12.

Suppose a laboratory has a 31 g sample of polonium-210. The half-life of polonium-210 is about 138 days. How many half-lives of polonium-210 occur in 966 days? How much polonium is in the sample 966 days later?

7; 0.24 g

7; 2,139 g

8; 0.12 g

7; 4.43 g

1 answer

To determine how many half-lives of polonium-210 occur in 966 days and how much polonium remains after 966 days, we can follow these steps:

1. **Calculate the number of half-lives:**

The half-life of polonium-210 is 138 days. To find the number of half-lives in 966 days, we use the formula:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} \]

\[ \text{Number of half-lives} = \frac{966 \text{ days}}{138 \text{ days}} \]

Calculate the number of half-lives:
\[ \text{Number of half-lives} = \frac{966}{138} \approx 7 \]

2. **Calculate the remaining amount of polonium-210:**

After \( n \) half-lives, the remaining quantity of a radioactive substance is given by:
\[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n \]

Given:
- Initial amount = 31 g
- \( n = 7 \) half-lives (as calculated above)

\[ \text{Remaining amount} = 31 \text{ g} \times \left(\frac{1}{2}\right)^7 \]

Calculate the remaining amount:
\[ \text{Remaining amount} = 31 \text{ g} \times \frac{1}{128} \]
\[ \text{Remaining amount} \approx 0.242 \text{ g} \]

So, the correct answers are:
- The number of half-lives: 7
- The remaining amount after 966 days: 0.24 g

Therefore, the correct choice is:

**7; 0.24 g**.