Lesson 7.2 The Bohr Theory of the Hydrogen Atom
Suggested Reading
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Zundahl Chapter 7 Sections 7.3 & 7.4
Essential Question

How does the Bohr model account for the stability of the hydrogen
atom and the line spectrum of the atom?
Learning Objectives



Distinguish between a continuous spectrum and a line spectrum.
Explain how the lines in the emission spectrum of hydrogen are
related to electron energy levels.
Calculate the energy associated with electron transition in the
hydrogen atom.
Introduction
According to Rutherford's nuclear model, the atom consists of a nucleus
which has most of the mass of the atom and a positive charge. Enough
electrons to make the atom electrically neutral orbit around the nucleus.
However, using Rutherford's theory, scientists showed that the electrons
would lose energy as they orbited the nucleus causing them to eventually
spiral into the nucleus. Thus, Rutherford's model could not explain the
stability of the atom. The atoms described by Rutherford's model fell apart!
A colleague of Rutherford, Neils Bohr came up with an alternative model
that was based upon the work of Planck and Einstein that you learned
about in the last lesson. Bohr applied quantum theory to the simplest atom,
hydrogen. This model still applies to hydrogen, but deteriorates for more
complicated atoms. Therefore, when we talk about Bohr's model, we must
remember that it only applies to hydrogen.
Before we can look at Bohr's model we need to discuss the line spectra of
atoms, so lets go.
Atomic Line Spectra
You learned when you carried out flame tests in Honors Chemistry that
heated chemicals can emit light. A heated tungsten filament in an old
fashioned light bulb is a familiar example of this. With a prism, we can
spread out the white light emitted by solids to give a continuous spectrum.
A continuous spectrum is a spectrum containing light of all visible
wavelengths, like that of a rainbow.
The light emitted by a gas gives a different result. Rather than a continuous
spectrum, with all of the colors of the rainbow, we obtain a line spectrum. A
line spectrum is a spectrum showing only certain colors or specific
wavelengths of light. When light from heated gaseous hydrogen is
separated by a prism, it gives a spectrum of lines. Each line corresponds to
a different wavelength of light. It has been determined that each element
has a characteristic line spectrum. Thus, spectra can be used to identify
elements. The spectra also tell us something about the structure of atoms,
and if you know something about the structures of atoms, you can explain
the formation of ions and molecules.
The line spectrum of hydrogen is especially simple. It consists of only four
lines with wavelengths corresponding to the visible region of the
electromagnetic spectrum (see figure to the left). In 1885, a scientist named
J.J. Balmer showed that the wavelengths in the line spectrum of hydrogen
could be reproduced by the following formula
where n is some whole number greater than 2, such as 3, 4, 5. Substituting
these values into the equation above gives wavelengths that correspond to
wavelengths found in the line spectrum of hydrogen. Although you will not
apply this equation directly, it is important because Bohr used Balmer's
equation in his model of the atom.
Bohr's Model
Bohr set down the following postulates to account for 1) the stability of the
hydrogen atom (that the atom exists) and 2) the line spectrum of hydrogen.
1) Energy-Level Postulate: An electron can have only specific energy
values in an atom. Bohr based this idea of quantization of energy
from Planck. Bohr was able to derive a mathematical equation that could
be used to calculate the energy of the electron in the hydrogen atom.
where RH is a constant withe the value 2.179 x 10-18 J. Substituting gives
Different values of the possible energies of the electron are calculated by
plugging in different values of n, which can have only the values 1, 2, 3,...
Here n is referred to as the principal quantum number, and the values 1,
2, 3,... are referred to as energy levels. The negative sign is used to
indicate that the energy of an electron bound in an atom is lower than the
energy of the unbound electron.
2) Transitions Between Energy Levels: An electron in an atom can
change energy only by going from one energy level to another energy level.
By
doing so, the electron undergoes a transition. The emission of light by
atoms to give a line spectrum is explained as follows. An electron in a
higher energy level (initial energy Ei) undergoes a transition to a lower
energy level (final energy Ef). In this process the electron loses energy,
which is emitted as a photon. This means that the initial energy of the
electron is equal to the final energy plus the energy of the photon. Apply
the law of conservation of energy to this gives, Ef +hv = Ei. This equation is
rearranged to give the energy of the emitted photon.
Energy of emitted photon = hv = Ei - Ef or ∆E = hv
Here, Bohr used Einstein's photon concept to explain the line spectra of
atoms. Bohr also combined Balmer's equation for wavelengths with his
equation for the energy levels to derive an equation that could be used to
calculate the change in energy associated with the transition of an electron
from one energy level to another. This equation is as follows.
However, this equation is not given the AP Exam while Bohr's equation for
the energy of the electron is. Recall that this is equation is as follows.
Because of this, you may want to get accustomed to using this equation
instead. The change in energy for a transition can be determined by
calculating the energy of the electron at each level given in the problem
and then subtracting, where
∆E = Efinal - Einitial
Example: Determining the Wavelength or Frequency of Hydrogen
Atom Transition
What is the wavelength of light emitted when the electron in a hydrogen
atom undergoes a transition from energy level n = 4 to level n =2?
Solution:
From the formula for the energy levels, you can determine the change in
energy for the transition as follows.
We can then use the relationship ∆E = hv to related the change in energy
to the wavelength of light emitted as follows.
The the relationship c = ⅄v can be used to find the wavelength where ⅄ =
c/v. Thus,
⅄ = 3.00 x 108 m/s ÷ 6.16 x 1014 /s = 4.86x10-7 m or 486 nm.
There are a lot of steps to a problem like this, but these types of problems
are pretty formulaic and vary little. If you master this problem, you should
be able to complete other similar problems..
According to the Bohr model, the emission of light from an atom occurs
when an electron undergoes a transition form an upper energy level to a
lower one. But how does an electron get to an upper level in the first place?
Well if energy is lost when electrons move to a lower energy level then
energy must be gained in order for an electron to transition to a higher
energy levle. Normally, the electron in a hydrogen atom exits in its lowest,
or n = 1 energy level. To get to a higher level, the electron must gain
energy, or be excited. One way this can happen is through the collision of
two hydrogen atoms. During this collision, some of the kinetic energy of
one atom can be gained by the electron of another atom. When this occurs,
the electron can be boosted from the n = 1 energy level to a higher energy
level.
Postulates 1 and 2 hold for atoms other than hydrogen except that the
energy levels can not be obtained by a simple formula. Wave functions,
which are beyond the scope of this course are used for these atoms.
Homework Problems:
Practice exercises 6.4-6.5
Book questions pg. 322 questions 45, 47, & 49