Solution Explanation:



1. Introduction:


In this problem, we are given the freezing point of a solution of acetic acid in benzene and we need to determine the equilibrium constant for the dimerization of acetic acid. To solve this problem, we will use the concept of freezing point depression and the equation for the freezing point depression constant.

2. Freezing Point Depression:


The freezing point depression is the difference between the freezing points of the pure solvent and a solution of the solute in that solvent. It is directly proportional to the molality of the solute in the solution. The equation for freezing point depression is:

ΔTf = Kf × m

Where:
ΔTf is the freezing point depression
Kf is the freezing point depression constant
m is the molality of the solute in the solution

3. Calculation of Molality:


To calculate the molality of the solute, we need to know the mole fraction of the solute, the freezing point of the solution, and the freezing point of the pure solvent. The formula for molality is:

m = (ΔTf / Kf)

Given:
ΔTf = 278.4 K - 277.4 K = 1 K
Kf = 5 K kg mol^-1

Using the formula, we can calculate the molality as:

m = 1 K / 5 K kg mol^-1 = 0.2 mol kg^-1

4. Relationship between Molality and Mole Fraction:


The mole fraction of a solute is defined as the ratio of the number of moles of the solute to the total number of moles of solute and solvent. In this case, the mole fraction of acetic acid is given as 0.02. The relationship between mole fraction and molality is:

m = (x / (1 - x)) × (M2 / M1)

Where:
x is the mole fraction of the solute
M1 is the molar mass of the solvent
M2 is the molar mass of the solute

5. Calculation of Molar Mass:


To determine the equilibrium constant for dimerization, we need to find the molar mass of acetic acid. The molar mass of acetic acid (CH3COOH) is:

M2 = (12.01 g/mol × 2) + (1.01 g/mol × 4) + (16.00 g/mol) + (1.01 g/mol) = 60.05 g/mol

6. Calculation of Equilibrium Constant:


Now, we can rearrange the equation for the relationship between mole fraction and molality to solve for the equilibrium constant (K):

K = (m × (1 - x) / x) × (M1 / M2)

Substituting the given values:

K = (0.2 mol kg^-1 × (1 - 0.02) / 0.02) × (78.11 g/mol / 60.05 g/mol)

Simplifying the expression:

K ≈ 3.19 kg mol^-1

Therefore, the equilibrium constant for dimerization of acetic acid