A radioactive element has rate of disintegration `10,000` disintegrations per minute at a particular instant. After four minutes it becomes `2500` disintegrations per minute. The decay constant per minute is

Options

A. `0.2log_e2`
B. `0.5log_e2`
C. `0.6log_e2`
D. `0.7log_e2`

Answer

To find the decay constant (λ) for the radioactive element, we will use the formula relating the activity of a radioactive substance at two different times.

### Step-by-Step Solution:

1. **Identify Initial and Final Activities**:
- Initial activity (R₀) = 10,000 disintegrations per minute.
- Activity after 4 minutes (R) = 2,500 disintegrations per minute.

2. **Use the Exponential Decay Formula**:
The relationship between the initial activity (R₀), the activity at time t (R), and the decay constant (λ) is given by:
\[ R = R₀ e^{-\lambda t} \]
We will apply this formula at two different times.

3. **Set Up the Equations**:
- At time t = 0 (initial time), we have:
\[ R₀ = 10,000 \]
- At time t = 4 minutes, we have:
\[ 2500 = 10,000 e^{-\lambda \cdot 4} \]

4. **Rearranging the Second Equation**:
Dividing both sides by 10,000 gives:
\[ \frac{2500}{10000} = e^{-4\lambda} \]
Simplifying the left side:
\[ 0.25 = e^{-4\lambda} \]

5. **Taking the Natural Logarithm**:
Taking the natural logarithm of both sides:
\[ \ln(0.25) = -4\lambda \]

6. **Expressing 0.25 in Terms of Logarithms**:
We know that:
\[ 0.25 = \frac{1}{4} = \frac{1}{2^2} = 2^{-2} \]
Therefore:
\[ \ln(0.25) = \ln(2^{-2}) = -2\ln(2) \]

7. **Substituting Back**:
Now substituting back into the equation:
\[ -2\ln(2) = -4\lambda \]

8. **Solving for λ**:
Dividing both sides by -4:
\[ \lambda = \frac{2\ln(2)}{4} = \frac{1}{2}\ln(2) \]

9. **Final Result**:
Thus, the decay constant (λ) is:
\[ \lambda = 0.5 \ln(2) \text{ per minute} \]