What is the value of `pK_(b)(CH_(3)COO^(c-))` if `wedge^(@)._(m)=390 S cm^(-1) mol ^(-1)` and `wedge_(m)=7.8 S cm^(2) mol^(-1)` for `0.04 M` of `CH_(3)COOH` at `25^(@)C`?

Options

A. `9.3`
B. `9.2`
C. `4.7`
D. `4.8`

Explanation

To find the value of \( pK_b \) for the acetate ion \( CH_3COO^- \), we will follow these steps:

### Step 1: Calculate the degree of dissociation (\( \alpha \))
Given:
- Molar conductivity of acetate ion, \( \Lambda_m = 390 \, \text{S cm}^2 \text{mol}^{-1} \)
- Conductivity of acetic acid solution, \( \kappa = 7.8 \, \text{S cm}^{-1} \)
- Concentration of acetic acid, \( C = 0.04 \, \text{M} \)

The degree of dissociation \( \alpha \) can be calculated using the formula:
\[ \alpha = \frac{\kappa}{\Lambda_m} \]
Substituting the values:
\[ \alpha = \frac{7.8 \, \text{S cm}^{-1}}{390 \, \text{S cm}^2 \text{mol}^{-1}} = 0.02 \]

### Step 2: Calculate the dissociation constant (\( K_a \))
The dissociation of acetic acid can be represented as:
\[ CH_3COOH \rightleftharpoons H^+ + CH_3COO^- \]
The equilibrium expression for the dissociation constant \( K_a \) is given by:
\[ K_a = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} \]
At equilibrium:
- Initial concentration of \( CH_3COOH = C = 0.04 \, \text{M} \)
- Change in concentration due to dissociation = \( C \alpha = 0.04 \times 0.02 = 0.0008 \, \text{M} \)

Thus, at equilibrium:
- \( [H^+] = C \alpha = 0.0008 \, \text{M} \)
- \( [CH_3COO^-] = C \alpha = 0.0008 \, \text{M} \)
- \( [CH_3COOH] = C - C \alpha = 0.04 - 0.0008 = 0.0392 \, \text{M} \)

Now substituting these values into the expression for \( K_a \):
\[ K_a = \frac{(0.0008)(0.0008)}{0.0392} = \frac{6.4 \times 10^{-7}}{0.0392} \approx 1.63 \times 10^{-5} \]

### Step 3: Calculate \( pK_a \)
Using the formula:
\[ pK_a = -\log K_a \]
Substituting the value of \( K_a \):
\[ pK_a = -\log(1.63 \times 10^{-5}) \approx 4.79 \]

### Step 4: Calculate \( pK_b \)
Using the relationship:
\[ pK_a + pK_b = 14 \]
We can find \( pK_b \):
\[ pK_b = 14 - pK_a = 14 - 4.79 \approx 9.21 \]

### Final Answer
The value of \( pK_b \) for the acetate ion \( CH_3COO^- \) is approximately **9.21**.

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