The hydrolyiss of an ester was carried out with `0.1 M H_(2)SO_(4)` and `0.1 M HCl` separately. Which of the following expresison between the rate constants is expected ? The rate expresison being rate `= k[H^(o+)]["ester"]`

Options

A. `k_(HCl) = k_(H_(2)SO_(4))`
B. `k_(HCl) gt k_(H_(2)SO_(4))`
C. `k_(HCl) lt k_(H_(2)SO_(4))`
D. `k_(H_(2)SO_(4)) = 2k_(HCl)`

Explanation

To solve the problem, we need to analyze the hydrolysis of an ester in the presence of two different acids: sulfuric acid (H₂SO₄) and hydrochloric acid (HCl). The rate of the reaction is given by the expression:

\[ \text{Rate} = k[H^+][\text{ester}] \]

### Step-by-Step Solution:

1. **Identify the dissociation of the acids**:
- For **H₂SO₄**: It is a strong acid and dissociates completely in solution. The dissociation can be represented as:
\[ \text{H}_2\text{SO}_4 \rightarrow 2 \text{H}^+ + \text{SO}_4^{2-} \]
Therefore, a 0.1 M solution of H₂SO₄ will produce:
\[ [H^+] = 2 \times 0.1 \, \text{M} = 0.2 \, \text{M} \]

- For **HCl**: It also dissociates completely in solution:
\[ \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \]
Thus, a 0.1 M solution of HCl will produce:
\[ [H^+] = 0.1 \, \text{M} \]

2. **Write the rate expressions for both reactions**:
- For **H₂SO₄**:
\[ R_1 = k_{H_2SO_4} [H^+] [\text{ester}] = k_{H_2SO_4} (0.2) [\text{ester}] \]

- For **HCl**:
\[ R_2 = k_{HCl} [H^+] [\text{ester}] = k_{HCl} (0.1) [\text{ester}] \]

3. **Assume the rates are equal**:
Since we are comparing the rate constants, we can set \( R_1 = R_2 \):
\[ k_{H_2SO_4} (0.2) [\text{ester}] = k_{HCl} (0.1) [\text{ester}] \]

4. **Cancel the common term**:
We can cancel the concentration of the ester from both sides (assuming it is constant):
\[ k_{H_2SO_4} (0.2) = k_{HCl} (0.1) \]

5. **Rearranging the equation**:
Rearranging gives:
\[ \frac{k_{H_2SO_4}}{k_{HCl}} = \frac{0.1}{0.2} = \frac{1}{2} \]

Thus,
\[ k_{H_2SO_4} = \frac{1}{2} k_{HCl} \]

6. **Conclusion**:
This means that the rate constant for the hydrolysis of the ester in the presence of H₂SO₄ is less than that in the presence of HCl:
\[ k_{HCl} > k_{H_2SO_4} \]

### Final Answer:
The expected expression between the rate constants is:
\[ k_{HCl} > k_{H_2SO_4} \]