Decay constant of radioactive isotope

Question

A radioactive isotope decays according to the formula A(t)=Aoe^kt, where t is time in years. The initial amount was 10g and 8g of the isotope remained after 5 years.

a) Find the decay constant, k

b)Calculate the amount remaining after 10 years to the nearest tenth of a gram.

c)Find thr half life to the nearest 0.1 years.

Answer 1

(a)

 

Set A = 8, A₀ = 10, t = 5, and solve for k.  It must come out negative.

 

8 = 10e^k(5)

 

0.8 = e^5k

 

k = ln(0.8) / 5 ≅ -0.0446287 years^-1

 

Test the answer.

 

(b)

 

A(10) = 10e^{-0.0446287(10)} g ≅ 6.4 g

 

(c)

 

Set e^kt = 1/2 and solve for t.

 

0.5 = e^-0.0446287t

 

t_1/2 = ln(0.5) / (-0.0446287) years ≅ 15.5314 years (round to 15.5 years)

 

Test the answer.