The following mechanism has been proposed for the exothermic catalyzed cmplex reaction:
`A + B overset("Fast")rarr IABoverset(k_(1))rarrAB+Ioverset(k_(2))rarrP+A`
If `k_(1)` is much smaller than `k_(2)`, the most suitable qualitative plot of potential energy `(PE)` versus reaction coordinates for the above reaction is

Options

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B.
C.
D.

Explanation

To solve the problem regarding the potential energy (PE) versus reaction coordinates for the given exothermic catalyzed complex reaction mechanism, we will analyze the steps involved in the reaction and the implications of the rate constants \( k_1 \) and \( k_2 \).

### Step-by-Step Solution:

1. **Understanding the Reaction Mechanism**:
The reaction is given as:
\[ A + B \overset{\text{Fast}}{\rightarrow} I_{AB} \overset{k_1}{\rightarrow} AB + I \overset{k_2}{\rightarrow} P + A \]
Here, \( I_{AB} \) is an intermediate formed in the first step, which is fast, and the second step leading to the product \( P \) is slower.

2. **Identifying the Nature of the Reaction**:
Since the reaction is exothermic, the energy of the products \( P \) will be lower than the energy of the reactants \( A + B \). This means that the potential energy diagram will show a decrease in energy from reactants to products.

3. **Analyzing Rate Constants**:
Given that \( k_1 \) is much smaller than \( k_2 \) (\( k_1 \ll k_2 \)), it indicates that the first step is much slower compared to the second step. This implies that the activation energy for the first step is relatively high, while the second step has a lower activation energy.

4. **Drawing the Potential Energy Diagram**:
- Start with the energy level of the reactants \( A + B \).
- The first step has a high activation energy barrier, leading to the formation of the intermediate \( I_{AB} \). This will be represented by a peak in the energy diagram.
- After the intermediate, the second step has a lower activation energy barrier, leading to the product \( P \). This will be represented by a smaller peak or a downward slope from the intermediate to the product.
- The final energy level of the products \( P \) will be lower than that of the reactants, indicating an exothermic reaction.

5. **Conclusion**:
The potential energy diagram will show:
- A high peak for the first step (due to the larger activation energy).
- A lower peak for the second step (due to the smaller activation energy).
- The overall trend will be a decrease in energy from reactants to products.

### Final Potential Energy Diagram:
The final diagram will look like this:

```
PE
|
| /\
| / \
| / \ (Intermediate I_AB)
| / \
| / \
| / \
| / \
| / \
|______________/ \__________ (Products P)
| (Reactants A + B)
```