Chem 14A

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Chem 14A

• Friday, June 29, 2007

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Lecture Outline

• Brief review of quantum theory experiments
• Photoelectric effect
• Wave-particle Duality
• Heisenberg Uncertainty Principle
• Schrödinger Equation
• Atomic Spectra and Energy

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Early Quantum Mechanics

• Max Planck postulated that energy is quantized
  based on blackbody radiation experiments where
  continuous energy absorption was not seen
• Einstein used this information to develop a
  theory that light is also quantized, or delivered
  in photons, particle-like energy packets with
  energy value Ehn

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Early Quantum Mechanics

• Millikan performed a series of experiments to
  prove Einsteins theory that light is quantized
  by using the photoelectric effect

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Photoelectric Effect

• A Metal surface is illuminated with incident
  radiation, exciting electrons which are ejected
  from the surface with kinetic energy

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Photoelectric Effect
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Photoelectric Effect

• Electrons were only ejected under certain
  conditions
• Radiation frequency must be above a certain level
  (threshold value)
• Electrons ejected immediately without regard to
  radiation intensity (number of photons)
• Kinetic energy of ejected electrons increases
  linearly with frequency of incident radiation

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Photoelectric Effect

• What do the results mean?
• If light is wave-like, the energy contained in
  one of those waves should depend only on its
  amplitude--that is, on the intensity of the
  light. Lower intensity light would give fewer
  electrons ejected more slowly.
• If light is particulate, changing the frequency
  will result in different energy values for the
  electrons, but a different intensity will have no
  effect, since each particle has the same energy
  value

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Photoelectric Effect

• As a result of the photoelectric effect,
  Einsteins theory that electromagnetic radiation
  is quantized was validated, and he was awarded
  the Nobel prize.
• Light now has a particulate nature!

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Work Function

• Photons of light collide with the metal surface
  electrons, ejecting them
• Energy required to eject en electron from the
  surface work function (F)
• Energy of the colliding photon Ehn

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Work Function

• The work function describes the difference in
  energy between the photon itself and the energy
  needed to eject the electron
• Elt F
• If the photons energy is smaller than the energy
  needed to eject the electron, the electron will
  never be ejected, regardless of the light
  intensity (number of photons)

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Work Function

• EgtF
• If the photons energy is greater than the energy
  needed to eject an electron, the electron is
  ejected with positive kinetic energy KE1/2mv2
• KE E- F

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Work Function

• KE E- F
• Substituting into the equation gives
• 1/2mv2hn- F

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Quick Exercise I.

• It takes 7.21x10-19J of energy to remove an
  electron from an iron atom. What is the maximum
  wavelength of light that can accomplish this?
• 1/2mv2hn- F
• ?c/n
• c 3.00x108 m/s

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Answers to Quick Exercise I.

• 1/2mv2hn- F
• We want to find the smallest possible frequency
  in order to get the largest possible wavelength,
  since ?c/n

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Answers to Quick Exercise I.

• Re-writing the equation in terms of frequency,
  and solving for the lowest possible frequency
  value by setting the velocity to zero

1/2mv2 F
1/2m(0) 7.21x10-19 J
n
n
6.626x10-34 Js
h
h
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Answers to Quick Exercise I.

• Solve for the lowest frequency value
• Then plug into equation for wavelength

n 7.21x10-19J
3.00x108 m/s
1.088x1015 s-1
?
1.088x1015 s-1
6.626x10-34 Js
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Answers to Quick Exercise I.

• Wavelength 2.76x10-7 m or 276 nm

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Wave-Particle Duality

• Electromagnetic Radiation also has evidence of
  wave-like properties
• Diffraction Light will show a diffraction
  pattern when passed through a diffraction grating

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Wave-Particle Duality

• What is diffraction?
• Diffraction is a property unique to waves, based
  on the addition and subtraction of several wave
  forms.
• Constructive Interference when waves are added
  together the amplitude increases
• Destructive Interference when waves are
  subtracted from each other the amplitude decreases

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Wave-Particle Duality
combinedwaveform
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Wave-Particle Duality

• The collection of high and low intensities
  formed from wave constructive and destructive
  interference produces a pattern of lines behind
  the diffraction grating

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Wave-Particle Duality

• Duality applies to electromagnetic radiation as
  seen through experiments, what about matter?
• DeBroglie proposed that all particles have
  wave-like properties too
• The DeBroglie Equation ?h/mv
• mvp linear momentum

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Wave-Particle Duality

• Wave-like properties of matter are strongly mass
  dependent
• Using the DeBroglie Equation
• Baseball wavelength
• ?h/mv
• 6.626x10-34/(.1kg)(35m/s)1.9X10-34 m

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Wave-Particle Duality

• Electron wavelength
• ?h/mv
• 6.626x10-34/(9.1x10-31kg)(1x107m/s)
• 7.3X10-11 m
• It is hard to observe wave properties of
  macroscopic objects, but they do exist

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Quick Exercise II.

• Calculate the de Broglie wavelength for
• A proton with a velocity 5 of the speed of light
• An electron with a velocity 15 of the speed of
  light
• c3x108 m/s ?h/mv
• me 9.1x10-31kg h 6.626x10-34
  Js
• mp 1.67x10-27kg

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Answers to Quick Exercise II.

• 1. 2.6x10-5 nm
• 2. 1.6x10-2 nm

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Heisenberg Uncertainty Principle

• Matter has both wave and particulate properties
• Waves are delocalized- Their exact location
  cannot be described
• Particles are localized- Their exact location is
  easily quantifiable

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Heisenberg Uncertainty Principle

• Since matter has both wave and particulate
  properties, how do we define its location and
  momentum?

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Heisenberg Uncertainty Principle

• We cannot know both the position and momentum of
  a particle to a high degree of precision at the
  same time
• The more precisely we know a particles position,
  the less precisely we know its momentum and vice
  versa
• ?x?ph/4p

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Heisenberg Uncertainty Principle

• For large particles, such as a baseball the level
  of uncertainty is small.
• For small particles, such as an electron, the
  level of uncertainty is large- we dont know the
  exact path an electron travels in around the
  nucleus (orbitals are 90 probability
  representations)

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Quick Exercise III.

• The hydrogen atom has a radius on the order of
  0.05 nm. Assuming we know the position of an
  electron to an accuracy of 1 of the radius,
  calculate the uncertainty in the velocity of the
  electron.
• me9.11x10-31 kg
• h 6.626x10-34 Js

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Answer to Quick Exercise III.

• ?x?ph/4p
• ?x 0.01x0.05nm 0.0005nm 5x10-13m
• ?ph/4p?x 1.05x10-22 kg m/s
• ?pm?v
• ?v ?p/m(1.05x10-22 kg m/s)/9.11x10-31kg
• ?v 1.15X108 m/s

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The Schrödinger Equation

• Austrian Physicist Erwin Schrödinger adopted the
  use of wave functions to describe the path of a
  particle, since it has wave-like character
• A wave function ? is a mathematical function of
  the position of a particle in three-dimensional
  space (x,y,z)

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The Schrödinger Equation

• A wave function squared describes the probablity
  density the probability of finding a particle
  within a certain space
• Schrodinger developed an equation for calculating
  wave functions
• H? E? where HHamiltonian operator

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The Schrödinger Equation

• Solutions to the Schrodinger Equation
• There are many different solutions to the
  Schrödinger equation
• Each solution to the Schrödinger equation
  represents an orbital
• For example, There are separate wave functions
  for the H 1s, 2s, and 2p orbitals

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Atomic Spectra and Energy Levels

• Atomic spectra give further evidence that
  energy is quantized
• Emission spectra are produced when light emitted
  by atoms is passed through a prism
• White light passed through a prism gives a
  continuous spectrum (rainbow)
• Light emitted by excited atoms (ie hydrogen)
  gives discrete spectral lines

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Atomic Spectra and Energy Levels

• Absorption spectra are created by shining light
  through an atomic vapor
• Absorption spectra exhibit discrete quantities of
  energy being absorbed by an atom and give the
  same spectra lines as emission spectra, only they
  are black against a rainbow background

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Atomic Spectra and Energy Levels
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Atomic Spectra and Energy Levels

• Because an atom can emit only certain wavelengths
  of light as indicated by the discrete spectral
  lines, energy is quantized
• Atoms lose energy in specific amounts, implying
  that electrons are present only in certain energy
  states
• During an electronic transition, an electron
  moves from a higher energy level to a lower
  energy level and releases energy in terms of a
  photon

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Atomic Spectra and Energy Levels

• Bohr Frequency Condition ?Ehn transition energy
  released as a photon
• Since each line on the spectrum corresponds to a
  particular electronic transition, the spectrum
  can be used to develop an energy-level diagram

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Atomic Spectra and Energy Levels

• Rydberg constant all spectral lines
• ?R(1/n12 1/n22) n1 1,2 n2 n11, n12
• RRydberg constant3.29x1015 Hz
• Balmer series visible spectral lines
• n12 and n23,4,
• Lyman series ultraviolet spectral lines
• n11 and n22,3,

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Atomic Spectra and Energy Levels

• Using the Rydberg equation to identify spectral
  lines
• Calculate the wavelength of radiation emitted by
  a hydrogen atom during an electronic transition
  with n23 and n12
• What color is the light?

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Atomic Spectra and Energy Levels

• nR(1/n12 1/n22) R3.29x1015 Hz
• nR(1/22 1/32) 4.57x1014 Hz
• ?c/n (3x108m/s)/(4.57x1014 Hz)
• ?6.57x10-7m or 657nm red light

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Quick Exercise IV.

• Calculate the wavelength for a transition between
  n25 and n11
• What region of the electromagnetic spectrum is
  the light found in ?
• nR(1/n12 1/n22) R3.29x1015 Hz
• ?c/n

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Answers to Quick Exercise IV.

• Wavelength 94.9nm Lyman Series
• Ultraviolet region
• nR(1/12 1/52) 3.158x1015 Hz
• ?c/n (3x108m/s)/(3.158x1015 Hz)