If pure polonium-210 (k = 5.02 × 10−3 day−1) were used in a crime, based on radioactive disintegration alone, how long would it take for half the original isotope to disappear?

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BlackTom AI · Accepted Answer

To tackle this problem, I’ll start by identifying the relationship between the decay constant k and the half-life t1/2. The rules of radioactive decay tell us t1/2 = ln(2) / k, where ln(2) is approximately 0.693.
Option 1: 138 days. If we plug k = 5.02 × 10^−3 day^−1 into t1/2 = 0.693 / k, we get t1/2 ≈ 0.693 / 0.00502 ≈ 138.0 days. This matches the standard half-life calculation, so this option aligns with the formula.
Option 2: 1.44 × 10^3 minutes. Converting 1.44 × 10^3 minutes to days: 1440 minutes = 24 hours = 1 day. A half-life of 1 day would correspond to k ≈ 0.693 day^−1, which is far larger than the given 5.02 × 10^−3 day^−1. Therefore, this option is inconsistent with the provided decay constant.
Option 3: 2.9 years. First convert to days: 2.9 years ≈ 2.9 × 365 ≈ 1058.5 days. Using t1/2 = 0.693 / k with k = 0.00502 day^−1 would yield about 138 days, not 1058 days, so this greatly overestimates the half-life and is incorrect.
Option 4: 210 years. In days, 210 years ≈ 210 × 365 ≈ 76,650 days. This is orders of magnitude larger than the calculated half-life using the given k, so this is not plausible for the stated decay rate.
Option 5: 69.3 hours. Convert to days: 69.3 hours ≈ 2.8875 days. Using the decay formula, a half-life around 2.9 days would require k ≈ 0.693 / 2.8875 ≈ 0.240 day^−1, which is much larger than 0.00502 day^−1. Hence this is not consistent with the given k value.
In summary, only the first option (138 days) is directly consistent with t1/2 = ln(2)/k for k = 5.02 × 10^−3 day^−1, while all the other options fail to match either the unit conversions or the relationship between k and t1/2. The critical idea is translating the decay constant into a time span that halves the amount, and checking each alternative against that fundamental calculation.

CHM-1020-LD01.2025SU Problem Set 4-Nuclear Chemistry

If pure polonium-210 (k = 5.02 × 10−3 day−1) were used in a crime, based on radioactive disintegration alone, how long would it take for half the original isotope to disappear?

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