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Basic Quantum Theory
• Why He and Ne are both non-reactive atoms?
• Why H2 (g) + Cl2 (g) → 2HCl (g)?
To answer these questions we have to understand the behavior of the e − in atoms
When atoms react, it is the e − that interact
Electronic structure of atoms: management of e − in atoms
↓
−
Distribution of e around nucleus and energies
Quantum Mechanics is the Physics that has been developed in order to describe atoms
correctly
→
→
Light: consists of waves of oscillating electric field ( E ) and magnetic field ( B ) that are
→
perpendicular to each other and to the direction of propagation ( k )
Wavelength (λ): distance between two
successive peaks or troughs provided that
the distance is reproducible from peak to
peak.
Amplitude: height of the wave
Frequency (ν): is the number of cycles
that pass the observer in a given time.
Hertz (Hz) is the unit of frequency, and
just means how many cycles per second.
υ=λν
Frequency and wavelength
c= λν
speed of light in vacuum: 2.99792458 x 108 m s-1
monochromatic light: light of single frequency or at a
single wavelength
Light travels more slowly
in a transparent material
than it does through a
vacuum due to interaction
of the light with the
electrons in the material.
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The electromagnetic spectrum
Example
A dental hygienist uses x-rays (λ = 1.00
Ǻ) to take a series of dental radiographs
while the patient listens to a radio station
(λ = 325 cm) and looks out the window at
the blue sky (λ = 473 nm). What is the
frequency (in s-1) of the electromagnetic
radiation from each source? (Assume that
the radiation travels at the speed of light,
3.00 x 108 m/s.)
c= λν⇒ν=c/λ
ν = (3.00x108 m/s/1.00 x 10-10m = 3x1018 s-1
ν = (3.00x108 m/s/325 x 10-2 m = 9.23x107 s-1
ν = (3.00x108 m/s/473 x 10-9 m = 6.34x1014 s-1
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The Paradoxes of Classical Physics
1st Paradox – The Blackbody Radiation
A black body is a theoretical object that absorbs 100% of the radiation that hits it.
Therefore it reflects no radiation and appears perfectly black.
In practice no material has been found to absorb all incoming radiation, but carbon in its
graphite form absorbs all but about 3%. It is also a perfect emitter of radiation. At a
particular temperature the black body would emit the maximum amount of energy
possible for that temperature. This value is known as the black body radiation. It would
emit at every wavelength of light as it must be able to absorb every wavelength to be
sure of absorbing all incoming radiation.
A bar of iron when it is heated first it becomes red, then orange, and as the temperature
increases it becomes firstly yellow and secondly white. In other words, a bar of iron
when it is heated, it emits radiation and the distribution of frequencies change with
temperature
 8πR  T
I =
 4
N

λ
(Classical Physics)
Classical Physics
predict that a
heated body
emits radiation
approaching
infinite intensity
at shorter
wavelengths. But
this doesn’t
happen. (1st
paradox)
ULTRAVIOLET
CATASTROPHE
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Max Planck (1900) solved the paradox of the blackbody radiation. Classical Physics
assumed that atoms and molecules could emit (or absorb) any arbitrary amount of
radiant energy. He proposed that this energy could be emitted or absorbed only in
discrete quantities. He gave the name of quantum to the smallest quantity of energy that
can be emitted or absorbed in the form of electromagnetic radiation.
E = hν (h = 6.62608 x 10-34 J s – Planck’s constant)
Energy is emitted in multiples of hν , ( hν, 2 hν , 3 hν …) but never as 1.67 hν …
In light of the new theory, at any given temperature, there is only a fixed amount of
thermal energy that is available to excite a given electromagnetic oscillation. In
Classical Physics one can put an arbitrary amount of energy that it can be distributed
evenly among the oscillations regardless of frequency.
In Planck’s model there is a minimum amount of energy that can be transferred into an
EM oscillation from the object and this minimum energy (quantum) increases with
increasing frequency.
For low-frequency EM waves, the quantum energy is much smaller than the average
amount of energy available for the excitation of the EM wave and this energy can be
evenly distributed among these oscillation modes as in Classical Physics. However, for
higher frequencies the quantum of energy is greater than the average available thermal
energy and excitation into high frequency modes is inhibited.
Black body radiation curves
showing peak wavelengths at
various temperatures
As the temperature increases, the
peak wavelength emitted by the
black body decreases
As temperature increases, the total
energy emitted increases, because
the total area under the curve
increases
Stefan’s Law: M = σ T4
M = emittance
σ = 5.67 x 10-8 W m-2 K-4 (Stefan –
Boltzmann’s constant)
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Therefore the Power radiated is proportional to T4 for an identical body which explains
why the area under the black body curves (the total energy) increases so much for a
relatively small increase in temperature. Sun at 6000K
2nd Paradox – The Photoelectric Effect (1905)
• If the incident wave frequency (ν) is
smaller than a certain value (νo), there is
no current flows.
• If ν > νo, the current flows
instantaneously.
Thus, there is minimum required photon energy
(hνo,) to overcome the work function of the
material, Φ
Φ = hν o
If the incident light energy is less than the work
function, the electron will not be freed from the
surface, and no photoelectric effect will be
observed.
Tmax =
1
meυ 2 = h(ν − ν o ) = hν − Φ
2
Observations:
1) The energy of photons is
determined by the light frequency,
not intensity
2) Material-specific “red boundary”
νo exists: no photocurrent at ν < νo
At ν < νo (hν < Φ) the photon
energy is insufficient to extract an electron from metal
Einstein suggested that light consists of a stream of packets of energy called photons
Therefore, light behaves like a particle (a stream of photons) and as a wave!
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3rd Paradox – Stability of Atoms
In Classical Physics atoms are constructed according to Rutherford’s model and they are
not stable. The motion of electrons in their orbits would cause them to radiate energy
and so quickly to spiral into the nucleus.
Different behaviors of waves and particles
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Atomic Spectra
Johann Balmer (1885) observed for excited H atoms:
1
4
ν = −
1 
(3.29 x1015 s −1 ) n = 3, 4, 5 … These lines are called Balmer series.
2 
n 
Other spectroscopists discovered additional series in the H-atom as the spectrographs
were improved.
 1
1 
ν =  2 − 2 (3.29 x1015 s −1 ) n1 > n2, n1 and n2, are integers
 n2 n1 
n2 = 1 Lyman (UV)
n2 = 2 Balmer (Vis)
n2 = 3 Paschen (IR)
n2 = 4 Brackett (IR)
n2 = 5 Pfund (IR)
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Atoms emit at discrete frequencies in regions of the EM spectrum quite remote from the
visible if excited properly.
Henry Moseley (1913): metallic elements emit X-rays of characteristic frequency when
excited with high-energy electrons
ν = ( Z − 1)(4.98 x10 7 s −1/ 2 ) , Z is an integer, different value for each element, same as
Rutherford’s atomic number
Squaring both sides of Moseley’s equation we get,
ν = ( Z − 1) 2 (2.48 x1015 s −1 )
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But 2.48 x10 =
3
3 1 1
(3.29 x1015 ) and = 2 − 2
4
4 1 2
The Bohr Model was an early attempt to formulate a quantum theory of matter
Niels Bohr made the following postulates for a single e − orbiting a nucleus of Z protons:
1. The e − moves in a circular orbit around the nucleus.
2. The energy of the e − can take on only certain well-defined values, that is quantized.
3. The only allowed orbits are those in which the magnitude of the angular momentum
of the e − is equal to an integer multiple of ħ.
→
→
→
The angular momentum is: L = r x p (cross product)
→
→
Linear momentum: p = m υ
4. The e − can only absorb or emit EM radiation when it moves
From one allowed orbit to another. The emitted radiation has energy of hν equal to the
difference in energy between the two orbits.
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L = meυr = n
h
2π
n = 1, 2, 3….
n2
ao , Z is the atomic number and ao, is
The allowed radius for each value of n: rn =
Z
Bohr’s radius, the predicted distance of the electron from the nucleus in the state n = 1
of the H-atom.
ao = 5.29177 x 10-11 m or 0.529 Å
Z2
Total energy of the e − in the allowed orbits: En = − 2
n

 h2

 2
2  n = 1, 2, 3….
8
π
m
a
e o 

1 Rydberg (Ry) = 2.179872 x 10-18 J (non-SI unit of energy)
Z2
Z2
En = − 2 R y = − 2 (2.18 x10 −18 J ) n = 1, 2, 3….
n
n
Ground-state: n = 1, state of lowest energy for the system of nucleus plus electron.
The Bohr explanation of the three series of spectral lines
Bohr’s Model introduces two important ideas:
• Electrons exist only at certain discrete energy levels ⇒ described by quantum
numbers
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• Energy is involved by moving an electron from one level to another
Limitations to Bohr’s Model
• Good explanation of H-atom; cannot explain spectra of other atoms
• It describes electron merely as a small particle circling around the nucleus.
However, the electron exhibits properties of waves (we will discuss this in
details)
Ionization Energy: is the minimum energy needed to remove an e − from the atom
when it is in its ground state and send it into infinity.
 12 
 12 
En = − 2  R y −  2  R y = 0 − (−1) R y = R y = 2.18 x10 −18 J
∞ 
1 
If we multiply this value by NA we get the ionization energy per mole of H atoms:
IE = (2.18x10-18J)(6.02x1023 mol-1) = 1.31 x 106 J mol-1 = 1310 kJ mol-1
Atomic Spectra
 Z2 
 1
Z2
1 
En = − 2 R y −  − 2  R y == − Z 2  2 − 2  R y = hν
n2
 n1 
 n2 n1 
 1
1 
(3.29 X 1015 s −1 ) n2 > n1 (absorption)
−
2
2 
 n1 n2 
 1
1 
ν == Z 2  2 − 2 (3.29 X 1015 s −1 ) n1 > n2 (emission)
 n2 n`1 
ν == Z 2 
The wave nature of e − and the particle nature of photons
Albert Einstein: E = mc (energy equivalent to a given amount of mass)
Max Planck: E = hν
2
mc 2 = hν = h
c
λ
⇒λ=
Louis de Broglie: λ =
h
mc
h
(wavelength of any particle of mass m moving at velocity
mυ
υ) – wave – particle duality of matter and energy. ( we will come back!)
10
λ=
h
h
h
= ⇒ p=
λ Shorter λ-photons have greater momentum
mυ p
Alfred Compton Effect (1923): directed a beam of X-ray photons at a sample of
graphite and the λ of the reflected photons increased ⇒ photons transferred some of
their momentum to the e − in the graphite ⇒ photons behave as particles with
momentum.
Uncertainty Principle (1924): it is impossible to know exactly the position and the
momentum of a particle simultaneously.
(∆x )(∆p x ) ≥
h
h
and since p = mυ (∆x )m(∆υ x ) ≥
4π
4π
(∆x ) = uncertainty in position
(∆p ) = uncertainty in momentum
(∆υ ) = uncertainty in velocity
From this principle we cannot assign fixed paths for e − (such as the circular paths in
Bohr’s model). Therefore, we need a better theory. This theory is the Quantum
Mechanical Model of Atom.
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Wave motion in restricted systems
The condition on the allowed
wavelengths:
nλ
=L
2
n = 1, 2, 3….
Fundamental or 1st harmonic: n = 1
2nd harmonic: n = 2
Regions of no vibration: nodes
(The ends do not count as
nodes)
The higher the number of
harmonics ⇒ the greater the
number of nodes
The shorter the wavelength, the
larger the frequency and the
higher the energy,
E = hν = h
c
λ
For a circular standing wave on a closed loop (Fig. 16-16 of textbook)
2πr = nλ
n = 1, 2, 3….
h
h
⇒ 2πr = n
2π
meυ
h
h
h
= nλ ⇒ λ =
=
--- de Broglie
Consequently, n
meυ
meυ p
Bohr: meυr = n
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The diffraction pattern caused by light passing through two adjacent slits
Davisson and Germer (1927): a crystal diffracts e − and de Broglie’s relationship
correctly predicts their wavelengths.
Comparing diffraction patterns of x-rays and electrons
X-ray diffraction
of aluminum
Electron diffraction
of aluminum
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Summary of the major observations and theories leading from classical theory to
quantum theory
The Schrödinger Equation (1925)
A wavefunction (Ψ) can describe in its entirety any physical system. Ψ incorporates
both wavelike and particlelike behavior of the e − .
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Max Born (1927) Ψ2 gives the probability density (at any point in space it represents the
probability that the e − will be found in space at that location.
To calculate the wavefunction for any particle we use Schrödinger Equation:
h 2 d 2ψ
d 2ψ
−
+ V ( x )ψ = Eψ The term
can be thought of as a measure of how
dx 2
2m dx 2
sharply the wavefunction will be. The left hand side term of the equation is written as
h2 d 2
−
+ V ( x ) = H and the SE becomes Hψ = Eψ . The H is called Hamiltonian
2m dx 2
and it is a set of mathematical operations that represent the total energy (kinetic and
potential) of the electron within the atom. The symbol E represents the total energy of
the electron.
2
A plot of ψ represents an orbital, a position probability distribution of the electron
Spherical Polar Coordinates
x = r cos φ sin θ
y = r sin φ sin θ
z = r cos θ
The wavefunction can be written as a function that depends only on r and another
function that depends on θ and φ, that is
ψ ( r,θ ,φ ) = R( r )Y (θ ,φ )
The function R(r ) is called the radial wavefunction and it tells us how the
wavefunction varies as we move away from the nucleus in any direction.
The function Y (θ ,φ ) is called the angular wavefunction and it tells us how the
wavefunction varies as the angles θ and φ change.
When the SE is solved it yields many solutions - many possible wavefunctions. The
wavefunctions are fairly complicated mathematical functions that we will not examine
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them here. Instead we will discuss graphical representations (or plots) of the orbitals
that correspond to these wavefunctions. An orbital is a one electron wavefunction.
Each orbital is specified by three interrelated quantum numbers, the principal quantum
number (n), the angular quantum number (l) – sometimes it is called azimuthal
quantum number and the magnetic quantum number (ml).
Electron probability density in the ground-state H atom
The principal quantum number (n) is an integer that determines the overall size and
energy of an orbital. Its possible values are n = 1, 2, 3,…
For a H-atom, the energy of an electron with quantum number n is given by:
12
En = − 2 ( 2.18 x10−18 J ) n = 1, 2, 3, …
n
The angular quantum number (l) is an integer that determines the shape of an orbital,
with possible values 0, 1, 2, … (n-1)
Value of l 0 1 2 3
orbital
s p d f
The magnetic quantum number (ml) is an integer that specifies the orientation of an
orbital in space and it has values ranging from –l ….. +l
Collection of orbitals with the same value of n is called a shell, (n = 3 third shell)
Collection of orbitals with the same values of l and ml is called a subshell
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Example
What values of the angular momentum (l) and magnetic (ml) quantum numbers are
allowed for a principal quantum number (n) of 3? How many orbitals exist for n = 3?
For n = 3, l = 0, 1, 2
For l = 0, ml = 0
For l = 1, ml = -1, 0, +1
For l = 2, ml = -2, -1, 0, +1, +2
There are 9 ml values and therefore, 9 orbitals with n = 3.
The number of ml values are 2l + 1 and the total number of orbitals is n2
Example
Give the name, magnetic quantum numbers, and number of orbitals for each sublevel
with the following quantum numbers
(a) n = 3, l = 2
(b) n = 2, l = 0
(c) n = 5, l = 1
d) n = 4, l = 3
Combine the n value and l designation to name the sublevel. Knowing l, we can find ml
and the number of orbitals
Question n, l Orbital(sublevel)
Number of orbitals
ml
a
3,2
3d
-2,-1,0,+1,+2
5
b
2,0
2s
0
1
c
5,1
5p
-1,0,+1
3
d
4,3
4f
-3,-2,-1,0,+1,+2,+3
7
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The 1s, 2s, and 3s orbitals
The 2p orbitals
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The 3d orbitals
One of the seven possible 4f orbitals
Radial Nodes: n – l - 1
Angular Nodes: l
Total Nodes: n – 1
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Observing the effect of electron spin: the Stern-Gerlach Experiment
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