Suppose you have a rock that, when it solidifies, contains 1 microgram of a radioactive isotope. how much of this isotope remains after five half-lives?
The decay equation is [tex]m(t) = m_{0} e^{-kt}[/tex] where m = mass remaining after time t, k = constant.
Let t = t₁ = time for half-life. Then [tex] \frac{m_{0}}{2} =m_{0} e^{-kt_{1}} \\\\ -kt_{1} = ln(1/2) \\\\ t_{1} = 0.6931/k[/tex]
When t = 5 half-lives, then t = 5*(0.6931/k) = 3.4657/k The mass remaining is [tex]m = m_{0} e^{-k( \frac{3.4657}{k})} =m_{0} e^{-3.4657} = 0.0313m_{0}[/tex]
The mass remaining after 5 half-lives is 0.0313 μg
