The half-life for the viral inactivation if in the beginning `1.5 %` of the virus is inactivated per minute is (Given : The reaction is of first order)

Options

A. `76 min`
B. `66 min`
C. `56 min`
D. `46 min`

Explanation

To solve the problem of finding the half-life for the viral inactivation given that 1.5% of the virus is inactivated per minute and that the reaction is first order, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that 1.5% of the virus is inactivated per minute.
- The reaction is of first order.

2. **Calculate the Rate Constant (k)**:
- For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
\[ k = \frac{\text{change in concentration}}{\text{time}} = \frac{\Delta A}{A_0 \cdot t} \]
- Here, \( \Delta A \) is the change in concentration (1.5% of the initial concentration), and \( A_0 \) is the initial concentration (which we can take as 100% for simplicity).
- Thus, we have:
\[ k = \frac{1.5}{100 \text{ min}^{-1}} = 0.015 \text{ min}^{-1} \]

3. **Use the Half-Life Formula for First-Order Reactions**:
- The half-life \( t_{1/2} \) for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
- Substituting the value of \( k \):
\[ t_{1/2} = \frac{0.693}{0.015} \text{ min} \]

4. **Calculate the Half-Life**:
- Performing the calculation:
\[ t_{1/2} = 46.2 \text{ min} \]
- We can round this off to 46 minutes.

5. **Final Answer**:
- Therefore, the half-life for the viral inactivation is approximately **46 minutes**.