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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
Electrons in Atoms
1
Announcement
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enrol in the lecture and lab, which are co-requisites.
Electrons in Atoms
2
Discovery of Electrons
J.J. Thomson (18561940) determined
the charge to
mass ratio for
electrons in 1897.
Robert Millikan’s oildrop experiment
determined the
charge of electrons.
Thus both the charge
and mass are known
Review Chapter 2
Electrons in Atoms
3
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
Electrons in Atoms
4
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 (s–1 or Hz) .
Be able to apply:
Speed of light
n = c / l;
l=c /n
c = l n = 3.0e8 m s–1
Electrons in Atoms
5
Frequency, Wavelength & Wave-numbers
A typical red light has a wavelength of 690 nm. What is its frequency?
Solution:
c
3e8 m s–1
n = ----- = ------------------ = __________s–1
l
690e–9 m
By the way, the wave_number is the number of waves per unit length.
1
wave_number = --- = __________ m–1
l
Electrons in Atoms
6
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 = ------l
where h is the Planck’s constant, and is the wavelength. Momentum in
any collision is conserved.
Electrons in Atoms
7
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
Electrons in Atoms
8
Interference
Combination or
superposition of waves
is called interference.
Electrons in Atoms
9
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
Electrons in Atoms
10
The Electromagnetic Spectrum
Electrons in Atoms
11
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 ( ---- – ----)
l
22
n2
(RH = 10973731.534 m-1
in wave_number)
This is the Balmer series
c
1
1
n = ---- = R’H (---- – ----) (R’H = 3.2881e15 s-1, in frequency)
l
22
n2 Electrons in Atoms
12
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
divided
into emission and absorption
spectroscopy, (AES and AAS).
Electrons in Atoms
13
Max Planck’s 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=hn
The proportional constant h is now known as Planck
constant, h = 6.62606876e-34 J s–1, 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.
Electrons in Atoms
Given one, be able to calculate the
others.
14
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 =
c
l
3e8 m s–1
= 656.3e-9 m = 4.57e14 s–1
energy E = h n = 6.626e –34 J s–1 * 4.57e14 s–1 = _______
wave number =
1
l
= _______
E=hc/l
Electrons in Atoms
See slide 12 and
complete the calculation
15
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/~cchieh/cact/c120/quantum.html for photoelectric effect
Electrons in Atoms
16
Einstein’s 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)
Electrons in Atoms
17
Graph Einstein’s Result
Vs = ½ m u2
kinetic energy
of electron
e Vs = e k n – e Vo
h n = e (Vo + Vs)
n0
threshold
n
Electrons in Atoms
18
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=hc/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
Electrons in Atoms
19
Significance of Einstein’s Result
Max Planck’s assumption is true, a proof.
Light indeed consists of photons (quanta of light, not continuous)
Quantity of energy in photon
E=hn
(energy of photon)
Photochemical reactions
O2 + h n  O + O
O2 + O  O 3
(formation of ozone)
Be able to calculate energy of photons, E, threshold, no, and
Electrons in Atoms
kinetic energy of electrons, in photoelectric
experiment.
20
The Bohr Atom
Bohr tried to interpret the hydrogen spectrum by applying Planck’s
quantum hypothesis and Rutherford’s atom, and he postulated:
The e revolves around the nucleus (Rutherford’s 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 Newton’s physics, he showed the energy level of the electron to
be
RH
En= –
n2
Electrons in Atoms
21
Bohr’s Energy Levels of Electrons in H
1/
En= –
RH
n2
– 1/25 R H
– 1/9 R H
– 1/4 R H

RH=0
– 1/36 R H
– 1/16 R H
R H = 2.179e-18 J
State transitions
– RH
Given RH and state of transition, nf, ni, be
Electrons in Atoms
able to calculate E, n, & l, of transition.
22
H-spectra & Energy Levels
Ionization
energy
Electrons in Atoms
23
Excitation and Ionization of H
1/
En= –
RH
n2
– 1/25 R H
– 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
suitable h n
excites an atom

RH=0
– 1/36 R H
– 1/16 R H
Absorption of a
photon with energy
equal or greater
than RH results in
ionization of H atom
– RH
Electrons in Atoms
24
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:
1 1
DEni – nf = Ei – Ef = RH (----2 – ----)
ni nf2
l = h c / DEni – nf
RH = 2.179e-18 J
h = 6.6262ee-34 J s
c = 3e8 m s-1
Electrons in Atoms
25
Photons & Transition
Electrons in Atoms
26
Wave-Particle Duality
E=hn=mc2
hn
= mc=p =
c
h=lp=lmv
h
l
E – energy of photon and particle
h – Planck constant
h/l, m v, m u, or p – momentum
c, u, v – velocity of photon or particle
n – wavelength of photon or particle
Louis de Broglie (1892-1987)
Nobel laureate 1929
A particle with momentum p = m v is a wave with wave length l
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.
Electrons in Atoms
27
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;
1 J = 1 N m = 1 kg m2 s–2
h = 6.626e-34 J s
1 eV = 1.6e-19 J;
E=½mv2
100 eV = 1.6e-17 J
v = (2 E / m)½ = (2 * 1.6e-17 kg m2 s–2 / 9.1e-31 kg)½ = 5.9e6 m s–1
m v = 9.1e-31 kg * 5.9e6 m s–1 = 5.4e-24 kg m s–1
l = h / p = 6.626e-34 1 kg m2 s–1 / 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
Electrons in Atoms
speed is given. Estimate p & l when an electron travels at 50% c
28
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-to300 eV electrons to bombard a
sample; a diffraction pattern is
shown.
Example of a LEED pattern
from the Si(111)7×7 surface.
Electrons in Atoms
29
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:
h
Dx Dp >
4p
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)
Electrons in Atoms
Read the arguments for the uncertainty
principle.
Heisenberg at3022
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) =
6.63e–34 kg m2 s–1
–1
=
5.3e-25
kg
m
s
4*3.1416*1e-10 m
Rest mass of electron = 9.1e-31 kg
Speed of e with p = 5.3 e-25 kg m s–1
Dve = 5.3e-25 kg m s–1 / 9.1e-31 kg = 5.8e5 m s–1
DEnergy = ½ * 9.1e-31 kg * (5.8e5 m s–1)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.
Morein Atoms
Electrons
accuracy will result in an electron in larger volume.
Discuss physical
meaning of results.
31
Announcement
International Exchange Information Night
Nov 5, 2003, 5:30-6:30pm in DC 1301 (Fishbowl)
Pizza and Beverages will be served.
This information session is geared mainly toward students in their
1st 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)
Electrons in Atoms
32
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.
Electrons in Atoms
www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html
33
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=
2L
;
n
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)
Electrons in Atoms
34
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) =  ( 2 ) sin ( n p x ),
L
L
n = 1, 2, 3, …
Please work out the following values for n = 1 and n = 2 yourself
(0) =
(L/4) =
(L/2) =
(3L/4) =
(L) =
n=1
___0_
_____
_____
_____
___0_
n=2
___0_
_____
_____
_____
___0_
Electrons in Atoms
(boundary)
(boundary)
35
An electron
viewed as a
wave in an
atom,
imagination
goes a long
way
Animation by
Naoki Watanabe
Electrons in Atoms
36
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 Broglie’s relationship p = h / l,
h2
p2
Ek = ---------- = -----------2 m l2
2m
2L
Recall l = ----n
h2
n2 h2
= -------------------------= -------------2
2 m (2 L / n)
8 m L2
The lowest energy of a (wave) particle is when n = 1, the zero point
energy
Electrons in Atoms
37
Energy Level of a Particle in a Box
n2 h2
En = -------------8 m L2
In a 3-dimensional space
E(nx, ny, nz)
n=3
n=2
nx 2
ny 2
nz 2
h2
= ------- ( ----- + ------ + ----- )
Lx
Ly
Lz
8m
n=1
What is the expression for E(nx, Electrons
ny, nz) ininAtoms
a 3-D cube?
38
Wavefunction of H atom
The wave function of hydrogen atom satisfies the Schrodinger equation:
2j
2j
2j
2j
h2



Ze
– ---------( ------- + ------- + ------- ) – --------= Ej
2
2
2
2
2
8p m
r
x
y
z
Solutions 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 m
n – the principle q.n. (dominate energy)
l – the orbital angular momentum q.n. (orbital momentum)
m – the magnetic q.n.
Electrons in Atoms
39
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 l
l = 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)
…
Electrons in Atoms
40
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.
Electrons in Atoms
41
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,
Zeff 2 me e4
En = – ---------------8e0h 2 n 2
Zeff2
= – 13.6 eV ------n2
The Zeff is the effective atomic
number (modified atomic number).
4s
– –––– 4f
3s ––
–– 4p
3p ––
–– ––
–– ––
–– 4d
3d ––––––
––-––
2s –– 2p –– –– ––
1s ––
Electrons in Atoms
Energy levels of H orbitals
42
Energy Levels of Many-electron atoms
4f ––
For many electron atoms, the energy
4d –– –– –– –– ––
levels of subshells change slightly
4p –– –– ––
due to Zeff
3d –– –– –– –– ––
4s ––
Zeff 2 me e4
En = – ---------------8e0h 2 n 2
Zeff2
= – 13.6 eV ------n2
3s –– 2p –– –– ––
2s ––
1s ––
Electrons in Atoms
Energy levels of many
electron atoms
43
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 see
www.if.ufrgs.br/~betz/quantum/SGPeng.htm
Electrons in Atoms
44
Electronic Configurations of Atoms
Electrons go to the lowest possible energy levels (minimize the energy
of the atom).
Pauli’s exclusion principle: No two electrons in an atom may have all
four quantum numbers alike
Hund’s 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.
Electrons in Atoms
45
The Modern Periodic Table
1s1
Work out the filling order of orbitals and the
electronic configurations from the periodic table
1s2
2s1-2
2p1 – 2p6
3s1-2
3p1 – 3p6
4s1-2
3d1 – – – 3d10
4p1 – 4p6
5s1-2
4d1 – – – 4d10
5p1 – 5p6
5d1 – – – 5d10
6p1 – 6p6
6s1-2
4f1 – – – – – 4f14
7s1-2
Th Pa U – – – 5f14
7p1 – 7p6
6d1 – – – 6d10
Quantum mechanical theory led to the modern periodic table, which
correlates chemical properties of elements nicely!
Electrons in Atoms
46
Writing Electronic Configuration
Z=
2
10
18
36
54
86
1s2 2s22p6 3s23p6 4s23d104p6 5s24d105p6 6s24f145d106p6
He
Ne
Ar
Kr
Xe
Rn
Electrons in Atoms
47
The Wavefunctions 1s & 2s
1s
2s
URL: optoele.ele.tottori-u.ac.jp/~abe/hyd/
Electrons in Atoms
48
The Wavefunctions, 2ps
2pz
2px
2py
Electrons in Atoms
49
Atomic orbitals
The Orbitron is a British website that gives wonderful views of the
atomic orbital, and its URL is
www . 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
Electrons in Atoms
50
Excitation and
de-excitation
of electrons of
the hydrogen
atom.
Animation by
Naoki Watanabe
Electrons in Atoms
51
Animation by Naoki
Watanabe
Fun with waves
An electric current viewed
as waves Just for fun
cms.phys.s.utokyo.ac.jp/~naoki/
research/review/indexe.html
Animations are done by
computational method for
large-scale and long-term fisrtprinciples 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.
Electrons in Atoms
52
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)
Electrons in Atoms
53
Quantum Review 1
Draw an energy level diagram
according to Bohr’s 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)
Electrons in Atoms
54
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 Broglie’s 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?
Electrons in Atoms
55
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 s–1
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.
Electrons in Atoms
56
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?
Electrons in Atoms
57