Resource Lesson
Derivation of Bohr's Model for the Hydrogen Spectrum
Printer Friendly Version
By the middle of the 19th century it was well known by chemists that excited hydrogen gas emitted a
distinct emission spectrum
. It was noted that the same lines were always present and that the spacing between these lines became smaller and smaller.
In 1885, the first person to propose a mathematical relationship for these lines was a Swiss high school physics teacher, J. J. Balmer. We now call hydrogen's visible spectrum the
Balmer series
. Balmer's empirical formula exactly matched the experimentalists' observed wavelengths.
Where
R
is called the
Rydberg constant
and has a well-established value of 1.0974 x 10
^{7}
m
^{-1}
.
It wasn't until 1913 that Niels Bohr developed a
theory of the atom
that explained why this formula worked.
Derivation
In an hydrogen atom, the centripetal force is being supplied by the coulomb force between it and the proton in the hydrogen nucleus.
Remember that
Z
represents the atomic number (the number of protons), that electrons and protons have the same magnitude charge,
±e
, and that a negative
F
_{electrostatic}
merely means that the electrostatic force is attractive. Also note that the values of
v
_{n}
of
r
_{n}
are unknowns in this equation.
As a means of evaluating these two unknowns,
Bohr first hypothesized
that the electron's
angular momentum was quantized
.
Upon solving the angular momentum equation for
v
_{n}
, substituting it into the centripetal force equation yields the following expression for
r
_{n}
.
For a ground state hydrogen electron,
or approximately half of an Angstrom.
Bohr's second hypothesis
in his model was that an electron only loses or releases energy (and therefore a photon) when it goes through de-excitation or drops from a higher energy state to a lower energy state. In order to determine the energy lost by the electron, an expression for an electron's total energy has to be developed.
Recall that the
electric potential energy
for an electron would equal
By extending the centripetal force relationship, an expression can also be derived for the electron's
kinetic energy
Thus, the
total energy, E
_{n}
,
of an electron equals
In this equation, notice that the
total energy is negative
. This is interpreting as meaning that the electron is trapped in an energy well about the nucleus; that is, it would take the addition of energy to ionize or free the electron.
Substituting in the value for r
_{1}
into this total energy expression yields a ground state energy of 2.18 x 10
^{-18}
Joules or -13.6 eV for a hydrogen atom. Using the fact that r
_{n}
= n
^{2}
r
_{1}
we can now generated the first four energy levels for hydrogen.
E
_{1}
= -13.6 eV
E
_{2}
= E
_{1}
/ 2
^{2}
= -3.4 eV
E
_{3}
= E
_{1}
/ 3
^{2}
= -1.51 eV
E
_{4}
= E
_{1}
/ 4
^{2}
= -0.85 eV
Bohr's second hypothesis combined with Planck's formula for quantized energy (E = hf) will now allow us to derive Balmer's equation. Remember that the energy released by the electron during de-excitation equals the energy of the emitted photon.
Let's begin by assuming that an electron is falling from E
_{j}
, a high energy state, to E
_{i}
, a lower energy state.
If we let i = 2, and j
{3, 4, 5, 6} then we have derived Balmer's empirical formula!
Related Documents
Lab:
Labs -
A Photoelectric Effect Analogy
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Aluminum Foil Parallel Plate Capacitors
Labs -
Basic Particles
Labs -
Calculation of "g" Using Two Types of Pendulums
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Electric Field Mapping
Labs -
Electric Field Mapping 2
Labs -
Experimental Radius
Labs -
Hydrogen Spectrum
Labs -
Hydrogen Spectrum
Labs -
Introductory Simple Pendulums
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Mass of an Electron
Labs -
Mass of the Top Quark
Labs -
Mirror Symmetry
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
Quantized Mass
Labs -
Radiation of a Metal Cylinder
Labs -
RC Time Constants
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Using Young's Equation - Wavelength of a Helium-Neon Laser
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
Water Springs
Resource Lesson:
RL -
A Comparison of RC and RL Circuits
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
An Outline: Dual Nature of Light and Matter
RL -
Atomic Models and Spectra
RL -
Capacitors and Dielectrics
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Continuous Charge Distributions: Charged Rods and Rings
RL -
Continuous Charge Distributions: Electric Potential
RL -
Coulomb's Law: Beyond the Fundamentals
RL -
Coulomb's Law: Suspended Spheres
RL -
Derivation: Period of a Simple Pendulum
RL -
Dielectrics: Beyond the Fundamentals
RL -
Electric Field Strength vs Electric Potential
RL -
Electric Fields: Parallel Plates
RL -
Electric Fields: Point Charges
RL -
Electric Potential Energy: Point Charges
RL -
Electric Potential: Point Charges
RL -
Electrostatics Fundamentals
RL -
Energy Conservation in Simple Pendulums
RL -
Energy-Level Diagrams
RL -
Excitation
RL -
Famous Discoveries and Experiments
RL -
Famous Discoveries: Bohr Model
RL -
Famous Discoveries: de Broglie Matter Waves
RL -
Famous Discoveries: The Franck-Hertz Experiment
RL -
Famous Discoveries: The Photoelectric Effect
RL -
Famous Experiments: Davisson-Germer
RL -
Famous Experiments: Michelson-Morley
RL -
Famous Experiments: Millikan's Oil Drop
RL -
Famous Experiments: The Compton Effect
RL -
Famous Experiments: The Discovery of the Neutron
RL -
Gauss' Law
RL -
Gravitational Energy Wells
RL -
Kepler's Laws
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Nuclear Reaction
RL -
Parallel Plate Capacitors
RL -
Period of a Pendulum
RL -
Rotational Kinematics
RL -
Shells and Conductors
RL -
SHM Equations
RL -
Simple Harmonic Motion
RL -
Spherical, Parallel Plate, and Cylindrical Capacitors
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Moment of Inertia
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
RL -
What is Mass?
REV -
Orbitals
Review:
REV -
Drill: Electrostatics
REV -
Electrostatics Point Charges Review
REV -
Review: Circular Motion and Universal Gravitation
Worksheet:
APP -
Big Al
APP -
Eternally Bohring
APP -
Nuclear Flu
APP -
Ring Around the Collar
APP -
The Birthday Cake
APP -
The Electrostatic Induction
APP -
The Satellite
APP -
The Science Fair
APP -
The Spring Phling
APP -
Timex
APP -
What's My Line
CP -
Atomic Nature of Matter
CP -
Atomic Nucleus and Radioactivity
CP -
Balancing Nuclear Equations
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Coulomb's Law
CP -
Electric Potential
CP -
Electrostatics: Induction and Conduction
CP -
Natural Transmutations
CP -
Nuclear Fission and Fusion
CP -
Radioactive Half Life
CP -
Satellites: Circular and Elliptical
CP -
The Atom and the Quantum
NT -
Atomic Number
NT -
Beta Decay
NT -
Binding Energy
NT -
Black Holes
NT -
Circular Orbits
NT -
Electric Potential vs Electric Potential Energy
NT -
Electrostatic Attraction
NT -
General Relativity
NT -
Helium Balloons
NT -
Hot Springs
NT -
Hydrogen Atom
NT -
Hydrogen Fusion
NT -
Lightning
NT -
Nuclear Equations
NT -
Pendulum
NT -
Photoelectric Effect
NT -
Potential
NT -
Radiant Energy
NT -
Radioactive Cookies
NT -
Rotating Disk
NT -
Spiral Tube
NT -
The Ax Handle
NT -
Uranium Decay
NT -
Uranium Fission
NT -
Van de Graaff
NT -
Water Stream
RL -
Chapter 3: Electrons
WS -
Atomic Models and Spectra
WS -
Basic Practice with Springs
WS -
Capacitors - Connected/Disconnected Batteries
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Combinations of Capacitors
WS -
Coulomb Force Extra Practice
WS -
Coulomb's Law: Some Practice with Proportions
WS -
Electric Field Drill: Point Charges
WS -
Electric Fields: Parallel Plates
WS -
Electric Potential Drill: Point Charges
WS -
Electrostatic Forces and Fields: Point Charges
WS -
Electrostatic Vocabulary
WS -
Energy Level Diagrams
WS -
Inertial Mass Lab Review Questions
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
More Practice with SHM Equations
WS -
Parallel Reading - The Atom
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice: SHM Equations
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
Rotational and Reflection Symmetries
WS -
SHM Properties
WS -
Standard Model: Particles and Forces
WS -
Static Springs: The Basics
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
TB -
38A: Atomic Physics
TB -
Advanced Capacitors
TB -
Basic Capacitors
TB -
Centripetal Acceleration
TB -
Centripetal Force
TB -
Electric Field Strength vs Electric Potential
TB -
Half-Life Properties
PhysicsLAB
Copyright © 1997-2017
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton