Quantum Theory and the Electronic Structure of Atoms
This chapter introduces the student to quantum theory and the importance of this theory in describing electronic behavior. Upon completion of Chapter 7, the student should be able to:
1. Explain how Planck’s theory challenged classical physics.
2. Define wavelength, frequency, and amplitude of waves.
3. Utilize the relationship between speed, wavelength, and frequency (hertz). 4. Describe Maxwell’s theory of electromagnetic radiation.
85. Recall from memory the speed of light (3.00 x 10 m/s).
6. Apply the metric unit of nano in calculations involving wavelength of light. 7. Classify various regions of the electromagnetic spectrum in terms of energy,
frequency and wavelength.
8. Use Planck’s equation to determine energy, frequency, or wavelength of
9. Describe the photoelectric effect as explained by Einstein using such terms as
threshold frequency, photons, kinetic energy, binding energy, light intensity and
number of electrons emitted.
10. Show how Bohr’s model of the atom explains emission, absorption and line
spectra for the hydrogen atom.
11. Compare Bohr’s model of the atom and that of the sun and surrounding planets.
12. Predict the wavelength (frequency) of electromagnetic radiation emitted (absorbed)
for electronic transitions in a hydrogen atom.
13. Use the terms ground state and excited state to describe electronic transitions. 14. Describe De Broglie’s relationship involving the wavelength of particles.
15. Explain why for common objects traveling at reasonable speeds the corresponding
wavelength becomes vanishingly small.
16. Explain the major components of a laser and list three properties that are
characteristic of a laser.
17. Describe Heisenberg’s uncertainty principle.
18. Contrast orbits (shells) in Bohr’s theory with orbitals in quantum theory.
19. Discuss the concept of electron density.
20. Recall from memory the four quantum numbers (n, ℓ, m, m) and their relationships. sℓ
21. Relate the values of the angular momentum quantum number, ℓ, to common names
for each orbital (s, p, d, f) and describe their shapes.
22. Account for the number of orbitals and number of electrons associated with each
value of ℓ, the angular momentum quantum number.
23. Categorize orbital energy levels in many-electron atoms in order of increasing
24. Write the four quantum numbers for all electrons in multi-electron atoms. 25. Predict the electron configuration and orbital diagrams for multi-electron atoms
using the Pauli exclusion principle and Hund’s rule.
26. Deduce orbital diagrams from diamagnetic and paramagnetic data.
27. Derive the ground state electron configuration of multi-electron atoms using the
28. List several exceptions to the expected electron configuration for common metals
(Cr, Mo, Cu, Ag, and Au).
Section 7.1 From Classical Physics to Quantum Theory
Max Planck started the “revolution” that suggests that classical physics does not govern atoms and molecules. He suggested that energy comes in discrete quantities, or quanta. Planck’s theory requires an understanding of waves which
include wavelength, frequency and amplitude. The speed of any wave is the product of
8its wavelength and its frequency. The speed of electromagnetic radiation is 3.00 ！ 10
meters per second or 186,000 miles per second. Your author introduces the concept of lasers in the Chemistry in Action featuring “Laser-the Splendid Light”. One of the
principles of lasers is that light bounces back and forth between mirrors inside the laser cavity. Since light travels at about 186,000 miles per second, it will travel about one
-9foot in 1 ！ 10 seconds. With this in mind, it is easy to see that light makes a huge number of passes between the mirrors in a typical laser cavity in a very short period of time.
Students find it interesting to determine the wavelength of their favorite radio station. For example, if a local station broadcasts at 109.5 megahertz, the wavelength can be determined.
c = ；？
6；；1x10/s8 ；3.00 x 10 m/s = ()(109.5 MHz)，~，~1 MHz：?
83.00 x 10 ； = m6109.5 x 10
； = 2.74 m
This is in contrast to green light which has a wavelength of 522 nm.
A direct result of Planck’s theory is that energy emitted or absorbed is directly proportional to frequency.
E = h？
-34Planck’s constant, h, has the value of 6.63 ！ 10 J(s. Sometimes it is easier for
students to use Planck’s relationship, if we think of Planck’s constant having the units of J(s/object where the term “object” may refer to photon or electron. For example, how much energy is emitted by 1.00 mole of photons having the frequency
14 equal to 5.75 ！ 10Hz?
14；；；；Js5.75 x 10，-34 ？E = h = 6.63 x 10，~，~，~photons：?：?
-19 E = 3.81 x 10J/photon
-19Each photon carries 3.81 ！ 10 J; therefore,
-19235 (3.81 x 10J/photon)(6.022 x 10photon/mol photon) = 2.29 x 10J/mol photon
5a mole of photons would represent 2.29 ！ 10 Joules.
Section 7.2 The Photoelectric Effect
The photoelectric effect, the generation of electricity by converting radiant energy to electrical energy, is demonstrated by light sensors. An industrial application of this effect is testing light sources used in the ultraviolet curing of polymers. Since no electrons are ejected below the threshold frequency, light sensors need to be selected for specific wavelengths of light being examined. For example, if one wishes to determine the intensity of a visible light source, it would be unwise to use an ultraviolet light sensor. This is because the threshold frequency for the ultraviolet sensor is higher than the frequency of visible light; therefore, no matter how intense the visible light is, it will not register on the ultraviolet sensor.
Section 7.3 Bohr’s Theory of the Hydrogen Atom
It is interesting to note that astronomy had a considerable influence on Bohr’s model of the atom. The logic used was if the planets revolve around the sun in specific orbits, then it would be reasonable that the electrons would revolve about the nucleus in specific orbits.
Students often wonder why we would even consider Bohr’s theory when it explains only the hydrogen atom. They should be reassured that even though the theory is limited, it is useful in gaining the foundation required to understand more complex theories. Bohr’s theory introduces the concepts of ground state, excited state, absorption and emission, for example.
-18An extension to Bohr’s theory is using Rydberg’s constant (2.18 ！ 10 J) and
；；11，~ ;E = - RH22，~nnif：?
to determine the energy that is absorbed or emitted by specific transitions. Once the energy is determined, both the frequency and wavelength can be determined using Planck’s relationship.
Section 7.4 The Dual Nature of the Electron
Louis de Broglie suggested that if light can have both wave nature and particle nature, then electrons can behave like waves even though they have a known mass. De Broglie’s relation is
h ； = mv
Students have difficulty accepting that particles can have wave nature and light can have particle nature. They should be reassured that their misgivings are founded on the fact that they have never “experienced” particles as small as electrons; therefore, they have to accept this concept as reasonable based on experimental evidence. The evidence is demonstrated by your author in the Chemistry in Action essay “Electron Microscopy".
Section 7.5 Quantum Mechanics
Bohr’s theory is successful in describing the spectra for hydrogen atoms, but
falls short for atoms with more than one electron. These shortcomings gave rise to development of the quantum mechanical description of atoms as suggested by Erwin Schrödinger. Quantum mechanics describe electron density which is the probability that an electron will be found in a particular region. This is a direct application of
Werner Heisenberg’s uncertainty principle, which suggests that it is not possible to know the exact position and the momentum of a particle simultaneously. Therefore, the concept of defined orbits for electrons yields to using atomic orbitals.
Section 7.6 Quantum Numbers
Our students understand that the equation y = mx + b describes a straight line. They know that m is the slope and b is the y- intercept. The quantum numbers are similar in that they are the mathematical solution to the Schrödinger equation. The principal quantum number, n, gives an indication of the total energy that an electron possesses. In a general way, it describes the average distance the electron is from the nucleus. The largest value n can be for a ground state electron is seven. The angular momentum quantum number, ℓ, describes the shape of the orbital. The magnetic
quantum number, m, sets the direction in space for each orbital. The electron spin ℓ
quantum number, m, suggests that each electron either spins clockwise or counter-s
clockwise. The rules for the possible values for each of the four quantum numbers are:
n = 1, 2, 3…
ℓ = 0, 1, 2, 3…, n - 1
m = --ℓ…0…+ℓ ℓ
m = + 1/2 or – 1/2 s
The larger the value of n, the further the electron is from the nucleus. Only positive integers are possible for n. The angular momentum quantum number starts with the value of zero and can be any integer up to and including one less than n. That is to say that if n equals five, then ℓ can be zero, one, two, three or four. It cannot be five or greater. The magnetic quantum number, m, can be negative as well as positive. The ℓ
possible values are from minus ℓ to zero to positive ℓ. Therefore, if ℓ is two, then m can ℓ
be minus two, minus one, zero, plus one and plus two. The electron spin quantum number can have the values of plus one-half or minus one-half. These two values correspond to spin up or spin down, as written by arrows, or clockwise and counter-clockwise.
Section 7.7 Atomic orbitals
In the previous section, we describe the four quantum numbers. In this section your author describes the shape of each of the orbitals. If ℓ equals zero (s orbital), then
the shape is spherical. The probability of finding an electron is within a certain shell at a given distance from the nucleus. If ℓ is one (p orbital), then the shape of the orbital is
much like a three-dimensional figure eight with the cross overpoint being the nucleus. When ℓ is two (d orbitals), the shape of the orbitals are two-fold. Four of the orbitals are like three-dimensional four-leaf clovers and the fifth is the shape of a three-dimensional figure eight with a doughnut surrounding its mid-section. See Figures 7.19, 7.20, and 7.21. The angular momentum quantum number, m, describes how each of ℓ
the above orbitals are oriented in space.
For hydrogen, a single-electron atom, all orbitals with the same principle quantum are the same energy. That is to say for a given value of n, s, p, and d are all the same energy. This is not true for multiple electron atoms where the s orbital is lower energy than p orbitals, which are lower energy than d orbitals for a given value of n.
Section 7.8 Electron Configuration
There are three common ways to describe each electron in an atom. The first is to use the notation
(n, ℓ, m, ms) ℓ
where n, ℓ, m, and mhave been described above. A second method is to use electron s ℓ
configuration. This is described by the example
where the two corresponds to the principal quantum number of two, the s represents the s orbital or ℓ of zero and the superscript one suggests that there is one electron with this value of ℓ. The third method is the “box diagram”. This is represented by
where the box is labeled as 2s and has one electron in that orbital with a plus spin. It should be noted that each box represents a single orbital. Each orbital can have a maximum of two electrons (spin up or spin down). Thus when ℓ equals one, there are
three boxes (three values of m) and they could be represented by ℓ
An interesting question for your students is to ask them to indicate what the maximum number of electrons a 3p orbital can have. They may very well indicate six, but the correct answer is two. (That is the maximum for all orbitals). The logic of answering six is that there are six electrons that can be labeled as 3p electrons. However, these six electrons occupy three orbitals with two electrons per orbital.
Pauli exclusion principle states that no two electrons within a given atom can have four identical quantum numbers. It is important to stress that this is for a given atom, because as we will see in the next section, an electron configuration for lithium uses the descriptions for the two electrons from helium when describing two of the three electrons.
It is useful to introduce the concepts of paramagnetism and diamagnetism. Paramagnetic materials have unpaired electrons. They have one or more orbitals (energy box) with a single electron. Diamagnetic materials have all orbitals containing two electrons (each energy box contains two electrons, one with a plus one-half spin and the other with a minus one-half spin). Paramagnetic materials are attracted by a magnet while diamagnetic materials are slightly repelled by a magnet. Since this is an experimental property, it can be used to justify Hund’s rule. Hund’s rule states that the most stable electron arrangement of electrons is the one which has the greatest number of parallel spins. For example, carbon has six electrons to account for. The
first two electrons (lowest energy electrons) fill the energy box labeled 1s.
The next two go into the next lowest energy level which is 2s.
We know that there are two electrons remaining to be described. Experimentally, we can determine that carbon is paramagnetic, thus it must have unpaired electrons. It follows then that the energy diagram for the 2p electrons must be
which would result in a diamagnetic material. With this in mind, it is apparent that the energy box diagram gives a bit of information that is not shown by standard electron configuration. For example, the electron configuration of oxygen is
422 1 2 2pss
The energy box diagram would be
which gives information about the paramagnetic properties of oxygen atoms. The electron configuration of oxygen could easily give the misconception that all electrons in oxygen are paired, but they are not.
Section 7.9 The Building-up Principle
The Aufbau principle (building-up principle) describes how we can “build” the
periodic table. In fact, we can use the periodic table to assist us in describing the correct electron configuration of each of the elements (certainly there are many exceptions; however, most common elements can be described by the following method).
First, we always use the previous noble gas as our core. For lithium the previous noble gas is He and for sodium the previous noble gas is Ne, etc. The core noble gas is placed in square brackets and the rest of the configuration follows. For example, sodium’s electron configuration is
1[Ne] core accounts for ten electrons and the 3s gives us the eleventh electron. There
are five exceptions that should be noted because these elements correspond to metals