Question

The half-life of polonium-218 is 3 min. if you start with 16 mg, how long will it be before only 1.0 mg remains?

Answer 1

The decay equation is of the form
$m(t) = m_{0} e^{-kt}$
where
m(t) = the mass remaining after t minutes
m₀ =  the initial mass
k = constant

Because the half-life is 3 minutes, therefore
$e^{-3k} = \frac{1}{2}$
Therefore
-3k = ln(0.5)
k = ln(0.5)/(-3) = 0.231

Because m₀ = 16 mg, the time, t, for the mass to decay to 1.0 mg is given by
$e^{-0.231t} = \frac{0.1}{16} =0.00625 \\\\ -0.231t = ln(0.00625) \\\\ t = \frac{ln(0.00625)}{-0.231} =21.97 \, min$

Answer: 22 min (nearest integer)