Initial active mass of a radioactive sample is 76 gm . The active mass at the end of two mean lives is approximately.

Options

A. 10 gm
B. 19 gm
C. 5.6gm
D. 25 gm

Answer

To solve the problem of finding the active mass of a radioactive sample at the end of two mean lives, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Mass**:
The initial active mass \( M_0 \) of the radioactive sample is given as 76 grams.

2. **Understand Mean Life**:
The mean life \( T_{mean} \) is defined as the average lifetime of a radioactive nucleus. It is related to the decay constant \( \lambda \) by the formula:
\[ T_{mean} = \frac{1}{\lambda} \]

3. **Determine Time for Two Mean Lives**:
The time for two mean lives is:
\[ T = 2 \times T_{mean} = \frac{2}{\lambda} \]

4. **Use the Decay Formula**:
The number of active nuclei remaining at time \( T \) can be expressed using the formula:
\[ N = N_0 e^{-\lambda T} \]
where \( N_0 \) is the initial number of active nuclei.

5. **Relate Mass to Number of Nuclei**:
The mass of the radioactive sample can be related to the number of nuclei:
\[ M = M_0 e^{-\lambda T} \]
Substituting \( T = \frac{2}{\lambda} \) into the equation gives:
\[ M = M_0 e^{-2} \]

6. **Calculate the Remaining Mass**:
Now substitute \( M_0 = 76 \) grams:
\[ M = 76 \times e^{-2} \]
The value of \( e^{-2} \) is approximately \( \frac{1}{e^2} \approx \frac{1}{7.39} \approx 0.1353 \).

7. **Final Calculation**:
Now calculate:
\[ M \approx 76 \times 0.1353 \approx 10.34 \text{ grams} \]

8. **Conclusion**:
The active mass at the end of two mean lives is approximately 10 grams.

### Summary of the Answer:
The active mass at the end of two mean lives is approximately **10 grams**.