Chapter 7: QUANTUM THEORY OF THE ATOM

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Chapter 7 QUANTUM THEORY OF THE ATOM

• Vanessa N. Prasad-Permaul
• Valencia Community College
• CHM 1045

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THE WAVE NATURE OF LIGHT

• Frequency, ? The number of wave peaks that pass
  a given point per unit time (1/s)
• Wavelength, ? The distance from one wave peak
  to the next (nm or m)
• Amplitude Height of wave
• Wavelength x Frequency Speed
• ?(m) x ?(s-1) c (m/s)
• The speed of light waves in a vacuum in a
  constant
• c 3.00 x 108 m/s

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THE WAVE NATURE OF LIGHT
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THE WAVE NATURE OF LIGHT

• EXAMPLE 7.1 WHAT IS THE WAVELENGTH OF
• THE YELLOW SODIUM EMISSION, WHICH HAS A FREQUENCY
  OF 5.09 X 1014 S-1?
• c nl
• c
• n
• l 3.00 x 108 m/s
• 5.09 x 1014 s-1
• 5.89 x 10 -7 m
• 589 x 10-9 m
• 589
  nm

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THE WAVE NATURE OF LIGHT
EXERCISE 7.1 The frequency of the strong red
line in the spectrum of potassium is 3.91 x 1014
s-1. What is the wavelength of this light in
nanometers?
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THE WAVE NATURE OF LIGHT

• EXAMPLE 7.2 WHAT IS THE FREQUENCY OF VIOLET
  LIGHT WITH A WAVELENGTH OF 408nm?
• c nl
• n c
• l
• n 3.00 x 108 m/s
• 408 X 10-9 m
• 7.35 x 1014 s-1

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THE WAVE NATURE OF LIGHT
EXERCISE 7.2 The element cesium was
discovered in 1860 by Robert Bunsen and Gustav
Kirchoff, who found to bright blue lines in the
spectrum of a substance isolated from a mineral
water. One of the spectral lines of cesium has a
wavelength of 456nm. What is the frequency?
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THE WAVE NATURE OF LIGHT
THE ELECTROMAGNETIC SPECTRUM

• Several types of electromagnetic radiation make
  up the electromagnetic spectrum

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QUANTUM EFFECTS PHOTONS

• Atoms of a solid oscillate of vibrate with a
• definite frequency
• E h ?
• E hc / ?
• h Plancks constant, 6.626 x 10-34 J s
• E energy
• 1 J 1 kg m2/s2
• When a photon hits the metal, its energy (hn) is
• taken up by the electron. The photon no longer
• exists as a particle and it is said to be absorbed

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QUANTUM EFFECTS PHOTONS

• Max Planck (18581947) proposed the energy is
  only emitted in discrete packets called quanta
  (now called photons).
• The amount of energy depends on the frequency
• E energy ? frequency
• ? wavelength c speed of light
• h plancks constant

hc
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-
E
h

h
6
.
626
10
J
s

n




l
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QUANTUM EFFECTS PHOTONS

• Albert Einstein (18791955)
• Used the idea of quanta to explain the
  photoelectric effect.
• He proposed that light behaves as a stream of
  particles called photons
• A photons energy must exceed a minimum threshold
  for electrons to be ejected.
• Energy of a photon depends only on the frequency.
• E h ?

THE PHOTOELECTRIC EFFECT The ejection of
electrons from the surface of a metal or from a
material when light shines on it
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QUANTUM EFFECTS PHOTONS
EXAMPLE 7.3 THE RED SPECTRAL LINE OF LITHIUUM
OCCURS AT 671nm (6.71 x 10-7m). CALCULATE THE
ENERGY OF ONE PHOTON OF THIS LIGHT. n c
3.00 x 108 m/s 4.47 x 1014 s-1
l 6.71 x 10-7 m E hn
6.63 x 10-34 J.s 4.47 x 1014 s-1
2.96 x 10-19 J
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QUANTUM EFFECTS PHOTONS

• EXERCISE 7.3 The following are representative
  wavelengths in the infrared, ultraviolet and
  x-ray regions of the electromagnetic spectrum,
  respectively
• 1.0 x 10-6 m, 1.0 x 10-8 m and 1.0 x 10-10 m.
• What is the energy of a photon of each
  radiation?
• Which has the greatest amount of energy per
  photon?
• Which has the least?

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THE BOHR THEORY OF THE HYDROGEN ATOM

• Atomic spectra Result from excited atoms
  emitting light.
• Line spectra Result from electron transitions
  between specific energy levels.
• Blackbody radiation is the visible glow that
  solid objects emit when heated.

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THE BOHR THEORY OF THE HYDROGEN ATOM

• BOHRS POSTULATE
• The stability of the atom (H2)
• The line spectrum of the atom
• ENERGY-LEVEL POSTULATE An electron can only
  have specific energy level values in an atom
  called ENERGY LEVELS
• E RH where n 1, 2, 3
• n2
• RH 2.179 x 10-18 J
• n principle quantum number

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THE BOHR THEORY OF THE HYDROGEN ATOM

• BOHRS POSTULATE
• The stability of the atom (H2)
• The line spectrum of the atom
• TRANSITIONS BETWEEN ENERGY LEVELS An electron
  in an atom can change energy only by going from
  one energy level to another energy level. By
  doing so, the electron undergoes a transition.
• An electron goes from a higher energy level (Ei)
  to a lower energy level (Ef) emitting light
• -DE -(Ef - Ei)
• DE Ei - Ef

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THE BOHR THEORY OF THE HYDROGEN ATOM
ENERGY LEVEL DIAGRAM OF THE HYDROGEN ATOM
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXAMPLE 7.4 WHAT IS THE WAVELENGTH OF THE
LIGHT EMITTED WHEN THE ELECTRON IN A HYDROGEN
ATOM UNDERGOES A TRANSITION FROM ENERGY LEVEL n
4 TO LEVEL n 2. Ei -RH
Ef - RH
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22 DE -RH - -RH
16 4 E
-4RH 16RH -RH 4RH 3RH
hn 64
16 16
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXAMPLE 7.4 Cont

• E 3RH 3 2.179 x 10-18 J
  6.17 x 1014 s-1
• h 16 h 16 6.626 x
  10-34 J.s
• l c 3.00 x 108 m/s 4.86 x 10-7 m
• n 6.17 x 10 14 s-1
• 486 nm
• (the color is blue-green)

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THE BOHR THEORY OF THE HYDROGEN ATOM
EXERCISE 7.4 Calculate the wavelength of
light emitted from the hydrogen atom when the
electron undergoes a transition from level 3 (n
3) to level 1 (n 1).
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THE BOHR THEORY OF THE HYDROGEN ATOM
EXERCISE 7.5 What is the difference in energy
levels of the sodium atom if emitted light has a
wavelength of 589nm?
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QUANTUM MECHANICS

• Louis de Broglie (18921987) Suggested waves
  can behave as particles and particles can behave
  as waves. This is called waveparticle duality.
• m mass in kg p momentum (mc) or (mv)

The de Broglie relation
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QUANTUM MECHANICS

• EXAMPLE 7.5
• CALCULATE THE l (in m) OF THE WAVE ASSOCIATED
  WITH A 1.00 kg MASS MOVING AT 1.00km/hr.
• v 1.00 km x 1000m x 1hr x 1min
  0.278m/s
• hr 1km
  60min 60 sec
• l h 6.626 x 10-34 kg.m2/s2.s
  2.38 x 10-33m
• mv 1.00kg 0.278m/s
  

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QUANTUM MECHANICS

• EXAMPLE 7.5 cont
• B) WHAT IS THE l (in pm) ASSOCIATED WITH
  AN ELECTRON WHOSE MASS IS 9.11 x 10-31kg
  TRAVELING AT A SPEED OF 4.19 X 106 m/s ?
• h 6.626 x 10-34 kg.m2/s2.s
  
• mv 9.11 x 10-31kg 4.19 x 106
  m/s
• 1.74 x 10-10 m
• 174pm

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QUANTUM MECHANICS
EXERCISE 7.6 Calculate the l (in pm)
associated with an electron traveling at a speed
of 2.19 x 106 m/s.
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QUANTUM MECHANICS
QUANTUM MECHANICS ( WAVE MECHANICS) The branch
of physics that mathematically describes the wave
properties of submicroscopic particles UNCERTAINT
Y PRINCIPLE A relation that states that the
product of the uncertainty in position and the
uncertainty in momentum (mass times speed) of a
particle can be no smaller than Plancks constant
divided by 4p. SCHRODINGERS EQUATION Y2 gives
the probability of finding the particle within a
region of space
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Quantum Mechanics

• Niels Bohr (18851962) Described atom as
  electrons circling around a nucleus and concluded
  that electrons have specific energy levels.
• Erwin Schrödinger (18871961) Proposed quantum
  mechanical model of atom, which focuses on
  wavelike properties of electrons.

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Quantum Mechanics

• Werner Heisenberg (19011976) Showed that it is
  impossible to know (or measure) precisely both
  the position and velocity (or the momentum) at
  the same time.
• The simple act of seeing an electron would
  change its energy and therefore its position.

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Quantum Mechanics

• Erwin Schrödinger (18871961) Developed a
  compromise which calculates both the energy of an
  electron and the probability of finding an
  electron at any point in the molecule.
• This is accomplished by solving the Schrödinger
  equation, resulting in the wave function

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QUANTUM NUMBERS
According to QUANTUM MECHANICS Each electron in
an atom is described by 4 different quantum
numbers (n, l, m1 and ms). The first 3 specify
the wave function that gives the probability of
finding the electron at various points in space.
The 4th (ms) refers to a magnetic property of
electrons called spin ATOMIC ORBITAL A wave
function for an electron in an atom
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Quantum Numbers

• Wave functions describe the behavior of
  electrons.
• Each wave function contains four variables called
  quantum numbers
• Principal Quantum Number (n)
• Angular-Momentum Quantum Number (l)
• Magnetic Quantum Number (ml)
• Spin Quantum Number (ms)

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QUANTUM NUMBERS

• PRINCIPLE QUANTUM NUMBERS (n)
• This quantum number is the one on which the
  energy of the electron in an atom principally
  depends it can have any positive value (1, 2, 3
  etc..)
• The smaller n, the lower the energy.
• The size of an orbital depends on n the larger
  the
• value of n, the larger the orbital.
• Orbitals of the same quantum number (n) belong
• to the same shell which have the following
  letters
• Letter K L M N
• n
  1 2 3 4

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Quantum Numbers

• ANGULAR MOMENTUM QUANTUM NUMBER (l) Defines the
  three-dimensional shape of the orbital.
• For an orbital of principal quantum number n, the
  value of l can have an integer value from
• 0 to n 1.
• This gives the subshell notation
• Letter s p d
  f g
• l 0 1 2
  3 4

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Quantum Numbers

• Magnetic Quantum Number (ml) Defines the spatial
  orientation of the orbital.
• For orbital of angular-momentum quantum number,
  l, the value of ml has integer values from l to
  l.
• This gives a spatial orientation ofl 0 giving
  ml 0 l 1 giving ml 1, 0, 1l 2 giving
  ml 2, 1, 0, 1, 2, and so on...

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Quantum Numbers

• Magnetic Quantum Number (ml) l to l
• S orbital
• 0
• P orbital
• -1 0 1
• D orbital
• -2 -1 0
  1 2
• F orbital
• -3 -2 -1 0
  1 2 3

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Quantum Numbers
Table of Permissible Values of Quantum Numbers
for Atomic Orbitals
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Quantum Numbers

• Spin Quantum Number ms
• The Pauli Exclusion Principle states that no two
  electrons can have the same four quantum numbers.

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QUANTUM MECHANICS

• EXAMPLE 7.6 State whether each of the following
  sets of quantum numbers is permissible for an
  electron in an atom. If a set is not permissible,
  explain.
• n 1, l 1, ml 0, ms 1/2
• NOT permissible The l quantum number is equal to
  n.
• IT must be less than n.
• b) n 3, l 1, ml -2, ms -1/2
• NOT permissible The magnitude of the ml quantum
• number (that is the ml value, ignoring its sign)
  must be
• greater than l.

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QUANTUM MECHANICS

• EXAMPLE 7.6 cont
• n 2, l 1, ml 0, ms 1/2
• Permissible
• n 2, l 0, ml 0, ms 1
• NOT permissible The ms quantum number can only
  be
• 1/2 or -1/2.

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QUANTUM MECHANICS
EXERCISE 7.7 Explain why each of the following
sets of quantum numbers is not permissible for an
orbital

• n 0, l 1, ml 0, ms 1/2
• n 2, l 3, ml 0, ms -1/2
• n 3, l 2, ml 3, ms 1/2
• n 3, l 2, ml 2, ms 0

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Electron Radial Distribution

• s Orbital Shapes Holds 2 electrons

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Electron Radial Distribution

• p Orbital Shapes Holds 6 electrons, degenerate

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Electron Radial Distribution

• d and f Orbital Shapes d holds 10 electrons
  and f holds 14 electrons, degenerate

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Effective Nuclear Charge

• Electron shielding leads to energy differences
  among orbitals within a shell.
• Net nuclear charge felt by an electron is called
  the effective nuclear charge (Zeff).
• Zeff is lower than actual nuclear charge.
• Zeff increases toward nucleus
• ns gt np gt nd gt nf

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Effective Nuclear Charge
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Example1 Light and Electromagnetic Spectrum

• The red light in a laser pointer comes from a
  diode laser that has a wavelength of about 630
  nm. What is the frequency of the light? c 3 x
  108 m/s

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Example 2 Atomic Spectra

• For red light with a wavelength of about 630 nm,
  what is the energy of a single photon and one
  mole of photons?

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Example 3 WaveParticle Duality

• How fast must an electron be moving if it has a
  de Broglie wavelength of 550 nm?
• me 9.109 x 1031 kg

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Example 4 Quantum Numbers

• Why cant an electron have the following quantum
  numbers?
• (a) n 2, l 2, ml 1
• (b) n 3, l 0, ml 3
• (c) n 5, l 2, ml 1

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Example 5 Quantum Numbers

• Give orbital notations for electrons with the
  following quantum numbers
• n 2, l 1
• (b) n 4, l 3
• (c) n 3, l 2