Question

What is radioactivity? Show that radioactive decay is exponential in nature. The half-life of a certain substance is $5$ days. How many days does this substance take for $31/{32}^{th}$ of the initial mass of disintegrate?

Answer 1

Radioactivity is the process by which a nucleus of a unstable atom loses energy by emitting radiation, including alpha particles, beta particles, gamma rays and conversion electrons.

Rate of decay of a nucleus is directly proportional to its quantity i.e. $\frac{-dN}{dt}\propto N$

Thus we get $\frac{dN}{dt}=-\lambda N$ where $\lambda =\frac{0.693}{T_{1/2}}$ is proportionality constant

Integrating both sides, ${\int }_{N_o}^N\frac{dN}{N}={\int }_o^t-\lambda t$ where $N_o$ is the amount of nucleus initially

$⟹\ln N{|}_{N_o}^N=-\lambda t{|}_o^t$

$⟹\ln \frac{N}{N_o}=-\lambda t$

$⟹\frac{N}{N_o}=e^{-\lambda t}$

$⟹$ Amount of nuclei left after time $t$, $N=N_o{e}^{-\lambda t}$

Given : Half-life of a substance $T_{1/2}=5$ days

As the substance gets disintegrated by $31/{32}^{th}$ of its initial mass in time $t$.

Thus amount of substance left, $N=(1-\frac{31}{32})N_o=\frac{1}{32}{N}_o$

From above equation:

$\frac{1}{32}{N}_o=N_o{e}^{-\lambda t}$

$\frac{1}{32}=e^{-\frac{0.693}{T_{1/2}}t}$

$\ln \frac{1}{32}=-\frac{0.693}{T_{1/2}}t$

$-3.4657=-\frac{0.693}{5}\times t$

$⟹t=25$ days