`t_(1//2) =` constant confirms the first order of the reaction as one `a^(2)t_(1//2) =` constant confirms that the reaction is of

Options

A. Zero order
B. First order
C. Second order
D. Third order

Explanation

To determine the order of the reaction based on the given information about the half-life, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life for Different Orders**:
- The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration.
- The relationship between half-life and the order of the reaction is as follows:
- **Zero Order**: \(t_{1/2} \propto [A]_0\) (directly proportional to initial concentration)
- **First Order**: \(t_{1/2} = \text{constant}\) (independent of initial concentration)
- **Second Order**: \(t_{1/2} \propto \frac{1}{[A]_0}\) (inversely proportional to initial concentration)
- **Third Order**: \(t_{1/2} \propto \frac{1}{[A]_0^2}\) (inversely proportional to the square of initial concentration)

2. **Analyzing the Given Information**:
- The first part states that \(t_{1/2} = \text{constant}\), which confirms that the reaction is of **first order**.
- The second part states that \(A^2 t_{1/2} = \text{constant}\). This implies that:
\[ t_{1/2} \propto \frac{1}{[A]^2} \]
- From this relationship, we can deduce that the half-life is inversely proportional to the square of the concentration of the reactant.

3. **Determining the Order of the Reaction**:
- From the relationship \(t_{1/2} \propto \frac{1}{[A]^2}\), we can equate this to the general formula for half-life:
\[ t_{1/2} \propto [A]^{-(n-1)} \]
- Setting the exponent equal to -2 gives:
\[ -(n-1) = -2 \implies n-1 = 2 \implies n = 3 \]
- Therefore, the order of the reaction is **third order**.

4. **Conclusion**:
- The correct option based on the analysis is that the reaction is of **third order**.

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