In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
(a) Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton.
(b) What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)?
(c) What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å separation?
The distance between electron-proton of a hydrogen atom, $d=0.53 \mathrm{~A}$
Charge on an electron, $q_{1}=-1.6 \times 10^{-19} \mathrm{C}$
Charge on a proton, $q_{2}=+1.6 \times 10^{-19} \mathrm{C}$
(a) Potential at infinity is zero.
Potential energy of the system, = Potential energy at infinity − Potential energy at distance d
$=0-\frac{q_{1} q_{2}}{4 \pi \in_{0} d}$
where,
$\epsilon_{0}$ is the permittivity of free space
$\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} \mathrm{C}^{-2}$
$\therefore$ Potential energy $=0-\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{0.53 \times 10^{-10}}=-43.47 \times 10^{-19} \mathrm{~J}$
$\because 1.6 \times 10^{-19} \mathrm{~J}=1 \mathrm{eV}$
$\therefore$ Potential energy $=-43.7 \times 10^{-19}=\frac{-43.7 \times 10^{-19}}{1.6 \times 10^{-19}}=-27.2 \mathrm{eV}$
Therefore, the potential energy of the system is −27.2 eV.
(b) Kinetic energy is half of the magnitude of potential energy.
Kinetic energy $=\frac{1}{2} \times(-27.2)=13.6 \mathrm{eV}$
Total energy = 13.6 − 27.2 = 13.6 eV
Therefore, the minimum work required to free the electron is 13.6 eV.
(c) When zero of potential energy is taken, $d_{1}=1.06 \mathrm{~A}$
$\therefore$ Potential energy of the system $=$ Potential energy at $d_{1}-$ Potential energy at $d$
$=\frac{q_{1} q_{2}}{4 \pi \epsilon_{0} d_{1}}-27.2 \mathrm{eV}$
$=\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{1.06 \times 10^{-10}}-27.2 \mathrm{eV}$
$=21.73 \times 10^{-19} \mathrm{~J}-27.2 \mathrm{eV}$
$=13.58 \mathrm{eV}-27.2 \mathrm{eV}$
$=-13.6 \mathrm{eV}$
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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,