A reacts to form P , A plot of the reciprocal of the concentration of A vs time is a straight line . When the initial concentration of A is `1.0 xx 10^(-2) M` , its half-life is found to be 20 min . When initial concentration of A is `3.0 xx 10^(-3) M` , the half -life will be

Options

A. 20 min
B. 40 min
C. 56 min
D. 67 min

Explanation

To solve the problem, we need to determine the half-life of substance A when its initial concentration is \(3.0 \times 10^{-3} \, M\), given that the reaction follows second-order kinetics.

### Step-by-Step Solution:

1. **Identify the Order of the Reaction**:
- The problem states that a plot of the reciprocal of the concentration of A vs. time is a straight line. This indicates that the reaction is second-order because for a second-order reaction, the integrated rate law is:
\[ \frac{1}{[A]} = kt + \frac{1}{[A]_0} \]

2. **Use the Half-Life Formula for Second-Order Reactions**:
- The half-life (\(t_{1/2}\)) for a second-order reaction is given by the formula:
\[ t_{1/2} = \frac{1}{k[A]_0} \]
- From the problem, when the initial concentration \([A]_0 = 1.0 \times 10^{-2} \, M\), the half-life is \(20 \, \text{min}\). We can rearrange the formula to find the rate constant \(k\):
\[ k = \frac{1}{t_{1/2} \cdot [A]_0} = \frac{1}{20 \, \text{min} \cdot 1.0 \times 10^{-2} \, M} \]

3. **Calculate the Rate Constant \(k\)**:
- Substitute the values into the equation:
\[ k = \frac{1}{20 \cdot 1.0 \times 10^{-2}} = \frac{1}{0.2} = 5 \, \text{M}^{-1}\text{min}^{-1} \]

4. **Calculate the Half-Life for the New Concentration**:
- Now, we need to find the half-life when \([A]_0 = 3.0 \times 10^{-3} \, M\):
\[ t_{1/2} = \frac{1}{k[A]_0} = \frac{1}{5 \cdot 3.0 \times 10^{-3}} \]

5. **Substitute and Solve**:
- Substitute \(k\) and \([A]_0\) into the half-life formula:
\[ t_{1/2} = \frac{1}{5 \cdot 3.0 \times 10^{-3}} = \frac{1}{0.015} \approx 66.67 \, \text{min} \]

6. **Final Answer**:
- Therefore, the half-life when the initial concentration of A is \(3.0 \times 10^{-3} \, M\) is approximately \(67 \, \text{min}\).

### Summary:
The half-life of A when the initial concentration is \(3.0 \times 10^{-3} \, M\) is approximately \(67 \, \text{min}\).