The product of half life `T_(1//2)` and the square of initial concentration of the reactant (a) is constant. Then the order of reaction is

Options

A. 2
B. 3
C. 0
D. 1

Explanation

To determine the order of the reaction based on the relationship between the half-life \( T_{1/2} \) and the square of the initial concentration of the reactant \( [A_0]^2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship**: We are given that the product of the half-life \( T_{1/2} \) and the square of the initial concentration \( [A_0]^2 \) is constant. This can be expressed mathematically as:
\[ T_{1/2} \times [A_0]^2 = \text{constant} \]

2. **Recall the formula for half-life**: The half-life \( T_{1/2} \) for different orders of reactions is given by:
- **Zero Order**:
\[ T_{1/2} = \frac{[A_0]}{2k} \]
- **First Order**:
\[ T_{1/2} = \frac{0.693}{k} \]
- **Second Order**:
\[ T_{1/2} = \frac{1}{k[A_0]} \]
- **Third Order**:
\[ T_{1/2} = \frac{3}{2k[A_0]^2} \]

3. **Substitute the half-life into the relationship**: We will analyze each order to see if the product \( T_{1/2} \times [A_0]^2 \) results in a constant:
- **Zero Order**:
\[ T_{1/2} \times [A_0]^2 = \left(\frac{[A_0]}{2k}\right) \times [A_0]^2 = \frac{[A_0]^3}{2k} \quad (\text{not constant}) \]
- **First Order**:
\[ T_{1/2} \times [A_0]^2 = \left(\frac{0.693}{k}\right) \times [A_0]^2 = \frac{0.693[A_0]^2}{k} \quad (\text{not constant}) \]
- **Second Order**:
\[ T_{1/2} \times [A_0]^2 = \left(\frac{1}{k[A_0]}\right) \times [A_0]^2 = \frac{[A_0]}{k} \quad (\text{not constant}) \]
- **Third Order**:
\[ T_{1/2} \times [A_0]^2 = \left(\frac{3}{2k[A_0]^2}\right) \times [A_0]^2 = \frac{3}{2k} \quad (\text{constant}) \]

4. **Conclusion**: The only case where the product \( T_{1/2} \times [A_0]^2 \) is constant is for a third-order reaction. Therefore, the order of the reaction is:
\[ \text{Third Order} \]

### Final Answer:
The order of the reaction is **third order**.