Question

If you have 100 grams of a radioactive isotope with a half-life of 10 years:
a. How much of the isotope will you have left after 10 years?
b. How much of the isotope will you have left after 20 years
c. How many half-lives will occur in 40 years

Answer 1

To solve these questions, we will use the concept of half-life. A half-life is the time required for half of the radioactive substance to decay.

Given:
- Initial amount (N₀) = 100 grams
- Half-life (t₁/₂) = 10 years

### a. How much of the isotope will you have left after 10 years?

After one half-life, half of the original amount will remain. Therefore, after 10 years:

\[
N = \frac{N₀}{2} = \frac{100 \text{ grams}}{2} = 50 \text{ grams}
\]

### b. How much of the isotope will you have left after 20 years?

In 20 years, two half-lives will have occurred (since the half-life is 10 years). After two half-lives:

\[
N = \frac{N₀}{2^2} = \frac{100 \text{ grams}}{4} = 25 \text{ grams}
\]

### c. How many half-lives will occur in 40 years?

To find the number of half-lives in 40 years, divide the total time by the duration of one half-life:

\[
\text{Number of half-lives} = \frac{40 \text{ years}}{10 \text{ years/half-life}} = 4
\]

### Summary of Answers:

a. 50 grams
b. 25 grams
c. 4 half-lives