A chemical reaction proceeds into the following steps
Step I, `2A hArrX` fast
Step II, `X+B rarr Y` slow
Step III, `Y+B hArr` product fast
The rate law for the overall reaction is

Options

A. Rate `=k[A]^2`
B. Rate `=k[B]^2`
C. Rate = k [A][B]
D. Rate `=k[A]^2[B]`

Answer

To determine the rate law for the overall reaction given the steps, we can follow these steps:

### Step 1: Identify the Rate-Determining Step
In the provided reaction steps:
- Step I: `2A ⇌ X` (fast)
- Step II: `X + B → Y` (slow)
- Step III: `Y + B ⇌ Products` (fast)

The slow step is Step II, which is the rate-determining step. The rate of the overall reaction is determined by this step.

### Step 2: Write the Rate Law for the Rate-Determining Step
The rate law for the rate-determining step can be expressed as:
\[ \text{Rate} = k [X][B] \]
where \( k \) is the rate constant for the slow step.

### Step 3: Express Intermediate in Terms of Reactants
Since \( X \) is an intermediate and not a reactant in the overall equation, we need to express \( X \) in terms of \( A \). From Step I, we have:
\[ 2A ⇌ X \]

At equilibrium, we can write the equilibrium constant \( K \) as:
\[ K = \frac{[X]}{[A]^2} \]
From this, we can express \( [X] \) as:
\[ [X] = K [A]^2 \]

### Step 4: Substitute \( X \) in the Rate Law
Now we substitute \( [X] \) back into the rate law:
\[ \text{Rate} = k [X][B] = k (K [A]^2) [B] \]
This simplifies to:
\[ \text{Rate} = k' [A]^2 [B] \]
where \( k' = kK \) is a new rate constant.

### Step 5: Final Rate Law Expression
Thus, the final rate law for the overall reaction is:
\[ \text{Rate} = k' [A]^2 [B] \]

### Conclusion
The overall rate law for the reaction is:
\[ \text{Rate} = k [A]^2 [B] \]