(B) The radioactive decay law is given by $N(t) = N_0 e^{-\lambda t}$.By definition,at half-life $T_{1/2}$,the number of radioactive nuclei remaining

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

$T_{1/2} = \frac{\ln 2}{\lambda}$

(B) The radioactive decay law is given by $N(t) = N_0 e^{-\lambda t}$.By definition,at half-life $T_{1/2}$,the number of radioactive nuclei remaining is half of the initial amount,i.e.,$N(T_{1/2}) = \frac{N_0}{2}$.Substituting this into the decay law: $\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$.$\frac{1}{2} = e^{-\lambda T_{1/2}}$.Taking the natural logarithm on both sides: $\ln(1/2) = -\lambda T_{1/2}$.$-\ln 2 = -\lambda T_{1/2}$.Therefore,$T_{1/2} = \frac{\ln 2}{\lambda}$.

Class 12 Physics Nuclei Law of Radioactivity by Rutherford and Soddy and Half Life and Mean Life

Easy

The relation between $\lambda$ and $T_{1/2}$ is ($T_{1/2} = \text{half-life}$,$\lambda = \text{decay constant}$)

AIPMT 2000 All AIPMT

A
$\left(\lambda + T_{1/2}\right) = \frac{\ln 2}{2}$
B
$T_{1/2} = \frac{\ln 2}{\lambda}$
C
$T_{1/2} \cdot \ln 2 = \lambda$
D
$T_{1/2} = \frac{1}{\lambda}$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Physics Questions

Chapter

Nuclei

Subtopic

Law of Radioactivity by Rutherford and Soddy and Half Life and Mean Life

Previous Year

AIPMT 2000 Paper All AIPMT years →

In a sample of radioactive material,what percentage of the initial number of active nuclei will decay during one mean life?

Medium

View Solution

The activity of a radioactive sample is $9750$ counts/minute at $t = 0$ and $975$ counts/minute at $t = 5$ minutes. The decay constant is .......... $min^{-1}$.

Medium

View Solution

At any instant,two elements $X_1$ and $X_2$ have the same number of radioactive atoms. If the decay constants of $X_1$ and $X_2$ are $10\lambda$ and $\lambda$ respectively,then the time when the ratio of their atoms becomes $\frac{1}{e}$ will be:

MediumIIT 2000

View Solution

$99\%$ of the nuclei of a radioactive element decay between which of the following times?

Medium

View Solution

The radioactive sources $A$ and $B$ have half-lives of $2 \ hr$ and $4 \ hr$ respectively,and initially contain the same number of radioactive atoms. At the end of $2 \ hr$,their rates of disintegration are in the ratio:

Difficult

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

The relation between $\lambda$ and $T_{1/2}$ is ($T_{1/2} = \text{half-life}$,$\lambda = \text{decay constant}$)

Similar Questions

In a sample of radioactive material,what percentage of the initial number of active nuclei will decay during one mean life?

The activity of a radioactive sample is $9750$ counts/minute at $t = 0$ and $975$ counts/minute at $t = 5$ minutes. The decay constant is .......... $min^{-1}$.

At any instant,two elements $X_1$ and $X_2$ have the same number of radioactive atoms. If the decay constants of $X_1$ and $X_2$ are $10\lambda$ and $\lambda$ respectively,then the time when the ratio of their atoms becomes $\frac{1}{e}$ will be:

$99\%$ of the nuclei of a radioactive element decay between which of the following times?

The radioactive sources $A$ and $B$ have half-lives of $2 \ hr$ and $4 \ hr$ respectively,and initially contain the same number of radioactive atoms. At the end of $2 \ hr$,their rates of disintegration are in the ratio:

Vedclass Test Series

Exam Paper Generator

Online Exam Module