Question

A radioactive isotope has a half-life of T. After how much time is its activity reduced to 6.25% of its original sample.

Answer 1

Hint:This is the problem of radioactive decay. Radioactivity states that a radioactive sample decays with time by the emission of appropriate rays. This is a natural phenomenon and with time the initial number of nuclei starts decreasing.

Complete step by step answer:
Given in the problem that the value of half-life is T.
It is given that the initial number of undecayed nuclei is \[{{N}_{0}}\] at time = t
Using the formula of radioactive decay \[N={{N}_{0}}{{[\dfrac{1}{2}]}^{n}}\]
Given that N= 6.25% of \[{{N}_{0}}\]
\[\begin{align}
& N={{N}_{0}}\times \dfrac{6.25}{100} \\
&\Rightarrow N=\dfrac{{{N}_{0}}}{16} \\
\end{align}\]
Using the above relation \[\dfrac{1}{16}={{\{\dfrac{1}{2}\}}^{n}}\]
Also, the relationship between n, T and \[{{T}_{1/2}}\]is \[n=\dfrac{t}{{{T}_{1/2}}}\]
\[\begin{align}
& \dfrac{1}{16}={{\{\dfrac{1}{2}\}}^{t/T}} \\
&\Rightarrow \dfrac{1}{2}={{\{\dfrac{1}{2}\}}^{t/T}} \\
&\Rightarrow \dfrac{t}{T}=4 \\
&\therefore t=4T \\
\end{align}\]
Time is 4 times the half-life. So, after 4T its activity reduced to 6.25% of its original sample.
Additional information:
Radium and polonium are examples of radioactive substances. The time in which half of the original number of nuclei decay is defined as the half-life. Half-life is an important parameter which describes the stability of the sample. Some radioactive samples have half-lives of thousands of years. Half life means that after that given time the number of undecayed nuclei will be one half of the initial number that were at the starting.

Note: Nuclear decay is an example of a purely statistical process. The activity of a radioactive substance is defined as the rate of the number of disintegrations per second. Its SI unit is Becquerel. Other units include Curie and Rutherford. They are bigger units of measuring the activity.