Half life for which of the following varies with initial concentration `[A_(0)]`

Options

A. zero order
B. first order
C. second order
D. third order

Answer

To determine which order of reaction has a half-life that varies with the initial concentration \([A_0]\), we can analyze the half-life expressions for different orders of reactions.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
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.

2. **Zero Order Reaction**:
- The half-life for a zero-order reaction is given by:
\[ t_{1/2} = \frac{[A_0]}{2k} \]
- Here, \(k\) is the rate constant. This shows that the half-life is directly proportional to the initial concentration \([A_0]\).

3. **First Order Reaction**:
- The half-life for a first-order reaction is given by:
\[ t_{1/2} = \frac{\ln 2}{k} \]
- This expression is independent of the initial concentration \([A_0]\).

4. **Second Order Reaction**:
- The half-life for a second-order reaction is given by:
\[ t_{1/2} = \frac{1}{k[A_0]} \]
- This indicates that the half-life is inversely proportional to the initial concentration \([A_0]\).

5. **Third Order Reaction**:
- The half-life for a third-order reaction can be expressed as:
\[ t_{1/2} = \frac{1}{k[A_0]^2} \]
- This shows that the half-life is inversely proportional to the square of the initial concentration \([A_0]\).

6. **Conclusion**:
- From the above analysis, we can conclude that the half-life varies with the initial concentration for zero-order, second-order, and third-order reactions. However, it does not vary for first-order reactions.

### Final Answer:
The half-life varies with initial concentration \([A_0]\) for:
- Zero Order
- Second Order
- Third Order