Question

A radioactive has a half-life of four months, three-fourth of the substance will decay in
A) Three months
B) Four months
C) Eight months
D) Twelve months

Answer 1

An element that emits radiations such as alpha rays, beta rays, and gamma rays are called radioactive elements. It decays on radiating radiations. This phenomenon is called radioactivity. The time required for the element to decay into half of its initial amount is called the half-life. According to the type of element and the initial amount the half-life of an element changes.

Formula used:
$T = N{t_{\tfrac{1}{2}}}$
Where, $T$=Time taken to decay, $N$=number of half-lives and ${t_{\tfrac{1}{2}}}$= half-life

Complete step by step answer:
Consider an initial amount as${N_o}$. We know that half-life is four months, so after four months ${N_o}$becomes$\dfrac{{{N_O}}}{2}$ . $$$$
For the next half-life, $\dfrac{{{N_O}}}{2}$is the initial amount. After the 2nd half-life, $\dfrac{{{N_O}}}{2}$becomes$\dfrac{{{N_O}}}{4}$.
$\dfrac{{{N_O}}}{4}$is the radioactive substance present after two half-lives means$\dfrac{3}{4}{N_O}$decayed.
$\Rightarrow {N_O} - \dfrac{{{N_0}}}{4} = \dfrac{3}{4}{N_O}$
We have the formula to calculate the time, as it requires 2 half-lives for the radioactive substance to decay three-fourth of it. Hence$N = 2$, and we know ${t_{\tfrac{1}{2}}} = 4$months.
Substituting the corresponding values in the formula,
$T = 2 \times 4$
$\Rightarrow T = 8$

Hence the radioactive substance takes eight months to decay three-fourth of it. Therefore, the correct option is C.

Additional information:
(i) The phenomenon of emission of radiation by the elements is called radioactivity. The elements of emitting radiation are called radioactive elements. It is due to the unstable nucleus.
(ii) The elements having a mass number greater than 210 are showing the radioactivity. For example, Uranium-235, Thorium-234, etc.
(iii) The radioactivity was first discovered by Henry Becquerel in 1896.
(iv) The radioactive elements radiate $\alpha$-rays, $\beta$-rays, and$\gamma$-rays. In the process of decaying, one element changes into another element as the atomic number of the element changes.
(v) Half-life is defined as the time required for half of the initial amount to decay. Half-life, ${t_{\tfrac{1}{2}}} = \dfrac{{0.693}}{\lambda }$ where $\lambda$ is decay constant.

Note:
We can also find the period for the decay using formula $N = {N_O}{e^{ - \lambda t}}$ if we know the decay constant. And the radioactive element takes an infinite time to decay completely. It can also be said that the radioactive element decays exponentially. And the last product of the process of radioactivity is mostly Lead.