Suppose you have 100 g of radioactive plutonium-239 with a half-life of 24,000 years. How many grams will remain after (a) 12,000 years (b) 24,000 years (c) 96,000 years?
To calculate the amount of radioactive plutonium-239 remaining after a given time, we can use the half-life formula. The formula is as follows:
N(t) = N₀ * (1/2)^(t / T₁/₂)
Where:
N(t) is the amount of radioactive material remaining after time t
N₀ is the initial amount of radioactive material
T₁/₂ is the half-life of the substance
t is the time passed
Now let's calculate the answers to the specific scenarios:
(a) After 12,000 years:
N(12,000) = 100 * (1/2)^(12,000 / 24,000)
N(12,000) = 100 * (1/2)^(1/2)
N(12,000) ≈ 100 * 0.7071
N(12,000) ≈ 70.71 grams
(b) After 24,000 years:
N(24,000) = 100 * (1/2)^(24,000 / 24,000)
N(24,000) = 100 * (1/2)^1
N(24,000) = 100 * 0.5
N(24,000) = 50 grams
(c) After 96,000 years:
N(96,000) = 100 * (1/2)^(96,000 / 24,000)
N(96,000) = 100 * (1/2)^4
N(96,000) = 100 * 0.0625
N(96,000) = 6.25 grams
Therefore, after (a) 12,000 years, approximately 70.71 grams will remain. After (b) 24,000 years, 50 grams will remain. After (c) 96,000 years, 6.25 grams will remain.