Question

A radioactive substance has \[{10^8}\] nuclei. Its half-life is \[30\,{\text{s}}\]. The number of nuclei left after \[15\,{\text{s}}\] is nearly
A. \[2 \times {10^5}\]
B. \[3 \times {10^6}\]
C. \[7 \times {10^7}\]
D. \[5 \times {10^8}\]

Answer 1

Hint:Use the formula for the population of the radioactive substance at any time t. This formula gives the relation between the population of the radioactive substance at any time, initial population of the radioactive substance, time and half life period of the radioactive substance. Substitute all the values in this formula and calculate the required answer.

Formula used:
The population of the radioactive substance \[N\] at any time is given by
\[N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}}\] …… (1)
Here, \[{N_0}\] is the initial population of the radioactive substance, \[t\] is the time and \[T\] is the half-life period of the radioactive substance.

Complete step by step answer:
We have given that the initial population of the radioactive nuclei is \[{10^8}\].
\[{N_0} = {10^8}\]
The half-life period of the radioactive substance is \[30\,{\text{s}}\].
\[T = 30\,{\text{s}}\]
We have asked to determine the population of the radioactive nuclei at time \[15\,{\text{s}}\].
\[t = 15\,{\text{s}}\]
Substitute \[{10^8}\] for \[{N_0}\], \[15\,{\text{s}}\] for \[t\] and \[30\,{\text{s}}\] for \[T\] in equation (1).
\[N = \left( {{{10}^8}} \right){\left( {\dfrac{1}{2}} \right)^{\dfrac{{15\,{\text{s}}}}{{30\,{\text{s}}}}}}\]
\[ \Rightarrow N = \left( {{{10}^8}} \right){\left( {\dfrac{1}{2}} \right)^{\dfrac{1}{2}}}\]
\[ \Rightarrow N = \left( {{{10}^8}} \right)\sqrt {\dfrac{1}{2}} \]
\[ \Rightarrow N = \dfrac{{{{10}^8}}}{{\sqrt 2 }}\]
\[ \Rightarrow N = \dfrac{{{{10}^8}}}{{1.414}}\]
\[ \Rightarrow N = 0.70 \times {10^8}\]
\[ \therefore N = 7 \times {10^7}\]

Therefore, the number of radioactive nuclei left after 15 seconds is nearly \[7 \times {10^7}\].Hence, the correct option is C.

Additional information:
The process in which the nucleus of a radioactive atom undergoes disintegration spontaneously is known as radioactive decay.The time required for half of the initial number of nuclei of the radioactive substance to decay is known as the half-life period of the radioactive substance.

Note:One can solve the same question by another method. One can determine the value of the decay constant from the half-life period of the radioactive substance and then use the decay rate formula to determine the number of nuclei left at any time after the disintegration process has started.