Transcript Components of the Atom

Chapter 1
Introduction and Background
to Quantum Mechanics
Slide 1
The Need for Quantum Mechanics in Chemistry
Without Quantum Mechanics, how would you explain:
•
Periodic trends in properties of the elements
•
Structure of compounds
e.g. Tetrahedral carbon in ethane, planar ethylene, etc.
•
Bond lengths/strengths
•
Discrete spectral lines (IR, NMR, Atomic Absorption, etc.)
•
Electron Microscopy
Without Quantum Mechanics, chemistry would be a purely
empirical science.
PLUS: In recent years, a rapidly increasing percentage of
experimental chemists are performing quantum mechanical
calculations as an essential complement to interpreting
their experimental results.
Slide 2
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
There is nothing new to be discovered in Physics now.
All that remains is more and more precise measurement.
Lord Kelvin (Sir William Thompson), ca 1900
Slide 3
Intensity
Blackbody Radiation
Heated Metal

Low Temperature: Red Hot
Intermediate Temperature: White Hot
High Temperature: Blue Hot
Slide 4
Rayleigh-Jeans (Classical Physics)
 ( , T ) 
8 kT 2
  const   2
3
c
Intensity
Assumed that electrons in metal oscillate about their equilibrium
positions at arbitrary frequency (energy). Emit light at oscillation frequency.

The Ultraviolet Catastrophe:


0
 ( , T ) d  
Slide 5
Max Planck (1900)
Arbitrarily assumed that the energy levels of the oscillating electrons
are quantized, and the energy levels are proportional to :
 = h(n)
He derived the expression:
 ( , T ) 
Intensity
n = 1, 2, 3,...
h = empirical constant
8 h 3
1

c3
e h / kT  1

Expression matches experimental data perfectly for
h = 6.626x10-34 J•s [Planck’s Constant]
Slide 6
The Photoelectric Effect
Kinetic Energy of ejected electrons can be
measured by determining the magnitude of
the “stopping potential” (VS) required to
stop current.
- VS +
A
Observations
Low frequency (red) light:  < o - No ejected electrons (no current)
K.E.
High frequency (blue) light:  > o - K.E. of ejected electrons  

o
Slide 7
Photons
K.E.
Einstein (1903) proposed that light
energy is quantized into “packets”
called photons.
Slope = h
Eph = h
Explanation of Photoelectric Effect

o
Eph = h =  + K.E.
 is the metal’s “work function”: the energy required to eject an
electron from the surface
K.E. = h -  = h - ho
o =  / h
Predicts that the slope of the graph
of K.E. vs.  is h (Planck’s Constant)
in agreement with experiment !!
Slide 8
Equations Relating Properties of Light
Wavelength/
Frequency:
Wavenumber:
   c
1

 (cm)

Energy:
c

 c
E ph  h 
Units: cm-1
1

hc

 c
c must be in cm/s
 hc
You should know these relations between the properties of light.
They will come up often throughout the course.
Slide 9
Atomic Emission Spectra
Sample
Heat
When a sample of atoms is heated up, the excited electrons emit
radiation as they return to the ground state.
The emissions are at discrete frequencies, rather than a continuum
of frequencies, as predicted by the Rutherford planetary model
of the atom.
Slide 10
Hydrogen Atom Emission Lines
UV Region:
(Lyman Series)
Visible Region:
(Balmer Series)
1
1 

 108, 680 1  2  cm  1

 n 
n = 2, 3, 4 ...
1
1 

 108, 680  0.25  2  cm  1

n 

n = 3, 4, 5, ...
 
 
Infrared Region:   1  108, 680  0.111  1  cm  1


n 2 
(Paschen Series)
n = 4, 5, 6 ...
General Form (Johannes Rydberg)
 
1 1
1
 R H  2  2  cm 1

 n1 n 2 
n1 = 1, 2, 3 ...
n2 > n 1
RH = 108,680 cm-1
Slide 11
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
Slide 12
The “Old” Quantum Theory
Niels Bohr (1913)
Assumed that electron in hydrogen-like atom moved in circular orbit,
with the centripetal force (mv2/r) equal to the Coulombic attraction
between the electron (with charge e) and nucleus (with charge Ze).
e
v2
1 Ze  e
f  ma  m 
r 4 0 r 2
r
Ze
He then arbitrarily assumed that the “angular momentum” is quantized.
L  rxp  m vr  n
n = 1, 2, 3,...
h
(Dirac’s Constant)

2
Why??
Because it worked.
Slide 13
v2
1 Ze  e
m 
r 4 0 r 2
m vr  n
It can be
shown
4 0
a0 
me 2
4 0 2 n 2
2 a0
n
r
2
Z
me Z
2
= 0.529 Å
(Bohr Radius)
2
Ze
1
E  K .E .  P .E .  1 mv 
2
4 0 r
2
me 4 Z 2 1

 18 1


2.181
x
10
Joules
E 2 2 2  2
2
n
8 0 h n
n
n = 1, 2, 3,...
Slide 14
nU
EU
nL
EL
E ph   E  E U  E L
 
E    2
 nU
   
   2 
  nL 
1
1 
E    2  2 
 nL nU 
 2.181x10
18
Lyman Series: nL = 1
1
1 
 2  2  Joules
 nL nU 
Balmer Series: nL = 2
Paschen Series: nL = 3
Slide 15
nU
EU
nL
EL
1
1 
E    2  2 
 nL nU 
1
1 
 2.181x10 18  2  2  Joules
 nL nU 
1
E
1 
 

 109, 800  2  2  cm  1
hc
hc
 n L nU 
E ph
Close to RH = 108,680 cm-1
Get perfect agreement if replace electron mass (m) by reduced
mass () of proton-electron pair.
Slide 16
The Bohr Theory of the atom (“Old” Quantum Mechanics) works
perfectly for H (as well as He+, Li2+, etc.).
And it’s so much EASIER than the Schrödinger Equation.
The only problem with the Bohr Theory is that it fails as soon
as you try to use it on an atom as “complex” as helium.
Slide 17
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
Slide 18
Wave Properties of Particles
The de Broglie Wavelength
Louis de Broglie (1923): If waves have particle-like properties (photons,
then particles should have wave-like properties.
Photon wavelength-momentum relation
E  h 

hc

and
E  mc2
hc
hc
h
h



E mc 2 mc p
de Broglie wavelength of a particle

h
h

p mv
Slide 19
What is the de Broglie wavelength of a 1 gram marble traveling
at 10 cm/s
h=6.63x10-34 J-s
 = 6.6x10-30 m = 6.6x10-20 Å (insignificant)
What is the de Broglie wavelength of an electron traveling
at 0.1 c (c=speed of light)?
c = 3.00x108 m/s
me = 9.1x10-31 kg
 = 2.4x10-11 m = 0.24 Å
(on the order of atomic dimensions)
Slide 20
Reinterpretation of Bohr’s Quantization
of Angular Momentum
L  rxp  m vr  n
n = 1, 2, 3,...
mvr 

h
(Dirac’s Constant)
2
nh
2
h
h
2 r  n
n
mv
p
The circumference of a Bohr
orbit must be a whole number
of de Broglie “standing waves”.
2 r  n 
Slide 21
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
Slide 22
Heisenberg Uncertainty Principle
Werner Heisenberg: 1925
It is not possible to determine both the position (x) and momentum (p)
of a particle precisely at the same time.
p  x 
2
p = Uncertainty in momentum
x = Uncertainty in position
There are a number of pseudo-derivations of this principle in various texts,
based upon the wave property of a particle. We will not give one of
these derivations, but will provide examples of the uncertainty principle
at various times in the course.
Slide 23
Calculate the uncertainty in the position of a 5 Oz (0.14 kg) baseball
traveling at 90 mi/hr (40 m/s), assuming that the velocity can be
measured to a precision of 10-6 percent.
h = 6.63x10-34 J-s
-34 J-s1
ħ
=
1.05x10
x = 9.4x10-28 m
Calculate the uncertainty in the momentum (and velocity) of an
electron (me=9.11x10-31 kg) in an atom with an uncertainty in
position, x = 0.5 Å = 5x10-11 m.
p = 1.05x10-24 kgm/s
v = 1.15x106 m/s (=2.6x106 mi/hr)
Slide 24
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
Slide 25
Math Preliminary: Trigonometry
and the Unit Circle
y axis
y
1

x
sin(0o) = 0
x axis
cos(180o) = -1
sin(90o) =
1
cos(270o) = 0
cos() = x
sin() = y
From the unit circle,
it’s easy to see that:
cos(-) = cos()
sin(-) = -sin()
Slide 26
Math Preliminary: Complex Numbers
i  1
Imag axis
Euler Relations
e i  c o s ( )  i s in ( )
y
e  i  c o s ( )  i s in ( )
R

x
Real axis
Complex number (z)
z  x  iy
where
or
z  R e i
x  R c o s (  ) a n d y  R s in (  )
Complex Plane
Complex conjugate (z*)
z *  x  iy
or z *  R e  i
Slide 27
Math Preliminary: Complex Numbers
z  x  iy
Imag axis
x  R c o s (  ) a n d y  R s in (  )
where
y
R

x
z  R e i
or
Magnitude of a Complex Number
Real axis
z 
z
2
z  z * z  ( x  iy )( x  iy )  x 2  y 2
2
or
Complex Plane
z  z * z  (R e i )(R e  i )  R 2
2
Slide 28
Outline
• Problems in Classical Physics
• The “Old” Quantum Mechanics (Bohr Theory)
• Wave Properties of Particles
• Heisenberg Uncertainty Principle
• Mathematical Preliminaries
• Concepts in Quantum Mechanics
Slide 29
Concepts in Quantum Mechanics
Erwin Schrödinger (1926): If, as proposed by de Broglie, particles display
wave-like properties, then they should satisfy a
wave equation similar to classical waves.
He proposed the following equation.
One-Dimensional Time Dependent Schrödinger Equation
2

 2
i

 V ( x, t )
2
t
2 m x

h
( D irac ' s C ons tan t )
2
m = mass of particle
V(x,t) is the potential energy
 is the wavefunction
||2 = *  is the probability of
finding the particle between
x and x + dx
Slide 30
Wavefunction for a free particle
-
+
V(x,t) = const = 0
  C cos(kx   t)
k
where
Classical Traveling Wave
2

and
For a particle:  

Unsatisfactory because
2
E  h  
h
p
 C 2 co s 2 ( kx   t )  co n stan t
The probability of finding the particle at any position
(i.e. any value of x) should be the same
  C e i ( kx  t )
Note that:

is satisfactory
2
  *   C 2  co n stan t
Slide 31
“Derivation” of Schrödinger Eqn. for Free Particle
  Ce
i ( kx  t )
where
k
2

E  h  
and

h
p
on
board
E  i
  Ce
i
( px  Et )

t
on board
2
p2
 2

2m
2 m x 2
2

 2
i

t
2 m x 2
Schrödinger Eqn.
for V(x,t) = 0
Slide 32
Note: We cannot actually derive Quantum Mechanics or the
Schrödinger Equation.
In the last slide, we gave a rationalization of how, if a
particle behaves like a wave and  is given by the de Broglie
relation, then the wavefunction, , satisfies the wave equation
proposed by Erwin Schrödinger.
Quantum Mechanics is not “provable”, but is built upon
a series of postulates, which will be discussed in the
next chapter.
The validity of the postulates is based upon the fact that
Quantum Mechanics WORKS. It correctly predicts the properties
of electrons, atoms and other microscopic particles.
Slide 33