Question

What will be the fraction which will remain after a time $\dfrac{T}{2}$, if the half life of radioactive is $T$?
A) $\dfrac{1}{{\sqrt 2 }}$
B) $\dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }}$
C) $\dfrac{1}{2}$
D) $\dfrac{3}{4}$

Answer 1

Hint: -The half – life is the time which is required to reduce the half of the value of the given quantity. It is also used to describe how quickly unstable atoms undergo or how long the stable atoms will survive the decaying of radioactive.

Complete step by step answer:
The half – life can be defined as the time required to reduce the half of the initial value of the given quantity. It is also used to describe how quickly unstable atoms undergo or how long the stable atoms will survive the decaying of radioactive. It can also determine type of exponential and non – exponential decay.
The formula for half – life in exponential decay is –
$
   \mapsto \dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}} \cdots \left( 1 \right) \\
   \mapsto \dfrac{N}{{{N_0}}} = {e^{\dfrac{{ - t}}{T}}} \cdots \left( 2 \right) \\
   \mapsto \dfrac{N}{{{N_0}}} = {e^{ - \lambda t}} \cdots \left( 3 \right) \\
 $
where, ${N_0}$ is the initial value of the substance which will decay
$N$ is the quantity which has not decayed even after time $t$ and is still remaining
${t_{\dfrac{1}{2}}}$ is the half – life of the quantity which will decay
$T$ is the average lifetime of the quantity which will decay
$\lambda $ is the decay constant
The exponential decay can be determined by using any of the above equivalent formulas.
So, for solving the question we will use the equation $\left( 1 \right)$ -
$
  \dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^n} \\
  \dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}} \\
 $
We know that from question, $t = \dfrac{T}{2}$
$
  \therefore \dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{{\dfrac{T}{2}}}{T}}} \\
  \dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{1}{2}}} \\
  \dfrac{N}{{{N_0}}} = \dfrac{1}{{\sqrt 2 }} \\
 $
Hence, the fraction remaining after time $\dfrac{T}{2}$ is $\dfrac{1}{{\sqrt 2 }}$.

Therefore, the correct option is (A).

Note: -Half – life becomes constant over the lifetime of an exponentially decaying quantity. Usually, it can also describe the discrete entities decay. The entity's decay can be radioactive atoms. Then, the half – life can also be defined as time required for the decay of half of the entities.