Question
State law of radioactive decay. Hence derive the relation $N = N _{ o } e ^{-\lambda t}$. Represent it graphically.
✓
Answer
Radioactive decay law :
The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at that instant.
Proof:
$\therefore \frac{ dN }{ dt }=-\lambda N ...........(1)$
Where λ - decayconst
$\therefore \frac{ dN }{ N }=-\lambda dt$
$\therefore$ integrating both sides
$\int \frac{ dN }{ N }=\int-\lambda dt$
${(e l l) N =\lambda t+C \ell^{\prime \prime} N ^{\prime \prime}=\lambda t + C ^{\prime}}..........(2)$
Where $C \rightarrow$ integration consta
$\therefore at t = 0 , N =N_0$
$\therefore \ell nN _0=C$
∴ Frome equation (2) and (3)
$\ln N=-\lambda t+\ell n N_0$
$\lambda t=\operatorname{In} N-\operatorname{In} N_0$
$-\lambda T =\ln \left(\frac{N}{N_0}\right)$
$\frac{N}{N_0}=e^{-(\lambda t)}$
$\therefore N=N_0 e^{-(\lambda t)}$
The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at that instant.
Proof:
- Let ‘N’ be the number of nuclei present at any instant ‘t’.
- Let 'dN’ be the number of nucleic that disintegrated in short interval of time ‘dt’.
- According to law
$\therefore \frac{ dN }{ dt }=-\lambda N ...........(1)$
Where λ - decayconst
$\therefore \frac{ dN }{ N }=-\lambda dt$
$\therefore$ integrating both sides
$\int \frac{ dN }{ N }=\int-\lambda dt$
${(e l l) N =\lambda t+C \ell^{\prime \prime} N ^{\prime \prime}=\lambda t + C ^{\prime}}..........(2)$
Where $C \rightarrow$ integration consta
$\therefore at t = 0 , N =N_0$
$\therefore \ell nN _0=C$
∴ Frome equation (2) and (3)
$\ln N=-\lambda t+\ell n N_0$
$\lambda t=\operatorname{In} N-\operatorname{In} N_0$
$-\lambda T =\ln \left(\frac{N}{N_0}\right)$
$\frac{N}{N_0}=e^{-(\lambda t)}$
$\therefore N=N_0 e^{-(\lambda t)}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the moment of inertia of a hydrogen molecule about its centre of mass if the mass of each hydrogen atom is $m$ and the distance between them is $R$.
→Describe biprism experiment to calculate the wavelength of a momochromatic light. Draw the necessary ray diagram.
In a circus, a motor-cyclist having mass of \(50 kg\) moves in a spherical cage of radius \(3 m\). Calculate the last velocity with which he must pass the highest point without losing contact. Also calculate his angular speed at the highest point.
The back emf in a motor is $100 V$ when operating . at $2500 rpm$. What would be the back emf at $1800 rpm$ ? Assume the magnetic field remains unchanged.
→State Pascal's law.
A solenoid $1.5 m$ long and $4 \ cm$ in diameter has $-10$ turns $/ \ cm$. A current of $5 A$ is passing through it. Calculate the magnetic induction $(i)$ inside $(ii)$ at one end on the axis of the solenoid.
→Define the SI and CGS units of coefficient of viscosity.
A circular race course track has a radius of $500 \mathrm{~m}$ and is banked at $10^{\circ}$. The coefficient of static friction between the tyres of a vehicle and the road surface is 0.25 . Compute
(i) the maximum speed to avoid slipping
(ii) the optimum speed to avoid wear and tear of the tyres.
→(i) the maximum speed to avoid slipping
(ii) the optimum speed to avoid wear and tear of the tyres.
What is a thermodynamic process ? Explain.
The optical path difference between identical waves from two coherent sources and arriving at a point is $87 \lambda$. What can you say about the resultant intensity at the point? If the path difference is $49.19 \mu m$, calculate the wavelength of light.
→