Download CH101 General Chemistry

Wave Theory of Light
Line Spectra of Atoms Quantum Hypothesis
Eu  E = E = h
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= mev 2 = h(  0 ) = h  
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Photoelectric Effect:
Particles of Light
Bohr’s Explanation
of Line Spectra
 particle
h
=
mv
de Broglie’s Matter
Waves
H=E
Bohr-de Broglie Model
of Hydrogen Atom
Schrodinger Equation:
A Wave Equation of Particles
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Chapter 2.
Quantum Revolution: Failure of Everyday
Notions to Apply to Atoms
2.1 Wave Theory of Light
wave –traveling or standing
wave
wave length λ – distance
between successive crests
or troughs
wave number :  =
1
cm 1 =

# of waves that fit into a 1-cm length
Wave amplitude A - distance
from the horizontal axis to
the crest
c
frequency:  = (cycles or hertz , Hz )
* c – speed of light (2.99792 x
1010 cm s-1)

= # of crests passing a point within 1 s
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Light wave
consist of traveling electromagnetic waves with the same
speed c, but different λ
oscillating electric field E
and magnetic field B
traveling wave
Spectroscopy - study of the interaction of light and matter
Diffraction and Interference of Light Waves
Interference pattern:- shining λ
through a pair of space slits,
diffraction interference pattern
(alternate bright and dark bands)
Huygens’ wave construction:
crests and troughs of the waves
coming from one slit alternately
reinforce or cancel those from the
other.
Cancellation – “out of phase” and call destructive interference.
Reinforcement- “in phase” and call constructive interference
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Spectrum of Electromagnetic Radiation
X-ray (λ=1Å) ~ order
of the spacing
between atoms
in a crystal
X-ray shines on a crystal
→ X-ray diffraction
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2.2 Line Spectra of Atoms
Emission and absorption spectra
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Fraunhofer’s solar spectrum
with dark (i.e., absent) lines
Na-D line (Flame spectra of
sodium salt)
gas-discharge spectrum
of Hydrogen
Electrical
discharge
emission
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Hydrogen Atom Spectrum
(difficult to explain with classical theory)
H-atom spectra → exhibit several distinct
groups (series) of spectral lines
H-atom
Each series can be expressed by Rydberg Formula
Rydberg constant (109,677.58 cm-1)
n1 =1 Lyman, n1 = 2 Balmer, n1 = 3 Paschen series
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2.3 Ultraviolet Catastrophe and Planck’s
Quantum Hypothesis
Black body
is an object that absorbs all electromagnetic
radiation that falls onto it. No radiation passes through it and none is
reflected. Despite the name, black bodies are not actually black as
they radiate energy as well. The amount and type of electromagnetic
radiation they emit is directly related to their temperature. Black
bodies below around 700 K (430 °C) produce very little radiation at
visible wavelengths and appear black (hence the name). Black bodies
above this temperature, however, begin to produce radiation at
visible wavelengths starting at red, going through orange, yellow, and
white before ending up at blue as the temperature increases.
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Blackbody radiation of heated metal
(failure of classical theory)
Precise measurement
Wave theory
Lummer &
Pringsheim (point)
Planck’s quantum
theory
(blue curve)
by Lummer & Pringsheim
→ a continuous spectrum with a peak
wavelength that becomes shorter and
shorter as the temperature increases.
However, classical electromagnetic wave
theory until then (1860s) → the spectral
intensity increases as λ decreases,
going to ∞ at very short λ (i.e., violet).
This disagreement is called
“ultraviolet catastrophe”.
The Quantum of Energy
Quantum Hypothesis (M. Planck, 1900): Nature
makes a jump !!
Radiant energy of the light →
the energy of waves could be described
as consisting of small packets, bundles
(calledquanta)
Light energy,
Classical energy
 = nh
n = 1, 2, 3, 4
Planck’s constant (6.6261 x 10-34J s)
(Appendix A: detailed formula)
where, I: spectral radiance or energy per unit time
Quantized energy
per unit surface area per unit solid angle per unit
frequency or wavelength
Max Planck was told in 1890s that there was nothing
new to be discovered in physics !!
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2.4 Photoelectric Effect: Particles of Light
(another clue to quantization of energy)
Shining the light on metal surface (photocathode) →
photoelectrons are ejected.
i)

of the incident light >
“threshold” 0 of photoelectron
Increase  of the incident light →
sudden increase of photoelectrons.
The amount of photoelectrons →
proportional to the brightness of
the light.
ii) The kinetic energy of electrons →
depends only on the  but not its intensity.
(i) Frequency and intensity dependence
0

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(ii) The kinetic energy of electrons vs frequency of light
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The result was a conflict with
the classical electromagnetic wave
Classical wave theory:



Energy of the light is proportional to its Intensity
only (intense red light > dim blue light)
Waves can have any amounts of energy
Kinetic energy of electrons should increase with
the light intensity
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Einstein’s Hypothesis
i) In order to overcome the conflict between the
photoelectric effect (Hertz’s data) and the
electromagnetic wave theory →
The energy quanta is composed of packets called
photons and the energy of each photon is given by,
 = h
Electron emission process→ a collision between a
photon and an electron embedded in the metal.
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Einstein’s Hypothesis
ii) A photoelectric equation was proposed,
K= final kinetic energy of
ejected electron
νo= threshold frequency
below which no
electrons can be
ejected
2
e
0
1
= m v = h(  ) = h  
2
Work Function - Energy to free an
 : electron from the metal
The light behaves as a wave (in diffraction) and as a
particle (in blackbody radiation and photoelectric
emission).
- wave-particle duality
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Millikan’s experimental data in Fig. 2.7
(10 years later)
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2.5 Nuclear Atom and the Quantum:
Bohr’s Explanation of Line Spectra
N. Bohr’s Stationary State Hypothesis
i) An atom could only exist in certain allowed state of specific
total energy, stationary state
ii) An atom could make a upward or a downward jump,
transition, Eu  E = E = h
by an absorption or an emission of a photon, respectively.
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Bohr’s Model for Hydrogen Atom
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The stationary state for the simplest atom corresponds to
circular orbits of various fixed radii of the single electron.
(ex. H, He+, Li2+, Be3+)
The model postulates the quantization of angular momentum L
as a criterion for fixing the radii of the orbits.
On the basis of the model, the Rydberg constant in
can be derived as
where
2 2 e4
RH =
h3c
μ is the “reduced mass” of the electron-proton pair
= memp/(me+ mp) = me/(1 + me/mp) = (0.999456 me)
1/ μ = 1/ me + 1/ mp
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However, Bohr’s Theory …
Couldn’t be extended into atoms with more
than one electron
Bohr's theory fits a lot of experimental results,
but it can’t explain why orbits are quantized
and how atoms behave the way they do.
If Newtonian mechanics governs the workings
of an atom, electrons would rapidly travel
towards and collide with the nucleus.
2.6 de Broglie’s Matter Waves: Beginning
of a New Mechanics
de Broglie Waves
i) The integers in Rydberg formula → only possible in a
constrained wave motion, as in a vibrating violin string.
ii) A string tied down at both ends → vibrate with certain
wavelength λ, and,

n  = L
2
n = 1, 2, 3,
L - length of string
n =1 : fundamental tone
n = 2, or over: overtones
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de Broglie Wavelength
From Planck’s formula E=hc/λ and Einstein E= mc2
h
wavelength for a photon:
Confirmed Experimentally
 photon =
mc
by Davisson and Germer (US)
and G.P. Thomson (England)
Similarly, wavelength for a particle:
 particle
h
=
mv
X-ray scattering by Al foil
Electron scattering by Al foil
Louis de Broglie (French)
B.S. History and Science
WW I, army service in radio
communications
Ph.D. Thesis (1924): matter wave
Creator of the Wave Mechanics
 Big Impact on the Modern Electron
microscopes !!
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The wave nature of the electron
must be invoked to explain the
behavior of electrons when they are
confined to dimensions on the
order of the size of an atom.
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Bohr-de Broglie Model of the Hydrogen
Atom
Bohr’s orbit → becomes circular
standing-wave vibration of the electron wave
→ λe = 2πr/n = h/mv → v = nh/2πmr
ne = 2 r n = 1, 2, 3,
de Broglie
Newton’s second law of motion for a circular orbit
in cgs-e & using the Coulomb potential energy.
Radii rn :
 v2
e2
 2 = me  
r
 r

2
2
 → v = e /mr

quantum number
Total energy:
1
e2
2
E = K  V = mev 
2
r
Details of derivation
 particle =
h
mv
ne = 2 r n = 1, 2, 3,
v=
h
m
 = 2 r / n
 v2 
e2
 2 = me   
r
 r 
2 2
2 2
e2
nh
mh
n
h
n
= mv 2 = m(
)2 = 2 2 2 = 2 2
r
2 mr
4 m r
4 mr
e2
h2n2
= 2
1 4 mr
h2
r = 2 2 n2
4 me
v=
nh
2 mr
2
1
e
E = K  V = mev 2 
2
r
 v2 
e2
 2 = me   
r
 r 
e2
= mv 2
r
1
e2
2
E = K  V = mev 
2
r
1 e2 e2
1 e2
=
 =
2 r
r
2 r
h2
r = 2 2 n2
4 me
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Energy Levels for Bound States
E=0
E = -ve
r → ∞ free state
r = na0 bound state
allowed energy En
Bohr radius a0
Eu = En2
Eu  E = E = h
E = h
El = En1
En2  En1 = E = h
2 mee4 1
2 mee4 1
hc

( 2 )  (
( 2 )) =
2
2
h
n2
h
n1

2 mee4 1
1
=
( 2  2)
2

h
n1 n2
hc
2 mee4 1
1
=
( 2  2)
3

h c n1 n2
1
2 2 mee4
RH =
h3c
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Hydrogen-Like Atoms
In terms of a0 and taking into account of the interaction between
the electron and the nucleus charge Ze ( ex. He+, Li2+)
Multiply both sides of Rydberg formula by hc
left hand side - △E = hv = hc/λ
right hand side – difference of
two energy levels
(Term values)
Bohr’s assignment of the Balmer
series of atomic hydrogen
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Hydrogen Atom Spectrum
H-atom spectrums → exhibit several distinct
groups (series) of spectral lines
H-atom
Each series can be expressed by Rydberg Formula
Rydberg constant (109,677.58 cm-1)
n1 =1 Lyman, n1 = 2 Balmer, n1 = 3 Paschen series
Hydrogen-Like Atoms
The Bohr explanation of the three series of
spectral lines.
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2.7 Schrödinger Equation: Wave Equation
for Particles
Schrödinger Equation: standing-wave motion of a
particle of mass m under the influence of a
potential V(x,y,z). The wave function Ψ(x,y,z)
should take the form of sin, cos or exp. Its second
derivative would take the same form as the
original form.
Ψ – wave function → orbital
E – total energy
V – potential energy
The Schrödinger Equation






De Broglie’s work attributes wave-like properties to electrons in
atoms, and the uncertainty principle shows that detailed
trajectories of electrons cannot be defined.
Consequently, we must deal in terms of the probability of
electrons having certain positions and momenta.
These ideas are combined in the fundamental equation of
quantum mechanics, the Schrodinger equation.
He reasoned that an electron (or any other particle) with wavelike properties should be described by a wave function that
has a value at each position in space.
This wave function [Ψ(x,y,z)] is the “height” of the wave at the
point in space defined by the set of Cartesian coordinates (x, y,
z).
Schrodinger wrote down the equation satisfied by y for a given
set of interactions between particles.
The meaning of wave function:
probability density


The square of the wave function Ψ2 for a particle
as a probability density for that particle.
In other words, [Ψ2(x, y, z)ΔxΔyΔz] is the
probability that the particle will be found in a
small volume ΔxΔdyΔz about the point (x, y, z).
Schrödinger Equation: wave equation of a particle
It plays a role analogous in quantum mechanics to Newton's
second law in classical mechanics.
The kinetic and potential energies are transformed into the
Hamiltonian which acts upon the wave function to generate the
evolution of the wave function in time and space
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Schrödinger Cat
Erwin Rudolf Josef Alexander Schrödinger
(1887-1962, Austria)
1926, Publication on the Equation
1933, Nobel Prize
An illustration of both states, a dead and
living cat. According to quantum theory, after
an hour the cat is in a quantum superposition
of coexisting alive and dead states. Yet when
we look in the box we expect to only see one
of the states, not a mixture of them
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A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, J.E. Verschaffelt,
W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin, P. Debye, M. Knudsen, W.L. Bragg, H.A.
Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, I. Langmuir, M.
Planck, Mme. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson,
O.W. Richardson
A Particle in a Box (Appendix B)
One-dimensional particle in a box with infinite energy barriers
 2 d 2 ( x)

= E ( x)
2
2m dx
or
d 2 ( x )
2mE
=  2  ( x)
2
dx

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 2 d 2 ( x)

= E ( x)
2
2m dx
General
solution
or
d 2 ( x)
2mE
=  2  ( x)
2
dx

 ( x) = A sin( kx)  B cos(kx)
With boundary
conditions
 (0) = A sin( 0)  B cos(0) = 0
A(0 )  B(1) = 0
therefore, B=0
 ( x ) = A sin( kx )
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d 2 ( x )
2mE
=  2  (x)
2
dx

 ( x ) = A sin( kx )
d2
dx
2
A sin( k x) = 
2mE

2
A sin( k x)
or
2
 Ak sin( k x) = 
2mE

2
 2mE 
k = 2 
  
A sin( k x)
1/ 2
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 2mE 
k = 2 
  
1/ 2
 ( x ) = A sin( kx )
 2mE 1 / 2 
 ( x ) = A sin  2  x 
  

With boundary condition x=L therefore (L)=0
 2mE  1 / 2 
 (L) = A sin  2  L = 0
  

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 2mE 1/ 2 
 ( L) = A sin 2  L = 0
  

A ≠ 0 for ψ to be meaningful. Therefore, inside [ ] = n.
1/ 2
 2mE 


2 
  
L = n , n = 1,2,3,...

Define En to be value of E for solutions for allowed values
of n
n 2 2  2
En =
2mL2
Correct h to  in the
lecture note.
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n 2 2 2
En =
2mL2
En =
=
n2h
2
h
2
, n = 1, 2, 3...
8m L
Quantization of energy states results from the imposition
of boundary conditions.
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Normalized wavefunctions for particle in a box
(see p.104 for normalization procedure)
2
 n (x) =  
L
1/ 2
 nx 
sin 
 n = 1, 2, 3...
 L 
Plots of  and 2 for the first
four energy levels
Which can absorb
shorter λ wave
between ethene
and butadiene?
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