Looking back at Electrons in Atoms

1
Looking back at Electrons in Atoms
Chemistry documented materials to restore health
(pharmacy). Atoms and elements were recognized
during 16th to 18th centuries. The discovery of
electrons in 1897 showed that there were more
fundamental particles present in the (Dalton)
atoms. Fourteen years later, Rutherford
discovered that most of the mass of an atom
resides in a tiny nucleus whose radius is only
1/100000 of that of an atom. In the mean time,
Max Planck (1858-1947) theorized that light beams
were made of photons that are equivalent to
particles of wave motion.
Explain waves and particles
2
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This should allow you to enrol in the lecture and
lab, which are co-requisites.
3
Discovery of Electrons
J.J. Thomson (1856-1940) determined the charge to
mass ratio for electrons in 1897. Robert
Millikans oil-drop experiment determined the
charge of electrons. Thus both the charge and
mass are known
Review Chapter 2
4
1-, 2-, 3-dimensional Waves
Demonstrate a single wave movement Explain
continuous set of 1-dimensional waves Water wave
and drum-skin movement as 2-dimensional
waves 3-dimensional waves sound waves seismic
waves
Explain wave motions
5
Wavelength, Frequency, and Speed
Electromagnetic waves are due to the oscillations
of electro- and magnetic-fields. Wavelength (l)
the diagram shows one whole wave, and note the
wavelength (in m, cm, nm, pm). Frequency (n) is
the number of waves passing a single point per
unit time (s1 or Hz) . Speed of light
Be able to apply n c / l l c / n c l
n 3.0e8 m s1
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Frequency, Wavelength Wave-numbers
A typical red light has a wavelength of 690 nm.
What is its frequency? Solution c 3e8 m s1
n ----- ------------------
__________s1 l 690e9 m By the way, the
wave_number is the number of waves per unit
length. 1wave_number --- __________ m1
l
7
Momentum of Photons
For a particle with restmass mo, its relative
mass m when moving with a velocity of u relative
to the speed of light c is mo m
----------------- ?1 (u/c)2 Light
particles, photons, have zero rest mass, travel
at the speed c. From this relationship, the
momentum of the photon, p, (Text p309) is h
p ------- ?where h is the Plancks
constant, and is the wavelength. Momentum in any
collision is conserved.
8
Superposition of Waves
A sine function y1 sin x is a typical wave
function. Plot y1 vs. x Plot y2 sin x. When
the two waves combine, what happens? y1
y2 y1 y2
9
Interference
Combination or superposition of waves is called
interference.
10
Radiation spectra
There are three types of spectra Continuous
spectrum (generated by hot solid) Line spectrum
(generated by hot gas, atoms) Absorption spectrum
(continuous spectrum with black lines)
Give the name of the spectrum from a known source
11
The Electromagnetic Spectrum
12
Hydrogen Emission Spectrum
The visible spectra of H consists of red (656.3
nm) green, (486.1 nm), blue, (343.0 nm), indigo
(410.1 nm), and violet (396.9 nm) lines.
Variation of wavelength follow this formula
1 1 1 --- RH ( ---- ----) (RH
10973731.534 m-1 l 22 n2 in
wave_number) This is the Balmer series
c 1 1n ---- RH
( ---- ----) (RH 3.2881e15 s-1, in
frequency) l 22 n2
13
Atomic Spectroscopy
The study of light emitted by or absorbed by
atoms is called atomic spectroscopy (AA). It
offers qualitative and quantitative analysis of
samples, because each element has a unique set of
lines. The simplest form is identify elements by
flame color.
Atomic spectroscopy can be dividedinto emission
and absorption spectroscopy, (AES and AAS).
14
Max Plancks Photon
Max Planck (1858 1947) proposed that light
consists of little quanta of energy, and he
called them photons. The energy of the photon E,
is proportional to its frequency n. E h n The
proportional constant h is now known as Planck
constant, h 6.62606876e-34 J s1, a universal
constant. His proposal or assumption was made
while studying the radiation from hot (black)
body.
Know relationships among frequency, n,
wavelength, l, wave number, speed, c, and energy
E. Given one, be able to calculate the others.
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Wavelength Frequency, Wave-number and Energy of
Photon
The red line in Balmer series of hydrogen has a
wavelength l 656.3 nm. Calculate the
frequency, wave number, and energy of the
photon. Solution frequency n
4.57e14 s1 energy E h
n 6.626e 34 J s1 4.57e14 s1 _______
wave number _______ E h c / l
cl
3e8 m s1 656.3e-9 m
1l
See slide 12 and complete the calculation
16
The Photoelectric Effect
In 1888, Hertz discovered that electrons are
emitted when light strikes a metal surface
photoelectric effect. In 1905, Einstein observed
and explained electrons are emitted only when
light frequency exceeds a particular value no,
he called these values threshold number of
electrons emitted is proportional to the
intensity of light kinetic energies of emitted
electrons depend on the frequency, n, not the
intensity of light See science.uwaterloo.ca/cchi
eh/cact/c120/quantum.html for photoelectric effect
17
Einsteins Experiment
Kinetic energy of electron (½ m u 2) is measured
by retarding potential Vs. ½ m u 2 e Vs Vs is
proportional to the light frequency, but
unrelated to light intensity. The frequency n
must be greater than certain threshold no, which
is metal dependent. Vs k (n no ) k n
k no k n Vo (substitute Vo for k no) e Vs
e k n e Vo ½ m u2 h n e Vo
(h e k, the Planck constant) h n e (Vs
Vo)
18
Graph Einsteins Result
Vs ½ m u2 kinetic energy of electron
h n e (Vo Vs)
e Vs e k n e Vo
n0 threshold
n
19
A typical problem
Radiation with wavelength of 200 nm causes
electron to be ejected from the surface of a
metal. If the maximum kinetic energy of electrons
is 1.5e-19 J, what is the lowest frequency of
radiation that can be used to dislodge electrons
from the surface of nickel? Solution Energy of
the photon E h c / l 6.6262e-34 J s
3e8 m / 200e-9 m _E1_ J Threshold
energy Eo E1 1.5e-19 J Threshold photon
wavelength, lo h c / Eo __please calculate __
m
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Significance of Einsteins Result
Max Plancks assumption is true, a proof. Light
indeed consists of photons (quanta of light, not
continuous) Quantity of energy in photon E h
n (energy of photon) Photochemical reactions O2
h n ? O O O2 O ? O3 (formation of ozone)
Be able to calculate energy of photons, E,
threshold, no, and kinetic energy of electrons,
in photoelectric experiment.
21
The Bohr Atom
Bohr tried to interpret the hydrogen spectrum by
applying Plancks quantum hypothesis and
Rutherfords atom, and he postulated The e
revolves around the nucleus (Rutherfords
atom) The electron has a set of allowed orbits
(angular momentum n h/2p, where n is an
integer) that are stable. Electron changes from
one state to another by absorbing or emitting a
photon. From Newtons physics, he showed the
energy level of the electron to be E n
R H n 2
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Bohrs Energy Levels of Electrons in H
1/? R H 0
1/36 R H
1/25 R H
R H n 2
1/16 R H
E n
1/9 R H
1/4 R H
R H 2.179e-18 J State transitions
R H
Given RH and state of transition, nf, ni, be able
to calculate E, n, l, of transition.
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H-spectra Energy Levels
Ionization energy
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Excitation and Ionization of H
1/? R H 0
1/36 R H
1/25 R H
R H n 2
1/16 R H
E n
1/9 R H
1/4 R H
R H 2.179e-18 J Excitation and ionization of an
atom differs from those of a molecule
Absorption of a photon with energy equal or
greater than RH results in ionization of H atom
Absorption of a suitable h nexcites an atom
R H
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A typical problem
The electron in a hydrogen atom undergoes a
transition from 4s to one of the 5p orbitals when
the atom absorbs a single photon. What is the
frequency of the absorbed photon? Solution DEni
nf Ei Ef RH (---- ----) RH
2.179e-18 J l h c / DEni nf h
6.6262ee-34 J s c 3e8 m s-1
1 1ni2 nf2
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Photons Transition
27
Wave-Particle Duality
h l p l m v
E h n m c 2 m c p
h l
h n c
E energy of photon and particleh Planck
constanth/l, m v, m u, or p momentumc, u, v
velocity of photon or particlen wavelength of
photon or particle A particle with momentum p m
v is a wave with wave length l
Louis de Broglie (1892-1987)Nobel laureate 1929
When in 1920 I resumed my studies ... what
attracted me ... to theoretical physics was ...
the mystery in which the structure of matter and
of radiation was becoming more and more enveloped
as the strange concept of the quantum, introduced
by Planck in 1900 in his researches into
black-body radiation, daily penetrated further
into the whole of physics.
28
Wavelength of Electrons
Estimate the velocity and wavelength of electrons
with kinetic energy of 100 eV. Solution (data
look up and background information required)Mass
of e me 9.1e-31 kg h 6.626e-34 J s E ½ m
v 2 1 J 1 N m 1 kg m2 s2 1 eV 1.6e-19 J
100 eV 1.6e-17 J v (2 E / m)½ (2
1.6e-17 kg m2 s2 / 9.1e-31 kg)½ 5.9e6 m s1 m
v 9.1e-31 kg 5.9e6 m s1 5.4e-24 kg m s1 l
h / p 6.626e-34 1 kg m2 s1 / 5.4e-24 kg m s
1 2.23e-10 m (approximately the diameter
of atoms)
Be able to calculate momentum and wavelength of
particle when its speed is given. Estimate p l
when an electron travels at 50 c
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Validity of Particle-Wave Duality
Electrons are usually considered particles. In
1927, a Davisson and Germer observed electron
diffraction by Ni surface. Low energy electron
diffraction (LEED) uses a beam of 30-to-300 eV
electrons to bombard a sample a diffraction
pattern is shown.
Example of a LEED pattern from the Si(111)77
surface.
30
The Heisenberg Uncertainty Principle
When electrons are considered particles, we
should be able to measure their positions (x) and
momenta (p) accurately, but Heisenberg showed
that is not the case. The arguments seem complex,
but the result is simple. The uncertainty of
position Dx and uncertainty in momentum Dp has
this relationship Dx Dp gt
h4p
The implications the position (location) is
fussy if we know the energy accurately. We are
concerned with the energy more than we are with
location probability of finding the electron
correlates to orbital (not orbit)
Read the arguments for the uncertainty principle.
Heisenberg at 22
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Quantities and the Uncertainty Principle
If the uncertainty of an electron is 1e-10 m (100
pm), what is the uncertainty of the momentum? Dp
h / (4p Dx) 5.3e-25 kg m s1
6.63e34 kg m2 s143.14161e-10 m
Rest mass of electron 9.1e-31 kg Speed of e
with p 5.3 e-25 kg m s1 Dve 5.3e-25 kg m
s1 / 9.1e-31 kg 5.8e5 m s1 DEnergy ½
9.1e-31 kg (5.8e5 m s1)2 1.5e-19 J (recall
1 eV 1.6e-19 J) Ionization energy of H is 13.6
eV DEnergy 0.9 / 13.6 7
If ionization energy of H is 7 accurate, the e
is some-where within 100 pm, size of H atom. More
accuracy will result in an electron in larger
volume.
Discuss physical meaning of results.
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Announcement International Exchange Information
NightNov 5, 2003, 530-630pm in DC 1301
(Fishbowl) Pizza and Beverages will be served.
This information session is geared mainly toward
students in their1st or 2nd year who are
interested in International Academic
Exchanges.Criteria to be accepted for an
International Exchange Program Completed two
years of University and maintained an overall
accumulative average of 70 Proficient in the
language of desired country of exchange (ie. must
be proficient in French to go to France)
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Standing Waves
A traveling wave can be of any wavelength. The
boundaries restrict a standing wave to some
integer times the half-wavelength as illustrated
by the 1-dimensional diagram. Note that the
points at the boundary are fixed.
www2.biglobe.ne.jp/norimari/science/JavaEd/e-wave
4.html
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Wavelength in Standing Waves
For standing waves with both ends fixed in a
length L, the wavelength l is limited to (where n
is an integer) l n 1, 2, 3, . In
quantum mechanics, electrons in atoms are treated
as standing waves confined by the electric field
due to the atomic nuclei. Thus, the electrons are
represented by wave functions. The wave, energy
etc are called state of the electron, and with
spin, each state accommodate 2 electrons. A state
is called an orbital (not orbit)
2 L n
35
Particle in a 1-Dimensional Box of Length L
The wave representing a particle in a box of
length L (variable x) can be represented by
?(x) ? ( ) sin ( ), n 1, 2,
3,
n p xL
2L
Please work out the following values for n 1
and n 2 yourself n 1 n 2 ?(0)
___0_ ___0_ (boundary) ?(L/4) _____ _____
?(L/2) _____ _____ ?(3L/4) _____ _____
?(L) ___0_ ___0_ (boundary)
36
An electron viewed as a wave in an atom,
imagination goes a long way
Animation by Naoki Watanabe
37
Energy of Waves
Kinetic energy of a particle with speed u Ek ½
m u 2 (m u)2 / 2 m p 2 / 2m de Broglies
relationship p h / l,
2 L n
h 22 m l2
p 22 m
Recall l -----
Ek ---------- ------------
-------------------------- --------------
h 2 2 m (2 L / n)2
n 2 h 28 m L2
The lowest energy of a (wave) particle is when n
1, the zero point energy
38
Energy Level of a Particle in a Box
n 2 h 28 m L2
En --------------
n 3
In a 3-dimensional spaceE(nx, ny, nz)
------- ( ----- ------ ----- )
n 2
nx 2Lx
ny 2Ly
nz 2Lz
h 28 m
n 1
What is the expression for E(nx, ny, nz) in a 3-D
cube?
39
Wavefunction of H atom
The wave function of hydrogen atom satisfies the
Schrodinger equation ---------- (
------- ------- ------- ) ---------
E jSolutions of this equation is beyond the
scope of this course, but a few points can be
made. This second order DE has many solutions and
by implying the boundary conditions in here,
these solutions are characteristic of three
quantum numbers n, l and mn the principle
q.n. (dominate energy)l the orbital angular
momentum q.n. (orbital momentum)m the magnetic
q.n.
h 28 p2 m
? 2j ?x2
? 2j ?y2
? 2j ?z2
Ze 2j r2
40
Properties of Quantum Numbers
Restrictions of quantum numbers are due to
physical and mathematical reasons. Thus, n 1,
2, 3, 4, (integer) l 0, 1, 2, 3, (n
1) m - l, - (l 1), - - (l 2) 0 (l 2),
- (l 1), l
The consequency Subshells are named according
to value of ll 0 (s-subshell) (1 state, m
0)l 1 (p-subshell) (3 states, m -1, 0, 1)l
2 (d-subshell) (5 states, m -2, -1. 0, 1,
2)l 3 (f-subshell) (7 states, m -3, -2, -1.
0, 1, 2, 3)
41
Wave (electron density) of Some Orbitals
Know the shape of 1s, 2s, 2p, 3s, 3p, 3d
orbitals for your bonding lessons in the
future. Know the sign of the orbitals in various
regions. Explain the significance of j vs. r
plots, j2 vs. r plots, r2 j2 vs. r plots
etc. Explain nodes, know how numbers of nodes
related to n and l. Animations are used during
the lecture to illustrate all the above points,
and these are subjects of tests and final exams.
42
Energy Levels of H-atoms
Solutions to the Schrodinger equation results in
expressions for the energy, which is essentially
the same as the one derived by Bohr, En
---------------- 13.6 eV
------- The Zeff is the effective atomic number
(modified atomic number).
Zeff 2 me e48e0h 2 n 2
4s 4p 4d 4f - -
3s 3p 3d
Zeff2n2
2s 2p
1s
Energy levels of H orbitals
43
Energy Levels of Many-electron atoms
4f
For many electron atoms, the energy levels of
subshells change slightly due to Zeff En
---------------- 13.6 eV -------
4d
4p
3d
4s
2p
Zeff 2 me e48e0h 2 n 2
3s
2s
Zeff2n2
1s
Energy levels of many electron atoms
44
Electron Spins
Stern-gerlach experiment In a magnetic field, a
beam of electrons splits into two beams, and a
beam of atoms are also splits into two beams. The
interpretation of this observation came after
some years is due to the spin of electrons, thus
a fourth quantum number s. For an applet
animation of this experiment seewww.if.ufrgs.br/
betz/quantum/SGPeng.htm
45
Electronic Configurations of Atoms
Electrons go to the lowest possible energy levels
(minimize the energy of the atom). Paulis
exclusion principle No two electrons in an atom
may have all four quantum numbers alike Hunds
rule electrons occupy singly in orbitals of
identical energy (degenerate orbitals p 3, d
5, f 7 etc.) The aufbau (build-up) process
illustrate this process during lecture and urge
students to do it. Understand the energy level is
the key.
46
The Modern Periodic Table
Work out the filling order of orbitals and the
electronic configurations from the periodic table
1s1 2s1-2 3s1-2 4s1-2 5s1-2 6s1-2 7s1-2
1s2 2p1 2p6 3p1 3p6 4p1 4p6 5p1 5p6
6p1 6p6 7p1 7p6
3d1 3d10 4d1 4d10 5d1
5d10 6d1 6d10
4f1 4f14 Th Pa U 5f14
Quantum mechanical theory led to the modern
periodic table, which correlates chemical
properties of elements nicely!
47
Writing Electronic Configuration
Z 2 10 18 36
54 86 1s2 2s22p6
3s23p6 4s23d104p6 5s24d105p6 6s24f145d106p6 He
Ne Ar Kr Xe
Rn
48
The Wavefunctions 1s 2s
1s
2s
URL optoele.ele.tottori-u.ac.jp/abe/hyd/
49
The Wavefunctions, 2ps
2pz
2px
2py
50
Atomic orbitals
The Orbitron is a British website that gives
wonderful views of the atomic orbital, and its
URL iswww . shef.ac.uk/chemistry/orbitron/
See Table 9.1 of your Text for A table of
wavefunctions of atomic orbitals of the hydrogen
atom. Please identify the wave function for the
following orbitals 1s 2s, 2px, 2py, 2pz 3s, 3px,
3py, 3pz 3dxy, 3dyz, 3dx2 y2, 3dz2
51
Excitation and de-excitation of electrons of the
hydrogen atom.
Animation by Naoki Watanabe
52
Fun with waves
Animation by Naoki Watanabe cms.phys.s.u-tokyo.ac
.jp/naoki/research/review/indexe.html Animations
are done by computational method for large-scale
and long-term fisrt-principles numerical
solutions of time-evolving quantum electronic
states. The base equation for this study is the
time-dependent Schroedinger equation. In
principle, by solving this partial differential
equation numerically, it would be possible to
analyze many kinds of quantum dynamic phenomena.
An electric current viewed as waves Just for fun
53
Quantum review 0
The longest wavelength of radiation that will
cause the emission of electrons from gold surface
is 257 nm. What is the enrgy per photon of this
radiation? (photoelectric effect)The threshold
photons for gold have a wavelength of 257 nm,
what is the threshold energy? E h c / l
Which of the following transitions for the H atom
produces radiation of the shortest wavelength?-
n from 2 to 3 3 to 2 5 to 6 6
to 5 2 to 1 (energy level diagram)
54
Quantum Review 1 Draw an energy level diagram
according to Bohrs atom. Identify the
transitions for red (656.3 nm) green, (486.1 nm),
blue, (343.0 nm), indigo (410.1 nm), and violet
(396.9 nm) lines of hydrogen. ( n c / l E
h n)
Evaluate n, wave_number, and energy of the red
and violet lines. Evaluate the wavelength of the
transition from n 2 to n 1 and compare with
that from n 10 to n 1 in the Lyman Series.
(transition and energy level diagram)
55
Quantum Review 2
What is the wavelength, frequency and energy of
the photon in the Balmer series for n 6?
Calculate these for the Lyman series. If
traveling at equal speed, which of the following
particles has the longest wavelength, and why
electron, proton, neutron, alpha particle (He2)?
(de Broglies theory) What are the quantum
numbers n, l, m for the states 2s, 3p, 4d,
5f? Write the electronic configuration for
uranium U.Review questions No. 24 of Chapter 9
in General Chemistry, 8th ed. How many photons
are emitted per second by an IR lamp consuming 95
W if 14 of the power is converted to photons of
wavelength of 1525 nm?
56
Quantum Review 3
The work function of mercury is 435 kJ / mole
(energy required to remove a mole of electrons
from Hg surface). (Photoelectric effect)What is
the threshold energy in eV per electron?What is
the wavelength, frequency and wave number for the
threshold photon?What is the kinetic energy of
electron if the light used has a wavelength of
215 nm? Energy per electron 4.35e5 J / 6.023e23
7.22e-19 J / e 1 eV / 1.6e-19 J 4.51
eV Frequency 7.22e-19 J / 6.63e-34 J s
1.09e21 s1 l h c / E E h n h c / l n
c / l Kinetic energy of electron h (n no)
h c (1/ l 1/lo) extra energy after
threshold.
57
Quantum review 4
Sketch the shape of these atomic orbitals
according to their electron density 1s, 2s, 3s,
2p, 3p, 3d, and what are the signs of the wave
functions in various lobs of the atomic
orbitals? What is the meaning of ? and ?2? What
do the plots of ?2 in a three-dimensional space
represent? What do the plots of ?2 and 4pr2?2
against r represent? How many nodal shells are
there in these atomic orbitals, 1s, 2s, 3s,
2p?How many nodal planes are there in these
atomic orbitals 2p, 3p, 3d? How many atomic
orbitals are there for n 5 and l 3?What are
the possible values of ml for these orbitals?How
many electrons do these orbitals accommodate?