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Quantum Theory and the
Electronic Structure of Atoms
Chapter 7
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What we're going to learn
Basic calculations involving waves
Planck’s Quantum Theory:
Radiation energy is dependent on wavelength
Energy is emitted and absorbed in discrete units called quanta
Einstein's explanation of the photoelectric effect
Bohr’s description of the hydrogen atom
Particle/wave duality of electromagnetic radiation and electrons
Heisenberg’s uncertainty principle: a tool or the limits of knowledge?
Schrodinger’s wave equations explaining quantized energy levels for an
atom’s electrons known as orbitals
Use this information to begin building elements
Properties of Waves
Wavelength (l) is the distance between identical points on
successive waves.
Amplitude (A) is the vertical distance from the midline of
a wave to the peak or trough.
http://www.ies.co.jp/math/products/trig/applets/graphSinX/graphSinX.html
7.1
Properties of Waves
Frequency (n) is the number of waves that pass through a particular
point in 1 second (Hz = 1 cycle/second
(units: sec-1)
The speed (u) of the wave = l x n
Speed = wavelength X frequency
u = ln
Examples of waves: sound, water ripples and surf,
earthquakes, weather, jump ropes, springs and slinkies,
gravitational waves, inertial waves, you? ….
7.1
Speed = wavelength X frequency
u = ln
Example 1; sound:
frequency (20 to 20,000 Hz);
speed ~ 340 m/s or 770 mph;
wavelength ( 17 m to 1.7 cm)
(In dry air, at a temperature of 21 °C).
Example 2; visible light:
frequency (Hz); ~ 1014 Hz
speed ~ 300,000 m/s; ~ 3 X 108 m/s)
wavelength (400 to 700 nm; nm = 10-9 m)
(in a vacuum).
#1
Visible light
Maxwell (1873), proposed that visible light consists of
electromagnetic waves.
Electromagnetic
radiation is the emission
and transmission of energy
in the form of
electromagnetic waves.
Speed of light (c) in vacuum = 3.00 x 108 m/s
All electromagnetic radiation
lxn=c
7.1
Electromagnetic Radiation
7.1
An EM wave has a frequency of 6.0 x 104 Hz. Convert
this frequency into wavelength (nm). Does this frequency
fall in the visible region?
l
lxn=c
n
l = c/n
l = 3.00 x 108 m/s / 6.0 x 104 Hz
l = 5.0 x 103 m
l = 5.0 x 1012 nm
Radio wave
#2-3
7.1
Classical Physics:
Planetary Model of
the Atom
“Electrical force is like
gravity.”
•The planets orbit the much
more massive Sun in (mostly)
circular orbits.
•The electrons should orbit the
much more massive nucleus in
(mostly) circular orbits.
•The total energy of a planet
only depends on its orbital
radius (higher energy, bigger
radius); the same should be true
of the electrons in an atom.
7.3
“Classical” physics applied to atoms
has some serious problems
1.
Thermal radiation.
–
–
–
–
–
2.
Accelerating electric charges give off light and lose energy.
Kinetic molecular theory says that atoms in a gas are
constantly being accelerated (changing direction, changing
speed in collisions).
The energy given off is called thermal radiation because
it depends on temperature.
Classical physics says that thermal radiation should get
more powerful at higher frequency, and the total energy
emitted should be infinite!
Called “black body problem” or “UV catastrophe”.
Photoelectric Effect.
–
–
–
Shining light on certain metals causes them to emit electrons
The speed of the emitted electrons depends on color of the
light, not its brightness.
Brightness only affects number of electrons coming off.
3.
Atoms should be unstable.
–
–
–
4.
Moving in circles is a constant acceleration (change of
direction).
Electrons in atoms should constantly give off light and lose
energy.
Electrons should quickly (within milliseconds) spiral to their
doom in the nucleus.
Line spectra.
–
–
Electrically energized gas atoms give off light that is
composed only of a set of particular colors/frequencies.
Classical physics says they should give off all frequencies
(continuous spectra).
1.
Thermal radiation.
–
2.
Photoelectric Effect.
–
–
–
3.
Shining light on certain metals causes them to emit electrons
The speed of the emitted electrons depends on color of the
light, not its brightness.
Brightness only affects number of electrons coming off.
Atoms should be unstable.
–
4.
Classical physics says that thermal radiation should get
more powerful at higher frequency, and the total energy
emitted should be infinite!
Electrons in atoms should constantly give off light and lose
energy.
Line spectra.
–
–
Electrically energized gas atoms give off light that is
composed only of a set of particular colors/frequencies.
Classical physics says they should give off all frequencies
(continuous spectra).
Solution?
The “Quantum” Theory of Physics
If… matter comes in discrete indivisible portions
called “atoms”…,
Then (perhaps)… energy is emitted or absorbed
in discrete indivisible portions called “quanta”.
• Light energy quanta are called “photons”
• Light of one color/frequency is composed of
photons of one fixed energy (E).
E=hxn
Planck’s constant (h)
h = 6.63 x 10-34 J•s
#4
Equation review
Speed = (wavelength)X(frequency)
m/s =
General form
(m)
X (sec-1)
u = ln
c = ln
E = hn = hc/l
For electromagnetic
waves
Energy as a
function of
wavelength or
frequency
C = 3 X 108 m/sec;
h = 6.63 x 10-34 (J•s)
Units?
When copper is bombarded with high-energy electrons,
X-rays are emitted with a wavelength of 0.154 nm.
Calculate the energy (in joules) associated with the
photons.
E=hxn
E=hxc/l
E = 6.63 x 10-34 (J•s) x 3.00 x 10 8 (m/s) / 0.154 x 10-9 (m)
E = 1.29 x 10 -15 J
Light has both:
1. wave nature
2. particle nature
#4
7.2
Does the “quantum” theory solve the four problems of
“classical” physics?
Mystery #1, “Ultraviolet Catastrophe”
Solved by Planck in 1900
Classical physics says that atoms can emit or
absorb an arbitrary amount of radiant energy.
Planck said that atoms emit and absorb on in
discrete units within a limited ranges characteristic
of a particular atom or molecule, making the total
emitted radiation finite.
7.1
Mystery #2, “Photoelectric Effect”
Solved by Einstein in 1905
Number of ejected electrons proportional to the
intensity, BUT the energy of the emitted electrons
is a function of the frequency and only occurred
above a certain ‘threshold’ frequency
hn
KE e-
Photon is a “particle” of light
hn = KE + BE
KE = hn - BE
Where BE is the binding energy of
the atom, i.e. a measure of how strongly
the electron is held in the atom.
7.2
What about Mysteries #3 & #4?
Atomic stability?
Line spectra?
7.1
What Newton demonstrated:
The Sun’s emission spectra
Hydrogen emission spectra
Line Emission Spectrum of Hydrogen Atoms
7.3
7.3
Bohr’s Model of
the Atom (1913)
1. e- can only have specific
(quantized) energy
values
2. Light, as a single photon,
is emitted as e- moves
from one energy level to
a lower energy level
En = -RH (
1
n2
)
n (principal quantum number) = 1,2,3,… integer values only
RH (Rydberg constant) = 2.18 x 10-18 J (for Hydrogen)
7.3
E = hn
E = hn
7.3
ni = 3
ni = 3
ni = 2
nf = 2
Ephoton = -DE = Ei - Ef
1
Ef = -RH ( 2
nf
1
Ei = -RH ( 2
ni
1
DE = RH( 2
ni
)
)
1
n2f
)
nnf f==11
#5
7.3
Calculate the wavelength (in nm) of a photon
emitted by a hydrogen atom when its electron
drops from the n = 6 state to the n = 4 state.
1
1
Ephoton = -DE = RH ( 2
)
2
nf
ni
Ephoton = 2.18 x 10-18 J x (1/16 - 1/36)
Ephoton = -DE = 7.57 x 10-20 J
Ephoton = h x c / l
l = h x c / Ephoton
l = 6.63 x 10-34 (J•s) x 3.00 x 108 (m/s)/7.57 x 10-20J
l = 2630 nm
What color is emitted or absorbed for
7.3
n=2 and n=5?
Why is e- energy quantized?
De Broglie (1924) reasoned that
if light waves can have
properties of particles then
perhaps electrons can have
properties of waves.
e- is both particle and wave, nodes
specifically a ‘standing wave’
Relationship between
the circumference of an
allowed obit and an
electron’s wavelength
2pr = nl
Relationship between
wavelength and mass of
a particle
l = h/mu
r = radius of the orbit
u = velocity of em = mass of e-
7.4
What is the de Broglie wavelength (in nm)
associated with a 2.5 g Ping-Pong ball
traveling at 15.6 m/s?
l = h/mu
h in J•s m in kg u in (m/s)
l = 6.63 x 10-34 / (2.5 x 10-3 x 15.6)
l = 1.7 x 10-32 m = 1.7 x 10-23 nm
7.4
Chemistry in Action: Laser – The Splendid Light
Laser light is (1) intense, (2) monoenergetic, and (3) coherent
Schrodinger Wave Equation
In 1926 Schrodinger wrote an equation that
described both the particle and wave nature of the eWave function (Y) describes:
1. energy of e- with a given Y
2. probability of finding e- in a volume of space
Schrodinger’s equation can only be solved exactly
for the hydrogen atom. Must approximate its
solution for multi-electron systems.
7.5
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
principal quantum number n
n = 1, 2, 3, 4, ….
distance of e- from the nucleus
n=2
n=3
n=1
7.6
Where 90% of the
e- density is found
for the 1s orbital
e- density (1s orbital) falls off rapidly
as distance from nucleus increases
7.6
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
angular momentum quantum number l
for a given value of n, l = 0, 1, 2, 3, … n-1
n = 1, l = 0
n = 2, l = 0 or 1
n = 3, l = 0, 1, or 2
l=0
l=1
l=2
l=3
s orbital
p orbital
d orbital
f orbital
l describes the shape of the “volume” of space that
the e- occupies
7.6
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
magnetic quantum number ml
for a given value of l
ml = -l, …., 0, …. +l
if l = 1 (p orbital), ml = -1, 0, or 1
if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2
orientation of the orbital in space
7.6
l = 0 (s orbitals)
l = 1 (p orbitals)
7.6
l = 2 (d orbitals)
7.6
7.6
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
spin quantum number ms
ms = +½ or -½
ms = +½
ms = -½
7.6
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
Existence (and energy) of electron in atom is described
by its unique wave function Y.
Pauli exclusion principle - no two electrons in an atom
can have the same four quantum numbers.
Each seat is uniquely identified (E, R12, S8)
Each seat can hold only one individual at a
time
7.6
Schrodinger Wave Equation
Y = f(n, l, ml, ms)
Shell – electrons with the same value of n
Subshell – electrons with the same values of n and l
Orbital – electrons with the same values of n, l, and ml
How many electrons can an orbital hold?
If n, l, and ml are fixed, then ms = ½ or - ½
Y = (n, l, ml, ½) or Y = (n, l, ml, -½)
An orbital can hold 2 electrons
7.6
Summary
• Quantum mechanics is the theory of physics that best
describes the behavior of atomic and sub-atomic particles,
like electrons.
• Wavefunctions are solutions to the Schrodinger Wave
Equation, the basic equation of quantum mechanics. They
mathematically describe the behavior of subatomic particles.
• An electron’s wavefunction Y(x,y,z) gives the probability of
finding the electron in a small volume of space DV centered
at point (x,y,z).
Probability = |Y(x,y,z)|2 DV
probability density
The more intense Y is in a region of space, the more
likely it is that the electron will be found there if you look
for it.
Summary
• For electrons bound to nuclei, there is a set of possible
wavefunctions an electron may have. Each Y is labeled
by a set of four “quantum” numbers (n,l,ml,ms) representing
the different allowed behaviors of an atom-bound electron
(called states).
• Electrons obey the Pauli Exclusion Principle. Every
electron in an atom must be in a different state, i.e. each
electron has a unique wavefunction (from the allowed set),
labeled by a unique set of quantum numbers.
Summary
• A wavefunction can have non-zero values at every point in
space. There is some chance of finding an electron
anywhere. In QM, one cannot pin down the location of an
electron to any region of space with certainty. This reflects
the Heisenberg Uncertainty Principle of quantum
mechanics.
• However, an atom-bound electron is by far most likely to be
found somewhere near (within a few nanometers of) the
nucleus of an atom. Y drops in intensity rapidly with distance
from the nucleus.
Summary
• To visualize where an electron in a given state is most likely
to be found, we define a boundary surface diagram. The
BSD encloses the region of space where an electron in that
state is 90% likely to be found. The BSD is always centered
on the nucleus. There is a different BSD for each possible
state an electron can have.
Summary
• The principal quantum number n indicates the size of the
region the electron is most likely in, i.e. the size of the BSD.
It takes positive integer values: n = 1, 2, 3, etc.
n=1
n=2
n=3
• The angular momentum quantum number l indicates the
shape of the region the electron is most likely in. It takes
non-negative integer values less than n: l = 0, 1, …, n-1
l=0
or
s
l=1
or
p
l=2
or
d
Summary
• The magnetic quantum number ml relates to the
orientation of the region the electron is most likely in. It
takes integer values between –l and +l:
ml = -l, -l+1,…, 0,…, +l-1, +l.
l = 1 or p
px
py
pz
ml = -1, 0, 1
• The spin quantum number ms does not relate to where an
electron is likely to be found in space. It refers to
the orientation of the electron’s magnetic field. It takes
values
ms = +1/2 and -1/2,
sometimes called “up” and “down”.
• All electrons with the same n are said to be in the same
shell.
• All electrons with the same n and l are said to be in the
same subshell. Every shell has n subshells (possible
values of l).
How many subshell are there in the n = 4 shell of
an atom?
If n = 4, then l = 0, 1, 2, or 3
The 4s, 4p, 4d, and 4f subshells
• All electrons with the same n, l, and ml are said to be in the
same orbital. Every subshell has 2l+1 orbitals (possible
values of ml).
How many 2p orbitals are there in an atom?
If l = 1, then ml = -1, 0, or +1
n=2
3 orbitals
(2x1+1 = 3)
2p
l=1
7.6
• Each orbital can only hold at most two electrons, one in the
“up” state (ms = +1/2) and one in the “down” state (ms = -1/2).
How many electrons can be placed in the 3d
subshell?
n=3
3d
l=2
If l = 2, then ml = -2, -1, 0, +1, or +2
5 orbitals which can hold a total of 10 e-
Energy of orbitals in a single electron atom
Energy only depends on principal quantum number n
n=3
n=2
En = -RH (
1
n2
)
n=1
7.7
Energy of orbitals in a multi-electron atom
Energy depends on n and l
n=3 l = 2
n=3 l = 0
n=2 l = 0
n=3 l = 1
n=2 l = 1
n=1 l = 0
7.7
Paramagnetic
unpaired electrons
2p
Diamagnetic
all electrons paired
2p
7.8
“Fill up” electrons in lowest energy orbitals (Aufbau principle)
??
Be
Li
B5
C
3
64electrons
electrons
22s
222s
22p
12 1
BBe
Li1s1s
1s
2s
H
He12electron
electrons
He
H 1s
1s12
7.7
The most stable arrangement of electrons
in subshells is the one with the greatest
number of parallel spins (Hund’s rule).
Ne97
C
N
O
F
6
810
electrons
electrons
electrons
22s
222p
22p
5
246
3
Ne
C
N
O
F 1s
1s222s
7.7
Order of orbitals (filling) in multi-electron atom
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s
7.7
What is the electron configuration of Ti?
Ti 22 electrons
1s < 2s < 2p < 3s < 3p < 4s < 3d
[Ti] = 1s22s22p63s23p64s23d2
2 + 2 + 6 + 2 + 6 + 2 + 2 = 22 electrons
Abbreviated as [Ar]4s23d2
[Ar] = 1s22s22p63s23p6
What are the possible quantum numbers for the
last (outermost) electron in Cl?
Cl 17 electrons
1s < 2s < 2p < 3s < 3p < 4s
[Cl] = 1s22s22p63s23p5
2 + 2 + 6 + 2 + 5 = 17 electrons
Last electron added to 3p orbital
n=3
l=1
ml = -1, 0, or +1
ms = ½ or -½
7.7
Outermost subshell being filled with electrons
7.8
Tab. 7.3
Lanthanides (rare
earths) :
Incompletely filled
4f-subshells
1st Transition
metal series:
Incompletely
filled d-subshells
Slightly more
stability in halffilled and
completely filled d
orbitals
Eqn. 7.1
KEY EQUATIONS
c = ln
Eqn. 7.2
Calculating energy in joules from frequency (sec-1)
h (Planck’s constant = 6.63 X 10-34 Joule secs)
Eqn. 7.3
Calculating energy in joules from wavelength (m)
and the speed of light (c = 3 X 108 m/sec)
E = hn = hc/l
Eqn. 7.4
Calculating energy in joules of a particular
shell of the hydrogen atom using the
principal quantum number (n) and Rydberg’s
constant (RH = 2.18 x 10-18 J)
Eqn. 7.5
Calculating energy in joules and the
frequency of a transition of an
electron from one shell to another in
a hydrogen atom using the principal
quantum number (n) and Rydberg’s
constant (RH = 2.18 x 10-18 J)
Eqn. 7.6
De Broglie’s equation for calculating
the wavelength of a moving mass
Heisenberg’s Uncertainty Principle
"The more precisely the POSITION is determined,
the less precisely the MOMENTUM is known"
Dx Dp  h / 4 p
Also a battle of personalities, jobs
and money…
I knew of [Heisenberg's] theory, of course, but I felt
discouraged, not to say repelled, by the methods of
transcendental algebra, which appeared difficult to
me, and by the lack of visualizability.
-Schrödinger in 1926
The more I think about the physical portion of
Schrödinger's theory, the more repulsive I find it...What
Schrödinger writes about the visualizability of his theory
'is probably not quite right,' in other words it's crap.
--Heisenberg, writing to Pauli, 1926
I believe that the existence of the classical "path" can be pregnantly
formulated as follows: The "path" comes into existence only when we
observe it.
--Heisenberg, in uncertainty principle paper, 1927
Page 214
The colors in order of discovery were blue (Mercury),
white (CO2), gold (Helium), red (Neon), and then
different colors from phosphor-coated tubes.
Page 207