Final answer:
To find the time it takes for a 200-gram sample of Actinium-227 to decay to less than 0.5 grams, we can calculate the number of half-lives required and then multiply by the half-life period. It will take approximately 38.72 years, which means the closest answer choice is 19 years.
Step-by-step explanation:
We need to calculate the amount of time it takes for a 200-gram sample of Actinium-227 to decay to less than 0.5 grams. The half-life of Actinium-227 is given as 2.2 years. We can use the concept of half-lives to determine how many half-lives it will take for the sample to decay to less than 0.5 grams.
After each half-life of 2.2 years, the amount of the substance is halved. So, after 2.2 years, we will have 100 grams; after another 2.2 years (4.4 years in total), we will have 50 grams; this pattern continues until we have < 0.5 grams remaining. This is a classic exponential decay problem, and we can solve it using the formula N(t) = N(0) × (1/2)^(t/T), where N(t) is the remaining amount after time t, N(0) is the initial amount, and T is the half-life. We need to rearrange this formula to find t when N(t) < 0.5 grams.
Let's solve for t:
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- N(t) = N(0) × (1/2)^(t/T)
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- 0.5 = 200 × (1/2)^(t/2.2)
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- (1/2)^(t/2.2) = 0.5 / 200
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- (1/2)^(t/2.2) = 1/400
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- t/2.2 = log2(400)
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- t = 2.2 × log2(400)
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- t ≈ 17.6 half-lives
Now, we need to multiply the number of half-lives by the half-life period:
17.6 half-lives × 2.2 years per half-life = 38.72 years
Therefore, it will take approximately 38.72 years for a 200-gram sample of Actinium-227 to decay to less than 0.5 grams. Since 38.72 years is not listed in the answer choices, we pick the closest one that still leads to a remaining mass less than 0.5 grams, which is option B. 19 years.