Question

The half life period of a radioactive substance is 5 minutes. The amount of substance decayed in 20 minutes will be?
a) $93.75\%$
b) $75\%$
c) $25\%$
d) $6.25\%$

Answer 1

Hint: We first need to find out the number of half-lives that will pass in the given time of decaying. Then using the formula, we will get the fraction of substance remaining and on subtracting it from one, we will get the required solution.

Formula used:
Fraction of a substance remaining after $n$ half-lives $=\dfrac{1}{2^n}$

Complete step by step solution:
We have been given the half-life of the radioactive substance, $t_{\dfrac{1}{2}}=5$ minutes.
And we have to find the percentage of substance decayed after 20 minutes.
Therefore, number of half-lives passed, $n=\dfrac{20}{5}=4$
Since, the fraction of a substance remaining after $n$ half-lives $=\dfrac{1}{2^n}$
Then the fraction of substance decayed $=1-\dfrac{1}{2^n}=1-\dfrac{1}{2^4}=1-\dfrac{1}{14}=\dfrac{15}{16}$
So, the percentage of substance decayed $=\dfrac{15}{16}\times 100\% =93.75\%$
Hence, option a is the correct answer.

Additional information:
The term half-life most commonly is used in nuclear physics to describe how long a stable atom survives. In biological terms it is also referred as biological half-life, whose conversion, doubling time is used to determine the time taken by a drug to spread.
Half-life is used for processes whose decays occur exponentially or approximately exponentially. There are processes in which the half-life changes and many often use the terms like first half-life, second half-life and so on for such types of processes.

Note: Half-life has a probabilistic nature and denotes the time in which, on average, about half of entities decays. Suppose if we have only one atom, then it is not like after one half-life, one half of the atom will decay. So, we can say that half-life just describes the decay of distinct entities