If the half life of Pu-239 is 2.4 x 10^4 years, how many years are required for a 20-gram sample of plutonium-239 to be reduced to a mass of 5 grams?
5 / 20 = 1/4 = (1/2)^2
2 half-lives
To answer this question, you need to understand the concept of half-life and how it applies to radioactive decay.
The half-life of a radioactive substance is the time it takes for half of the initial amount to decay. In this case, the half-life of Pu-239 is 2.4 x 10^4 years, which means that after this time, half of the substance will have decayed.
To calculate the number of half-lives required for a certain amount of Pu-239 to decay, you can use the following formula:
Number of half-lives = (log(remaining mass / initial mass)) / (log(0.5))
Let's apply this formula to the given problem:
Remaining mass = 5 grams
Initial mass = 20 grams
Number of half-lives = log(5/20) / log(0.5)
Using a calculator, you can determine that log(5/20) ≈ -1 and log(0.5) ≈ -0.301.
Number of half-lives ≈ -1 / -0.301 ≈ 3.32 (approx.)
Since we cannot have a fraction of a half-life, we take the next highest whole number. So, it will take approximately 4 half-lives for a 20-gram sample of plutonium-239 to be reduced to a mass of 5 grams.
Now, you need to calculate the total time required for 4 half-lives to occur. To do this, multiply the half-life by the number of half-lives:
Total time = Half-life x Number of half-lives
Total time ≈ (2.4 x 10^4 years) x 4
Total time ≈ 9.6 x 10^4 years
Therefore, it would take approximately 9.6 x 10^4 years for a 20-gram sample of plutonium-239 to decay to a mass of 5 grams.