Misconceptions & Issues
in Quantum Theory
AP* Chemistry & AP Physics
Original text taken from an article posted on
AP Central by George E. Miller
AP® is a registered trademark of the College Board. The College Board was not involved in the
production of and does not endorse this product.
Dr. George E. Miller
University of California, Irvine
Dr. George E. Miller is senior lecturer SOE emeritus
in the Department of Chemistry at the University of
California, Irvine, where he is also the principal
scientist and the reactor supervisor of UCI's nuclear
reactor and faculty director for science education
programs at UCI's Center for Educational
Partnership, including Faculty Outreach
Collaborations Uniting Scientists, Students and
Schools (FOCUS). He was a member and chair of
the AP Chemistry Development Committee and is
currently a member of the College Board Subject
Test in Chemistry Development Committee. He has
been a Reader, Table Leader, and Question Leader
for AP Chemistry.
Introduction
Students seem to grasp the structure of atoms better than
many other subjects in the AP Chemistry or AP Physics
curriculum.
Still, there is always room for improvement.
We will review a few key issues regarding atomic
structure and give suggestions for pedagogy to enhance
student understanding.
Introduction
From the 2008-2008 AP® Chemistry Course Description:
A. Atomic theory and atomic structure
1. Evidence for the atomic theory
2. Atomic masses; determination by chemical and
physical means
3. Atomic number and mass number; isotopes
4. Electron energy levels: atomic spectra, quantum
numbers, atomic orbitals
5. Periodic relationships including, for example,
atomic radii, ionization energies, electron affinities,
oxidation states
Introduction
From the 2008-2009 AP® Physics B Course Description:
V. Atomic and Nuclear Physics . . . . . . . . . . . 10%
A. Atomic physics and quantum effects
1. Photons, the photoelectric effect,
Compton scattering, x-rays
2. Atomic energy levels
3. Wave-particle duality
B. Nuclear physics
1. Nuclear reactions (including
conservation of mass number
and charge)
2. Mass–energy equivalence
7%
3%
Background
In Pre-AP Chemistry, students should
receive a modest historical
introduction reviewing the ideas and
experiments of Dalton, J. J. Thomson
and Rutherford, and Bohr that
established the bases of atomic theory
and structure as matter consisting of
atoms, each with a central positive
nucleus surrounded by negatively
charged electrons.
Students must firmly master this
notion before proceeding.
John Dalton (September 6, 1766 – July 27, 1844)
John Dalton (September 6, 1766 – July 27, 1844)
He was a Fellow of the Royal
Society and an English chemist,
meteorologist and physicist.
He is best known for his pioneering
work in the development of modern
atomic theory.
John Dalton (September 6, 1766 – July 27, 1844)
Five main points of Dalton's Atomic Theory
1. Elements are made of tiny particles called atoms.
2. All atoms of a given element are identical.
3. The atoms of a given element are different from those of any
other element; the atoms of different elements can be
distinguished from one another by their respective relative
weights.
4. Atoms of one element can combine with atoms of other elements
to form chemical compounds; a given compound always has the
same relative numbers of types of atoms.
5. Atoms cannot be created, divided into smaller particles, nor
destroyed in the chemical process; a chemical reaction simply
changes the way atoms are grouped together.
John Dalton (September 6, 1766 – July 27, 1844)
TWO MAJOR EXCEPTIONS:
To be sure, the conviction that atoms cannot be subdivided,
created, or destroyed into smaller particles when they are
combined, separated, or rearranged in chemical reactions is
inconsistent with the existence of nuclear fusion and nuclear
fission, but such processes are nuclear reactions and not
chemical reactions.
In addition, the idea that all atoms of a given element are
identical in their physical and chemical properties is not precisely
true, as we now know that different isotopes of an element have
slightly varying weights due to differing numbers of neutrons.
Sir Joseph John “J. J.” Thomson
Order of Merit, Fellow of the Royal Society (18 December 1856 – 30 August 1940)
Thomson was a British physicist and Nobel
laureate, credited for the discovery of the
electron and of isotopes, and the invention of
the mass spectrometer.
He was awarded the 1906 Nobel Prize in
Physics for the discovery of the electron and
his work on the conduction of electricity in
gases.
“J. J.” Thomson’s First Experiment
In his first experiment, he investigated whether or not the negative
charge could be separated from the cathode rays by means of
magnetism.
He constructed a cathode ray tube ending in a pair of cylinders with slits in
them. These slits were in turn connected to an electrometer. Thomson found
that if the rays were magnetically bent such that they could not enter the slit,
the electrometer registered little charge. Thomson concluded that the
negative charge was inseparable from the rays.
“J. J.” Thomson’s Second Experiment
In his second experiment, he investigated whether or not
the rays could be deflected by an electric field (something
that is characteristic of charged particles).
“J. J.” Thomson’s Second Experiment
Previous experimenters had failed to observe this, but
Thomson believed their experiments were flawed
because they contained trace amounts of gas.
Thomson constructed a cathode ray tube with a
practically perfect vacuum, and coated one end with
phosphorescent paint. Thomson found that the rays did
indeed bend under the influence of an electric field, in a
direction indicating a negative charge.
Deducing the Behavior of the Electron
External Magnetic Field Applied
External Electric Field Applied
BOTH Fields Applied
“J. J.” Thomson’s Third Experiment
In his third experiment, Thomson
measured the mass-to-charge ratio of
the cathode rays by measuring how
much they were deflected by a
magnetic field and how much energy
they carried.
He found that the mass to charge ratio
was over a thousand times lower than
that of a hydrogen ion (H+), suggesting
either that the particles were very light
or very highly charged.
“J. J.” Thomson’s Conclusions
Thomson's conclusions were bold: cathode rays
were indeed made of particles which he called
"corpuscles", and these corpuscles came from within
the atoms of the electrodes themselves, meaning
that atoms are in fact divisible. The "corpuscles"
discovered by Thomson are identified with the
electrons which had been proposed by
G. Johnstone Stoney.
Thomson imagined the atom as being made up of
these corpuscles swarming in a sea of positive
charge; this was his plum pudding model. This model
was later proved incorrect when Ernest Rutherford
showed that the positive charge is concentrated in
the nucleus.
Earnest Rutherford
OM, PC, FRS (30 August 1871 – 19 October 1937)
A New Zealand physicist who became
known as the father of nuclear physics.
He pioneered the orbital theory of the
atom through his discovery of Rutherford
scattering off the nucleus with his gold
foil experiment. He was awarded the
Nobel Prize in Chemistry in 1908.
The Geiger-Marsden Experiment
Also called the Gold foil experiment or the Rutherford
experiment.
Actually an experiment done by Hans Geiger and Ernest
Marsden in 1909, under the direction of Ernest Rutherford
at the Physical Laboratories of the University of
Manchester which led to the downfall of the plum pudding
model of the atom.
The Geiger-Marsden Experiment
They measured the deflection of alpha particles (helium
ions with a positive charge) directed normally onto a sheet
of very thin gold foil. Under the prevailing plum pudding
model, the alpha particles should all have been deflected
by, at most, a few degrees.
However they observed that a very small percentage of
particles were deflected through angles much larger than
90 degrees. From this observation Rutherford concluded
that the atom contained a very physically-small (as
compared with the size of the atom) positive charge, which
could repel the alpha particles if they came close enough.
These conclusions were subsequently developed into the
Bohr model.
Rutherford Quote
"It was almost as incredible as if you fired a fifteen-inch
shell at a piece of tissue paper and it came back and hit
you“.
Electromagnetic Radiation
It is also essential that students have a firm grasp of the properties of
electromagnetic radiation, since the results of its interaction with atoms
is crucial to understanding the experimental evidence that led to the
development of the modern picture of atomic structure. Students should
have fully mastered the relationships between wavelength, frequency,
and energy, and they should know, for example, the relative energy and
frequency ranges of ultraviolet vs. visible vs. infrared radiation.
Electromagnetic Radiation
c =  =f = 3108 m/s
E = h = hf = mc2
Electromagnetic Radiation
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
It is helpful if students can view a
hydrogen discharge lamp
through a diffraction grating to
see for themselves the evidence
of electronic energy state
transitions that led Bohr to
propose his model of the
electronic structure of the atom.
They must understand that the
energies (and hence frequency
and wavelength) of emitted light
correspond to the differences
between the outer energy state of
an electron and the inner
electronic state into which it can
move.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Make sure students identify that
they are considering a hydrogen
atom, so only a single electron
exists, and no other electrons
need to be considered or can
"get in the way" of the one
electron's transitions.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Confusion exists in many students' minds because of the rather unusual
use of language such as "higher" and "lower" with respect to position
and energy states, and the way in which energy states are represented
in diagrams, with energy clearly marked as increasing toward the "top"
of the diagram.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Thus we may read that an
electron with a higher energy
state (say n = 5) "falls" into an
n = 2 state and releases blue
light as it does so.
In fact, the n = 2 electron is
closer to the nucleus, in a
more strongly attracted and
hence "higher" energy
condition than the n = 5
electron.
The change in potential
energy has converted into
electromagnetic energy, and
the atom is now in a more
stable condition.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
The discrepancy arises because we define the hydrogen atom's
potential energy as zero when the electron is at an infinite distance
away from the nucleus, and the potential energy becomes increasingly
negative as the electron moves closer and closer to the nucleus, due to
a greater attraction (Coulomb force).
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
So, in energy diagrams, zero is at the "top," and energies become less
negative, not more positive as one moves "up" on the diagram. This
information is rarely included in text diagrams. Students will need
careful guidance to steer their way through these seeming
contradictions.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
What’s
wrong with
this one?
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Make sure students identify that they are considering a hydrogen atom,
so only a single electron exists, and no other electrons need to be
considered or can "get in the way" of the one electron's transitions.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
An exam question may ask students to compute electromagnetic radiation
energies (Eemr) from changes in positions in hydrogen atoms (never in any
multiple-electron atoms).
The exam booklet provides a version of the Bohr equation containing the
appropriate Rydberg constant (En = -2.178  10-18/n2 joule).
Eemr = ΔEatom = Efinal state - Einitial state gives the electromagnetic radiation energy.
This equation generates a negative value for ΔEatom, correctly indicating
that the atom system has transferred energy to its surroundings (exoergic
process). However, the electromagnetic radiation energy cannot be
negative because it is the energy transferred out, so we ignore the sign.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
If the process involved transfer of electromagnetic radiation
into the atom (excitation of the electron, moving the electron
further from the nucleus), then the ΔEatom would be positive.
Note that we would still not consider the electromagnetic
radiation as having negative energy. Hydrogen atom spectra
lines involving absorption of electromagnetic radiation are
observed in the sun's spectrum as the cooler outer layers of
hydrogen in the sun absorb electromagnetic radiation emitted
from the hotter inner core.
Energy in Electronic States of Hydrogen
Atoms: The Bohr Model
Increasing energy
Establishing the underlying principles
of these transitions firmly in students'
minds will serve them well in future
chemistry studies.
They will see many more examples of
emission and absorption spectroscopy
and spectrometry in all the wavelength
regions of electromagnetic radiation,
from gamma radiation to radio waves.
Beyond the Bohr Model
The Bohr model was extremely
successful in modeling
observations with hydrogen atoms.
However, it failed to meet the same
challenge for all other atoms.
Attempts to "tweak" the model were
not too successful until
Schrödinger, de Broglie, and
Heisenberg contributed (in different
ways) to a greater understanding of
the behavior of electrons in atoms.
Beyond the Bohr Model
The net result is the
quantum picture we now
use, where electrons still
have discrete energies
associated with their
atomic positions but
occupy positions called
orbitals, the shapes of
which we can generate
with computer modeling
but will probably never
observe.
Beyond the Bohr Model
It is unfortunate that the term
"orbital" bears a close similarity
to "orbit," coined for the defined
pathway of objects rotating
around other objects, since
beginning students still tend to
think of orbitals as resembling
orbits.
Beyond the Bohr Model
Beyond the Bohr Model
If possible, encourage your students to think of orbitals as
"something completely different" (with apologies to Monty
Python).
Electrons in atoms (not necessarily in outer space) take on
predictable but unusual shapes, best identified as
"probability distributions" that retain highly specific potential
energy states, known because transitions between such
states still result in electromagnetic radiation line spectra.
Beyond the Bohr Model
However, we cannot predict the
energies from simple formulas when
more than a single electron is present
because of the interactions between the
electrons as well as between the
electrons and the nucleus.
We still need complex approximations
(except in the very simplest cases) to
get the "right" answers from computer
models.
Beyond the Bohr Model
This much has evolved from the original Schrödinger model:
each electron's energy equation is characterized by a set of four
integers called quantum numbers.
Rules (clearly described in textbooks) establish the allowed
ranges of these numbers.
No two electrons in any atom can have the same values for all
four numbers.
Beyond the Bohr Model
The rules, combined with a well-established filling order
(aufbau principle) determine the structure of neutral groundstate atoms, as well as the possible excited states where
electrons can occupy previously unoccupied orbitals.
).
Beyond the Bohr Model
When writing electron configuration listings for an
atom, we use symbols that represent particular sets of
quantum numbers.
These symbols evolved, unfortunately, from empirical
work by spectroscopists carried out before
Schrödinger's work established a more workable
model.
The basic shape of the orbital derives mostly from the
value of the angular momentum quantum number ().
Thus we identify the simplest electron group (or
subshell) with  = 0 by its overall spherical shape, and
we refer to it with the symbol s (for single).
).
Beyond the Bohr Model
If  = 1, the symbol is p (for principal); when = 2, the symbol is d
(diffuse); and for  = 3, the symbol is f (fundamental). When  = 4,
used only in the actinide elements discovered much later, the symbol
is g (the letter after f), which at least makes some sense.
Tests included with textbooks and former AP Exams give many
examples of how to use these symbols, the aufbau principle, and
other quantum number rules in writing electron configurations for
atoms.
The most common error that students make is the easily avoidable
one of not counting the correct number of electrons (equal to the
atomic number for a neutral atom). They also fail to take into account
the loss or gain of electrons if the subject species is an ion.
).
Beyond the Bohr Model
).
Beyond the Bohr Model
).
Beyond the Bohr Model
).
Beyond the Bohr Model
).
How Atomic Structure Affects Atomic Properties
The chemical behavior of an atom is entirely determined by
its electronic structure.
To learn how atomic properties change in a systematic way,
the easiest way is to understand and master the
relationship between the electronic structure of an atom
and its location in the periodic table.
How Atomic Structure Affects Atomic Properties
Some Clarifying Definitions
Ionization energy -- The energy change associated with removing one
electron from a neutral ground-state (usually) atom: The electron is moved
an infinite distance away. This energy is always a positive number since
energy must be added to the atom system to remove an electron (always an
endoergic process).
How Atomic Structure Affects Atomic Properties
Electron affinity -- The energy change associated with adding one electron to a neutral
(usually) ground-state atom: The electron comes from infinity. By proper sign convention, this
would be a negative number if the result is energy transferred from the atom system to the
surroundings (exoergic) and a positive number if energy from the surroundings is transferred
to the atom (endoergic; energy is necessary to keep the electron on the atom). Some older
texts may use the opposite sign convention, defining the electron affinity slightly differently.
How Atomic Structure Affects Atomic Properties
Electronegativity -- Relative ability of an atom to attract electrons
when bonding with another atom in a molecule: Scientists have tried
different ways to compute such values. The most popular is that of
Pauling, who compared the bond energy of a molecule Q-X with that of
the average bond energy of Q-Q and X-X molecules. In some sense,
this single parameter reflects the comparison of ionization energy and
electron affinity as applied strictly to chemical bonding.
How Atomic Structure Affects Atomic Properties
Atomic radius -- The value of the effective radius of a neutral ground-state atom
as estimated by some form of measurement: The distance between atoms in
various compounds is the covalent atomic radius.
If the atoms do not readily form useful compounds, we must use other techniques.
Metallic radius represents the distance between atoms in a solid metal. The most
difficult to determine are the noble gases, for obvious reasons.
The numbers are rarely as accurate (three significant figures!) as they are listed in
some texts; they should be used only in comparing similar elements (as they will
have been determined using the same method).
We assume the values represent the approximate size of the electronic "cloud" and
it is roughly spherical, surrounding the nucleus of the atom. Recent use of atomic
force microscopy tends to confirm that these assumptions are reasonable.
).
How Atomic Structure Affects Atomic Properties
A "deep" explanation goes right to the core of the atom. As one example:
Why is the first ionization energy of sodium (495 kJ/mol) different from that
of potassium (419 kJ/mol)?
Point 1: The electronic configuration of a neutral ground-state sodium atom
(Z = 11) is 1s22s22p63s1; potassium (Z = 19) is 1s22s22p63s23p64s1.
Point 2: The outermost electron is the one removed first as any atom is
ionized.
Point 3: The first electron removed is 3s for sodium and 4s for potassium.
How Atomic Structure Affects Atomic Properties
Point 4: The 3s electron in sodium is attracted by a nucleus with 11 protons;
the 4s electron in potassium is attracted by a nucleus with 19 protons.
Point 5: We might first think that the potassium electron is harder to remove
(more protons attracting it).
This is obviously not correct, so what else is changing?
).
How Atomic Structure Affects Atomic Properties
Point 6: Atomic radius issue -- the 4s electron in potassium
is further from the nucleus than the 3s electron in sodium
(the data support this: the radius of Na is 186 nm, and the
radius of K is 227 nm).
The Coulomb force of attraction (and the energy resulting)
is less as the distance increases between the charges.
).
How Atomic Structure Affects Atomic Properties
This is the most misused and abused argument!
Point 7: Shielding issue -- the presence of other electrons influences
the property of each electron considered. Electrons are negatively
charged, so the presence of inner shells of negative electrons tends to
shield or "screen" the outermost electron from the effects of the
nucleus.
Strictly what the inner electrons do is lower the effective nuclear
charge and thus the Coulomb force of attraction on the outer electron
resulting from the nucleus. Students are allowed to state that the inner
electrons "help to neutralize" the nuclear charge, as long as that
doesn't imply that these electrons reduce the actual charge on the
nucleus. It would be a bit like putting nasty-tasting JELL-O® between
yourself and an attractive dessert. The attraction of the dessert seems
much less if you have to fight through the JELL-O to get it.
How Atomic Structure Affects Atomic Properties
Shielding should ONLY be used when discussing elements
from different periods. Shielding is ineffective within a period
and that argument will gain no points!
Point 8: In summary, the combined issues in points (7) and (8)
must overcome the effect discussed in point (5) so that the
first ionization energy of K is significantly lower than that of
Na. This explains the trend in the entire family of alkali metals
in Group 1 of the periodic table.
Students do not need to use all of the above arguments for full credit on
the AP Exam. However, a student who can think through all eight points
will have truly mastered how to explain in terms of atomic structure,
observed trends, and differences in atomic properties in relation to the
periodic table.
The Mathematics Associated with
Quantum Mechanics on Each AP Exam
The Mathematics Associated with
Quantum Mechanics on Each AP Exam