Please note links to Figures and Tables
from the text will not work. Due to
copyright restrictions, figures from the
text are not shown. Links to external
sites are active.
Chapter 2 - Atomic structure
Chapter opener, two d orbitals a dxz (?) and a dz2
2.1 Historical Development of Atomic Theory?
I include more events than the text to tell a more coherent story. Dates
are approximate

440 BC - Democritus - Develops the idea of the indivisible atom making up all
matter

1704 - Newton - Establishes the laws of mechanic that describes how matter
behaves as particles

1777 - Lavoisier - Studies gasses and stated the Law of conservation of Matter

1800 - Proust - Law of Definite composition

1800 - Avogadro - Equal volumes of a gas have equal number of particles

1804 - Dalton - First modern atomic theory
o summary of results Dalton used
o Statements of Dalton's theory

1869 - Mendeleev and Meyer - Periodic table. Arranged by atomic mass.
Mendeleev made prediction for unknown element.
o Mendeleev's predictions for undiscovered elements
o Modern periodic Table
o My favorite periodic table

1873 - Maxwell - Maxwell's equations that describes how energy behaves like a
wave

1885 - Balmer - Studies emission spectra and records the visible emission
spectrum of hydrogen gas.
o Dark side of the moon album cover
o Emission and Absorption spectra

1896 - Bequerel- Discovers radioactivity

1897 - Thomson - Studies cathode rays and calls the particles electrons. Found
Charge/mass)
o Cathode ray tube used to study the electron
o Plum Pudding model

1900 - Rydberg - develops a mathematical formula for the Balmer spectra and
other spectra of hydrogen
o Rydberg's equation

1900 - The Wall - Physics is over(?)
Matter = Particle understood by Newton's Laws
________________________________________________________________
_________________________
Energy = Wave understood by Maxwell's equations

1900 - Planck - black Body Radiation. Proposes matter is quantized = comes in
bundles of hν

1905 - Einstein - Explained the photoelectric effect using photons (light particles)
quantized as Planck proposed
o Photoelectric effect

1909 - Milikan - Oil drop experiment found charge on electron. With Thomson's
results determined mass of electron
o Milikan's set up

1911 - Rutherford - Gold foil
o Set up of Rutherford's experiment
o Interpretation of Rutherford's Experiment

1914 - Mosley - Periodic Table should be arranged by number of protons.

1913 - Bohr - Circular planetary model. Explained emission spectra for hydrogen.
Failed for helium and larger atoms
o a Derivation of Bohr's atom
o Java applet explaining emission and absorption of light by an atom
o Bohr's explanation of spectra (scroll and look at entire page
o

1924 - de Broglie - Proposed wavelength for matter
o de Broglie wavelength
o baseball and electron wavelengths
o Calculate a de Broglie wavelength

1925 - Heisenberg - Stated the uncertainty principle
o Uncertainty principle

1930 - Schr&oml;edinger and Heisenberg - Developed quantum mechanics in
wave and matrix form respectively
o Time independent Schrödinger's Equation
o
o
o
o
o

Schrodinger's Equation in 1-D
Schrodinger's Equation in 3-D (Cartesian coordinates)
Spherical polar coordinates
Schrodinger's Equation in 3-D (Spherical Polar coordinates)
Wave function
 Description
 properties
 Constraints
 Probability
1932 - Chadwick - Discovered the neutron
2.1 Historical Development of Atomic Theory?
Covered and Expanded in Section 2.0 above
2.1.1 The Periodic Table

Figure 2.1 Layout of the Modern Periodic Table

Groups Names to Know
o 1 - Alkali Metals
o 2 - Alkaline Earths
o 17 - Halogens
o 18 - Noble Gases

Sections to Know

Groups Names to Know
o Periods = Rows (1-7) and Groups = Columns (also called Families) (1-18)
o Main Group Elements (Groups 1,2,13-18)
o Transition Metals (Groups 3-12)
o 1Lanthanides and Actinides (also called Lanthanoids and Actinoids
2.1.2 Discovery of Subatomic Particles and the Bohr
Atom
Theme 1

Table 2.1 Review of discoveries leading to Bohr's atom

Rydberg equation
wavenumber form 1/λ
RH = 1.097 x 10-7 1/m
Energy form
RH = -2.197 x 10-18 J


ΔE = hν and νλ = c, so λ = hc/ΔE
o λ = wavelength
o ν = frequency
-34
o h = Planck's constant = 6.626 x 10
J
+8 m
o c = speed of light = 2.998 x 10
/s

Figure 2.2 Bohr Energy levels

Working a Bohr transition problem on the board. What is the wavelength of the
transition from n = 4 to n = 2?

Problems with the Bohr model
o Did not work for He or higher elements
o Modifications - elliptical orbits and relativity did not help
o Why? de Broglie wavelength and uncertainty principle
o Must treat electron as a wave = no specific orbits but orbitals with
probabilities

What is good about the Bohr model?
o Energy levels are correct for hydrogen
o introduced the idea of quantum states
o quantum numbers are expanded in the quantum mechanical atom
o nuclear model is correct
2.2 The Schrödinger Equation
Theme 1

Simplest form HΨ = EΨ
o H = Hamiltonian Operator include Kinetic and Potential Energy
o E = Energy value of different states
o Ψ = The Wavefunction

All About the Wave function (similar to Page 23 points 1-5)
o Description
o properties
o Constraints
o Probability

The Schrödinger Equation in detail (on the board see page 22)
2.2.1 The Particle in a box
A System that can be Solved Exactly to Illustrate the Properties of
Wavefunctions

Figure 2.3 set up of the Particle in a Box

Equation and Solution on the Board

Boundary conditions at x = 0 and x = a

Figure 2.4 Wavefunctions and the Squares for the Particle in a Box
2.2.2 Quantum Numbers and Atomic Wavefunctions

Solutions for the Schrödinger Equation for Hydrogen

Table 2.2 Quantum Numbers

Figure 2.5 Spherical Polar Coordinates


Separation of variables
Ψ(n, ℓ, ℓ) = [R(n,ℓ)(r)]•[Θ(ℓ,
ℓ)(θ)]•[Φ(ℓ,
ℓ)(φ)]
Separation of variables, but combining the two angular parts
Ψ(n, ℓ, ℓ) = [R(n,ℓ)(r)]•[Y(ℓ, ℓ)(θ, φ)]

Table 2.3 Hydrogen atom angular Part of the Wavefunction

Figure 2.4 Math Formula for Hydrogen Atom Radial Psrt of the Wavefunction

Another Table of the Math Equations of the Radial Part Emphasizing the
equation as (Constant) x (Polynomial) x (Decaying Exponential)

Figure 2.6 Shape of the Angular Part of the Wavefunction

Table 2.5 Radial nodes and surfaces (planes and cones)
o
The first time a specific orbital occurs → zero nodes (1s, 2p, 3d, 4f, ...)
o
The first time a specific orbital occurs → zero nodes (2s, 3p, 4d, 5f, ...)
o
The first time a specific orbital occurs → zero nodes (3s, 4p, 5d, 6f, ...)
o
etc.

Figure 2.7 Radial part (top) and Radial probability distribution (bottom) for
hydrogen

More Radial parts and Radial Probability distributions lined up to show clustering
by n as a function of distance from the nucleus.

Figure 2.8 Cross sectional views of the full electron density = (Radial probability
function) x (Angular part squared) Individual slides will follow.
o
Figure 2.8 (detail) Cl 3s
o
Figure 2.8 (detail) Cl 2pz
o
Figure 2.8 (detail) Cl 3pz
o
o
o
Figure 2.8 (detail) Ti3+ 3dz2
Figure 2.8 (detail) Ti3+ 3dx2- y2 Perpendicular to the yz plane
Figure 2.8 (detail) Ti3+ 3dx2- y2 Perpendicular to the xy plane
2.2.3 The Aufbau Principle
 Before determining electron configurations for atom, we need to learn some rules:




Electrons in atoms fill the lowest available orbitals first.
Electrons follow the Pauli Exclusion Principle - No two electrons in an atom
may have all four quantum numbers the same.
A consequence of the Pauli Exclusion Principle and the fact that s has only two
values is that an orbital may hold no more than two electrons and if two electrons
are in the orbital, the spins must be paired.
Electrons follow Hund's Rule - When electrons are placed in equivalent energy
orbitals the most stable arrangement has the maximum number of parallel spins
also called the maximum multiplicity.

Table 2.6 Illustration of Hund's Rule for p3. State 3 has maximum multiplicity and is most
stable.
 Explanation of Hund's Rule. There are two types of energy interactions if there are
multiple electrons in degenerate (same energy) orbitals

A Coulombic Interaction (Πc) if Two electrons are in the same orbital →
destabilizing


An Exchange Interaction (Πe) if Two electrons of the same spin have a pairwise
exchange → stabilizing
The relative destabilization of a Coulombic interactions is about equal and
opposite of the stabilization of an exchange interaction.

(Page 35) Example of an exchange interaction.

(Page 35) Possible state for electron configuration p2. Left has a Coulombic interaction,
Right has an Exchange interaction, Center has neither.

(Page 35) Relative energies of a p2 electron configuration.

(Page 36) Oxygen has p4 electron configuration. We will look at two possible states.
This is state A.
 Analysis of Electron Configuration A





Left most orbital shows one Coulombic interaction
The single down spin has no exchanges
The three up spins have three pairwise exchanges.
Overall Πc + 3Πe

Oxygen has p4 electron configuration. We will look at two possible states. This is state
B.
 Analysis of State B




Two orbital shows Coulombic interaction
The two down spin has one exchanges
The two up spins have one pairwise exchanges.
Overall 2Πc + 2Πe
 The net result is State A is more stable
(1 destabilization + 3 stabilizations) > (1 destabilization + 3 stabilizations)
than state B

Figure 2.9 The most convenient way to obtain the order of filling of orbitals is the
periodic table itself.

Figure 2.10 The energy levels of a multielectron atom.

Figure 2.10 Electron Configurations of the Elements. You should be able to write these.
 Two exception to the normal filling rules you should remember are Cr and Cu. There
are other, but remember these two.
2.2.4 Shielding
 Shielding takes into effect the screening of the nuclear charge by electrons closer to
the nucleus.
 Z* or Zeff = Z - S


Z* = Effective Atomic Number - amount of nuclear charge felt by an electron.
Z = Atomic Number

S = Shielding

Shielding may be estimated by Slater's Rules. Exact calculation is by a
Computer.

Using Slater’s Rules for calculating S listed below:
1. An electron does not shield itself. (Not in the book, but a common error)
2. The electronic structure of the atom is written in groupings as follows:
(1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p), and so on.
3. Electrons in higher orbitals (to the right in the list above) do not shield
those in lower orbitals.
4. For ns or np valence electrons:
a. Electrons in the same ns, np level contribute 0.35,
except the 1s, where 0.30 works better.
b. Electrons in the n-1 level contribute 0.85.
c. Electrons in the n-2 or lower levels contribute 1.00.
5. For d and nf valence electrons:
a. Electrons in the same nd or nf level contribute 0.35.
b. Electrons in groupings to the left contribute 1.00.
6. After calculating S, then subtract from Z to get Z* Z* = Z - S

Calculation of several examples on the board.

Slater's rules and shielding can explain why transition metals fill the (n+1)s
before the nd, but lose the (n+1)s electrons first.

For Example, Fe = [Ar]4s23d6 but Fe3+ = [Ar]3d5.
The 4s electrons are lost first. The 4s electron have a lower Z*.

Figure 2.11 Slater's rules can explain the anomalous electron configurations in
the periodic table.
There are many, but you only need to remember Cr and Cu.

Skip the explanation of one and two electron orbital energies on pages 42 and
43.
2.3.1 Periodic Properties: Ionization Energy

Definition: Energy of A(g) → A(g)+ + eIE1
Removal of an electron from a gaseous atom.

General Trend

Figure 2.13 Plots of Ionization Energy and Electron Affinity. Noitice Anomalies at
Be-B and N-O against the general trend.

"Anomalies may be explained by examining the electron configurations
o Be (removing 2s electron) vs B (removing 2p electron)
3
4
o N is 2p and O is 2p . Removing electron from O removes a Columbic
interaction.

Successive ionization energies. Example for Al.
→ Al (g) + e
Al (g) → Al (g) + e
Al (g) → Al (g) + e
Al (g) → Al (g) + e
Al(g)
+
-
IE1 = 577.5 kJ/mol
+
2+
-
IE2 = 1,816.7kJ/mol
2+
3+
-
IE3 = 2,744.8kJ/mol
3+
4+
-
IE4 = 11,577kJ/mol (Huge jump, core electron)
2.3.2 Periodic Properties: Electron Affinity
 Definition: Energy of A(g)+ → A(g) + eEA1
Note: Your book defines Electron Affinity Opposite of Nearly Every Other Book in the
World. Always double check the definition in the book you are using.
 The Trend is not Clear Cut. See Figure under Ionization Energy above.
2.3.3 Periodic Properties: Covalent and Ionic Radii
o
Radius of atom treated like a sphere in a bond. Radius A + Radius B = AB bond distance.
o
Table 2.8 Numerical Values for Covalent Radii
o
o

General Trend
Plot from www.webelements.com
Ionic Radii
o
Cations have fewer electrons and are smaller that their parent atom. Ca 2+
< Ca
o
Anions have more electrons and are larger that their parent atoms. Br- >
Br
o
Table 2.9 Crystal Radii for Selected Ions
o
Comparison of atomic and ionic radii from http://www.chem.uwec.edu.
o
To Compare Ions across the Periodic Table one should compare
isoelectronic ions, and consider the number of Protons.
o
Table 2.11 Crystal Radii Going down a Group
o
Table 2.12 Crystal Radii as the Charge on an Ion of a Specific Element
Increases
o
Table 2.10 Crystal Radii for Isoelectronic Ions with Differing Nuclear
Charges