At time $\mathrm{t}=0$, a material is composed of two radioactive atoms $A$ and $B$, where $\mathrm{N}_{\mathrm{A}}(0)=2 \mathrm{~N}_{\mathrm{B}}(0)$. The decay constant of both kind of radioactive atoms is $\lambda$. However, A disintegrates to $B$ and $B$ disintegrates to $\mathrm{C}$. Which of the following figures represents the evolution of $\mathrm{N}_{\mathrm{B}}(\mathrm{t}) / \mathrm{N}_{\mathrm{B}}(0)$ with respect to time $\mathrm{t}$ ?
$\left[\begin{array}{l}\mathrm{N}_{\mathrm{A}}(0)=\text { No. of } \mathrm{A} \text { atoms at } \mathrm{t}=0 \\ \mathrm{~N}_{\mathrm{B}}(0)=\text { No. of } \mathrm{B} \text { atoms at } \mathrm{t}=0\end{array}\right]$
Correct Option: , 3
$\mathrm{A} \rightarrow \mathrm{B}, \mathrm{B} \rightarrow \mathrm{C}$
$\frac{\mathrm{dN}_{\mathrm{B}}}{\mathrm{dt}}=\lambda \mathrm{N}_{\mathrm{A}}-\lambda \mathrm{N}_{\mathrm{B}}$
$\frac{\mathrm{dN}_{\mathrm{B}}}{\mathrm{dt}}=2 \lambda \mathrm{N}_{\mathrm{B}_{0}} \mathrm{e}^{-\lambda \mathrm{t}}-\lambda \mathrm{N}_{\mathrm{B}}$
$\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{N}_{\mathrm{B}} \mathrm{e}^{\lambda \mathrm{t}}\right)=2 \lambda \mathrm{N}_{\mathrm{B}_{0}}$, on integrating
$\mathrm{N}_{\mathrm{B}} \mathrm{e}^{\lambda \mathrm{t}}=2 \lambda \mathrm{t} \mathrm{N}_{\mathrm{B}_{0}}+\mathrm{N}_{\mathrm{B}_{0}}$
$\mathrm{N}_{\mathrm{B}}=\mathrm{N}_{\mathrm{B}_{0}}[1+2 \lambda \mathrm{t}] \mathrm{e}^{-\lambda \mathrm{t}}$
$\frac{\mathrm{dN}_{\mathrm{B}}}{\mathrm{dt}}=0$ at $-\lambda[1+2 \lambda \mathrm{t}) \mathrm{e}^{-\lambda \mathrm{t}}+2 \lambda \mathrm{e}^{-\lambda \mathrm{t}}=0$
$\mathrm{N}_{\mathrm{B}_{\max }}$ at $\mathrm{t}=\frac{1}{2 \lambda}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,