Question

After $20\min $, the amount of certain radioactive substance disintegrates ${{\left( 15/16 \right)}^{th}}$ of the original amount.
What is the half-life of the radioactive substance?
(A) 7
(B) 4
(C) 5
(D) 10

Answer 1

Hint: Consider the half life and average life of radioactive substances. Use the half-life formula to find the half-life, consider the given data in the question. Radioactive irrespective of other factors always follow First order kinetics in which the rate of disintegration at any time for a substance is proportional to its concentration at that point of time.

 Complete step by step answer:
By the law of radioactive decay which state that when a radioactive substance goes either $\alpha $or$\beta $ or $\gamma $ decay, the number of nucleus going under the decay per unit time is depended to the total number of nucleus in the sample given.
The radioactive disintegration happens when the radioactive substance nucleus breaks simultaneously which is happening here in this too, so we are going to use the formula of Half-life.
Let use consider the amount left after$n$half-life be $\left( N \right)$
Where $\left( N \right)=\dfrac{{{N}_{{\mathrm O}}}}{{{2}^{n}}}$
Here $\left( N \right)$=Amount of radioactive substance left after ${{n}^{th}}$hour
         ${{N}_{{\mathrm O}}}$= Initial concentration of radioactive substance
          $n$ = Time
 And we can write it as:
\[\dfrac{N}{{{N}_{{\mathrm O}}}}=\dfrac{1}{{{2}^{n}}}\]
Now putting the data which is already given to us
\[\dfrac{1}{16}=\dfrac{1}{{{2}^{n}}}\]
Or
\[{{\left( \dfrac{1}{2} \right)}^{4}}={{\left( \dfrac{1}{2} \right)}^{n}}\]
\[\therefore n=4\]
Now for finding the half-life we can say that
\[n=\dfrac{TotalTime}{{{t}_{1/2}}}\]………………(2)
Now putting the values in (2) equation we get
\[{{t}_{1/2}}=\dfrac{20\min }{4}\]
${{t}_{1/2}}=5$
We expect that after a lifetime the radioactive elements will live longer than the half life of it. If that radioactive element (substance) gets half decayed after a half-life, some well defined average life expectancy can be assumed which is also the mean life of the atom (element).
A Radioactive substance undergoes decay or disintegration due to their higher instability and they keep decay or disintegrate to get the stability in them.

NOTE:
-The half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value which is given.
-The radioactive substances always follow first order kinetics, irrespective of any other factors.
- According to the first order kinetics the rate of disintegration of a substance at any time is directly proportional to its concentration at that point.