Question

The isotope 239 Pu 94 has a half life of 24,400 years. If a sample of nuclear waste originally contained 12kg of 239 Pu 94, how much time must pass before the amount falls to 3 kg?

Answer 1

Equation for half-lives:

Nt = No x (1/2)^n

No = initial amount
Nt = final amount after t years
n = number of half lives = t/(single half-life)
t = years

Nt = 3/12 = 0.25
No = 12/12 = 1.00
n = t/(24400)

3/12 = (12/12) x (0.5)^(t/24400)
(0.25) = 1.00 x (0.5)^(t/24400)
0.25/1.00 = 0.5^(t/24400)
ln(0.25) = ln(0.5^(t/24400))
ln(0.25) = (t/24400)*ln(0.5)
ln(0.25)/ln(0.5) = (t/24400)
2 = t/24400
2*24400 = t
t = 48800 yrs

answer is t = 48,800 yrs