Title: George Mason University
1George Mason University General Chemistry
211 Chapter 7 Quantum Theory and Atomic
Structure Acknowledgements Course Text
Chemistry the Molecular Nature of Matter and
Change, 6th edition, 2011, Martin S. Silberberg,
McGraw-Hill The Chemistry 211/212 General
Chemistry courses taught at George Mason are
intended for those students enrolled in a science
/engineering oriented curricula, with particular
emphasis on chemistry, biochemistry, and biology
The material on these slides is taken primarily
from the course text but the instructor has
modified, condensed, or otherwise reorganized
selected material.Additional material from other
sources may also be included. Interpretation of
course material to clarify concepts and solutions
to problems is the sole responsibility of this
instructor.
2Quantum Theory of The Atom
- How are electrons distributed in space?
- What are electrons doing in the atom?
- The nature of the chemical bond must first be
approached by a closer examination of the
electrons - Electrons are involved in the formation of
chemical bonds between atoms - Quantum theory explains more about the electronic
structure of atoms
3Quantum Theory of The Atom
- Origin of Atomic Theory
- When burned in a flame metals produce colors
characteristic of the metal. - This process can be traced to the behavior of
electrons in the atom.
4Quantum Theory of The Atom
- Emission (line) Spectra of Some Elements
- When elements are heated in a flame and their
emissions passed through a prism, only a few
color lines exist and are characteristic for each
element. Atoms emit light of characteristic
wavelengths when excited (heated)
5Quantum Theory of The Atom
- Electrons and Light Wave Nature of Light
- Light moves (propagates) along as a wave (similar
to ripples from a stone thrown in water) - Light consists of oscillations of electric and
magnetic fields that travel through space - All electromagnetic radiation
- Visible light, Microwaves, Radio Waves,
Ultraviolet Light, X-rays, Infrared Light - consists of energy propagated by means of
electric and magnetic fields that alternately
increase and decrease in intensity as they move
through space
6Quantum Theory of The Atom
- A Light Wave is Propagated as an Oscillating
Electric Field (Energy) - The Wave properties of electromagnetic radiation
are described by two independent variables
wavelength and frequency
? wavelength, equiv. crest to crest distance ?
c/? frequency, cycles per second (Hz) c
speed of electromagnetic radiation (3 x 108 m/s)
7Quantum Theory of The Atom
- Wave Nature of Light
- Wavelength the distance between any two
adjacent identical points in a wave (given the
notation ? - Frequency number of wavelengths that pass a
fixed point in one unit of time (usually per
second, given the notation ?). The common unit
of freq is hertz (Hz, /s) - Propagation (Velocity) of an Electromagnetic wave
is given as - c ??
- c velocity 3.0 x 108 m/s in a vacuum
- c is independent of ? or ? in a vacuum
8Quantum Theory of The Atom
- Relationship Between Wavelength and Frequency
- Wavelength and frequency (c ??) are inversely
proportional
9The electromagnetic spectrum
Frequency (?)
High
Low
Energy (E)
High
Low
Wavelength (?)
Short
Long
10Distinction between Energy Matter
- At the macro scale level everyday life energy
matter behave differently
11Distinction between Energy Matter
12Distinction between Energy Matter
- The diffraction pattern caused by waves passing
through two adjacent slits - Constructive and destructive interference occurs
as water waves viewed from above pass through two
adjacent slits in a ripple tank - As light waves pass through two closely spaced
slits, they also emerge as circular waves and
interfere with each other - They create a diffraction (interference) pattern
of bright regions where crests coincide in phase
and dark regions where crests meet troughs (out
of phase) cancelling each other out
13Quantum Theory of The Atom
- Quantum Effects Wave-Particle Duality
- Wave-Particle Duality is a central concept in
Chemistry Physics - All matter and energy exhibits both wave-like and
particle-like properties - Duality applies to macroscopic (large scale)
objects, microscopic objects (atoms and
molecules), and quantum objects (elementary
particles protons, neutrons, quarks, mesons) - As atomic theory evolved, matter was generally
thought to consist of particles - At the same time, light was thought to be a wave
14Quantum Theory of The Atom
- Quantum Effects - Wave-Particle Duality
- Christiaan Huygens proposed the wave theory of
light - Huygens wave theory was displaced by Isaac
Newtons view that light consisted of a beam of
particles - In the early 1800s Young and Fresnel showed that
light, like waves, could be diffracted and
produce interference patterns, confirming
Huygens view - In the late 1800s James Maxwell developed
equations, later verified by experiment, that
explained light as a propagation of
electromagnetic waves - At the turn of the 20th century, physicists began
to focus on 3 confounding phenomena to explain
Wave-Particle Duality - Black Body radiation
- The Photoelectric Effect
- Atomic Spectra
15Quantum Theory of The Atom
- Quantum Effects Wave-Particle Duality
- Black Body Radiation - As the temperature of an
object changes, the intensity and wavelength of
the emitted light from the object changes in a
manner characteristic of the idealized
Blackbody in which the temperature of the body
is directly related to the wavelengths of the
light that it emits - In 1901, Max Planck developed a mathematical
model that reproduced the spectrum of light
emitted by glowing objects - His model had to make a radical assumption (at
that time) - A given vibrating (oscillating) atom can have
only certain quantities of energy and in turn can
only emit orabsorb only certain quantities of
energy
16Quantum Theory of The Atom
- Quantum Effects Wave-Particle Duality
- Plancks Model
- E (Energy of Radiation)
- v (Frequency)
- n (Quantum Number) 1,2,3
- h (Plancks Constant, a Proportionality
Constant) - 6.626 x 10-34 J ? s)
- 6.626 x 10-34 kg ? m2/s
- Atoms, therefore, emit only certain quantities of
energy and the energy of an atom is described as
being quantized - Thus, an atom changes its energy state by
emitting (or absorbing) one or more quanta
17Quantum Theory of The Atom
- Wave-Particle Duality The Photoelectric
Effect - The Planck model views emitted energy as waves
- Wave theory associates the energy of the light
with the amplitude (intensity) of the wave, not
the frequency (color) - Wave theory predicts that an electron would break
free of the metal when it absorbed enough energy
from light of any color (frequency) - Wave theory would also imply a time lag in the
flow of electric current after absorption of the
radiation - Both of these observations are at odds with the
- Photoelectric Effect
18Quantum Theory of The Atom
- Wave-Particle Duality The Photoelectric
Effect - Photoelectric Effect
- Flow of electric current when monochromatic light
of sufficient frequency shines on a metal plate - Electrons are ejected from the metal surface,
only when the frequency exceeds a certain
threshold characteristic of the metal. - Radiation of lower frequency would not produce
any current flow no matter how intense - Violet light will cause potassium to eject
electrons, but no amount of red light (lower
frequency) has any effect - Current flows immediately upon absorption of
radiation
19Quantum Theory of The Atom
- Wave-Particle Duality The Photoelectric
Effect - Einstein resolved these discrepancies
- He reasoned that if a vibrating atom changed
energy from nhv to (n-1)hv, this energy would be
emitted as a quantum (hv) of light energy he
called a photon - He defined the photon as a Particle of
Electromagnetic energy, with energy E,
proportional to the observed frequency of the
light. - The energy (hv) of an impacting photon is taken
up (absorbed) by the electron and ceases to exist - The Wave-Particle Duality of light is regarded as
complimentary views of wave and particle pictures
of light
20Quantum Theory of The Atom
- In 1921 Albert Einstein received the Nobel Prize
in Physics for discovering the photoelectric
effect
- Electrons in metals exist in different and
specific energy states - Photons whose frequency matches or exceeds the
energy state of the electron will be absorbed - If the photon energy (frequency) is less than the
electron energy level, the photon is not absorbed - The electron moves to a higher energy state and
is ejected from the surface of the metal - The electrons are attracted to the positive anode
of a battery, causing a flow of current
21Practice Problem
- Light with a wavelength of 478 nm lies in the
blue region of the visible spectrum. - Calculate the frequency of this light
- Speed of Light 3 x 108 m/s
- Ans
22Practice Problem
- The green line in the atomic spectrum of Thallium
(Tl) has a wavelength of 535 nm. - Calculate the energy of a photon of this light?
23Practice Problem
- At its closest approach, Mars is 56 million km
from earth. How many minutes would it take to
send a radio message from a space probe of Mars
to Earth when the planets are at this closest
distance?
24Quantum Theory of The Atom
- Atomic Line Spectra
- When light from excited (heated) Hydrogen atoms
or other atoms passes through a prism, it does
not form a continuous spectrum, but rather a
series of colored lines (Line Spectra) separated
by black spaces - The wavelengths of these lines are characteristic
of the elements producing them - The spectra lines of Hydrogen occur in several
series, each series represented by a positive
integer, n
25Quantum Theory of The Atom
n 1
n 2
n 3
26Quantum Theory of The Atom
- Atomic Line Spectra
- In 1885, J. J. Balmer showed that the
wavelengths, l, in the visible spectrum of
Hydrogen could be reproduced by a Rydberg
Equation - where R The Rydberg Constant
- ? wavelength of the spectral
line - n1 n2 are positive integers and
n2 gt n1
27Quantum Theory of The Atom
- Atomic Line Spectra
- For the visible series of lines the value of n1
2 - The known wavelengths of the four visible lines
for hydrogen correspond to values of n2 3, n
4,n 5, and n 6 - The Rydberg equation becomes
- The above equation and the value of R are based
on data rather than theory - The following work of Niels Bohr makes the
connection between the data model and Theory
with n2 3, 4, 5, 6.
28Quantum Theory of The Atom
- Bohr Theory of the Hydrogen Atom
- Prior to the work of Niels Bohr, the stability of
the atom could not be explained using the
then-current theories - How can electrons (e-) lose energy and remain in
orbit? - Bohr in 1913 set down postulates to account for
(1) the stability of the hydrogen atom and (2)
the line spectrum of the atom - Energy level postulateAn electron can have only
specific energy levels in an atom - Transitions between energy levelsAn electron in
an atom can change energy levels by undergoing a
transition from one energy level to another
29Quantum Theory of The Atom
- Bohr Theory
- Transitions of the Electron in theHydrogen Atom
30Quantum Theory of The Atom
- Bohr Theory of the Hydrogen Atom
- Bohrs Postulates
- Bohr derived the following formula for the energy
levels of the electron in the hydrogen atom
31Quantum Theory of The At
- Bohr Theory of the Hydrogen Atom
- Bohrs Postulates
- For the Hydrogen atom, Z 1
- For the energy of the ground state (n 1)
32Quantum Theory of The Atom
- Bohr Theory of the Hydrogen Atom
- Bohrs Postulates
- When an electron undergoes a transition from a
higher energy level (ni) to a lower one (nf), the
energy is emitted as a photon
33Quantum Theory of The Atom
- Bohr Theory of the Hydrogen Atom
- Bohrs Theory vs. Rydberg Data model
- If we make a substitution into the previous
equation that states the energy of the emitted
photon, h?, equals hc/? - Thus, from the classical relationships of charge
and motion combined with the concept of discreet
energy levels theory matches data
versus
Bohr (theory)
Rydberg (data)
34Practice Problem
- From the Bohr model of the hydrogen atom we can
conclude that the energy required to excite an
electron from n 2 to n 3 is ___________ the
energy to excite an electron from n 3 to n 4 - a. less than b. greater than
- c. equal to d. either equal to or less than
- e. either equal to or greater than
Greater Than
35Practice Problem
- An electron in a hydrogen atom in the level n 5
undergoes a transition to level n 3. What is
the wavelength of the emitted radiation? - (R 2.179 x 10-18 J)
Note For computation of frequency and
wavelength the negative sign of the
energy value can be ignored
36Quantum Theory of The Atom
- Bohr Theory of the Hydrogen Atom
- Bohrs Postulates
- Bohrs theory explains not only the emission of
light, but also the absorption of light - When an electron falls from n 3 to n 2 energy
level, a photon of red light (wavelength, 685 nm)
is emitted - When red light of this same wavelength shines on
a hydrogen atom in the n 2 level, the energy is
gained by the electron that undergoes a
transition to n 3
37Quantum Theory of The Atom
- Quantum Mechanics
- Bohrs theory established the concept of atomic
energy levels but did not thoroughly explain the
wave-like behavior of the electron - Current ideas about atomic structure depend on
the principles of quantum mechanics, a theory
that applies to subatomic particles such as
electrons. Electrons show properties of both
waves and particles
38Quantum Theory of The Atom
- Quantum Mechanics
- The first clue in the development of quantum
theory came with the discovery of thede Broglie
relation - In 1923, Louis de Broglie reasoned that if light
exhibits particle aspects, perhaps particles of
matter show characteristics of waves - He postulated that a particle with mass m and a
velocity v has an associated wavelength - The equation ? h/mv is called the
- de Broglie relation
39Quantum Theory of The Atom
- Quantum Mechanics
- If matter has wave properties, why are they not
commonly observed? - The de Broglie relation shows that a baseball
(0.145 kg) moving at about 60 mph (27 m/s) has a
wavelength of about 1.7 x 10-34 m. -
- This value is so incredibly small that such waves
cannot be detected. - Electrons have wavelengths on the order of a few
picometers (1 pm 10-12 m).
40Practice Problem
- At what speed must an neutron (1.67 x 10-27 kg)
travel to have a wavelength of 10.0 pm?
41Quantum Theory of The Atom
- Quantum mechanics is the branch of physics that
mathematically describes the wave properties of
submicroscopic particles - We can no longer think of an electron as having a
precise orbit in an atom - To describe such an orbit would require knowing
its exact position and velocity, i.e., its motion
(mv) - In 1927, Werner Heisenberg showed (from quantum
mechanics) that it is impossible to
simultaneously measure the present position while
also determining the future motion of a particle,
or of any system small enough to require quantum
mechanical treatment
42Quantum Theory of The Atom
- Max Born stated in his Nobel Laureate speech
- To measure space coordinates and instants of
time, rigid measuring rods and clocks are
required. - On the other hand, to measure momenta and
energies, devices are necessary with movable
parts to absorb the impact of the test object and
to indicate the size of its momentum (mass x
velocity). - Paying regard to the fact that quantum mechanics
is competent for dealing with the interaction of
object and apparatus, it is seen that no
arrangement is possible that will fulfill both
requirements simultaneously.
43Quantum Theory of The Atom
- Mathematically, the uncertainty relation between
position and momentum, i.e., the variables,
arises due to the fact that the expressions of
the wavefunction in the two corresponding bases
(variables) are Fourier transforms of one
another. - According to the uncertainty principle of
Heisenberg, if the two operators representing a
pair of variables do not commute, then that pair
of variables are mutually complementary, which
means they cannot be simultaneously measured or
known precisely. - In the mathematical formulation of quantum
mechanics, changing the order of the operators
changes the end result, i.e., the operators are
non-commuting, and are subject to similar
uncertainty limits.
44Quantum Theory of The Atom
- Quantum Mechanics
- Heisenbergs uncertainty principle is a relation
that states that the product of the uncertainty
in position (Dx) and the uncertainty in momentum
(mDvx) of a particle can be no smaller than -
- When m is large (for example, a baseball) the
uncertainties are very small, but for electrons,
high uncertainties disallow defining an exact
orbit
45Practice Problem
- Heisenberg's uncertainty principle can be
expressed mathematically as - Where ?x is the uncertainty in position
- ?p ( m?v) is the uncertainty in Momentum
- h is Planck's constant (6.626 x 10-34 kg ? m2/s)
- What would be the uncertainty in the position
(?x) of a fly (mass 1.245 g) that was traveling
at a velocity of 3.024 m/s if the velocity has an
uncertainty of 2.72?
Plancks Constant h 6.626 x 10-34 J ? s 1 J 1
kg ? m2/s2 h 6.626 x 10-34 kg ? m2/s
46Quantum Theory of The Atom
- Quantum Mechanics
- Acceptance of the dual nature of matter and
energy (E mc2) and the Uncertainty Principle
culminated in the field of Quantum Mechanics - Wave Nature of objects on the Atomic Scale
- Erwin Schrodinger developed quantum mechanical
model of the Hydrogen atom, where - An Atom has certain allowed quantities of energy
- An Electrons behavior is wavelike, but its exact
location is impossible to know - The Electrons Matter-Wave occupies 3-dimentional
space near nucleus - The Matter-Wave experiences continuous, but
varying influence from the nuclear charge
47Quantum Theory of The Atom
- Quantum Mechanics
- Schrodinger Equation
- H? ??
- ? Energy of the atom
- ? Wave Function
- H Hamiltonian Operator Mathematical
operations that when carried out on a
particular wave yields the allowed energy
value - Each solution of the wave equation is associated
with a given atomic orbital, which bears no
resemblance to an orbit in the Bohr model - An Orbital is a mathematical function, which
like a Bohr Orbit, represents a particular energy
level of the orbiting electron, but it has no
direct physical meaning
48Quantum Theory of The Atom
- Quantum Mechanics
- Heisenberg's uncertainty principle says we cannot
precisely define an electrons orbit - The wave function (atomic orbital) has no direct
physical meaning - The square of the wave function, ? 2, however,
is defined as the probability density, a measure
of the probability that the electron can be found
within a particular tiny volume of the atom
49Quantum Theory of The Atom
- Probability of Finding an Electron in a Spherical
Shell About the Nucleus
50Quantum Theory of The Atom
- Quantum Numbers and Atomic Orbitals
- According to quantum mechanics each electron is
described by 4 quantum numbers - Principal Quantum Number (n)
- Angular Momentum Quantum Number (l)
- Magnetic Quantum Number (ml)
- Spin Quantum Number (ms)
- The first three quantum numbers define the wave
function of the electrons atomic orbital - The fourth quantum number refers to the spin
orientation of the 2 electrons that occupy an
atomic orbital
51Quantum Theory of The Atom
- Quantum Numbers and Atomic Orbitals
- The Principal Quantum Number (n) represents the
Shell Number in which an electron resides - It represents the relative size of the orbital
- Equivalent to periodic chart Period Number
- Defines the principal energy of the electron
- The smaller n is, the smaller the orbital
- The smaller n is, the lower the energy of the
electron - n can have any positive value from
- 1, 2, 3, 4 ?
- (Currently, n 7 is the maximum known)
52Quantum Theory of The Atom
- Quantum Numbers and Atomic Orbitals (Cont)
- The Angular Momentum Quantum Number (l)
distinguishes sub shells within a given shell - Each main shell, designated by quantum number
n, is subdivided into - l n - 1 sub shells
- (l) can have any integer value from 0 to n - 1
- The different l values correspond to thes, p,
d, f designations used in the electronic
configuration of the elements - Letter s p d f
- l value 0 1 2 3
53Quantum Theory of The Atom
- Quantum Numbers and Atomic Orbitals (Cont)
- The Magnetic Quantum Number (ml) defines atomic
orbitals within a given sub-shell - Each value of the angular momentum number (l)
determines the number of atomic orbitals - For a given value of l, ml can have any
integer value from -l to l - ml -l to l
- Each orbital has a different shape and
orientation (x, y, z) in space - Each orbital within a given angular momentum
number sub shell (l) has the same energy
54Quantum Theory of The Atom
- Quantum Numbers and Atomic Orbitals (Cont)
- The Spin Quantum Number (ms) refers to the two
possible spin orientations of the electrons
residing within a given atomic orbital - Each atomic orbital can hold only two (2)
electrons - Each electron has a spin orientation value
- The spin values must oppose one another
- The possible values of ms spin values are
- 1/2 and 1/2
55Summary of Quantum Numbers
Name
Symbol
Permitted Values
Property
principal
n
positive integers (1, 2, 3, )
orbital energy (size)
angular momentum
l
integers from 0 to n -1
orbital shape The l values 0, 1, 2, and 3
correspond to s, p, d, and f orbitals,
respectively
magnetic
ml
integers from-l to 0 to l
orbital (x,y,z) orientation
spin
ms
1/2 or -1/2
e- spin orientation
56Quantum Numbers and Atomic Orbitals
The Hierarchy of Quantum Numbers for Atomic
Orbitals
Note n gt 7 l gt 3 not defined for the current
list of elements in the Periodic Table
57Quantum Numbers and Atomic Orbitals
- Using calculated probabilities of electron
position, the shapes of the orbitals can be
described - The s (n 1) sub shell orbital (there is only
one) is spherical - The p (n 2) sub shell orbitals (there are
three) are dumbbell shape - The d (n 3) sub shell orbitals (there are
five) are a mix of cloverleaf and dumbbell shapes
58Quantum Numbers and Atomic Orbitals
- Cross-sectional Representations of the
Probability Distributions of s Orbitals
(spherical)
59Quantum Numbers and Atomic Orbitals
- Cutaway Diagrams Showing the Spherical Shape of
s Orbitals
60Quantum Numbers and Atomic Orbitals
- Radial Probability Distributionof the Three 2p
Orbitals (dumbell shapes)
n 2 l 2 1 1 (p) ml -1
0 1
61Quantum Numbers and Atomic Orbitals
- Radial Probability Distribution
- of the Five 3d Orbitals (Cloverleaf Dumbells)
n 3 l 3 1 2 (d) ml -2
-1 0 1 2
62Quantum Numbers and Atomic Orbitals
- Radial Probability Distribution
- of the Seven 4f Orbitals
n 4 l 4 1 3 (f) ml -3 -2
-1 0 1 2 3
63Quantum Numbers and Atomic Orbitals
- Orbital Energies of the Hydrogen Atom
64Practice Problems
- If the n quantum number of an atomic orbital is
4, what are the possible values of the l quantum
number? - Ans (l) can have any integer value from 0 to n
1 - l n - 1 4 1 3
- ? Values of l 0 1 2 3
65Practice Problem
- If the l quantum number is 3, what are the
possible values of ml? - Ans ml can have any integer value from -l to
l -
- Since l 3
- ml -3 -2 -1 0 1 2 3
66Practice Problem
- State which of the following sets of quantum
numbers would be possible and which impossible
for an electron in an atom? - a. n 0, l0, ml 0, ms 1/2
- b. n 1, l0, ml 0, ms 1/2
- c. n 1, l0, ml 0, ms -1/2
- d. n 2, l1, ml -2, ms 1/2
- e. n 2, l1, ml -1, ms 1/2
- Ans Possible
- Impossible
- Impossible
-
b c e
a n must be positive 1, 2, 3...
d ml can only be -1 0 1
67Summary Equations
- Light c ?? c 3 x
108 m/s - Plancks Model
- Photoelectric
- Balmer Rydberg
- Bohr Model
- Bohr Postulate
- De Broglie
- Heisenberg