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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

Announcement Please enroll Chem123 and related

physics using Quest by doing the following First

screen - add LAB class No. - DO NOT PRESS

CONTINUE! Press INSERT CLASS (Again) - add

LECTURE class No. - then press CONTINUE You will

see two boxes. Update your attributes (add your

tutorials where applicable) and then SUBMIT.

This should allow you to enrol in the lecture and

lab, which are co-requisites.

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

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

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

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

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.

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

Interference

Combination or superposition of waves is called

interference.

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

The Electromagnetic Spectrum

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

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).

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.

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

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

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)

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

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

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.

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

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.

H-spectra Energy Levels

Ionization energy

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

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

Photons Transition

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.

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

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.

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

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.

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)

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

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

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)

An electron viewed as a wave in an atom,

imagination goes a long way

Animation by Naoki Watanabe

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

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?

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

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)

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.

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

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

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

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.

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!

Writing Electronic Configuration

Z 2 10 18 36

54 86 1s2 2s22p6

3s23p6 4s23d104p6 5s24d105p6 6s24f145d106p6 He

Ne Ar Kr Xe

Rn

The Wavefunctions 1s 2s

1s

2s

URL optoele.ele.tottori-u.ac.jp/abe/hyd/

The Wavefunctions, 2ps

2pz

2px

2py

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

Excitation and de-excitation of electrons of the

hydrogen atom.

Animation by Naoki Watanabe

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

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)

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)

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?

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.

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?