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

Answer:

$\boxed{\textbf{1700 yr}}$

Step-by-step explanation:

$-\dfrac{\text{d}L}{\text{d}t} = kL\\\\\dfrac{dL}{L}=-kdt\\\\\ln L = -kt + C\\\text{At t = 0, L = L _{0} , so C = \ln L_{0} }\\\ln L = -kt + \ln L_{0}\\\ln L_{0} - \ln L = kt\\\\\ln \dfrac{L_{0}}{L} =kt$

Data:

L₀ = 24 g

L = 20 g

k = 0.000 11 yr⁻¹

Calculation:

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