Which of the following is correct for a first order reaction ?

Options

A. `t_(1//2) prop a`
B. `t_(1//2) prop 1//a`
C. `t_(1//2) prop a^(0)`
D. `t_(1//2) prop (1)/(a^(2))`.

Explanation

To determine which statement is correct for a first-order reaction, we need to analyze the half-life (T_half) of such reactions and its relationship with concentration.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions**:
- A first-order reaction is characterized by a rate that is directly proportional to the concentration of one reactant. The general form of the rate law for a first-order reaction can be written as:
\[ \text{Rate} = k[A] \]
where \( k \) is the rate constant and \([A]\) is the concentration of reactant A.

2. **Expression for Half-Life (T_half)**:
- The half-life of a first-order reaction is defined as the time required for the concentration of the reactant to decrease to half of its initial value. The formula for the half-life of a first-order reaction is given by:
\[ T_{half} = \frac{0.693}{k} \]
or equivalently,
\[ T_{half} = \frac{\ln 2}{k} \]

3. **Dependency of T_half on Concentration**:
- From the expression for T_half, we can see that it is independent of the initial concentration of the reactant. This means that regardless of how much of the reactant is present initially, the half-life remains constant as long as the rate constant \( k \) does not change.

4. **Comparing with Other Orders**:
- For zero-order reactions, the half-life is directly proportional to the initial concentration:
\[ T_{half} = \frac{[A]_0}{2k} \]
- For second-order reactions, the half-life is inversely proportional to the initial concentration:
\[ T_{half} = \frac{1}{k[A]_0} \]
- This shows that as the order of the reaction increases, the dependency of T_half on concentration changes from direct to inverse.

5. **Conclusion**:
- Therefore, the correct statement regarding the half-life of a first-order reaction is that it is independent of the initial concentration. This corresponds to the option that states T_half is directly proportional to \( 1^{0} \) (which is 1), indicating no dependency on concentration.

### Final Answer:
The correct option is **C**: T_half is directly proportional to \( 1^{0} \).