Problem 80 - Chemistry The Central Science

Question

Lauryl alcohol is obtained from coconut oil and is used to make detergents. A solution of 5.00 g of lauryl alcohol in 0.100 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 calculated using the freezing point depression formula. Given a mass of 5.00 g of lauryl alcohol in 0.100 kg of benzene and a freezing point depression of 4.1°C, the molality of the solution is found to be \(m = \frac{4.1\,^{\circ}\mathrm{C}}{5.12\,^{\circ}\mathrm{C\,kg/mol}} = 0.800\,\mathrm{mol/kg}\). Using the rearranged formula \(Molar\ mass\ of\ lauryl\ alcohol = \frac{Mass\ of\ lauryl\ alcohol}{(molality)(kg\ of\ solvent)}\), the molar mass of lauryl alcohol is approximately 62.5 g/mol.

Step 1: Write down the given information

We are given the following information: - Mass of lauryl alcohol: 5.00 g - Mass of benzene: 0.100 kg - Freezing point depression: 4.1°C

Step 2: Calculate the molality of the solution

First, we need to calculate the molality (m) of the solution. Molality is defined as the number of moles of solute per kg of solvent. The formula to calculate molality is: \(m = \frac{moles\ of\ solute}{kg\ of\ solvent}\) By re-arranging this formula, we will get: \(Molar\ mass\ of\ lauryl\ alcohol = \frac{Mass\ of\ lauryl\ alcohol}{(molality)(kg\ of\ solvent)}\) We will now use the formula to calculate the molality of the solution: \(\Delta T_f = K_f \cdot m\) Where: - ΔTf is the freezing point depression (4.1°C) - Kf (benzene) is the cryoscopic constant of benzene (5.12 °C kg/mol) - m is the molality of the solution Rearranging the formula, we get: \(m = \frac{\Delta T_f}{K_f}\) Now, plug in the values: \(m = \frac{4.1\,^{\circ}\mathrm{C}}{5.12\,^{\circ}\mathrm{C\,kg/mol}} = 0.800\,\mathrm{mol/kg}\)

Step 3: Calculate the molar mass of lauryl alcohol

Now, we will use the rearranged formula from Step 2 to calculate the molar mass of lauryl alcohol: \(Molar\ mass\ of\ lauryl\ alcohol = \frac{Mass\ of\ lauryl\ alcohol}{(molality)(kg\ of\ solvent)}\) Plugging in the values, we get: \(Molar\ mass\ of\ lauryl\ alcohol = \frac{5.00\,g}{(0.800\,\mathrm{mol/kg})(0.100\,kg)} = \frac{5.00\,g}{0.0800\,mol} = 62.5\,g/mol\) So, the molar mass of lauryl alcohol is approximately 62.5 g/mol.

Key Concepts

Freezing Point Depression

Freezing point depression is an important colligative property used to determine how the addition of a solute affects the freezing point of a solvent. When a solute is dissolved in a solvent, the freezing point of the solution is lower than that of the pure solvent. This happens because the solute molecules interfere with the formation of the solid lattice structure, meaning the system requires a lower temperature to freeze.
Freezing point depression can be calculated using the formula:
\[\Delta T_f = K_f \cdot m\]where:

• \( \Delta T_f \) is the change in freezing point.
• \( K_f \) is the cryoscopic constant of the solvent.
• \( m \) is the molality of the solution.

By measuring how much the freezing point decreases, you can determine other properties like the molality or the molar mass of the solute, as seen in this exercise!

Cryoscopic Constant

The cryoscopic constant, often denoted as \( K_f \), is a specific value for each solvent that relates the molality of a solution to its freezing point depression. It is expressed in units of \( \text{°C kg/mol} \) and remains constant for a given solvent. For example, benzene has a cryoscopic constant of 5.12 °C kg/mol.
In the formula \( \Delta T_f = K_f \cdot m \), the cryoscopic constant effectively provides a way to link freezing point depression with the concentration of solute in the solution. Therefore, knowing \( K_f \) and measuring \( \Delta T_f \) enables us to calculate the molality \( m \), which is essential for other calculations such as determining the molar mass of the solute.

Molality

Molality (\( m \)) is a concentration term that expresses the number of moles of solute per kilogram of solvent, given in mol/kg. Unlike molarity, molality doesn't change with temperature since it is based on the mass of the solvent, not volume.
Calculating molality involves using the mass and amount of solute dissolved. The formula for molality is:
\[m = \frac{\text{moles of solute}}{\text{kg of solvent}}\]To use freezing point depression for calculating the molality, rearrange the freezing point formula to solve for \( m \):
\[m = \frac{\Delta T_f}{K_f}\]This gives you a precise value of molality based on how much the freezing point is affected. Knowing the molality is fundamentally helpful in calculating the molar mass of the solute.

Molar Mass Determination

Determining the molar mass of a solute can be efficiently achieved via freezing point depression data. Once you've found the molality using the formula \( m = \frac{\Delta T_f}{K_f} \), you can then find the molar mass of the solute.
The following rearranged formula helps in calculating the molar mass:
\[\text{Molar mass} = \frac{\text{Mass of solute (g)}}{m \cdot \text{kg of solvent}}\]Knowing the mass of solute and the molality, you can determine how many grams are in one mole of the substance, giving you its molar mass. In the example exercise, you're given the mass of lauryl alcohol and the freezing details for benzene, which leads to finding its molar mass as approximately 62.5 g/mol. This concept is crucial for understanding how thermodynamic properties can reveal the molecular structure of unknown compounds.