There is 50% dimer formation of benzoic acid `(C_(6)H_(5)COOH)_(2)` in benzene solution. `2C_(6)H_(5)COOHhArr (C_(6)H_(5)COOH)_(2)`. Hence abnormal molecular weight of benzoic acid (theorectical value `=122g mol^(-1))` is

Options

A. 61 g `mol^(-1)`
B. 244 g `mol^(-1)`
C. 163 g `mol^(-1)`
D. 81 g `mol^(-1)`

Explanation

To solve the problem of calculating the abnormal molecular weight of benzoic acid when there is 50% dimer formation, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Dimer Formation**:
The reaction given is:
\[ 2 \text{C}_{6}\text{H}_{5}\text{COOH} \rightleftharpoons \text{(C}_{6}\text{H}_{5}\text{COOH)}_{2} \]
This indicates that two moles of benzoic acid (C₆H₅COOH) combine to form one mole of dimer (C₆H₅COOH)₂.

2. **Defining the Degree of Dimerization**:
Given that there is 50% dimer formation, this means that out of every 2 moles of benzoic acid, 1 mole will exist as a dimer and 1 mole will remain as monomer.

3. **Setting Up the Equilibrium Expression**:
Let the initial concentration of benzoic acid be \( n \) moles. At equilibrium, we will have:
- Moles of monomer = \( n - x \)
- Moles of dimer = \( \frac{x}{2} \)
Here, \( x \) is the amount that dimerizes. Since 50% dimerization occurs, \( x = n \).

4. **Calculating Moles at Equilibrium**:
Since 50% of the benzoic acid dimerizes:
- Moles of monomer = \( n - \frac{n}{2} = \frac{n}{2} \)
- Moles of dimer = \( \frac{n}{2} \)

5. **Calculating the Abnormal Molecular Weight**:
The abnormal molecular weight (M₀) can be calculated using the formula:
\[ \frac{1}{2} = \frac{M_{0} - 122}{M_{0}} \cdot \left(\frac{1}{2} - 1\right) \]
Rearranging gives:
\[ \frac{1}{2} = \frac{M_{0} - 122}{M_{0}} \cdot \left(-\frac{1}{2}\right) \]
This simplifies to:
\[ 1 = \frac{M_{0} - 122}{M_{0}} \]

6. **Cross Multiplying**:
Cross multiplying gives:
\[ M_{0} = 4(M_{0} - 122) \]
Expanding this:
\[ M_{0} = 4M_{0} - 488 \]

7. **Solving for M₀**:
Rearranging gives:
\[ 3M_{0} = 488 \]
Thus:
\[ M_{0} = \frac{488}{3} \approx 162.67 \text{ g/mol} \]

### Final Answer:
The abnormal molecular weight of benzoic acid is approximately **163 g/mol**.