Problem: Americium-241 is used in smoke detectors. It has a first order rate constant for radioactive decay of k = 1.6 10-3-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1.How much of a 1.00 mg sample of americium remains after 3 half-lives?

We’re being asked to calculate the amount of americium (Am) remaining after 3 half-lives if we start with 1 mg sample. 

Recall that half-life is the time needed for the amount of a reactant to decrease by 50% or one-half. One way to determine the amount remaining after x half-lives is:

$\boxed{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{t}}\mathbf{=}\frac{\left[A\right]}{{\mathbf{2}}^{\mathbf{x}}}}$

where [A]_t = concentration after x half-lives, [A]₀ = initial concentration and x = number of half-lives.