Question

If a substance has a half-life of one million years, how much of it will be left after three million years? After four million years?

Answer 1

Hint: First we need to understand what is half-life. The time taken by any substance to reduce to half of its original quantity is known as the half-life of that substance. It is usually used to describe any form of decay be it exponentially or non-exponentially.

Complete answer: When we talk about the decaying of a substance, it is usually the exponential decay of a substance. When a substance decays or decreases at a rate that is in proportion to its current value, it is known as the exponential decay of a substance.
For a substance that decays exponentially, its half-life is constant throughout its lifetime. This helps us to determine the exponential decay equation and the decrease in the quantity of a substance when a number of half-lives have passed.
The fraction of substance remaining when n number of half-lives have passed is given by $\dfrac{1}{{{2}^{n}}}$, and the percentage of the substance remaining is given by $\dfrac{100}{{{2}^{n}}}$.
It is given to us that the half-life of the substance is 1 million years.
So, after 3 million years, the number of half-lives passed will be n=3.
Hence the fraction of substance remaining is
$\dfrac{1}{{{2}^{n}}}=\dfrac{1}{{{2}^{3}}}=\dfrac{1}{8}$
And the percentage of substance remailing is
$\dfrac{100}{{{2}^{n}}}=\dfrac{100}{{{2}^{3}}}=12.5%$
Similarly, after 4 million years, the number of half-lives passed will be n=4.
Hence the fraction of substance remaining is
$\dfrac{1}{{{2}^{n}}}=\dfrac{1}{{{2}^{4}}}=\dfrac{1}{16}$
And the percentage of substance remailing is
$\dfrac{100}{{{2}^{n}}}=\dfrac{100}{{{2}^{4}}}=6.25%$

Note: It is important to note that when the half-life is given for discrete entities like radioactive atoms, it describes the probability of a single unit of the entity decaying within its half-life time and not the time taken to decay the single entity in half as that is not possible.