Quantum Theory of the Atom

1
Quantum Theory of the Atom

• Chapter 7

2
Introduction

• Elements that exhibit similar chemical properties
  were placed together in the same column of the
  periodic table. When metal compounds heated in
  flame, they emit bright colors. Li(IA) and
  Sr(IIA) red, Ba(IIA) green
• The red lights of Li and Sr can be resolved by
  prism into lines of different colors which
  distinguishes two elements. How does each atom
  emit particular color of light? Does the line
  spectrum tell us the structure ot the atom?
• We need to understand the arrangement of
  electrons in an atom (called electronic
  structure of an atom)

3
Topics of Discussion

• The Wave Nature of Light
• Quantum Effects and Photons
• The Bohr Theory of Hydrogen Atom
• Quantum Mechanics
• Quantum Numbers and Atomic Orbitals

4
Properties of wave

• Wavelength, ?, is the distance between any two
  adjacent point of a wave

distance

• The frequency, ?, of a wave is the number of
  wavelengths that pass a fixed point in a unit
  time, usually second. Unit is Hz (1/s).

?4 Hz
?8 Hz
time
5
Wave property of light
E
B

• Light is electromagnetic wave. Electric and
  magnetic fields oscillate synchronously in planes
  perpendicular to each other.
• Speed of light in vacuum, c3x108m/s
• c??

6
Diffraction of light wave property

• Light is diffracted when the refractive index, n,
  of medium changes.
• c ?? (in vacuum)
• v c/n (in medium)

nair1
nwater1.5
7
Electromagnetic Spectrum

• Electromagnetic spectrum covers the range of
  frequencies and wavelengths of electromagnetic
  radiation.

8
Characteristic frequency of Cs atom

• What is the frequency of one of the spectral
  lines of Cs with a wave length of 456 nm?

9
Continuous Spectrum

• Continuous Spectrum contains light of all
  colors(wavelengths)
• Sunlight or light from an incasdescent lamp
  through the prism

10
Particle and wave nature of light

• Max Plank (German Physicist) found a formula
  to explain the blackbody radiation from heated
  objects.

• Atoms in a hot body oscillate with a definite
  frequency, ?,
• E nh?, n 1, 2, 3, ..
• h 6.63x10-34 J.s (Planks constant)
• Later, Albert Einstein postulated that light
  consists of quanta (photon) or particles of
  electromagnetic energy with energy E h?
• Photoelectric effect, Albert Einstein, 1905

11
Photoelectric Effect

• Photoelectric Effect is the ejection of electrons
  from the surface of a metal when light shines on
  it. Results
• Electrons are ejected, only when ? exceeds
  threshold value of ?0
• Number of electrons emitted directly proportional
  to the intensity of light
• Kinetic energy of electron is linearly
  proportional to the ?
• Ek h? - h?0 where h 6.63x10-34 J.s Plank
  constant and E h? energy of a photon and w h?0
  is called the work function of metal and ?0
  threshold ?.

12
Line spectrum
H
400 500 600
700

• Line Spectrum contains set of single or large
  number of very closed lines
• Electric discharge of a gas creates plasma where
  energetic electrons excite atoms or molecules
  which emit light.

13
Balmers H-line spectrum

• In 1885 J.J.Balmer showed that the line spectrum
  of hydrogen atom in the visible region could be
  reproduced by

• where, n 3, 4, ..
• What is the walelength of light when n 4 in
  Balmer series?

?656 nm (red line)
14
Stability of electrons in classical theory

• Rutherfords atom model Electron with negative
  charge rotates around positively charged nucleus
• According to electromagnetic theory electron
  moving in an electric fiels will radiate energy,
  and collapse.

15
Bohr Model of Hydrogen Atom

• Bohrs Postulated H atom model where electrons
• are stable
• 1. Energy postulate An electron can have only
  specific energy values in an atom, which are
  called energy levels.

• 2. Transition between energy levels An electron
  in an atom can change energy only by going from
  one energy level to another energy level.

16
The Wavelength of Transition

• When an electron in a higher energy level (Ei)
  undergoes a transition to a lower energy level,
  Ef, the energy which is lost by electron, is
  emitted as a photon, Ef h? Ei

17
Hydrogen atomic lines
0
Pashen series (IR)
-1/16RH
n3
-1/9RH
Balmer series (VIS)
-1/4RH
n2
Lyman series (UV)
Energy
-RH
n1
nf 1 Lyman (UV), nf 2 Balmer (VIS), nf
3 Paschen (IR)
18
Emission and Absorption of Light

• When electron undergoes a transition from an
  upper energy level to a lower one, light (photon)
  is emitted.

• If a photon(light) is absorbed, energy is gained
  by the electron and electron undergoes a
  transition from lower energy level to a higher one

19
Hydrogen emission line

• Calculate the wavelength of light emitted from
  the hydrogen atom when electron goes a transition
  from level n3 to level n1.

?103 nm (UV), 2nd line in Lyman Series
20
Ionization energy of hydrogen atom

• What is the wavelength of photons to ionize a
  hydrogen atom,
• in ground state and,
• in excited state with n3?
• Solution

1. ni1, nf?
?91.1nm

• ni3, nf ?

?820nm
21
Energy difference of atomic levels

• What is the difference in energy levels of the
  sodium atom if emitted light has a wavelength of
  589 nm?

• Solution
• Ehc/?
• E6.63x10-34J.s 3x108m.s-1/589x10-9m
• E3.38x10-19J

22
Colors of Material

• Materials have a color because of the absorption
  of light. For example, when white light falls on
  a substance that absorbs red light, the yellow
  and blue light are reflected. The substance
  appears blue-green

23
Bohr model for elements other than hydrogen

• Can you use Bohr Model for the electrons of
  vanadium atom?
• No. Multi-electron atoms can not be explained by
  Bohr model. Electron-electron interactions are
  not included.

• How about 23V22 ?
• Yes

24
Quantum Mechanics

• The concept of atomic energy levels is
  established by Bohrs Model. But neither the
  atomic structure nor the atomic energy levels for
  atoms other than hydrogen atom can be predicted
  by this model.
• Quantum Mechanics describes the wave properties
  of submicroscopic particles such as electrons,
  and predicts the atomic structure and the atomic
  energy levels.

25
De Broglie Relation

• Atomic structure depend on principles of quantum
  mechanics. Sub microscopic particles (electrons)
  have particle and wave nature.
• Energy of photon Eh?
• Momentum of photon E/c h?/c h/?
• Wavelength of photon ? h/momentum
• For electron ?h/mv
• h is Plancks constant, m is mass and v is the
  speed of particle

26
Wavelength of an electron

• What is the wavelength (in pm) of an electron
  travelling with a speed of 2.19x106 m/s?
• Solution

? 332 pm
27
Wavelength of radiation and microscopy

• The spatial resolution of a microscope is limited
  by the wavelength of radiation.
• Light microscope can see objects with sub
  micrometer size. Wavelength of green light 520 nm
  or 0.5 ?m.
• Wavelength of an electron moving with a speed of
  2x106 m/s is 0.3 nm. The resolution of an
  electron microscope is sub nanometer.

28
Electron in an atom

• De Broglie relation applies to small particles,
  like electrons in a force free environment (free
  electrons).
• An electron moving in an atom is subject to
  electric forces.
• The branch of physics that mathematically
  describes the wave properties of submicroscopic
  particles is called quantum mechanics of wave
  mechanics.

29
Wave Function of an Electron

• In 1926 Erwin Schrödinger proposed a general wave
  equation for a particle.
• Wave function (?) contains information about an
  electron in a given energy. ( de Broglie
  relation)
• ?2 is the probability of finding electron at a
  certain point in hydrogen atom.( Heisenberg
  Uncertainty Principle)
• 4?r2 ?2 is the probability of finding an electron
  within a shell (radial prob.)

30
Probability of finding electron

• Quantum mechanics does not allow us to describe
  the electron in atom moving in orbits.
• It allows us to make statistical statements about
  where the electron is in the atom.
• Wave function and its square, ?2 defines a
  probability function.

31
Radial probability function

• The probability of finding electron in shells
  about the nucleus can be used.
• Radial probability function

4?r2
r
r
32
Heisenbergs Uncertainty Principle

• Heisenbergs uncertainty principle states that
  the product of the uncertainty in position and
  the uncertainty in momentum of a particle can
  not be smaller than h/4?
• Position x ? ?x, Momentum px ? ?px

• Quantum Mechanics It is not possible to measure
  the precise position and the precise momentum a
  particle simultaneously.
• Bohr Model Electron orbits around the nucleus
  (?x 0, ?px 0)

33
Double slit experiment
photon or electron

• An interference pattern forms when monochromatic
  light or electrons pass through a closely spaced
  double slit

34
Quantum Numbers

• Quantum Mechanics
• Electron waves in three dimension ?three QN ( all
  integers)
• n is the principal QN which identifies SHELL..
• l is the angular momentum QN which identifies
  SUBSHELL.
• ml is the magnetic QN which identifies ORBITAL.

35
Quantization of electron motion in atom,
principle quantum number
n is the number of nodes, principle quantum numbel
electron in an atom
36
Orbital and spin quantum numbers, ml and ms
37
Interpretation of Quantum Numbers

• Two possible orientations of spin axis of an
  electron is given by ms it can have values of
  1/2 and -1/2.

38
Principle quantum number

• The size of atomic orbital and the energy of an
  electron in an atom depends principally on n it
  can have any integer value 1(K), 2(L), 3(M) ...

• How many times the size of an hydrogen atom
  increases when it is excited from ground state to
  n10
• 100

39
Orbital angular momentum

• The shape of orbital is determined by orbital
  angular momentum QN l it can have any integer
  value from 0 to n-1
• l 0(s), 1(p), 2(d), 3(f), 4(g) ...

l1
ml1

• The orientation of the orbitals relative to each
  other in space is designated by ml it can have
  integers from l to l

ml0
ml-1
40
Quantum Numbers

• Explain why each of the following set of quantum
  numbers is not permissible for an orbital
• n 0, l 1, ml 0, ms 1/2 ? n ? 0
• n 2, l 3, ml 0, ms -1/2 ? l ? n-1
• n 3, l 2, ml 3, ms 1/2 ? -l ? ml ? l
• n 3, l 2, ml 2, ms 0 ? ms ? 1/2
• What orbitals would be found in the subshell of
  4d?
• 4d? n 4, l 2, ml -2, -1, 0, 1, 2
• 5 orbitals of 4d subshell.

41
Orbitals for the Hydrogen Atom

• All orbitals with the same principal quantum
  number n have the same energy in hydrogen atom.
  They are said to be degenerate
• Number of orbitals for a given shell n is n2
• Number of orbitals for a given subshell l is 2l1

42
Atomic Orbitals Shapes

• The probability of finding electron at a given
  point is ?2
• Electron Cloud is the dot-density diagram which
  shows the way of the probability of finding
  electron varies in space
• Boundary surface (Contour) diagram indicate the
  volume in which the total probability of finding
  electron is 90 (or 99)

43
Atomic Orbital Shapes

• s-subshell(l0) ? ml 0 ? one orbital
• p-subshell(l1) ? ml -1, 0, 1 ? three orbitals
  with the same shape different orientation in
  space
• d-subshell(l2) ? ml -2, -1, 0, 1, 2 ? five
  orbitals with the same shape different
  orientation in space
• f-subshell(l3) ? ml -3, -2, -1, 0, 1, 2, 3
  ? seven