Question

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?

Answer 1

The original mass is m₀ = 1 μg.

The decay equation is

`m(t) = m_(0) e^(-kt)`
where
m = mass remaining after time t,
k = constant.

Let
t = t₁ = time for half-life.
Then

`(m_(0))/(2) =m_(0) e^{-kt_(1)} \\\\ -kt_(1) = ln(1/2) \\\\ t_(1) = 0.6931/k`

When t = 5 half-lives, then
t = 5*(0.6931/k) = 3.4657/k
The mass remaining is

`m = m_(0) e^{-k( (3.4657)/(k))} =m_(0) e^(-3.4657) = 0.0313m_(0)`

The mass remaining after 5 half-lives is
0.0313 μg

Answer: 0.0313 μg