If one of the two electrons of a $\mathrm{H}_{2}$ molecule is removed, we get a hydrogen molecular ion $\mathrm{H}_{2}^{+}$. In the ground state of an $\mathrm{H}_{2}^{+}$, the two protons are separated by roughly $1.5 \AA$, and the electron is roughly $1 \AA$ from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.
The system of two protons and one electron is represented in the given figure.
Charge on proton $1, q_{1}=1.6 \times 10^{-19} \mathrm{C}$
Charge on proton 2, q2 = 1.6 ×10−19 C
Charge on electron, q3 = −1.6 ×10−19 C
Distance between protons 1 and 2, d1 = 1.5 ×10−10 m
Distance between proton 1 and electron, d2 = 1 ×10−10 m
Distance between proton 2 and electron, d3 = 1 × 10−10 m
The potential energy at infinity is zero.
Potential energy of the system,
$V=\frac{q_{1} q_{2}}{4 \pi \epsilon_{0} d_{1}}+\frac{q_{2} q_{3}}{4 \pi \epsilon_{0} d_{3}}+\frac{q_{3} q_{1}}{4 \pi \epsilon_{0} d_{2}}$
Substituting $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} \mathrm{C}^{-2}$, we obtain
$V=\frac{9 \times 10^{9} \times 10^{-19} \times 10^{-19}}{10^{-10}}\left[-(16)^{2}+\frac{(1.6)^{2}}{1.5}+-(1.6)^{2}\right]$
$=-30.7 \times 10^{-19} \mathrm{~J}$
$=-19.2 \mathrm{eV}$
Therefore, the potential energy of the system is $-19.2 \mathrm{eV}$.
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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,