A dilute solution contains 'x' moles of solute A in 1 kg of solvent with molal elevation constant `K_(b)`. The solute dimerises in the solution according to the following equation. The degree of association is (a) :
`2A hArrA_(2)`
The van't Hoff factor will be:

Options

A. I = 1 -2a
B. I = 1 `-a/2`
C. I = 1 `+a/2`
D. I = 1 + a

Answer

To find the van't Hoff factor for the dimerization of solute A in a dilute solution, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Dimerization Reaction**:
The dimerization of solute A can be represented as:
\[ 2A \rightleftharpoons A_2 \]
This means that two moles of A associate to form one mole of dimer A₂.

2. **Define Initial Conditions**:
Let’s assume we start with \( x \) moles of solute A in the solution. Therefore, initially, the number of moles of A is \( x \) and the number of moles of A₂ is 0.

3. **Define Degree of Association**:
Let \( \alpha \) be the degree of association. This means that a fraction \( \alpha \) of the initial moles of A dimerizes. Thus, the amount of A that dimerizes is \( \alpha x \).

4. **Calculate Moles at Equilibrium**:
At equilibrium:
- The moles of A that remain unassociated will be:
\[ \text{Moles of A at equilibrium} = x - \alpha x = x(1 - \alpha) \]
- The moles of A₂ formed will be:
\[ \text{Moles of A₂ at equilibrium} = \frac{\alpha x}{2} \]
- Therefore, the total moles at equilibrium will be:
\[ \text{Total moles at equilibrium} = \text{Moles of A} + \text{Moles of A₂} = x(1 - \alpha) + \frac{\alpha x}{2} \]

5. **Simplify Total Moles at Equilibrium**:
Combining the terms gives:
\[ \text{Total moles at equilibrium} = x(1 - \alpha) + \frac{\alpha x}{2} = x\left(1 - \alpha + \frac{\alpha}{2}\right) = x\left(1 - \frac{\alpha}{2}\right) \]

6. **Calculate the van't Hoff Factor (i)**:
The van't Hoff factor \( i \) is defined as the ratio of the total number of moles at equilibrium to the initial number of moles:
\[ i = \frac{\text{Total moles at equilibrium}}{\text{Initial moles}} = \frac{x\left(1 - \frac{\alpha}{2}\right)}{x} \]
Simplifying this gives:
\[ i = 1 - \frac{\alpha}{2} \]

### Final Answer:
The van't Hoff factor \( i \) for the dimerization of solute A is:
\[ i = 1 - \frac{\alpha}{2} \]