each molecule can cover. The given cross-sectional area of stearic acid molecules is \(4.6*10^{-15} cm^2\). Thus, the total area that 10g of stearic acid can cover is \(2.1077*10^{22} * 4.6*10^{-15} cm^2 = 97 m^2\)

Calculate the length of a stearic acid molecule

To find the length of a stearic acid molecule, we need to convert the mass of stearic acid to a volume, by using the density formula. The volume of stearic acid is \(10.0g / 0.85 g/cm^3 = 11.76 cm^3\). Because stearic acid forms a monolayer, the volume of stearic acid is approximately equal to the area covered by the monolayer multiplied by the thickness of the monolayer: \(11.76 cm^3 = 97 m^2 * Thickness\). Solving for Thickness, we get \(Thickness = 11.76 cm^3 / 97 m^2 = 1.21 * 10^{-7}m = 1.21 nm\)

Calculate Avogadro's number

Finally, we can calculate Avogadro's number based on the information given for oleic acid. With 0.10mL of oleic acid solution spread in a monolayer covering 85 cm^2 area, we first calculate the number of molecules from the known cross-sectional area: \(85 cm^2 / 4.6*10^{-15} cm^2 = 1.843*10^{16}\) molecules. The moles of oleic acid in 0.10mL of solution can be determined from the dilution and the original volume, yielding 0.00317 moles of oleic acid. Avogadro's number is the number of molecules in one mole, therefore \(1.843*10^{16} molecules / 0.00317 mol = 5.81*10^{23} /mol\)

Key concepts

Stearic Acid

Stearic acid is a saturated fatty acid that has a waxy texture and is often used in cosmetics, soaps, and detergents. It is known for forming a monolayer on water surfaces, which means it creates a single molecule-thick layer. This property is exploited in practical applications such as reducing water evaporation in ponds.

The chemical formula for stearic acid is \( \text{C}_{18}\text{H}_{36}\text{O}_{2} \), and its molar mass is about \( 284.5 \ g/mol \). The molecules of stearic acid are long and consist of a hydrophilic carboxyl group \( (\text{-COOH}) \) and a long hydrophobic hydrocarbon tail. When in a monolayer, these molecules stand upright with their tails away from the water. This arrangement minimizes the surface area and results in a cohesive layer.

To determine the area that a given mass of stearic acid covers, you need to know its cross-sectional area and the number of molecules. Using the given density of stearic acid, the mass can be converted into volume, and from volume to thickness of a single molecule in the monolayer, you can calculate its approximate length.

Oleic Acid

Oleic acid is an unsaturated fat found in olive oil and various other oils. It's characterized by having one double bond in its structure, distinguishing it from saturated fats like stearic acid. Its chemical formula is \( \text{C}_{18}\text{H}_{34}\text{O}_{2} \), and it has a molar mass of approximately \( 282.5 \ g/mol \).

Like stearic acid, oleic acid can form a monolayer on water. In experimental contexts, you might start with a solution of oleic acid in a solvent like pentane and repeatedly dilute it until the desired concentration is achieved. When this solution is spread onto water, the oleic acid molecules arrange themselves similar to stearic acid molecules, covering a measurable surface area.

To find out how many oleic acid molecules are in a specific area, such as a monolayer, use the molecule's cross-sectional area and the covered area. This kind of calculation is essential when estimating values like Avogadro's number, as it gives a way to connect micro-scale phenomena with macro-scale observations.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry, representing the number of units (usually atoms or molecules) in one mole of a substance. It is approximately \( 6.022 \times 10^{23} \ mol^{-1} \). This number links the macroscopic scale of materials we can observe and measure directly, with the nanoscale quantities that compose them: atoms and molecules.

In the context of monolayer experiments with substances like oleic acid, Avogadro's number can be used to calculate the number of molecules in a given volume or surface area. By measuring how much area a known quantity of a substance covers in a monolayer, you can determine how many molecules are involved. This is then divided by the number of moles present to estimate Avogadro's number.

This calculation can be tricky, as it depends on precise measurements of volume, density, and molecular dimensions. However, understanding and accurately calculating Avogadro's number is crucial for bridging the gap between laboratory measurements and theoretical chemistry concepts.