Question

A radioactive element has a half-life period of 600 years. After 3000 years, what amount will remain?
[A] $\dfrac{1}{2}$
[B] $\dfrac{1}{8}$
[C] $\dfrac{1}{16}$
[D] $\dfrac{1}{32}$

Answer 1

Hint: Half-life of the element is the time required for the substance to reduce to its half. To solve this, use the relation between the half-life of an element and the initial amount of the element. The relation is \[{{N}_{\circ }}{{\left( \dfrac{1}{2} \right)}^{n}}=N\].

Complete step by step solution:
We know that the spontaneous disintegration of certain nuclei to form new element gives us active radiation which effects photographic plate and affects the electric and magnetic field. This is known as radioactivity.
The half-life period of a radioactive element is the time required for the substance to reduce to half of its initial value.
The general equation that we can use here to find out the amount of substance left after 3000 years is-
     \[{{N}_{\circ }}{{\left( \dfrac{1}{2} \right)}^{n}}=N\]
Where, \[{{N}_{\circ }}\] is the initial amount of the element and N is the amount of the element left after n half-life time. ‘n’ is the number of half-life.
Here, the half-life of the element is given to us which is 600 years. After 3000 years, its half-life will become 3000 times the actual half-life. To understand this, let us take an example. Let us say we have 1g of an element with a half-life of 2 years. So, after two years the element will be half of its actual amount and after 4 years, it will be one-fourth.
Similarly, here the half-life is 600 years so after 3000 years, it will become 3000 times of ${1}/{600}\;$.
Therefore, n = $3000\times \dfrac{1}{600}$ = 5.
We can write the equation as \[\dfrac{{{N}_{\circ }}}{N}={{\left( \dfrac{1}{2} \right)}^{5}}=\dfrac{1}{32}\]
As we can see that after 3000 years, $\dfrac{1}{32}$ amount of the radioactive element would be left.

Therefore, the correct answer is option [D] $\dfrac{1}{32}$.

Note: According to the radioactive disintegration law, the rate of disintegration at any time is proportional to the number of atoms present at that time. This law gives us a relation between the number of atoms present at a particular time. The relation is-
     \[n={{n}_{\circ }}{{e}^{-\lambda t}}\]
From here, we can find the half-life of an element which comes out to be-
     \[{{t}_{{1}/{2}\;}}=\dfrac{0.693}{\lambda }\]
Where, lambda $\left( \lambda \right)$ is the decay constant. We can see from the above equation that there is no term of the number of atoms present. It means that half-life of any element does not depend on the number of atoms initially.