The initial rate of hydrolysis of methyl acetate (1M) by a weak acid `(HA,1M)` is `1//100th` of that of a strong acid `(HX, 1M),` at `25^(@)C`. The `K_(a)(HA)` is

Options

A. `1xx10^(-4)`
B. `1xx10^(-5)`
C. `1xx10^(-6)`
D. `1xx10^(-3)`

Answer

To solve the problem, we need to find the acid dissociation constant \( K_a \) of the weak acid \( HA \) given the initial rates of hydrolysis of methyl acetate in the presence of a weak acid and a strong acid.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The initial rate of hydrolysis of methyl acetate with weak acid \( HA \) (1 M) is \( \frac{1}{100} \) of that with strong acid \( HX \) (1 M).
- This means:
\[ \text{Rate}_{HA} = \frac{1}{100} \times \text{Rate}_{HX} \]

2. **Expressing the Rates**:
- The rate of hydrolysis in the presence of a strong acid \( HX \) can be expressed as:
\[ \text{Rate}_{HX} = k \cdot [H^+]_{HX} \]
- For the strong acid \( HX \), since it is a strong acid, it completely dissociates:
\[ [H^+]_{HX} = 1 \text{ M} \]
- Thus,
\[ \text{Rate}_{HX} = k \cdot 1 = k \]

3. **Calculating the Rate for Weak Acid \( HA \)**:
- For the weak acid \( HA \), the concentration of \( H^+ \) ions is less than the concentration of the acid due to partial dissociation. Let \( \alpha \) be the degree of dissociation.
- The concentration of \( H^+ \) from the weak acid is:
\[ [H^+]_{HA} = C \cdot \alpha = 1 \cdot \alpha = \alpha \]
- Therefore, the rate of hydrolysis with weak acid is:
\[ \text{Rate}_{HA} = k \cdot [H^+]_{HA} = k \cdot \alpha \]

4. **Relating the Rates**:
- From the information given:
\[ k \cdot \alpha = \frac{1}{100} k \]
- Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ \alpha = \frac{1}{100} \]

5. **Calculating \( K_a \)**:
- The dissociation constant \( K_a \) for the weak acid \( HA \) is given by:
\[ K_a = C \cdot \alpha^2 \]
- Substituting \( C = 1 \) M and \( \alpha = \frac{1}{100} = 0.01 \):
\[ K_a = 1 \cdot (0.01)^2 = 1 \cdot 10^{-4} = 10^{-4} \]

6. **Final Answer**:
- Therefore, the \( K_a \) of the weak acid \( HA \) is:
\[ K_a = 10^{-4} \]