One mole of a solute A is dissolved in a given volume of a solvent. The association of the solute take place as follows: `nArarrA_(n)` If `a` is the degree of association of A, then van't hoff factor `i` is expressed as:

Options

A. I = 1 - a
B. `i=1+a/n`
C. `i=(1-a+(a)/(n))/1`
D. I = 1

Answer

To solve the problem, we need to express the van't Hoff factor \( i \) in terms of the degree of association \( \alpha \) and the number of molecules \( n \) that associate to form a single molecule.

### Step-by-Step Solution:

1. **Understanding the Association Reaction**:
The solute \( A \) associates to form \( A_n \). The reaction can be represented as:
\[ nA \rightleftharpoons A_n \]

2. **Initial Concentration**:
Initially, we have 1 mole of solute \( A \) dissolved in a given volume of solvent. Therefore, the initial concentration \( C \) of \( A \) is:
\[ C = 1 \text{ mole} \]

3. **Degree of Association**:
Let \( \alpha \) be the degree of association. This means that a fraction \( \alpha \) of the solute has associated to form \( A_n \). Thus, the concentration of \( A \) that remains unassociated at equilibrium is:
\[ C - C\alpha = C(1 - \alpha) \]

4. **Concentration of Associated Species**:
The concentration of the associated species \( A_n \) formed is:
\[ \frac{C\alpha}{n} \]
This is because \( n \) molecules of \( A \) combine to form one molecule of \( A_n \).

5. **Total Concentration at Equilibrium**:
The total concentration at equilibrium will be the sum of the concentration of unassociated \( A \) and the concentration of associated \( A_n \):
\[ \text{Total Concentration} = C(1 - \alpha) + \frac{C\alpha}{n} \]

6. **Calculating the Van't Hoff Factor \( i \)**:
The van't Hoff factor \( i \) is defined as the ratio of the total concentration at equilibrium to the initial concentration:
\[ i = \frac{\text{Total Concentration}}{\text{Initial Concentration}} = \frac{C(1 - \alpha) + \frac{C\alpha}{n}}{C} \]
Simplifying this expression:
\[ i = 1 - \alpha + \frac{\alpha}{n} \]

7. **Final Expression for \( i \)**:
Therefore, the van't Hoff factor \( i \) can be expressed as:
\[ i = 1 - \alpha + \frac{\alpha}{n} \]