Homework Problems

Chapter 6 Homework Problems: 1, 6, 8, 16, 18, 30, 34, 41, 48, 51, 54, 58, 66, 68, 69, 76, 86, 91, 96, 108, 130

CHAPTER 6

Quantum Theory and the Electronic Structure

of Atoms

Pre-Quantum Physics

Scientists have generally believed that the behavior of objects in the universe can be summarized in terms of a small number of fundamental physical laws. By 1900, most scientists believed that the major laws of physics had been found.

Conservation laws (conservation of mass, conservation of energy)

Thermodynamics (First law, second law, third law)

Laws of motion (Newton’s laws)

Newton’s theory of gravity

Maxwell’s equations for electricity and magnetism.

Atomic theory

Light

Light is a general term used for electromagnetic radiation. Although some scientists (like Newton) believed that light was a particle phenomenon, by 1900 most scientists were convinced that light was a wave phenomenon, for reasons discussed below.

Properties of Waves

The following general terms are used for waves.

wavelength () – The distance between successive peaks of the wave (SI units are m)

frequency ()– The number of peaks that pass a given point per unit time (SI units are s-1, sometimes called Hertz (Hz)).

Wavelength, Frequency, and Wave Velocity

There is a general relationship between the wavelength, frequency, and velocity (c, the speed) of a wave.

= c where c is the speed of the wave (SI unit = m/s)

Light

Experimentally it is found that the speed at which light travels in a vacuum is independent of the wavelength or frequency of the light.

For light in vacuum, c = 2.998 x 108 m/s.

If we know either the frequency or the wavelength, we can use the relationship

= c

to find the missing quantity.

Example: A sodium lamp emits yellow light at a wavelength = 589.3 nm. What is the frequency of the light?

Example: A sodium lamp emits yellow light at a wavelength = 589.3 nm. What is the frequency of the light?

Since = c, it follows that = c/, so

= (2.998 x 108 m/s) = 5.087 x 1014 s-1

589.3 x 10-9 m

Wavelength and Intensity

The color of light (for visible light) depends on the wavelength of the light, while the intensity (energy per unit time) of light depends on the amplitude of the light.

Electromagnetic Spectrum

All light is fundamentally the same, and we often use the word “light” to mean any electromagnetic radiation. However, it is convenient to divide the spectrum into regions based on wavelength or frequency.

Wave Properties of Light

Classically, light was considered a wave phenomena. This was based on experimental observations such as the interference pattern for light observed in the two slit experiment.

Interference – The increase or decrease in amplitude that occurs when two waves of the same wavelength are combined together, due to constructive or destructive interference.

Constructive and Destructive Interference

When two waves with the same wavelength combine together they can be in phase (peaks line up with one another) or out of phase (peaks of one wave line up with troughs of the other wave), or somewhere between the two. When the waves are in phase, adding the waves together increases the amplitude. This is called constructive interference. When the waves are out of phase, adding the waves together decreases the amplitude. This is called destructive interference.

Two Slit Interference

In the two slit experiment monochromatic light passes through two small openings in a barrier. A diffraction pattern is observed.

If light were a particle, then no diffraction pattern should occur. Instead, there should be only two spots on the film corresponding to the two slits.

white spots - places where light has illuminated the film

dark spots - places where no light has illuminated the film

Particle Properties of Light

While it was generally accepted that light behaved as a wave, experiments at the end of the 19th century gave evidence that under some conditions light behaved like a particle.

1) Blackbody radiation (explained by Max Planck in 1900).

2) Photoelectric effect (explained by Albert Einstein in 1905).

monochromatic light - light of a single color (wavelength)

critical wavelength (0) – If > 0, no electrons are detected

Einstein Theory For the Photoelectric Effect

Einstein explained the photoelectric effect as follows:

1) Light consists of particles (now called photons) which have an energy given by the Planck expression

Ephoton = h = hc/ h = 6.626 x 10-34 J.s

2) There is a minimum energy (W, the binding energy or work function for the metal) required for an electron to escape from the metal.

3) When an electron in the metal absorbs a photon, the energy of the photon is transferred to the electron. There are two possibilities

a) If the energy is less than the work function for the metal, the electron does not have enough energy to escape.

b) If the energy is greater than the work function for the metal, the electron can escape and be detected.

0 then represents the boundary between these two regions

Predictions From Einstein’s Theory

Several predictions follow from Einstein’s theory for the photoelectric effect.

1) The critical wavelength 0 is related to the binding energy W.

Ephoton = h = hc/0 = W for minimum photon energy

0 = hc/W

2) If < 0 then electrons are produced instantaneously, even for dim light. If > 0 no photons are produced.

3) The maximum kinetic energy of the ejected electrons can be found from conservation of energy

Ephoton = h = KEmax + W KEmax = maximum KE of electron

KEmax = h - W

Therefore, a plot of KEmax vs should have a slope equal to h.

The x-intercept in the above plot can be used to find the work function for the metal.

Ephoton = h = KEmax + W The minimum value for KEmax is 0

h0 = W , the work function for the metal.

CONCLUSION – Light has both wave and particle properties.

Atomic Spectra (Experimental)

When a sample of an element is heated to high temperature it will emit light. The observed light emission is called a line spectrum.

Atomic Spectra (Observed Results)

1) Light emission occurs only at particular values of wavelength.

2) Different elements emit light at different wavelengths (and so this light emission can be used to identify the presence of an element in a sample, and even determine the concentration of element present).

3) An element will absorb light at the same wavelengths at which it emits light.

Hydrogen Spectrum

For most elements the pattern of wavelengths where light is emitted or absorbed is complicated. For hydrogen, however, the pattern fits a simple equation called the Rydberg formula

1 = R 1 - 1 R = 0.01097 nm-1 ni, nf = 1, 2, 3, … nf

2 ni2 ni > nf

Example: At what wavelength (in nm) will light be emitted for the transition ni = 3 --> nf = 2?

Example: At what wavelength (in nm) will light be emitted for the transition ni = 3 --> nf = 2?

1/ = (0.01097 nm-1) [ (1/22) - (1/32)] = 1.524 x 10-3 nm-1

= 1 (1.524 x 10-3 nm-1)

= 656.3 nm

ni = 6 5 4 3

The concept of energy levels can be used to explain the pattern of light absorption and emission for atoms. For example, for hydrogen the energy levels are given by the equation:

En = - 2.18 x 10-18 J n = 1, 2, …

n2

Other atomic spectra can also be explained in terms of energy levels, though there is no simple formula for their location as there is for hydrogen.

However, there is no expla-nation for the origin of these energy levels by classical physics.

Wave Properties of Matter

Since light has “particle-like” properties it is reasonable to consider the possibility that matter might, under some conditions, have “wave-like” properties. The first person to explore this idea was Louis de Broglie, in 1924.

E = mc2 (Einstein) E = hc (Planck)

If we set these equal to one another, then

mc2 = hc or = hc = h mc2 mc

For a particle, replace c (speed of light) with v (speed of the particle), to get

deBroglie = h the de Broglie wavelength mv

Interpretation of de Broglie Wavelength

What does the de Broglie wavelength mean? It is interpreted to mean the length scale at which particles can exhibit wave behavior.

X-ray diffraction electron diffraction

Aluminum foil

Using the de Broglie Equation

Example: What is the speed of an electron whose de Broglie wavelength is = 0.100 nm (approximate spacing between particles in a crystal)?

Example: What is the speed of an electron whose de Broglie wavelength is = 0.100 nm (approximate spacing between particles in a crystal)?

Since deBroglie = h mv

then v = h = (6.626 x 10-34 J.s) m(deBroglie) (9.109 x 10-31 kg) (0.100 x 10-9 m)

= 7.27 x 106 m/s

This is approximately 2.4 % of the speed of light.

Electron diffraction was first observed experimentally in 1927 by the American physicists Clinton Davisson and Lester Germer.

Summary

From the previous discussion we get the following important points:

1) Light has both wave-like and particle-like properties, which can be brought out by different experiments.

2) Matter also has both wave-like and particle-like properties, which can also be brought out in different experiments.

3) Many systems, such as atoms, behave as if they can only have certain values for energy (energy levels).

Quantum Mechanics

Quantum mechanics is the theory developed to account for the above observations. It is based on solving the Schrodinger equation.

[ (- 2/2m) 2 + V(x) ] n(x) = Enn(x)

Schrodinger Wavefunction Probability

equation Energy

By solving the Schrodinger equation for a system we can find the possible values for energy and the probability of finding the particles making up the system at a particular location in space.

Uncertainty Relationship

One general property of systems described by quantum mechan-ics is that they must satisfy an uncertainty principle (Heisenberg, 1927).

(x) (p) = (x) (mv) (h/4) p = mv = momentum

What this means is that it is not possible to assign a definite position for a particle in a system. All that can be given is the probability of finding the particle at a particular location. This is why, for example, we describe the electrons in an atom as a “cloud” of charge surrounding the nucleus.

Hydrogen Atom

The solutions to the Schrodinger equation for the hydrogen atom are given in terms of quantum numbers, a series of numbers used to label the solutions and which describe the solutions. For an electron in a hydrogen atom there are four quantum numbers, each which gives information about the state the electron is in.

Quantum Numbers

The quantum numbers and their possible values are as follows.

n = 1, 2, 3, …

= 0, 1, …, (n-1)

m = 0, 1, 2, …,

ms = 1/2

Principal quantum number. Determines the energy and average distance between the electron and the nucleus, and orbital size.

Angular momentum quantum number. Determines the shape of the electron cloud (orbital shape).

Magnetic quantum number. Determines the orientation of the orbital.

Spin quantum number. Determines the orientation of the electron spin.

Example

If n = 2, what are the possible values for ?

If = 2, what are the possible values for m?

Are these possible sets of quantum numbers for an electron?

n = 2, = 1, m = -1, ms = ½

n = 3, = 0, m = 1, ms = ½

Example

If n = 2, what are the possible values for ?

Answer: Since ranges in value from = 0 up to = (n – 1), the possible values for are 0 or 1.

If = 2, what are the possible values for m?

Answer: Since m = 0, ±1, …, ± , the possible values for m are m = 2, 1, 0, -1, and -2.

Are these possible sets of quantum numbers for an electron?

n = 2, = 1, m = -1, ms = ½ yes

n = 3, = 0, m = 1, ms = ½ no

Relationship Between n and EnergyFor the hydrogen atom the energy of the electron depends only

on the value of the quantum number n.

En = (- 2.18 x 10-18 J) n = 1, 2, 3, ... n2

Angular Momentum (Orbital) Quantum Number

We use letters to designate the various electron orbitals, which describe the shape of the region where the electron is most likely to be found.

orbital m

0 s orbital (1) 0

1 p orbital (3) 1, 0, -1

2 d orbital (5) 2, 1, 0, -1, -2

3 f orbital (7) 3, 2, 1, 0, -1, -2, -3

Note that the total number of orbitals corresponding to a particular value of is equal to the number of possible values of m, or (2 + 1).

We usually label the orbitals by both their value for n and their value for . So, for example, if n = 3 and = 1, we have the 3p orbitals.

Orbital Shape (s and p Orbitals)

For each value of there is a set of orbitals with distinctive shapes. These orbitals represent the region of space where it is most likely to find the electron.

= 0 (s-orbital) (1)

= 1 (p-orbitals) (3)

Orbital Shape (d Orbitals)

= 2 (d-orbital) (5)

Orbitals With Different Values of nThe value of n does not affect the shape of an orbital. Therefore

a 1s and a 2s orbital have the same shape, and the 2p and 3p orbitals have the same shape.

There are two effects that n has on the orbitals:

1) The larger the value of n, the larger the orbital. This leads to the electron being further away from the nucleus.

2) Orbitals have radial nodes, or distances from the nucleus where the probability of finding the electron goes to zero. The number of radial nodes is given by the relationship

# nodes = (n - ) - 1

Size and Nodes For s-Orbitals 2s 3s

# radial nodes = 1 # radial nodes = 2

Spin Quantum Number (ms)The spin quantum number indicates whether the electron is

spinning clockwise or counterclockwise, and can take on two values, +1/2 and - 1/2.

Direct evidence for this quantum number was first found experimentally by Stern and Gerlach.

Multi-electron Atoms

There are two differences between hydrogen and other atoms, which have more than one electron.

1) Each electron in a multi-electron atom has its own set of four quantum numbers. These electrons must obey the Pauli exclusion principle.

2) The energy for an electron in a multi-electron atom depends on both n and . We label the various energy levels for the electrons by their value for n and the letter used to represent the value for .

Example: n = 2, = 0 2s orbital

n = 3, = 0 3s orbital

n = 3, = 2 3d orbital

This dependence of energy on both n and is a consequence of electron-electron repulsion.

The Pauli Exclusion Principle

Each electron in a multi-electron atom has its own set of four quantum numbers. An early observation concerning these quantum numbers was made by Wolfgang Pauli, and is called the Pauli Exclusion Principle.

No two electrons in an atom or ion can have the same set of four quantum numbers.

Note that as many as three of the quantum numbers can be the same, but at least one quantum number must be different.

As we will see, the arrangement of elements in the periodic table and their periodic properties are in a real sense a consequence of the exclusion principle.

Energy Ordering For Orbitals

The usual ordering of energy for orbitals is as indicated below. Notice that the ordering is not strictly in terms of the quantum number n

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f ...

Mnemonic Device For Energy Ordering

We may use the following mnemonic device for the order in energy of the orbitals. The order in which the labels are crossed out below is the order of energies.

1s

2s 2p

3s 3p 3d

4s 4p 4d 4f

5s 5p 5d 5f …

6s 6p 6d 6f …

So 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f ...

Rules For Adding Electrons to Atoms

There are three rules that must be followed when adding electrons to a multielectron atom to find the lowest energy state (ground state) of the atom.

1) Pauli principle - No two electrons can have the same set of four quantum numbers.

2) Aufbau principle - Electrons add to the lowest energy available orbital until that orbital is filled.

3) Hund’s rule - Electrons add in such a way as to make as many of the electrons as possible “spin up” (ms = 1/2).

Electron configuration - A list of each electron containing orbital, in order of energy. A superscript to the right of the orbital is used to indicate how many electrons are present in the orbital.

Element n m ms electron configuration

H (1 e-) 1 0 0 1/2 1s1

He (2 e-) 1 0 0 1/2 1s2

1 0 0 - 1/2

Li (3 e-) 1 0 0 1/2 1s22s1

1 0 0 - 1/2

2 0 0 1/2

Element n m ms electron configuration

O (8 e-) 1 0 0 1/2 1s22s22p4

1 0 0 - 1/2

2 0 0 1/2

2 0 0 - 1/2

2 1 1 1/2

2 1 0 1/2

2 1 -1 1/2

2 1 1 - 1/2

Maximum number of electrons in an orbital s-orbital 2

= 2(2 + 1) p-orbitals 6

d-orbitals 10

f-orbitals 14

Orbital Filling Diagram

An orbital filling diagram is a picture where the orbitals for an atom are indicated by lines, and electrons by arrows pointing up (“spin up”) or down (“spin down”). Such a diagram is useful in identifying the total number of unpaired electron spins in an atom.

Example: (Note – Can also use boxes for the orbitals.)

O(8 e-) 1s22s22p4 __ __ __ __ __

1s 2s 2p

2 unpaired electron spins

Notice that if we are only interested in identifying unpaired electron spins we need only examine those orbitals that are partially filled. All electron spins will be paired up for filled orbitals.

Also notice how we use Hund’s rule in filling the orbitals.

Magnetic Properties of Atoms

An atom placed in a magnetic field will be attracted or repelled by the field depending on the number of unpaired electron spins in the atom. There are two cases

diamagnetic - Weakly repelled by an external magnetic field. Occurs when there are no unpaired electron spins.

paramagnetic - Strongly attracted by an external magnetic field. Occurs when there are one or more unpaired electron spins. The number of unpaired spins can be determined by measuring the strength of the attraction felt by the atoms.

Examples

1) Give the electron configuration for Sn (50 e-).

2) Consider the following elements: Li, Be, N.

How many unpaired electron spins occur in atoms of each of the following elements? Which atoms are diamagnetic and which are paramagnetic?

1) Give the electron configuration for Sn (50 e-).

Filling order: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<…

Sn (50e-) 1s22s22p63s23p64s23d104p65s24d105p2

2) Consider the following elements: Li, Be, N.

How many unpaired electron spins occur in atoms of each of the following elements? Which atoms are diamagnetic and which are paramagnetic? 1s 2s 2p

Li 1s22s1 __ __ __ __ __ 1 unpaired

Be 1s22s2 __ __ __ __ __ 0 unpaired

N 1s22s22p3 __ __ __ __ __ 3 unpaired

Li and N have at least one unpaired spin and so are paramagnetic. Be has no unpaired electron spins and so is diamagnetic. Note that we only need to look at the partially filled orbitals to count unpaired spins.

Shorthand Notation For Electron Configurations

Electron configurations can be very long. For example

Sn (50 e-) 1s22s22p63s23p64s23d104p65s24d105p2

We may use the following shorthand notation for cases where there are a large number of electrons. We represent the electron configuration as the configuration for the closest noble gas with a smaller number of electrons than the atom, plus the additional electrons.

In the above case, we would say

Sn (50e-) [Kr]5s24d105p2

Example: Give the electron configurations for O (8 e-), Ti (22 e-), and Te (52 e-). Give these using both the long notation and the shorthand notation.

Example: Give the electron configurations for O (8 e-), Ti (22 e-), and Te (52 e-). Give these using both the long notation and the shorthand notation.

Filling order: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<…

O (8 e-) 1s22s22p4 = [He]2s22p4

Ti (22 e-) 1s22s22p63s23p64s23d2 = [Ar]4s23d2

Te (52 e-) 1s22s22p63s23p64s23d104p65s24d105p4 = [Kr]5s24d105p4

For atoms with a large number of electrons the shorthand notation saves time and space.

Core and Valence Electrons

We may divide the electrons in an atom into two categories.

Valence electrons - The total number of electrons found in all orbitals having the highest value of n.

Core electrons - All other electrons in the atom.

Examples:

O(8e-) 1s2 2s2 2p4 = [He]2s22p4 6 valence, 2 core

Ti(22 e-) 1s2 2s2 2p6 3s2 3p6 4s2 3d2 = [Ar]4s23d2 2 valence, 20 core

The chemical properties of a main group element are primarily determined by the number of valence electrons per atom. Atoms in the same group in the periodic table have the same configuration for their valence electrons.

Chemical bonding takes place almost exclusively by the transfer or sharing of valence electrons.

The Periodic Table

The periodic table arranges atoms of elements in such a way that each element in a particular group in the table has the same number of valence electrons. Since it is the valence electrons that are primarily responsible for chemical properties of an element, that means that elements in the same group will have similar chemical properties.

Example: Group 2A

Be (4 e-) 1s22s2 = [He]2s2

Mg (12 e-) 1s22s22p63s2 = [Ne]3s2

Ca (20 e-) 1s22s22p63s23p64s2 = [Ar]4s2

Sr (38 e-) 1s22s22p63s23p64s23d104p65s2 = [Kr]5s2

Ba (56 e-) 1s22s22p63s23p64s23d104p65s24d105p66s2 = [Xe]6s2

All group 2A elements have 2 valence electrons.

So what similarities in properties do we expect?

For example, consider the ionic compounds group 2A elements make with the nonmetal fluorine.

BeF2 beryllium fluoride

MgF2 magnesium fluoride

CaF2 calcium fluoride

SrF2 strontium fluoride

BaF2 barium fluoride

Of course, there are other metals that form compounds with the same stoichiometry (CuF2, copper (II) fluoride, for example). But if we look at a large number of chemical and physical properties for the group 2A elements we find that they are similar to one another and different from elements in other groups.

Electron Configuration for the Elements

The electron configurations for the main group and transition metals are shown below. Note the similarities for elements in the same group.

Anomalous Electron Configurations

The rules we have given for predicting electron configurations for atoms work most of the time. However, there are occasional cases where the actual electron configuration is different from the predicted configuration.

Example:

Cr(24 e-) [Ar]4s23d4 predicted

[Ar]4s13d5 actual

These deviations from prediction occur because the rules developed for predicting electron configurations are based on approximate solutions to Schrodinger’s equation. We detect these anomalous configurations experimentally.

You need to be aware that there are occasional deviations from prediction, but not the actual deviations that occur.

End of Chapter 6“Louis de Broglie made the astonishing suggestion that if light is

a particle as well as a wave, perhaps electrons and other particles also behave as waves. He proposed this in a 1924 PhD thesis that did not impress his examiners, and would have failed without the endorsement of Einstein.” - Lee Smolin, The Trouble With Physics

“It’s a particle; it’s a wave.” - Keanu Reeves

“We virtually ignore the astonishing range of scientific and practical applications that quantum mechanics undergirds: today an estimated 30 percent of the U.S. gross national product is based on inventions made possible by quantum mechanics, from semiconductors in computer chips to lasers in compact-disc players, magnetic resonance imaging in hospitals, and much more.”

Max Tegmark and John Wheeler Scientific American, Feb. 2001.