Answer to Question #200365 in Atomic and Nuclear Physics for Lee Agustin

Question

A certain radioactive isotope decays to one fourth of its original amount in 4.0 hours. What is its half life? A. 1.4 hours B. 2.0 hours C. 2.5 hours D. 3.4 hours

Answer 1

Let us consider the equation of the radioactive decay. If $\tau_{0.5}$ is the half-life time, then after t hours the amount of radioactive atoms will be

$N(t) =N_0\cdot2^{-t/\tau_{0.5}},$ in our case

$0.25 N_0 =N_0\cdot2^{-t/\tau_{0.5}},$ or

$0.25=2^{-t/\tau_{0.5}}, \\ 2^{-2}=2^{-t/\tau_{0.5}}, \\ 2=t/\tau_{0.5}, \;\; t=4.$

So $2=4/\tau_{0.5},\;\; \tau_{0.5}=2.$

Therefore, the half-life time is 2 hours.