Question

How much of a radioactive kind of thorium will be left after 14,680 years if you start with 35,912 grams and the half-life is 7,340 years?
grams

Answer 1

To calculate the amount of a radioactive substance left after a certain period of time, we can use the formula for exponential decay based on half-lives:

\[
N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}}
\]

where:
- \( N \) is the remaining quantity of the radioactive substance,
- \( N_0 \) is the initial quantity of the substance,
- \( t \) is the elapsed time,
- \( t_{1/2} \) is the half-life of the substance.

Given:

- \( N_0 = 35,912 \) grams
- \( t_{1/2} = 7,340 \) years
- \( t = 14,680 \) years

First, we need to determine how many half-lives fit into the time period of 14,680 years:

\[
\frac{t}{t_{1/2}} = \frac{14,680 \text{ years}}{7,340 \text{ years}} = 2
\]

This means that 14,680 years is equivalent to 2 half-lives.

Now we can substitute this value back into the decay formula:

\[
N = 35,912 \left( \frac{1}{2} \right)^{2}
\]

Calculating \( \left( \frac{1}{2} \right)^{2} \):

\[
\left( \frac{1}{2} \right)^{2} = \frac{1}{4}
\]

Now substituting this back into the equation:

\[
N = 35,912 \times \frac{1}{4} = \frac{35,912}{4} = 8,978
\]

Thus, after 14,680 years, there will be approximately **8,978 grams** of the radioactive thorium remaining.