Question

Plutonium-240 decays according to the function L where o

represents the quantity remaining after t years and k is the decay constant,
0.00011... How long will it take 24 grams of plutonium-240 to decay to 20
grams?

Answer 1

`\boxed{\textbf{1700 yr}}`

Explanation:

`-\frac{\text{d}L}{\text{d}t} = kL\\\\(dL)/(L)=-kdt\\\\\ln L = -kt + C\\\text{At t = 0, L = L$_(0)$, so C = $\ln L_(0)$}\\\ln L = -kt + \ln L_(0)\\\ln L_(0) - \ln L = kt\\\\\ln (L_(0))/(L) =kt`

Data:

L₀ = 24 g

L = 20 g

k = 0.000 11 yr⁻¹

Calculation:

`\ln (24)/(20) =0.000 11t\\\\\ln 1.2 = 0.000 11t\\\\0.1823 = 0.000 11t\\\\t = (0.1823)/(0.000 11) = \textbf{1700 yr}\\\\\text{It will take } \boxed{\textbf{1700 yr}}\text{ for the polonium to decay to 20 g}`