A radioactive sample contains `10^(-3)` kg each of two nuclear species A and B with half-life 4 days and 8 days respectively . The ratio of the amounts of A and B after a period of 16 days is

Options

A. `1 : 2`
B. `4 : 1`
C. `1 : 4`
D. `2 : 1`

Explanation

To solve the problem, we need to determine the remaining amounts of two radioactive species A and B after 16 days, given their respective half-lives. We will then find the ratio of these amounts.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- Initial mass of species A, \( m_A = 10^{-3} \) kg
- Initial mass of species B, \( m_B = 10^{-3} \) kg
- Half-life of species A, \( t_{1/2,A} = 4 \) days
- Half-life of species B, \( t_{1/2,B} = 8 \) days

2. **Calculate the number of half-lives for species A**:
- Total time = 16 days
- Number of half-lives for A:
\[ n_A = \frac{16 \text{ days}}{4 \text{ days}} = 4 \]

3. **Calculate the remaining amount of species A after 16 days**:
- The remaining amount of A after \( n_A \) half-lives is given by:
\[ m_A' = m_A \left(\frac{1}{2}\right)^{n_A} = 10^{-3} \left(\frac{1}{2}\right)^{4} \]
- Calculating this:
\[ m_A' = 10^{-3} \times \frac{1}{16} = \frac{10^{-3}}{16} \text{ kg} \]

4. **Calculate the number of half-lives for species B**:
- Total time = 16 days
- Number of half-lives for B:
\[ n_B = \frac{16 \text{ days}}{8 \text{ days}} = 2 \]

5. **Calculate the remaining amount of species B after 16 days**:
- The remaining amount of B after \( n_B \) half-lives is given by:
\[ m_B' = m_B \left(\frac{1}{2}\right)^{n_B} = 10^{-3} \left(\frac{1}{2}\right)^{2} \]
- Calculating this:
\[ m_B' = 10^{-3} \times \frac{1}{4} = \frac{10^{-3}}{4} \text{ kg} \]

6. **Find the ratio of the remaining amounts of A and B**:
- The ratio \( R \) of the amounts of A and B is given by:
\[ R = \frac{m_A'}{m_B'} = \frac{\frac{10^{-3}}{16}}{\frac{10^{-3}}{4}} = \frac{1/16}{1/4} = \frac{1}{16} \times \frac{4}{1} = \frac{4}{16} = \frac{1}{4} \]

7. **Final result**:
- The ratio of the amounts of A and B after 16 days is:
\[ \text{Ratio of A to B} = 1 : 4 \]