A sample contains `16 gm` of radioactive material, the half-life of which is two days. After `32` days, the amount of radioactive material left in the sample is

Options

A. `lt 1mg`
B. `1/4 gm`
C. `1/2 gm`
D. `1 gm`

Answer

To solve the problem of how much radioactive material is left after 32 days when starting with 16 grams and a half-life of 2 days, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Initial amount of radioactive material, \( n_0 = 16 \) grams.
- Half-life of the material, \( t_{1/2} = 2 \) days.
- Total time elapsed, \( t = 32 \) days.

2. **Calculate the Number of Half-Lives**:
- The number of half-lives that have passed can be calculated by dividing the total time by the half-life:
\[ n = \frac{t}{t_{1/2}} = \frac{32 \text{ days}}{2 \text{ days}} = 16 \]
This means that 16 half-lives have passed in 32 days.

3. **Use the Formula for Remaining Amount**:
- The amount of radioactive material left after \( n \) half-lives can be calculated using the formula:
\[ n = n_0 \left(\frac{1}{2}\right)^n \]
Substituting the values we have:
\[ n = 16 \left(\frac{1}{2}\right)^{16} \]

4. **Calculate the Value**:
- First, calculate \( \left(\frac{1}{2}\right)^{16} \):
\[ \left(\frac{1}{2}\right)^{16} = \frac{1}{65536} \]
- Now, substitute this back into the equation:
\[ n = 16 \times \frac{1}{65536} = \frac{16}{65536} = \frac{1}{4096} \text{ grams} \]

5. **Convert to Decimal**:
- To find the decimal value, calculate:
\[ \frac{1}{4096} \approx 0.000244140625 \text{ grams} \]

6. **Express in Scientific Notation**:
- This can be expressed in scientific notation:
\[ n \approx 2.441 \times 10^{-4} \text{ grams} \]

### Final Answer:
The amount of radioactive material left after 32 days is approximately \( 2.441 \times 10^{-4} \) grams.