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/ of that of an atom.
In the mean time, Max Planck ( ) theorized that light beams were made of photons that are equivalent to particles of wave motion.
Explain waves and particles
Electrons in Atoms

2. Announcement
Please enroll Chem123 and related physics using Quest by doing the following:
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Electrons in Atoms

3. Discovery of Electrons
J.J. Thomson ( ) determined the charge to mass ratio for electrons in 1897.
Robert Millikan’s oil-drop experiment determined the charge of electrons. Thus both the charge and mass are known
Review Chapter 2
Electrons in Atoms

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
Electrons in Atoms
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 (s–1 or Hz) .
Speed of light
Be able to apply: n = c / l; l = c / n
c = l n = 3.0e8 m s–1
Electrons in Atoms

6. Frequency, Wavelength & Wave-numbers
A typical red light has a wavelength of 690 nm. What is its frequency?
Solution:
c 3e8 m s– n = = = __________s–1 l 690e–9 m
By the way, the wave_number is the number of waves per unit length wave_number = --- = __________ m–1 l
Electrons in Atoms

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 Planck’s constant, and is the wavelength. Momentum in any collision is conserved.
Electrons in Atoms

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 y y2
Electrons in Atoms

9. Interference
Combination or superposition of waves is called interference.
Electrons in Atoms

10. Give the name of the spectrum from a known source
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

11. The Electromagnetic Spectrum
Electrons in Atoms

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
= RH ( ---- – ----) (RH = m-1 l 22 n in wave_number) This is the Balmer series
c n = ---- = R’H ( ---- – ----) (R’H = e15 s-1, in frequency) l n2
Electrons in Atoms

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 divided into emission and absorption spectroscopy, (AES and AAS).
Electrons in Atoms

14. 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 = h n
The proportional constant h is now known as Planck constant, h = e-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.
Given one, be able to calculate the others.
Electrons in Atoms

15. Wavelength Frequency, Wave-number and Energy of Photon
The red line in Balmer series of hydrogen has a wavelength l = nm. Calculate the frequency, wave number, and energy of the photon.
Solution:
frequency n = = = e14 s–1 energy E = h n = 6.626e –34 J s–1 * 4.57e14 s–1 = _______ wave number = = _______
E = h c / l
c l
3e8 m s– e-9 m
1 l
See slide 12 and complete the calculation
Electrons in Atoms

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/~cchieh/cact/c120/quantum.html for photoelectric effect
Electrons in Atoms

17. 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 u = h n – e Vo (h = e k, the Planck constant)
h n = e (Vs + Vo)
Electrons in Atoms

18. Graph Einstein’s Result
Vs = ½ m u2 kinetic energy of electron
h n = e (Vo + Vs)
e Vs = e k n – e Vo
n0 threshold n
Electrons in Atoms

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 = e-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

20. 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 = 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.
Electrons in Atoms

21. 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
E n = –
R H
n 2
Electrons in Atoms

22. Bohr’s 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.
Electrons in Atoms

23. H-spectra & Energy Levels
Ionization energy
Electrons in Atoms

24. 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 n excites an atom
– R H
Electrons in Atoms

25. 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 = ee-34 J s c = 3e8 m s-1
ni2 nf2
Electrons in Atoms

26. Photons & Transition
Electrons in Atoms

27. Wave-Particle Duality
h = l p = l m v
E = h n = m c 2
= m c = p =
h n c
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
A particle with momentum p = m v is a wave with wave length l
Louis de Broglie ( ) 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.
Electrons in Atoms

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 s–2 1 eV = 1.6e-19 J;
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 speed is given. Estimate p & l when an electron travels at 50% c
Electrons in Atoms

29. 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)7×7 surface.
Electrons in Atoms

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 >
h 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)
Read the arguments for the uncertainty principle.
Heisenberg at 22
Electrons in Atoms

31. 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 s–1
6.63e–34 kg m2 s–1 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. More accuracy will result in an electron in larger volume.
Discuss physical meaning of results.
Electrons in Atoms

32. Pizza and Beverages will be served.
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

33. 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

34. 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
Electrons in Atoms

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, …
2 L
n p x L
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)
Electrons in Atoms

36. An electron viewed as a wave in an atom, imagination goes a long way
Animation by Naoki Watanabe
Electrons in Atoms

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 Broglie’s relationship p = h / l,
2 L n
p 2 2 m
h 2 2 m l2
Recall l = -----
Ek = = = =
h m (2 L / n)2
n 2 h 2 8 m L2
The lowest energy of a (wave) particle is when n = 1, the zero point energy
Electrons in Atoms

38. Energy Level of a Particle in a Box
n 2 h 2 8 m L2
En =
n = 3
In a 3-dimensional space E(nx, ny, nz) = ( )
n = 2
h 2 8 m
nx 2 Lx
ny 2 Ly
nz 2 Lz
n = 1
What is the expression for E(nx, ny, nz) in a 3-D cube?
Electrons in Atoms

39. Wavefunction of H atom
The wave function of hydrogen atom satisfies the Schrodinger equation: – ( ) – = E j 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.
h 2 8 p2 m
 2j x2
 2j y2
 2j z2
Ze 2j r2
Electrons in Atoms

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 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

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.
Electrons in Atoms

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 e4 8e0h 2 n 2
4s –– 4p –– –– –– 4d – – – – – 4f - -
3s –– 3p –– –– –– 3d –– –– –– –– ––
Zeff2 n2
2s –– 2p –– –– ––
1s ––
Energy levels of H orbitals
Electrons in Atoms

43. Energy Levels of Many-electron atoms
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 e4 8e0h 2 n 2
3s ––
2s ––
Zeff2 n2
1s ––
Energy levels of many electron atoms
Electrons in 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 see
Electrons in Atoms

45. 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

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!
Electrons in Atoms

47. Writing Electronic Configuration
Z= s2 2s22p6 3s23p6 4s23d104p6 5s24d105p6 6s24f145d106p6 He Ne Ar Kr Xe Rn
Electrons in Atoms

48. The Wavefunctions 1s & 2s 1s 2s
URL: optoele.ele.tottori-u.ac.jp/~abe/hyd/
Electrons in Atoms

49. The Wavefunctions, 2ps
2pz
2px
2py
Electrons in Atoms

50. 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

51. Excitation and de-excitation of electrons of the hydrogen atom.
Animation by Naoki Watanabe
Electrons in Atoms

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
Electrons in Atoms

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; to 2; 5 to 6; to 5; 2 to 1 (energy level diagram)
Electrons in Atoms

54. 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

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 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

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 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

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?
Electrons in Atoms