Question

A radioactive isotope has a half-life of $T$ years. How long will it take the activity to reduce to (a) $3.125\%$ (b) $1\%$ of its original value?

Answer 1

(a) Let the half-life be $T$ years.

original amount of isotope is $N_o$
after decay, the remaining amount is $N$
now as per given values, $N/N_o=3.125\%=1/32$
also, $N/N_o=e^{-\lambda t}$
where $\lambda$ is decay constant
also $\lambda =0.693/T$
so, $t=3.4657/\lambda =5T$ years.

(b) Let the half-life be $T$ years.
original amount of isotope is $N_o$
after decay, the remaining amount is $N$
now as per given values, $N/N_o=1\%=1/100$
also, $N/N_o=e^{-\lambda t}$
where $\lambda$ is decay constant
also $\lambda =0.693/T$
so, $t=4.6052/\lambda =6.645T$ years.