Activity of a substance changes from `700 s^(–1)` to `500 s^(–1)` in 30 minute. Find its half-life in minutes

Options

A. 72 min
B. 66 min
C. 52 min
D. 62 min

Answer

To find the half-life of a substance given its activity changes from \(700 \, s^{-1}\) to \(500 \, s^{-1}\) in 30 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between activity and decay constant**:
The activity \(A\) of a radioactive substance at time \(t\) is given by the equation:
\[ A(t) = A_0 e^{-\lambda t} \]
where \(A_0\) is the initial activity, \(\lambda\) is the decay constant, and \(t\) is the time.

2. **Set up the equation with the given values**:
Here, \(A_0 = 700 \, s^{-1}\), \(A(t) = 500 \, s^{-1}\), and \(t = 30 \, \text{minutes} = 1800 \, \text{seconds}\).
Thus, we can write:
\[ 500 = 700 e^{-\lambda \cdot 1800} \]

3. **Rearrange the equation**:
Dividing both sides by 700 gives:
\[ \frac{500}{700} = e^{-\lambda \cdot 1800} \]
Simplifying the left side:
\[ \frac{5}{7} = e^{-\lambda \cdot 1800} \]

4. **Take the natural logarithm of both sides**:
\[ \ln\left(\frac{5}{7}\right) = -\lambda \cdot 1800 \]

5. **Solve for the decay constant \(\lambda\)**:
Rearranging gives:
\[ \lambda = -\frac{\ln\left(\frac{5}{7}\right)}{1800} \]

6. **Calculate the half-life \(t_{1/2}\)**:
The half-life is given by the formula:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
Substituting \(\lambda\) from the previous step:
\[ t_{1/2} = \frac{\ln(2)}{-\frac{\ln\left(\frac{5}{7}\right)}{1800}} = \frac{1800 \cdot \ln(2)}{-\ln\left(\frac{5}{7}\right)} \]

7. **Calculate the numerical value**:
Using approximate values:
- \(\ln(2) \approx 0.693\)
- \(\ln\left(\frac{5}{7}\right) \approx \ln(5) - \ln(7) \approx 1.609 - 1.946 = -0.337\)
Thus:
\[ t_{1/2} \approx \frac{1800 \cdot 0.693}{0.337} \approx 61.8 \, \text{seconds} \]

8. **Convert to minutes**:
Since \(61.8 \, \text{seconds} \approx 1.03 \, \text{minutes}\), we can round this to approximately \(1 \, \text{minute}\).

### Final Answer:
The half-life of the substance is approximately **1 minute**.