If one starts with one curie of radioactive substance `(T_(1//2) = 12 hrs)` the activity left after a period of `1` week will be about

Options

A. 1 curie
B. 120 microcurie
C. 60 microcurie
D. 8 millicurie

Answer

To find the activity left after one week from an initial activity of one curie of a radioactive substance with a half-life of 12 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Convert One Week to Hours:**
- One week consists of 7 days.
- Each day has 24 hours.
- Therefore, the total time in hours for one week is:
\[ \text{Total time} = 7 \text{ days} \times 24 \text{ hours/day} = 168 \text{ hours} \]

2. **Determine the Number of Half-Lives:**
- The half-life \( T_{1/2} \) of the substance is given as 12 hours.
- To find the number of half-lives in 168 hours, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{168 \text{ hours}}{12 \text{ hours}} = 14 \]

3. **Calculate the Remaining Activity:**
- The formula for the remaining quantity of a radioactive substance after \( n \) half-lives is:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
- Here, \( N_0 \) is the initial quantity (1 curie), and \( n \) is the number of half-lives (14).
- Thus, the remaining activity after 14 half-lives is:
\[ N = 1 \text{ curie} \times \left(\frac{1}{2}\right)^{14} \]

4. **Calculate the Numerical Value:**
- We can compute \( \left(\frac{1}{2}\right)^{14} \):
\[ \left(\frac{1}{2}\right)^{14} = \frac{1}{16384} \]
- Therefore, the remaining activity is:
\[ N = 1 \text{ curie} \times \frac{1}{16384} = \frac{1}{16384} \text{ curie} \]

5. **Convert to Microcuries:**
- Since \( 1 \text{ curie} = 10^6 \text{ microcuries} \):
\[ N = \frac{1}{16384} \times 10^6 \text{ microcuries} = \frac{10^6}{16384} \text{ microcuries} \]
- Calculating this gives:
\[ N \approx 60.1 \text{ microcuries} \]

6. **Final Result:**
- The activity left after one week is approximately:
\[ \text{Activity left} \approx 60 \text{ microcuries} \]

### Final Answer:
The activity left after one week will be about **60 microcuries**.