Question

Calculate the half-life period of a radioactive element which remains only 1/16 of its original amount in 4740 years:
A.1185 years
B.2370 years
C.52.5 years
D.none of these

Answer 1

Hint: We know that half life is the time required for the concentration of a reactant to decreases to half of its initial concentration. It is represented by the symbol ${t_{1/2}}$. The formula to calculate half life is $\dfrac{{0.693}}{k}$, here, k is rate constant.

Complete step by step answer:
Let’s take the original concentration of a substance as $\left[ {{A_0}} \right]$. After first half life, the concentration becomes$\dfrac{{\left[ {{A_0}} \right]}}{2}$, after second half life concentration becomes $\dfrac{{\left[ {{A_0}} \right]}}{4}$, after third half life concentration becomes $\dfrac{{\left[ {{A_0}} \right]}}{4}$and after fourth half life concentration becomes $\dfrac{{\left[ {{A_0}} \right]}}{{16}}$.

$\left [A _{0} \right ]\overset{t_{1/2} (1)}{\rightarrow}\dfrac{[A_{0}]}{2}\overset{t_{1/2} (2)}{\rightarrow}\dfrac{[A_{0}]}{4}\overset{t_{1/2}(3)}{\rightarrow}\dfrac{[A_{0}]}{8}\overset{t_{1/2} (4)}{\rightarrow}\dfrac{[A_{0]}}{16}$


Now, come to the question. 1/16 of the original substance is left in 4740 years, that means, 4 half lives is equal to 4740 years as 1/16 of the original substance remains in four half lives.

$ \Rightarrow 4{t_{1/2}} = 4740$
$ \Rightarrow {t_{1/2}} = \dfrac{{4740}}{4} = 1185\,{\rm{years}}$

Therefore, half life of the substance is 1185 years.

So, the correct answer is Option C.

Additional Information:
Let’s learn about nuclear reactions. Nuclear chemistry is the branch of chemistry that deals with the structure of nuclei of atoms and the changes atomic nuclei undergoes. Nuclear reactions involve a very large change of energy. In nuclear reactions, combination, splitting and emission of nuclei occurs. Catalyst, temperature, pressure do not affect the reaction rates of nuclear reactions.


Note: Radioactivity is the phenomenon in which a substance emits radiation. The process radioactivity is named by chemist Marie Curie. There are three ways by which emission of radiation takes place. They are alpha radiation, beta radiation and gamma radiation. In alpha radiation, alpha particles releases, in beta radiation, beta particles releases and in gamma radiation high energy electrons (photons) releases.