
The ratio activity of an element becomes \[\dfrac{1}{{64}}\,th\] of its original value in 60 sec. Then the half life period is
A. 5 Sec
B. 10 Sec
C. 20 Sec
D. 30 Sec
Answer
162.6k+ views
Hint: Activity of a radioactive sample is defined as the number of disintegration occurring per second in the sample and it is measured in Becquerel (Bq) or Curie (Ci). For a sample containing a number of elements, activity is equal to the sum of individual values of each element. Relation between initial \[({A_0})\]and final activity (A) is given by \[A = {A_0}{\left( {\dfrac{1}{2}} \right)^n}\].
Formula used:
\[A = {A_0}{\left( {\dfrac{1}{2}} \right)^n}\]and \[n = \dfrac{t}{T}\]
Here, A = Final activity of substance after time t, \[{A_0} = \]Initial activity of substance
T = Half life and n = Number of half lives
Complete answer:
Given,
Ratio of activity \[\dfrac{A}{{{A_0}}} = \dfrac{1}{{64}}\] and t =60 sec
To calculate half life T
As we know that ratio of final and initial activity of a radioactive substance is given by,
\[A = {A_0}{\left( {\dfrac{1}{2}} \right)^n} \Rightarrow \dfrac{A}{{{A_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}}\]
Substituting given values in above equation we get,
\[\dfrac{1}{{64}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{{60}}{T}}}\]
Above equation can be rewritten as,
\[{\left( {\dfrac{1}{2}} \right)^6} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{{60}}{T}}}\]
As the base in both side of equation is equal, therefore exponent on both sides will also be equal,
i.e. \[6 = \dfrac{{60}}{T} \Rightarrow T = 10\,\sec \]
Hence, the half life period of the given element will be 10 sec.
Therefore, option B is the correct option.
Note: Alternatively this problem can be solved by using the equation \[A = {A_0}{e^{ - \lambda t}}\], where\[\lambda \] is the decay constant. Initial activity of the element is directly proportional to final activity of the element which means that if initial activity increases final activity will also increase or decrease.
Formula used:
\[A = {A_0}{\left( {\dfrac{1}{2}} \right)^n}\]and \[n = \dfrac{t}{T}\]
Here, A = Final activity of substance after time t, \[{A_0} = \]Initial activity of substance
T = Half life and n = Number of half lives
Complete answer:
Given,
Ratio of activity \[\dfrac{A}{{{A_0}}} = \dfrac{1}{{64}}\] and t =60 sec
To calculate half life T
As we know that ratio of final and initial activity of a radioactive substance is given by,
\[A = {A_0}{\left( {\dfrac{1}{2}} \right)^n} \Rightarrow \dfrac{A}{{{A_0}}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}}\]
Substituting given values in above equation we get,
\[\dfrac{1}{{64}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{{60}}{T}}}\]
Above equation can be rewritten as,
\[{\left( {\dfrac{1}{2}} \right)^6} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{{60}}{T}}}\]
As the base in both side of equation is equal, therefore exponent on both sides will also be equal,
i.e. \[6 = \dfrac{{60}}{T} \Rightarrow T = 10\,\sec \]
Hence, the half life period of the given element will be 10 sec.
Therefore, option B is the correct option.
Note: Alternatively this problem can be solved by using the equation \[A = {A_0}{e^{ - \lambda t}}\], where\[\lambda \] is the decay constant. Initial activity of the element is directly proportional to final activity of the element which means that if initial activity increases final activity will also increase or decrease.
Recently Updated Pages
Fluid Pressure - Important Concepts and Tips for JEE

JEE Main 2023 (February 1st Shift 2) Physics Question Paper with Answer Key

Impulse Momentum Theorem Important Concepts and Tips for JEE

Graphical Methods of Vector Addition - Important Concepts for JEE

JEE Main 2022 (July 29th Shift 1) Chemistry Question Paper with Answer Key

JEE Main 2023 (February 1st Shift 1) Physics Question Paper with Answer Key

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

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Degree of Dissociation and Its Formula With Solved Example for JEE

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

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

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

Charging and Discharging of Capacitor

Wheatstone Bridge for JEE Main Physics 2025

Formula for number of images formed by two plane mirrors class 12 physics JEE_Main

In which of the following forms the energy is stored class 12 physics JEE_Main
