Question

let us assume that the half-lives of a radioactive element for $\alpha $ and $\beta $ decay are $4$ years and $12$ years respectively, then find the percentage of the element that remains after $12$ years?
$\begin{align}
  & A.6.25\% \\
 & B.12.5\% \\
 & C.25\% \\
 & D.50\% \\
\end{align}$

Answer 1

Hint: The number of half-lives completed due to the $\alpha -decay$ and the $\beta -decay$ is to be calculated first. Then the total number of half-lives that happened is to be found. The percentage of element remaining after $12$ years will be the reciprocal of the ${{2}^{n}}$ which is then multiplied by a hundred. This information will help you in solving this question.

Complete step-by-step solution
First of all, let us calculate the number of half-lives completed due to $\alpha -decay$. This can be found by taking the ratio of the total time after which the half-life is being calculated to the period of one half-life. That is,
$\text{number of half life=}\dfrac{\text{total time period}}{\text{half life}}$
The half-life period of $\alpha -decay$ is given as,
${{\tau }_{alpha}}=4yrs$
The total time period can be mentioned as,
$t=12yrs$
Substituting the values in it will give,
${{n}_{\alpha }}=\dfrac{12}{4}=3$
Similarly, we have to find the number of half-lives of the $\beta -decay$.
The half-life period of $\beta -decay$ is given as,
${{\tau }_{\beta }}=12yrs$
Substituting the values in it will give,
${{n}_{\beta }}=\dfrac{{{\tau }_{\beta }}}{t}$
Substituting the values in it will give,
${{n}_{\beta }}=\dfrac{12}{12}=1$
Therefore the total number of the half-lives can be written as,
$\tau ={{\tau }_{\alpha }}+{{\tau }_{\beta }}$
Substituting the values in it will give,
$\tau =3+1=4$
Hence the percentage of the element that remains after $12years$ will be calculated as,
$P=\dfrac{1}{{{2}^{\tau }}}\times 100$
Substituting the values in it will give,
$P=\dfrac{1}{{{2}^{4}}}\times 100=\dfrac{100}{16}=6.25\%$
Therefore the answer is calculated as per the question. It has been given as option A.

Note: Half-life is a common term in radioactivity, which is the interval of time needed for one-half of the atomic nuclei of a radioactive sample to decay. This will change spontaneously into other nuclear species by the emission of particles and energy. The element with the shortest half-life is francium.