We’re being asked to **calculate the time it takes (in years) for the sample to be reduced to 1.00 mg of Rn-222** if there are 36.0 g of Rn-222 initially.

Recall that ** radioactive/nuclear decay of isotopes** follows first-order kinetics, and the integrated rate law for first-order reactions is:

$\overline{){\mathbf{ln}}{\mathbf{}}{\mathbf{\left[}\mathbf{N}\mathbf{\right]}}_{{\mathbf{t}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{kt}}{\mathbf{+}}{\mathbf{ln}}{\mathbf{}}{\mathbf{\left[}\mathbf{N}\mathbf{\right]}}_{{\mathbf{0}}}}$

where:

**[N] _{t}** = concentration at time t

**k** = decay constant

**t** = time

**[N] _{0}** = initial concentration.

Also, recall that ** half-life** is the time needed for the amount of a reactant to decrease by 50% or one-half.

The half-life of a first-order reaction is given by:

$\overline{){{\mathbf{t}}}_{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}{\mathbf{=}}\frac{\mathbf{ln}\mathbf{}\mathbf{2}}{\mathbf{k}}}$

The half-life of ^{222}Rn is 3.80 days. If a sample contains 36.0 g of Rn-222, how many years will it take for the sample to be reduced to 1.00 mg of Rn-222?

a. 58.28

b. 0.1824

c. 19.908

d. 0.1597

e. 10.49

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