
Which is the correct expression for half-life
A. \[{(t)_{1/2}} = \log 2\]
B. \[{(t)_{1/2}} = \lambda /\log 2\]\[\]
C. \[{(t)_{1/2}} = (\lambda /\log 2)2.303\]
D. \[{(t)_{1/2}} = (2.303\log 2)/\lambda \]
Answer
164.7k+ views
Hint: According to this question, we need to find the correct expression for half-life. So first, we need to define the half-life. It is significant that the formula for the half-life of a reaction varies depending on the order of the reaction.
Complete step by step solution:
Another characteristic of each radionuclide is its half-life. Half-life represents the length of time required for half of the radioactive atoms of a particular radionuclide to decay. A good rule of thumb is that, after seven half-lives, one will have less than one percent of the initial amount of radiation. Based on the radionuclide, this process could be quick or time-consuming – radioactive half-lives can vary from milliseconds to hours, days, and then sometimes millions of years.
For a zero-order reaction, the mathematical expression that can be employed to define the half-life is,
\[{t_{1/2}} = {R_0}/2k\]
For a first-order reaction, the half-life is calculated by
\[{t_{1/2}} = 0.693/k\]
For a second-order reaction, the formula for the half-life of the reaction is,
\[1/k{[R]_0}\]
where \[{t_{1/2}}\] is the half-life of the reaction, \[{[R]_0}\] is the initial reactant concentration, and k is the rate constant of the reaction.
For first-order kinetics,
\[k = \dfrac{{2.303}}{t}{\log _{10}}\dfrac{a}{{a - x}}\]
At \[{t_{1/2}}\],
\[k = \dfrac{{2.303}}{{{t_{1/2}}}}{\log _{10}}\dfrac{a}{{a - \dfrac{a}{2}}}\]
\[\therefore {t_{1/2}} = \dfrac{{2.303}}{k}{\log _{10}}2\]
Therefore the correct answer is option D.
Additional Information: The half-life formula is used to find the half-life of a material that is reducing or decaying in quantity. A substance that is decaying has various rates of decay for various amounts of the substance. As the quantity of the substance decreases the rate of decay also slows down, so it is very hard to find the life of a decaying substance. Therefore, the half-life formula is used to offer the right metrics to describe the life of decaying substances.
Note: Students must know the correct half-life expression. So that they can easily find out the correct expression. If they are unable to remember the half-life expression, they are unable to find correct and accurate expressions.
Complete step by step solution:
Another characteristic of each radionuclide is its half-life. Half-life represents the length of time required for half of the radioactive atoms of a particular radionuclide to decay. A good rule of thumb is that, after seven half-lives, one will have less than one percent of the initial amount of radiation. Based on the radionuclide, this process could be quick or time-consuming – radioactive half-lives can vary from milliseconds to hours, days, and then sometimes millions of years.
For a zero-order reaction, the mathematical expression that can be employed to define the half-life is,
\[{t_{1/2}} = {R_0}/2k\]
For a first-order reaction, the half-life is calculated by
\[{t_{1/2}} = 0.693/k\]
For a second-order reaction, the formula for the half-life of the reaction is,
\[1/k{[R]_0}\]
where \[{t_{1/2}}\] is the half-life of the reaction, \[{[R]_0}\] is the initial reactant concentration, and k is the rate constant of the reaction.
For first-order kinetics,
\[k = \dfrac{{2.303}}{t}{\log _{10}}\dfrac{a}{{a - x}}\]
At \[{t_{1/2}}\],
\[k = \dfrac{{2.303}}{{{t_{1/2}}}}{\log _{10}}\dfrac{a}{{a - \dfrac{a}{2}}}\]
\[\therefore {t_{1/2}} = \dfrac{{2.303}}{k}{\log _{10}}2\]
Therefore the correct answer is option D.
Additional Information: The half-life formula is used to find the half-life of a material that is reducing or decaying in quantity. A substance that is decaying has various rates of decay for various amounts of the substance. As the quantity of the substance decreases the rate of decay also slows down, so it is very hard to find the life of a decaying substance. Therefore, the half-life formula is used to offer the right metrics to describe the life of decaying substances.
Note: Students must know the correct half-life expression. So that they can easily find out the correct expression. If they are unable to remember the half-life expression, they are unable to find correct and accurate expressions.
Recently Updated Pages
Uniform Acceleration - Definition, Equation, Examples, and FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

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

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

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

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

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

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

Degree of Dissociation and Its Formula With Solved Example for JEE

Wheatstone Bridge for JEE Main Physics 2025

Instantaneous Velocity - Formula based Examples for JEE
