Bohr Model of Hydrogen Atom
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both in
the U.S. and worldwide. Using an open-source, collaborative, and
web-based compilation model, CK-12 pioneers and promotes the
creation and distribution of high-quality, adaptive online textbooks
that can be mixed, modified and printed (i.e., the FlexBook®
textbooks).
Copyright © 2016 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/about/
terms-of-use.
Printed: March 6, 2016
www.ck12.org
C HAPTER
Chapter 1. Bohr Model of Hydrogen Atom
1
Bohr Model of Hydrogen
Atom
The Bohr Model of the Atom
Niels Bohr attempted to join the nuclear model of the atom with Einstein’s quantum theory of light and with his own
idea of electron energy levels to explain the electron arrangement within the atom. Bohr started with the planetary
model of electron arrangement but postulated that electrons in stable orbits would not radiate energy even though the
electrons were being accelerated by traveling in circular paths. Bohr hypothesized that the electrons were organized
into stepwise energy levels within the electron cloud and only radiated energy when the electrons moved from one
energy level to another. Bohr’s hypothesis suggested that the energy of atomic electrons came in packages and only
whole packages could be absorbed or emitted. This quantization of energy allowed electrons to only absorb or emit
exact amounts of energy to move from one energy level to another.
The quantization of energy is not apparent in everyday experience. If we could observe molecular sized automobiles
traveling down miniature roads, we would see cars traveling at 7 km/hr or 14 km/hr, or 21 km/hr, but never at 9 or
17 km/hr. The quantization of energy means that energy comes in packages and when energy is added to an object,
whole packets of energy must be added. This is the explanation for why atomic electrons are only allowed to have
certain amounts of energy and therefore, occupy certain energy levels. The lowest energy level for an electron is
near the nucleus and each quanta of energy added moves the electron to the next distant energy level. Einstein’s
theory says that each light photon has an energy of h f , where h is Planck’s constant, and f is the frequency. The
emission of a photon of light from an atom indicates a change in energy level for an electron such that
h f = Ehigher − Elower
The energy of an orbit is related to the inverse of the square of the orbit number. The energy of an electron in a given
energy level of hydrogen is calculated by
−18
En = (−2.17 × 10
1
J) 2
n
The radius of an orbit is related to the square of the orbit number.
Bohr calculated allowed electron energy levels for the hydrogen atom and found the emission spectrum of hydrogen
to match perfectly with particular electron transitions between his suggested energy levels. Other electron transitions
predicted electromagnetic frequencies outside the visible range and when those were looked for, they were present
and also matched precisely with theoretical calculations.
Example 1
For the hydrogen atom, determine
a) the energy of the innermost energy level (n = 1).
b) the energy of the second energy level.
c) the difference between the first and second energy levels.
1
www.ck12.org
Solution:
a) En = (−2.17 × 10−18 J)
1
= −2.17 × 10−18
12
b) En = (−2.17 × 10−18 J)
E2 = (−2.17 × 10−18 J) 212 = −5.43 × 10−19
c) E2−1 = (−5.43 × 10−19 J) − (−2.17 × 10−18 J) = 1.63 × 10−18 J
1
n2
1
n2
E1 = (−2.17 × 10−18 J)
J
J
This is the amount of energy that would need to be added to a 1st energy level electron to raise it to the second energy
level.
Example 2
According to the Bohr model, how many times larger is the second level hydrogen orbit compared to the first level
hydrogen orbit?
Solution:
4 times
Summary
• Rutherford, using Coulomb’s law and Newton’s laws, found that the results of his ’gold foil experiment’ could
be explained only if all the positive charge of the atom were concentrated in a tiny, central core, now called
the nucleus.
• The atom is 10,000 times as large as the nucleus and is mostly empty space.
• The energies of light frequencies in the emission spectrum of atoms is much too small to be involved in
the nucleus of atoms, therefore, any explanation of these wavelengths would have to involve the electron
arrangement.
• Bohr hypothesized that the electrons were organized into stepwise energy levels within the electron cloud in a
planetary model and only radiated energy when the electrons were changing from one energy level to another.
• Bohr suggested that the energy levels were quantized, that is, the energy held by the atomic electrons came in
packages and only whole packages could be absorbed or emitted.
• Einstein’s theory says that each light photon has an energy of h f , where h is Planck’s constant, and f is the
frequency. The emission of a photon of light from an atom indicates a change in energy level for an electron
such that h f = Ehigher − Elower .
• The energy of an electron in a given energy level of hydrogen is calculated by En = (−2.17 × 10−18 J) n12 .
Practice
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/64136
Watch the video at: http://www.youtube.com/watch?v=hpKhjKrBn9s
Follow up questions:
1. Why does the Rutherford planetary model predict that the electron would collapse into the nucleus?
2
www.ck12.org
Chapter 1. Bohr Model of Hydrogen Atom
2. Bohr used Planck’s suggestion that energy emitted by matter was not emitted in continuous quantities but in
_________ bundles.
3. Bohr’s calculations were based on the atoms of which element?
Review
1. Why did Rutherford suggest that the positive charge in an atomic nucleus is concentrated in a tiny region rather
than spread evenly throughout the atom?
2. The absorption spectrum of an element has the same frequencies as the emission spectrum. How does Bohr’s
theory explain this?
3. Why do scientists say the planetary model can’t be correct because the electrons would collapse into the nucleus?
4. If you were trying to explain the idea of quantization to younger students, do you think you should use water or
money as an example? Why?
The Bohr Atom
Objectives
The student will:
•
•
•
•
Understand the role that atomic spectra had in explaining the internal structure of the atom.
Solve problems involving the Balmer series.
Understand Bohr’s atomic theory.
Solve problems using the equation for the Bohr atom.
Vocabulary
• Balmer series: The results for emitted hydrogen spectra, given by Balmer’s equation.
• Bohr radius: The smallest orbital radius of the hydrogen atom, r1 = 0.529 × 10−10 m.
• Emission spectrum: The distinct set of sharp bright lines produced when the light from a heated element is
refracted in a prism, also known as line spectrum.
• Ground state: The lowest-energy state of an atom.
• Stationary states: A situation in which electrons are existing within specific orbits around a nucleus. An
electron in each stationary state possesses a definite energy.
• Transition: Electrons that have undergone a change in energy are said to have undergone a transition.
Emission Spectrum of Hydrogen
Heating different elements creates different kinds of light. This is usually done by taking the gaseous form of an
element, putting it in a sealed glass tube, and applying high voltage to the gas. The high voltage ionizes the gas and
it gives off a distinct color depending on the element. For example, neon gas glows a bright red-orange color (the
natural color of neon signs), while sodium vapor glows yellow and hydrogen glows pink. As discussed earlier, when
sunlight is refracted through a prism, we see a rainbow spectrum. When the light from a heated element is refracted
in a prism, however, we don’t see a smooth rainbow spectrum. Instead, we see a distinct set of sharp bright lines,
called the emission spectrum (or line spectrum) of that element. The diagram below shows a simple model of a
spectroscope to measure the emission spectrum of hydrogen.
3
www.ck12.org
FIGURE 1.1
A simple model of a spectroscope.
In an actual spectroscope, a diffraction grating may be used instead of a triangular glass prism, but the effect is the
same – to separate colors in proportion to their wavelength. A detailed view of this is explained in the video in the
link below.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/171288
Watch the video at: http://www.youtube.com/watch?v=2tZ6plRHLMg
Hydrogen has only one electron per atom and possesses the simplest line spectrum. The spectrum is shown in detail
in the Figure 1.2.
FIGURE 1.2
Hydrogen gas emits line spectra.
Each line is a specific color with a distinct, constant wavelength.
TABLE 1.1:
Intensity
Brightest
Faintest
Color and Position
Red, on right
Light blue, in middle
Dark blue, on left
Violet, farthest left
Wavelength
656 nm
486 nm
434 nm
410 nm
In 1885, Swiss high-school teacher Johann Balmer (1825-1898) put forth a mathematical description of the line
spectra of the hydrogen atom. He derived an empirical formula that describes the observed wavelength λ for each
line in the hydrogen spectrum. A different integer n is associated with each emission line in the Figure 1.2, starting
with n = 3 for the red emission line.
4
www.ck12.org
Chapter 1. Bohr Model of Hydrogen Atom
1
1
1
= R 2 − 2 , n = 3, 4, . . .
λ
2
n
The letter R is known as the Rydberg constant, after Swedish physicist Johannes Robert Rydberg (1854-1919), who
generalized Balmer’s empirical result.
R = 1.097 × 107 m−1
Balmer’s equation gives results for emitted hydrogen spectra known today as the Balmer series. See the simulation
below to try the Balmer series.
• http://demonstrations.wolfram.com/SpectralSeriesOfTheHydrogenAtom/
Check Your Understanding
1. Show that for n = 3, the Balmer series predicts a wavelength of 656 nm.
Solution:
1
1
1
1
5
1
1
7 −1
= R 2 − 2 = = R 2 − 2 = (1.097 × 10 m )
→
λ
2
n
λ
2
3
36
λ = 6.56 × 10−7 m = 656 nm
Could you have predicted that n = 3 corresponded to red light? What happens to the wavelength as n grows larger?
2. As n increases, do the spectral lines fall closer together or farther part from one another?
Solution:
As n increases, n12 → smaller, so the lines must fall closer together. As n becomes increasingly larger, the separation
between lines becomes so small that the spectrum begins to look continuous.
It was later found that Balmer’s formula could be extended to include electromagnetic radiation of smaller and larger
wavelengths (as anticipated by Rydberg) than those of visible light.
The Lyman Series, for example, describes the ultraviolet region of the electromagnetic spectrum for the hydrogen
atom, and has the same form as the Balmer series, as shown below.
1
1
1
= R 2 − 2 , n = 2, 3, . . .
λ
1
n
The Paschen series, as well, describes the infrared region of the electromagnetic spectrum for the hydrogen atom,
and also has the same form as the Balmer series, as shown below.
1
1
1
= R 2 − 2 , n = 4, 5, . . .
λ
3
n
5
www.ck12.org
Example 1
What line spectrum frequency of light does the Lyman series predict for n = 4?
Solution:
The formula for the Lyman series is:
1
1
1
= R 2 − 2 , n = 2, 3, . . . →
λ
1
n
1
1
1
15
16 1
=R 2 − 2 = R→λ=
λ
1
4
16
15 R
λ = 9.723 × 10−8 m (97.2 nm)
But, c = f λ → f =
c
λ
=
3.00×108 ms
9.723×10−8 m
= 3.085 × 1014 Hz → 3.09 × 1014 Hz
Bohr’s Model
A 27-year-old Danish physicist Niels Bohr (1885-1962) was working in Rutherford’s laboratory, and attempted to
explain inconsistencies in the model where electrons were in orbit around a tiny, dense, positive nucleus.
According to classical electromagnetic theory, a charge spinning in a circle – like an orbiting electron – is being
accelerated back and forth, and it should radiate out electromagnetic waves just by orbiting. This did not happen.
Hydrogen by itself radiated no light. Instead, it gave off light when energized, but only in specific lines as described
by Balmer. Bohr suspected that Rutherford’s atomic model would require some form of quantization similar to the
one described in the works of Planck and Einstein.
He postulated (that is, stated without proof) that electrons could not lose energy in a continuous fashion but rather
only when going from one energy state to another, as described by the Planck equation:
E = nh f
The energy of the atom must, therefore, be quantized.
Bohr also assumed that electrons could only take on specific (allowed) orbits about the nucleus, so the orbit radii
must be quantized as well.
Electrons radiate or absorb energy only when changing from one “allowed orbit” to another. He called the allowed
orbits stationary states. An electron in each stationary state possesses a definite energy. Electrons, Bohr hypothesized, could “jump” from one stationary state to another when absorbing or emitting a single photon. Not just any
photon would do, of course. A photon of a specific energy had to be absorbed by the electron in order for the electron
to move into a stationary state of higher energy. The electron would then be in an excited stationary state and would
quickly emit the energy it had absorbed and fall back into its previous lower stationary state. Energy could, therefore,
be emitted or absorbed only in going from one stationary state to another within the atom. That idea could explain
the reason for a discrete line spectrum and do away with Maxwell’s prediction that all accelerating charges must
radiate electromagnetic energy.
Bohr made several important assumptions in order to (mathematically) derive a model of the hydrogen atom based
on quantization. We state his assumptions (“postulates”) below:
1. Energy of the electrons within an atom is quantized.
2. Momentum of the electrons within an atom is quantized.
6
www.ck12.org
Chapter 1. Bohr Model of Hydrogen Atom
3. The orbits of the electrons are circular.
Using these assumptions, Bohr was able to derive the following:
The allowed radii for the hydrogen atom are
rn = n2 r1 , n = 1, 2, . . ., Equation A,
where r1 is the smallest orbital radius of the hydrogen atom, commonly referred to as the Bohr radius. The Bohr
radius is r1 = 0.529 × 10−10 m. The result is in good agreement with the previous estimates of the radius of the atom.
The allowable energy levels, or stationary states, of the Bohr atom are,
E=
E1
,n
n2
= 1, 2, . . ., Equation B,
where E1 is called the ground state of the atom. The ground state is the lowest-energy state of the atom. In the
hydrogen atom, the ground state energy is about
E1 = −13.6 eV
The negative sign indicates that the electron must absorb 13.6 eV in order to be ejected from the atom. This energy
is called the ionization energy of the ground state of the hydrogen atom.
Energies E2 , E3 , ... represent higher-energy, or “excited” states of the electron. Notice that the higher the energy
level, the less negative is the corresponding energy. For example,
E2 > E1 , where E2 = −3.40 eV
The difference between allowable energy levels can be expressed as
h f = Eh − El
where Eh is a higher energy state of the electron and El is a lower energy state of the electron. When an electron
undergoes a change in energy, we often say it has undergone a transition.
Bohr was able to derive the results of the Balmer, Lyman, and Paschen series based on his assumptions. A slightly
more generalized version of these series given by Bohr is
1
1
1
=R 2 − 2
λ
nl nh
Where nl is the lower state and nh is the higher state.
• http://demonstrations.wolfram.com/AbsorptionAndEmissionOfRadiationByAtoms/
Example 2
What is the wavelength of a photon emitted when a transition occurs between n = 4 to n = 3 in a hydrogen atom?
Solution:
We will solve this problem two ways:
Method 1
1
1
1
1
1
7 −1
= R 2 − 2 = (1.097 × 10 m ) 2 − 2 = 533, 263.9 → l = 1.875 × 10−6 m = 1, 875 nm
λ
3
4
nl nh
7
www.ck12.org
Method 2
1
E1 E1
1
h f = Eh − El = 2 − 2 = (−13.6 eV ) 2 − 2 = 0.6611 eV → 1.06 × 10−19 J →
4
3
nh nl
c
h f = 1.05777 × 10−19 J → h = 1.05777 × 10−19 J →
λ
(6.626 × 10−34 J − s) 3.00 × 108 ms
λ=
= 1.875 × 10−6 m = 1, 875 nm
1.06 × 10−19 J
We have kept the same number of significant digits for comparison to Method 1.
Method 2 directly exhibits the quantized energy levels within the hydrogen atom.
For more information on the Bohr atom follow the links below.
• http://demonstrations.wolfram.com/BohrsOrbits/
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/171290
Watch the video at: http://www.youtube.com/watch?v=KPozYveIaK4&feature=related
References
1. User:Merikanto/Wikimedia Commons and User:Adrignola/Wikimedia Commons. http://commons.wikimedia
.org/wiki/File:Emission_spectrum-H.svg .
8