Question

The half-life period of a radioactive element is 140 days. After 560 days, 1 g of the element will reduce to:
A. $0.5$ g
B. $0.25$ g
C. $\dfrac{1}{8}$ g
D. $\dfrac{1}{16}$ g

Answer 1

Hint: The half-life refers to the amount of time that it requires for the original activity to get reduced by half. For example, let us suppose that we have a total of ‘N’ atoms, then the half-life will be the time it takes for half of the atoms to decay. This means that after one half-life, we will have $\dfrac{N}{2}$atoms and after two half-lives, we will have $\dfrac{1}{2} \times \dfrac{N}{2} = \dfrac{N}{4}$atoms, and so on.

Complete answer:
 The half-life period actually measures the time for a given amount of an element to reduce by half as a result of decay, and thus the emission of radiation. We know that:
$T = n \times {t_{1/2}}$
Where, T = time period, $t_{1/2 }$ = half-life time, n = number of half-lives
In the question, we are provided with the following information:
T = 560 days (Given)
$t_{1/2 }$ = 140 days (Given)
Substitute these values in the aforementioned formula to get the value of ‘n’.
$
  560 = n \times 140 \\
\Rightarrow n = 4
$
Now, we know that:
${N_t} = {N_o}{\left( {\dfrac{1}{2}} \right)^n}$
Here, $N_t$ = amount of remaining radioactive element, $N_o$ = amount of the original radioactive element, $n$ = number of half-lives
In the question we are given the value of $N_o$ and we have to calculate the value of $N_t$
$N_o$= 1 g (Given)
Substituting the values, we get:
${N_t} = 1{\left( {\dfrac{1}{2}} \right)^4} = \dfrac{1}{{16}}$
Therefore, after 560 days, 1 g of the element will reduce to $\dfrac{1}{{16}}g$.

Hence, the correct answer is Option D.

Note:
Having information about the half-lives is very important since it allows you to determine the time period when a radioactive sample material is actually safe to handle. A sample is usually safe when its radioactivity drops below the detection limits which mostly occurs at 10 half-lives.