Question

A human body requires $0.01\,m$activity of radioactive substance after $24\,hours$ . Half-life of radioactive substance is $6\,hours$ . Then injection of maximum activity of radioactive substances that can be injected.
A. $0.08$
B. $0.04$
C. $0.16$
D. $0.32$

Answer 1

In questions of radioactivity, there is always a substance which becomes half of its amount in a given time period that is its half-life. Here $0.01\,m$ is the remaining amount of that radioactive substance and its half-life is also given. Use the relation between the initial amount and remaining amount.

Complete step-by-step answer: There are radioactive substances which decompose of their initial amount in a specific time period, we called it as half-life of that substance. If we talk about uranium it is also radioactive in nature and decomposes as by its half-life period. It is given in the question that radioactive substance in the human body decomposes and the amount which remains after that is $0.01\,m$ it is given in activity terms, it decomposes in $6\,hours$. It means in every $6\,hours$ it will become half of its initial amount, we know that there is a relation between initial amount and remaining amount.
Let’s write it as- Remaining amount= $initial\,activity{\left( {\dfrac{1}{2}} \right)^n}$
Where, $n = \dfrac{{Total\,time\,(T)}}{{Half - life\,}}$ $= \,\dfrac{{24\,hours}}{{6\,hours}} = \,4\,$
We get a value of $4$ so putting it in the above equation,
Now putting the values in this equation we get, $0.01\,m = \,initial\,amount\,{\left( {\dfrac{1}{2}} \right)^4}$
$0.01\,m = \,\dfrac{{initial\,amount\,}}{{16}}$
$initial\,amount\, = 0.01\,m\,\, \times \,16\,$
$0.01\,m\,\, \times \,16\, = \,0.16\,m$
We get $0.16$ molar of initial concentration, it means that radioactive substance is present as $0.16\,m$ and it decomposes and becomes $0.01\,m$ after $24\,hours$ .

Option C is correct.

Note: There are two time period given, one is the total time and other is half-life period. Half-life is represented as ${t_{\dfrac{1}{2}}}$ and total time period is represented by capital $T$. You have to put $24\,hours$ in place of capital $T$ in the formula and $6\,hours$ is the half life. Solve the equation by taking initial concentration as $0.01\,m$ .