A radioactive substance is decaying exponentially. If there are initially 15 grams of the substance and after 3 hours there are 5 grams, what is the approximate half life of the substance, in hours?

Answer 1

Radioactive decay follows the exponentially decaying function

F(t) = Ae^-λt

So that over time the amount remaining at any specific time is less than the initial amount (t = 0).

F(0) = Ae^-0 = A = 15 grams

So,

F(t) = 15e^-λt

Given the fact that after three hours the amount remaining is 5 grams we can compute λ giving the decay constant...

F(3) = 15e^-3λ = 5

1/3 = e^-3λ

ln(1/3) = -3λ

λ = -ln(1/3) / 3 = ln(3)/3 ≈ 0.366

Now...

F(t) ≈ 15e^-0.366t

If we want to compute the half-live (half of the sample remains in how long) we can use our initial function...

F(t) = 7.5 for half of the initial sample

7.5 = 15e^-0.366t

1/2 = e^-0.366t

ln(1/2) = -ln(2) = -0.366t = -λt

Therefore t_1/2 = λ/ln(2) ≈ 0.366/0.693 ≈ 0.531 hours (about half and hour)