Consider the reaction,
`Cl_(2)(aq) + H_(2)S(aq) rarr S(s) + 2H^(+) (aq) + 2Cl^(-)(aq)`
The rate equation for this reaction is,
Rate `=k [Cl_(2)][H_(2)S]`
Which of these mechanisms is`//`are consistent with this rate equation ?
(I) `Cl_(2) + H_(2)S rarr H^(+) + Cl^(-) + Cl^(+) + HS^(-)` (slow)
`Cl^(+) + HS^(-) rarr H^(+) + Cl^(-) + S` (fast)
(II) `H_(2)S hArr H^(+) +HS^(-)` (fast equilibrium)
`Cl^(+) + HS^(-) rarr 2 Cl^(-) + H^(+) + S` (slow)

Options

A. B only
B. Both A and B
C. Neither A nor B
D. A only

Explanation

To determine which of the proposed mechanisms is consistent with the given rate equation for the reaction:

\[ \text{Cl}_2(aq) + \text{H}_2\text{S}(aq) \rightarrow \text{S}(s) + 2\text{H}^+(aq) + 2\text{Cl}^-(aq) \]

with the rate equation:

\[ \text{Rate} = k[\text{Cl}_2][\text{H}_2\text{S}] \]

we will analyze each mechanism step by step.

### Mechanism I:
1. **Step 1 (Slow Step):**
\[ \text{Cl}_2 + \text{H}_2\text{S} \rightarrow \text{H}^+ + \text{Cl}^- + \text{Cl}^+ + \text{HS}^- \]

Since this is the slow step, the rate of the reaction for this mechanism can be expressed as:
\[ \text{Rate} = k[\text{Cl}_2][\text{H}_2\text{S}] \]

This matches the given rate equation.

2. **Step 2 (Fast Step):**
\[ \text{Cl}^+ + \text{HS}^- \rightarrow \text{H}^+ + \text{Cl}^- + \text{S} \]

This step does not affect the overall rate since it is fast and follows the slow step.

### Conclusion for Mechanism I:
- The overall rate law derived from the slow step matches the given rate equation. Therefore, Mechanism I is consistent with the rate equation.

### Mechanism II:
1. **Step 1 (Fast Equilibrium):**
\[ \text{H}_2\text{S} \rightleftharpoons \text{H}^+ + \text{HS}^- \]

This step establishes an equilibrium, and we can express the equilibrium constant \( K \):
\[ K = \frac{[\text{H}^+][\text{HS}^-]}{[\text{H}_2\text{S}]} \]

2. **Step 2 (Slow Step):**
\[ \text{Cl}^+ + \text{HS}^- \rightarrow 2\text{Cl}^- + \text{H}^+ + \text{S} \]

The rate for this step can be expressed as:
\[ \text{Rate} = k'[\text{Cl}^+][\text{HS}^-] \]

To relate this to the concentrations of \(\text{Cl}_2\) and \(\text{H}_2\text{S}\), we need to substitute \([\text{HS}^-]\) using the equilibrium expression from Step 1.

From the equilibrium expression:
\[ [\text{HS}^-] = \frac{K[\text{H}_2\text{S}]}{[\text{H}^+]} \]

Substituting this into the rate expression gives:
\[ \text{Rate} = k'[\text{Cl}^+]\left(\frac{K[\text{H}_2\text{S}]}{[\text{H}^+]}\right) \]

This results in:
\[ \text{Rate} = k''[\text{Cl}^+][\text{H}_2\text{S}][\text{H}^+]^{-1} \]

This does not match the given rate equation, which is in terms of \([\text{Cl}_2]\) and \([\text{H}_2\text{S}]\).

### Conclusion for Mechanism II:
- The derived rate law does not match the given rate equation. Therefore, Mechanism II is not consistent with the rate equation.

### Final Answer:
Only Mechanism I is consistent with the rate equation.