Problem 80 - Chemistry: The Central Science in SI Units, Global Edition

Question

Lauryl alcohol is obtained from coconut oil and is used to make detergents. A solution of \(5.00 \mathrm{~g}\) of lauryl alcohol in \(0.100 \mathrm{~kg}\) of benzene freezes at \(4.1^{\circ} \mathrm{C}\). What is the molar mass of lauryl alcohol from this data?

Step-by-Step Solution

Answer

The molar mass of lauryl alcohol can be determined using the freezing point depression data. The freezing point depression ΔT is calculated as 1.4°C. Using the formula ΔT = k_f × molality × i, molality is calculated as 0.2734375 mol/kg. The number of moles of lauryl alcohol is then obtained by multiplying molality by the mass of the solvent (benzene) in kg, resulting in 0.02734375 mol. The molar mass is calculated as \(\frac{5.00 g}{0.02734375 mol}\), yielding a molar mass of approximately 182.9 g/mol for lauryl alcohol.

Step 1: Find the freezing point depression

First, calculate the freezing point depression by subtracting the freezing point of the pure solvent (benzene) from the freezing point of the solution. The normal freezing point of benzene is 5.5°C. Freezing point depression = ΔT = T (solvent) - T(solution) ΔT = 5.5 - 4.1 = 1.4°C

Step 2: Use the formula for freezing point depression

Now, use the formula: ΔT = k_f × molality × i where: ΔT is the freezing point depression k_f is the cryoscopic constant for the solvent (benzene) which is 5.12°C·kg/mol molality is the moles of solute per kilogram of solvent i is the van 't Hoff factor, which is the number of particles the solute splits into upon dissolving in the solvent (1 for nonelectrolytes like lauryl alcohol)

Step 3: Calculate molality

In this case, i = 1, and ΔT = 1.4°C. Rearrange the formula to find molality: molality = \(\frac{ΔT}{k_f × i}\) molality = \(\frac{1.4}{5.12 \times 1}\) molality = 0.2734375 mol/kg

Step 4: Determine the number of moles

Now that we have molality, we can find the number of moles of lauryl alcohol. Multiply the molality by the mass of the solvent (benzene) in kg. Number of moles = Molality × mass (solvent) Number of moles = 0.2734375 mol/kg × 0.100 kg Number of moles = 0.02734375 mol

Step 5: Calculate the molar mass

Now we have the moles of lauryl alcohol and the given mass of lauryl alcohol, so we can calculate the molar mass. Molar mass = \(\frac{mass}{moles}\) Molar mass = \(\frac{5.00 g}{0.02734375 mol}\) Molar mass ≈ 182.9 g/mol Therefore, the molar mass of lauryl alcohol is approximately 182.9 g/mol.

Key Concepts

Molar Mass Calculation

To determine the molar mass of a substance, you need to divide the mass of the sample by the number of moles present. In the context of freezing point depression, once we've found out how many moles of a solute we have, thanks to changes in the solvent's freezing point, we can use this formula:
\[ \text{Molar mass} = \frac{\text{mass of solute (g)}}{\text{number of moles}} \]
For instance, in the problem with lauryl alcohol, knowing the moles allows us to calculate the molar mass, which came out to approximately 182.9 g/mol. This helps us understand the composition of the substance and is crucial for chemical analysis.

Cryoscopic Constant

The cryoscopic constant \(k_f\) is an important factor in freezing point depression, representing the change in the freezing point per molal concentration of a solute in a solvent. It varies among different solvents and helps quantify how a solute affects the freezing point. For benzene, the cryoscopic constant is 5.12°C·kg/mol.
This value tells us how sensitive benzene is to added solutes, allowing us to determine other properties like the molar mass of substances dissolved in it by analyzing the temperature drop in the freezing point.

Molality

Molality is a measure of concentration specifically for solutions and is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, molality is independent of temperature because it doesn't involve volume, which can change with temperature.
To calculate molality, use the formula:
\[ \text{Molality} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]
In our exercise, the molality was calculated as 0.273 mol/kg, reflecting how much lauryl alcohol is dissolved in benzene and helping us find the solution's freezing point characteristics.

Nonelectrolyte Solutions

Nonelectrolyte solutions contain solutes that do not dissociate into ions when dissolved. Such solutions have a van 't Hoff factor \(i\) of 1, meaning the solute particles remain intact in solution.
Lauryl alcohol is a nonelectrolyte, so the freezing point depression formula simplifies to:
\( \Delta T = k_f \times \text{molality} \)
This simplification helps in accurately calculating the effect of solutes like lauryl alcohol on solvents without considering ion dissociation, making the analysis straightforward for compounds that don't ionize.