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

72,300 years.

Step-by-step explanation:

Answer 2

72,300 years.

Step-by-step 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 (63\; g)/(504\; g) = (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* 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_{(1)/(2)}{(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((1)/(8)\right)}}{\displaystyle \ln{\left((1)/(2)\right)}}`.