Question

The 3.8 days is the half-life period of a sample. After how many days, the sample will become \[{(1/8)^{th}}\] of the original substance
A. 11.4
B. 3.8
C. 3
D. None of these

Answer 1

Hint: In this question, we need to find the days taken by the sample to become \[{(1/8)^{th}}\] of the original substance. By using the formula of half life and radioactive decay law, we will find the days.

Formula Used
The following formula is used for solving the given question.
The amount of substance that will decay is given by
\[\dfrac{N}{{{N_0}}}\]= \[{(\dfrac{1}{2})^n}\]
Here, \[{N_0}\] is the amount of substance that will initially decay and \[N\] is the quantity that still remains.
\[t = n\]x \[{T_{1/2}}\]
Here, \[{T_{1/2}}\] is half-life, n is the number of half-life, and t is the time taken by the sample.

Complete step by step solution:
Half-Life is normally described as the time required by a radioactive substance (or by one-half the atoms) to transform or break into various substances. The principle was first found by Ernest Rutherford in 1907. It is usually signified by the symbol Ug or \[{T_{1/2}}\].

Given,
Half-life \[{T_{1/2}}\] = 3.8 days.
\[\dfrac{N}{{{N_o}}}\]=\[1/8\]
We know that the amount of substance that will decay is
\[\dfrac{N}{{{N_o}}}\]= \[{(\dfrac{1}{2})^n}\]

By substituting \[\dfrac{N}{{{N_o}}}\]=\[1/8\], we get
\[\dfrac{1}{8}\]= \[{(\dfrac{1}{2})^n}\]
\[\Rightarrow {(\dfrac{1}{2})^3}\] = \[{(\dfrac{1}{2})^n}\]
Therefore, Number of half-life n = 3

To find the days taken by the sample to become \[{(1/8)^{th}}\] of the original substance, the below formula is used.
\[t = n\] x \[{T_{1/2}}\]
By substituting n=3, we get
$t= 3 \times 3.8$
Therefore, t= 11.4 days i.e. the days taken by the sample to become \[{(1/8)^{th}}\] of the original substance is 11.4 days.

Therefore the correct answer is Option A.

Additional information: Half-Life or formerly known as Half-Life Period is one of the common terminologies used in Physics to define the radioactive decay of a certain sample or element within a specified period of time. Though, this concept is also widely used to define different kinds of decay processes, particularly non-exponential and exponential decay.

If the radioactive element is taken in a case in which half of the atoms have degenerated after half a lifetime, it would be suitable to assume that they have a well-defined average, namely life expectancy. Atoms with a mean life that is significantly longer than their half-life. This would mean the mean life would be the half-life divisible by 2 which is the standard algorithm. Meanwhile, the half-life is often depicted in probability terms.

The formulas for half-life are used to define disintegration in substances.
$N(t) = \dfrac{{{N_o}(1/2)t}}{{{t_{1/2}}}}$
$\Rightarrow N(t) = {N_o}e - t/r$
$\Rightarrow N(t) = {N_o}e - \lambda t$
Here we consider the following,
\[{N_o}\]= initial amount of the substance
N(t) = the quantity that remains
\[{T_{1/2}}\] = half-life
r = mean a lifetime of the decaying quantity
\[\lambda \] = decay constant

Note: To solve this question, it is necessary to remember the formula and define the half-life of the substance. In the decay chain, students can see a half-life period of a sample of 3.8 days. If they make mistakes here, then the result will be wrong. So, it is necessary to do the calculations carefully.