Question

Plutonium-239 is a radioactive isotope commonly used as fuel in nuclear reactors. The half-life of plutonium-239 is 24,100 years. About how long would it take 504 grams of plutonium-239 to decay until there were only 63 grams plutonium-239 left?

Answer 1

Answer:

72,300 years.

Explanation:

• Initial mass of this sample: 504 grams;

• Current mass of this sample: 63 grams.

What's the ratio between the current and the initial mass of this sample? In other words, what fraction of the initial sample hasn't yet decayed?

$\displaystyle \frac{\text{Current Mass}}{\text{Initial Mass}} = \rm \frac{63\; g}{504\; g} = \frac{1}{8}$.

The value of this fraction starts at 1 decreases to 1/2 of its initial value after every half-life. How many times shall 1/2 be multiplied to 1 before reaching 1/8? $2^{3} = 8$. It takes three half-lives or $3\times 24100 = 72300$ years to reach that value.

In certain questions the denominator of the fraction is large. It might not even be an integer power of 2. The base-x logarithm function on calculators could help. Evaluate

$\displaystyle \log_{\frac{1}{2}}{\frac{1}{8}} = 3$ to find the number of half-lives required. In case the base-x logarithm function isn't available, but the natural logarithm function $\ln()$ is, apply the following expression (derived from the base-changing formula) to get the same result:

$\displaystyle \frac{\displaystyle\ln{\left(\frac{1}{8}\right)}}{\displaystyle \ln{\left(\frac{1}{2}\right)}}$.