
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
161.1k+ views
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.
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.
Recently Updated Pages
JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

Young's Double Slit Experiment Step by Step Derivation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Displacement-Time Graph and Velocity-Time Graph for JEE

Uniform Acceleration

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

If a wire of resistance R is stretched to double of class 12 physics JEE_Main
