Question

If ${N_0}$ is the initial number of nuclei, the number of nuclei remaining un-decayed at the end of ${n^{th}}$ half-time is:
(a) ${2^{ - n}}{N_0}$
(b) ${2^n}{N_0}$
(c) ${n^{ - 2}}{N_0}$
(d) ${n^2}{N_0}$

Answer 1

First we have to know what is the decay constant of any radioactive particles and the formula used to find the number of nuclei remaining un-decayed at the end of ${n^{th}}$ half-time.

Formula used:
Decay constant formula:
$\lambda = \dfrac{{2.303}}{t}\log \dfrac{{{N_0}}}{{{N_t}}}$
Where, $\lambda$ is decay constant, $t$ is the time, ${N_t}$ is the concentration of the radioactive particle at time and ${N_0}$ is the initial concentration of the radioactive particles.
Half-life formula,
For half-life, (Half-life: The time at which reaction is half completed i.e, ${N_t} = \dfrac{{{{[N]}_O}}}{2}$ )
${t_{\dfrac{1}{2}}} = \dfrac{{\ln (2)}}{\lambda } = \dfrac{{0.693}}{\lambda }$
Where, $\lambda$ is decay constant and ${t_{\dfrac{1}{2}}}$ is half-life time.

Complete step-by-step answer
We start with the definition of the decay constant:
Decay constant: The decay constant is a term which states that the proportionality between the size of the total radioactive atoms and the rate at which the radioactive atoms decrease because of radioactive decay. This is also known as disintegration constant.
We have to find the number of nuclei remaining un-decayed at the end of ${n^{th}}$ half-time is,
The decay constant formula is:
$\lambda = \dfrac{{2.303}}{t}\log \dfrac{{{N_0}}}{{{N_t}}}$
At half-life time, ${t_{\dfrac{1}{2}}} = \dfrac{{\ln (2)}}{\lambda } = \dfrac{{0.693}}{\lambda }$
So, ${N_t} = {N_0}{\left( {\dfrac{1}{2}} \right)^n}$
So, the number of nuclei present in the un-decayed at the end of ${n^{th}}$ half-time is,
${N_t} = {N_0}{\left( 2 \right)^{ - n}}$
Where, $n$ is the number of half-life and ${N_0}$ is the initial concentration.
Hence, the correct option is (a) ${2^{ - n}}{N_0}$ .

Additional Information
Radioactive decay: The process in which the low stability atomic nuclei loses their energy by radiation is called Radioactive decay. There are three main types of the emissions of the decay, they are alpha decay, gamma decay and beta decay.

Note
The radioactive reactions are always first order reactions because the rate of reaction is directly proportional to the radioactive atoms and the rate of decay of a sample of radioactive particle is the decrease in the number of radioactive nuclei in per unit time.