Which of the following expresison is correct for first order reaction (`c_(0)` refers to initial concentration of reactant)

Options

A. `t_(1//2)propc_(0)`
B. `t_(1//2)propc_(0)^(-1)`
C. `t_(1//2)propc_(0)^(-2)`
D. `t_(1//2)propc_(0)^(0)`

Answer

To determine the correct expression for the half-life of a first-order reaction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**: The half-life (T_half) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration.

2. **General Formula for Half-Life**: For a general nth order reaction, the expression for half-life is given by:
\[ T_{1/2} \propto C_0^{(1-n)} \]
where \( C_0 \) is the initial concentration of the reactant and \( n \) is the order of the reaction.

3. **Identifying the Order of Reaction**: In this case, we are dealing with a first-order reaction, which means \( n = 1 \).

4. **Substituting the Order into the Formula**: Plugging \( n = 1 \) into the general formula gives:
\[ T_{1/2} \propto C_0^{(1-1)} = C_0^0 \]

5. **Simplifying the Expression**: Since any number raised to the power of 0 is 1, we have:
\[ T_{1/2} \propto 1 \]
This indicates that the half-life of a first-order reaction is independent of the initial concentration \( C_0 \).

6. **Conclusion**: Therefore, the correct expression for the half-life of a first-order reaction is that it is constant and does not depend on the initial concentration.

### Final Answer:
The correct expression for the half-life of a first-order reaction is independent of the initial concentration \( C_0 \).