Question

If the half- life of a radioactive substance is $T$, then the fraction of its initial mass that remains after time $T/2$ will be
A) $\dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }}$
B) $\dfrac{3}{4}$
C) $\dfrac{1}{2}$
D) $\dfrac{1}{{\sqrt 2 }}$

Answer 1

Hint
Here we will use the formula of the fraction that is decay after $n$ half live which is equal to ${\left( {\dfrac{1}{2}} \right)^n}$, where $n$ is the ratio of the time $(t)$ after which we have to calculate the fraction of substance and the half life of substance $(T)$ i.e. $n = \dfrac{t}{T}$. Now substituting the values from the given question, we will get the fraction of substance after the time $T/2$.

Complete step by step solution
As it is given that, Half life of the substance is $T$.
Therefore, the fraction of its initial mass that remains after time $T/2$ is ${\left( {\dfrac{1}{2}} \right)^n}$
Here, $n$ is the number of years which is the ratio of $t$ and $T$ i.e. $n = \dfrac{t}{T}$.
Where, $t$ is the time after which we have to calculate the fraction of the substance and $T$ is the half life of the given radioactive substance.
As, it is given that $t = \dfrac{T}{2}$and $T = T$
Then, fraction of initial mass that remains after time $T/2$ is $= {\left( {\dfrac{1}{2}} \right)^{\dfrac{{T/2}}{T}}} = {\left( {\dfrac{1}{2}} \right)^{0.5}} = \dfrac{1}{{\sqrt 2 }}$
Hence, (D) option is correct.

Note
In this question we have come across the term half- life. For better understanding we should know about this term. So, half - life is the amount of time that is taken by the atoms of the given amount of the radioactive substance to disintegrate. Half life is also known as the biological half- life. This is known as so because it also signifies the amount of time that is taken for an activity of the substance in the body to lose half of the initial effectiveness.