Question

We have $ 230g $ of substances with a half-life of $ 0.75years. $ How much will remain after $ 3 $ years?

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. The formula we will use will be fraction of a substance remaining after $ n $ half-lives $ =\dfrac{1}{{{2}^{n}}} $

Complete answer:
Each time a period of time goes by that is equal to the half-life; there will be only $ \dfrac{1}{2} $ of the previous amount of material remaining. In this case, three years represents $ \dfrac{3}{0.75}=4 $ half-lives.
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. So, the amount that remains has been halved four times:
 $ \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}={{\left( \dfrac{1}{2} \right)}^{4}}=\dfrac{1}{16} $
Thus we have, there is $ \dfrac{1}{16} $ of the original amount, which is; $ 230\times \dfrac{1}{16}=14.375g $

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.