During nuclear explosion, one of the products is with half-life of 28.1 years. If of was absorbed in the bones of a newly born baby instead of calcium, how much of it will remain after 10 years and 60 years if it is not lost metabolically.
(after_10_years) 0.781 μg
(after_60_years) 0.228 μg
Explanation
This is a first-order radioactive decay problem. The decay of radioactive isotopes follows first-order kinetics, where the half-life is constant and independent of initial concentration. Using the relationship between half-life and rate constant, we first determine k, then apply the integrated rate law to find remaining amounts at different time intervals.
Solution Steps
Step 1: Calculate rate constant (k) from half-life using k = 0.693/t₁/₂
k = 0.693/28.1 years = 0.02466 years⁻¹
Step 2: Calculate amount remaining after 10 years using k = (2.303/t) log([R]₀/[R])
log([R]₀/[R]) = kt/2.303 = ()/2.303 = 0.1071
[R]₀/[R] = antilog(0.1071) = 1.28
[R] = 1 μg/1.28 = 0.781 μg
Step 3: Calculate amount remaining after 60 years
log([R]₀/[R]) = kt/2.303 = ()/2.303 = 0.6426
[R]₀/[R] = antilog(0.6426) = 4.39
[R] = 1 μg/4.39 = 0.228 μg