The mechanism for the overall reaction is
`A_(2) + B rarr C`
`A_(2) rarr 2A` (slow)
`2A + B rarr X` (fast)
If a catalyst `D` changes the mechanism to
`A_(2) + D rarr A_(2) D` (slow)
`A_(2) D rArr 2A+D` (fast)
`2A +B rarr C` (fast)
Which is the rate expresison for the reaction with and without a catalyst ?

Options

A. `r = k'[A_(2)][D], r = k[A_(2)]`
B. `r = k[A_(2)], r = k'[A_(2)][D]`
C. `r = k'[A_(2)D], r = k[A_(2)][B]`
D. `r = k[A_(2)][B], r = k'[A_(2)D]`

Explanation

To determine the rate expressions for the reaction with and without a catalyst, we need to analyze the given mechanisms step by step.

### Step 1: Identify the Slow Step
In both mechanisms, the slow step is the rate-determining step (RDS).

- **Without Catalyst**: The slow step is:
\[ A_2 \rightarrow 2A \quad \text{(slow)} \]
- **With Catalyst**: The slow step is:
\[ A_2 + D \rightarrow A_2D \quad \text{(slow)} \]

### Step 2: Write the Rate Expression Without Catalyst
For the reaction without the catalyst, the rate expression is based on the concentration of the reactants in the slow step. The rate expression can be written as:
\[ \text{Rate} = k [A_2]^1 \]
where \(k\) is the rate constant for the reaction without the catalyst.

### Step 3: Write the Rate Expression With Catalyst
For the reaction with the catalyst, the rate expression is based on the concentration of the reactants in the slow step of the new mechanism. The rate expression can be written as:
\[ \text{Rate} = k' [A_2]^1 [D]^1 \]
where \(k'\) is the rate constant for the reaction with the catalyst.

### Summary of Rate Expressions
- **Without Catalyst**:
\[ \text{Rate} = k [A_2] \]
- **With Catalyst**:
\[ \text{Rate} = k' [A_2][D] \]