Half-life of a radioactive substance is `20` minutes. Difference between points of time when it is `33 %` disintegrated and `67 %` disintegrated is approximate.

Options

A. 10 min
B. 20 min
C. 30 min
D. 40 min

Explanation

To solve the problem of finding the difference between the times when a radioactive substance is 33% disintegrated and when it is 67% disintegrated, we can follow these steps:

### Step 1: Determine the Decay Constant
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{0.693}{T_{1/2}} \]
where \(T_{1/2}\) is the half-life of the substance. Given that the half-life is 20 minutes:
\[ \lambda = \frac{0.693}{20} \approx 0.03465 \text{ min}^{-1} \]

### Step 2: Calculate Time for 67% Disintegration
When the substance is 67% disintegrated, 33% remains. Thus, if we take \(N_0\) as the initial amount, then:
\[ N = 0.33 N_0 \]
Using the formula for radioactive decay:
\[ N = N_0 e^{-\lambda t} \]
Substituting for \(N\):
\[ 0.33 N_0 = N_0 e^{-\lambda t_1} \]
Dividing both sides by \(N_0\):
\[ 0.33 = e^{-\lambda t_1} \]
Taking the natural logarithm:
\[ \ln(0.33) = -\lambda t_1 \]
Thus:
\[ t_1 = -\frac{\ln(0.33)}{\lambda} \]
Substituting the value of \(\lambda\):
\[ t_1 = -\frac{\ln(0.33)}{0.03465} \approx 11.6 \text{ minutes} \]

### Step 3: Calculate Time for 33% Disintegration
When the substance is 33% disintegrated, 67% remains. Thus:
\[ N = 0.67 N_0 \]
Using the same decay formula:
\[ 0.67 N_0 = N_0 e^{-\lambda t_2} \]
Dividing both sides by \(N_0\):
\[ 0.67 = e^{-\lambda t_2} \]
Taking the natural logarithm:
\[ \ln(0.67) = -\lambda t_2 \]
Thus:
\[ t_2 = -\frac{\ln(0.67)}{\lambda} \]
Substituting the value of \(\lambda\):
\[ t_2 = -\frac{\ln(0.67)}{0.03465} \approx 32 \text{ minutes} \]

### Step 4: Calculate the Difference in Time
Now, we find the difference between the two times:
\[ \Delta t = t_2 - t_1 = 32 - 11.6 = 20.4 \text{ minutes} \]

### Conclusion
The difference between the points of time when the substance is 33% disintegrated and 67% disintegrated is approximately **20.4 minutes**.