Which of the following relations is not correctly matched with the formula ?

Options

A. In case of association, `alpha=(i-1)/(1/n-1)`
B. In case of dissociation, `alpha=(i-1)/(n+1)`
C. Relative lowering of vapour pressure `=(p_A^@ - p_A)/p_A^@ =i(n_B)/(n_A+n_B)`
D. Elevation in boiling point, `DeltaT_b=K_b xx(W_Bxx1000)/(M_BxxW_A)`

Explanation

To solve the question regarding which of the following relations is not correctly matched with the formula, we will analyze each option step by step.

### Step 1: Analyze Option A
**Relation:** In case of association, α value is given by \( i - 1 \) where \( i \) is Van't Hoff factor divided by \( \frac{1}{n} - 1 \).

**Derivation:**
1. Let the initial number of moles of A be 1.
2. Let α be the degree of association. Thus, the amount of A that associates is α.
3. The remaining amount of A after association is \( 1 - \alpha \).
4. If n molecules of A associate to form 1 molecule, then α moles of A will yield \( \frac{\alpha}{n} \) moles of the product.
5. Total moles after association = \( (1 - \alpha) + \frac{\alpha}{n} = 1 - \alpha + \frac{\alpha}{n} \).
6. The Van't Hoff factor \( i \) is given by:
\[ i = \frac{\text{Total moles after association}}{\text{Initial moles}} = \frac{1 - \alpha + \frac{\alpha}{n}}{1} \]
7. Simplifying gives:
\[ i = 1 - \alpha + \frac{\alpha}{n} \]
8. Rearranging leads to:
\[ \alpha = \frac{i - 1}{1 - \frac{1}{n}} = \frac{i - 1}{\frac{n - 1}{n}} = \frac{n(i - 1)}{n - 1} \]
This matches the given relation, so **Option A is correct**.

### Step 2: Analyze Option B
**Relation:** In case of dissociation, α is given by \( i - 1 \) divided by \( n + 1 \).

**Derivation:**
1. Let the initial number of moles of \( A_n \) be 1.
2. Let α be the degree of dissociation. Thus, the amount that dissociates is α.
3. The remaining amount after dissociation is \( 1 - \alpha \).
4. If \( A_n \) dissociates into n moles of A, then α moles of \( A_n \) will yield \( n \alpha \) moles of A.
5. Total moles after dissociation = \( (1 - \alpha) + n\alpha = 1 - \alpha + n\alpha = 1 + (n - 1)\alpha \).
6. The Van't Hoff factor \( i \) is given by:
\[ i = \frac{1 + (n - 1)\alpha}{1} \]
7. Rearranging gives:
\[ \alpha = \frac{i - 1}{n - 1} \]
This does not match \( n + 1 \) in the denominator, so **Option B is incorrect**.

### Step 3: Analyze Option C
**Relation:** Relative lowering of vapor pressure is given by \( \frac{P_0 - P}{P_0} = \frac{i \cdot n_{\text{solute}}}{n_{\text{total}}} \).

**Derivation:**
1. The relative lowering of vapor pressure is defined as:
\[ \frac{P_0 - P}{P_0} = \text{Mole fraction of solute} \]
2. When the solute dissociates, we need to consider the Van't Hoff factor \( i \).
3. Thus, the correct relation should incorporate \( i \) as:
\[ \frac{P_0 - P}{P_0} = i \cdot \text{Mole fraction of solute} \]
This means that the relation does not match the given expression, so **Option C is incorrect**.

### Step 4: Analyze Option D
**Relation:** Elevation in boiling point is given by \( K_b \cdot M \).

**Derivation:**
1. The elevation in boiling point formula is:
\[ \Delta T_b = K_b \cdot m \]
where \( m \) is the molality.
2. Molality \( m \) is defined as:
\[ m = \frac{n_{\text{solute}}}{\text{mass of solvent in kg}} \]
3. The expression \( K_b \cdot M \) is correct if \( M \) is defined properly as molality, so **Option D is correct**.

### Conclusion
The relations that are not correctly matched with their formulas are **Option B and Option C**.