Question

Starting with a sample of ${}^{66}Cu$, $\dfrac{7}{8}$ of it decays into $Zn$ in 15 minutes. The corresponding half-life is-
(A). $10\,\min$
(B). $15\,\min$
(C). $5\,\min$
(D). $7\dfrac{1}{2}\,\min$

Answer 1

A radioactive substance gives out radiation due to which its decomposition takes place along with the change in its properties. The time taken by the substance to decompose to half the amount is called the half life of a radioactive substance. The half life depends on the amount of substance as well as the decay constant. First we calculate the number of half-lives that have been finished and divide the time taken by the number of half lives finished.

Complete answer:
Half life of a radioactive substance is the time taken by it to decompose in half of the total remaining amount of it. Its SI unit is seconds ($\sec$).
It is given by-
$N={{N}_{0}}{{e}^{-\lambda t}}$
Here, $N$ is the amount of substance left
${{N}_{0}}$ is the initial amount of substance
$\lambda$ is the decay constant of the substance
$t$ is the time taken
Let the initial amount of ${}^{66}Cu$ be $N$.
The amount of copper that decays in one half life is $\dfrac{N}{2}$.
Therefore, the remaining amount is $\dfrac{N}{2}$.
Amount of copper decayed in two half lives is $\dfrac{\dfrac{N}{2}}{2}=\dfrac{N}{4}$.
Therefore, amount of copper left is $\dfrac{N}{2}-\dfrac{N}{4}=\dfrac{N}{4}$
Amount decayed in three half lives is $\dfrac{\dfrac{N}{4}}{2}=\dfrac{N}{8}$
Therefore, the total amount of copper decayed is-
$\dfrac{N}{2}+\dfrac{N}{4}+\dfrac{N}{8}=\dfrac{7N}{8}$
Therefore, we know that time taken for copper to decay to $\dfrac{7N}{8}$ is 15 mins
15minutes has three half lives, so one half life will be- $\dfrac{15}{3}=5\,\min$.
Therefore, the half life of copper is $5\,\min$.

Hence, the correct option is (C).

Note:
Total time taken for a radioactive substance to completely vanish is called its decay constant. Radioactive substances are highly unstable as a result of which during radioactive reactions, they give out alpha, beta and gamma rays due to which their properties change completely.