Matter and Measurement

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Light oscillating electric and magnetic fields -
electromagnetic (EM) radiation - travelling
wave Characterize a wave by its wavelength, l, or
frequency, n c l n Black-body
radiation Planck vibrating atoms in a heated
object give rise to the emitted EM radiation
these vibrations are quantized Energy of
vibration h n Photoelectric effect Einstein
light consists of massless particles called
photons Ephoton h n h c / l E of a mole of
photons No h n
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Calculate the energy of each photon of blue light
of frequency 6.40 x 1014 Hz. What is the energy
of a mole of photons of this frequency? E per
photon h n (6.626 x 10-34 J s) (6.40 x 1014
s-1) 4.20 x 10-19 J E for a mole of
photons 4.20 x 10-19 J x 6.023 x
1023/mole 2.53 x 105 J/mole or 253 kJ/mol
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Atomic line spectra
(nm)
For the H atom
n1 3, 4, n2 n1 1, n1 2, ...
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The Bohr Model of the Atom explains spectra of
one-electron atoms such as H, He, Li2. Also
accounts for the stability of the atom.
Classical physics prediction
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Bohr theory 1) Quantization of angular momentum
For electrons, only those orbits, or energy state
that have certain values of angular momentum are
allowed Allowed orbits angular momentum n h /
2 p (n 1, 2, 3, ..) 2) As long as an electron
stays in an allowed orbit it does not absorb or
emit energy 3) Emission or absorption occurs only
during transitions between allowed orbits. The
emission and absorption are observed in spectra.
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• From the Bohr model
• angular momentum
• Radius of allowed orbitals
• Z positive charge on the nucleus
• ao Bohr radius 5.29177 x 10-11 m
• Total energy of electron in a stable orbital

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Electrons can move from one allowed orbit to
another, changing n is the energy absorbed or
released equals the energy difference between
allowed orbits of two different n values
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• An electron in an atom can exist only in a series
  of discrete levels
• When an electron makes a transition, its energy
  changes from one of these levels to another.
• The difference in energy DE Eupper - Elower is
  carried away as a photon
• The frequency of the photon emitted, n DE/h

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Energy from light of frequency n can also be
absorbed by the H atoms if h n DE Eupper -
Elower In this case, an electron in a lower
energy level is excited to an upper energy
level. If h n ? Eupper - Elower the electron
cannot undergo a transition. The frequency of
an individual spectral line is related to the
energy difference between the two levels h n
DE Eupper - Elower
animation
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The observation of discrete spectral lines
suggest that an electron in an atom can have only
certain energies/ Electron transitions between
energy levels result in emission or absorbption
of photons in accord with the Bohr frequency
condition.
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Lyman series n1 1 Balmer series n1
2 Paschen series n1 3 Brackett series n1 4
Pfund series n1 5
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Problems with Bohrs theory Could not be used to
determine energies of atoms with more than one
electron. Unable to explain fine structure
observed in H atom spectra Cannot be used to
understand bonding in molecules, nor can it be
used to calculate energies of even the simplest
molecules. Bohrs model based on classical
mechanics, used a quantization restriction on a
classical model.
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Wave-Particle Duality

• Light is a traveling wave.
• Wave properties demonstrated through interference
  and diffraction

Interference of light
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Diffraction of light
Einsteins experiments are explained by the
particle nature of light - photons Light
behaves as both a wave and a particle
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Wave-Particle Duality of Matter

• Louis de Broglie suggested that all particles
  should have wave like properties.
• Wavelength associated with a particle of mass m
  and speed v

where p m v, the linear momentum of the particle
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For a golf ball of mass 1.62 ounces (0.0459 kg),
propelled at an average speed of 150. Mi/hr (67.1
m/s), the deBroglie wavelength is 2.15 x 10-34 m
too small to be measured Wavelength of an
electron (me 9.11 x 10-31 kg) moving at a speed
of 3.00 x 107 m/s is 2.42 x 10-11 m corresponds
to the X-ray region of the electromagnetic
spectrum
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Electron diffraction pattern (Davisson, Germer,
Thompson, 1927)
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The Uncertainty Principle

• In classical mechanics a particle has a definite
  path, or trajectory, on which location and linear
  momentum are specified at each instant.
• However, if a particle behaves as a wave, its
  precise location cannot be specified.
• Heisenbergs uncertainty principle impossible to
  fix both the position of an electron in an atom
  and its energy with any degree of certainty if
  the electron is described by a wave

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• Heisenberg uncertainty principle if the location
  of a particle is known to within an uncertainty
  of Dx, then the linear momentum parallel to the x
  axis can be known only to within Dp, where

Max Born If the energy of an electron in an atom
is known with a small uncertainty, then there
must be a large uncertainty in its position. Can
only assess the probability of finding an
electron with a given energy within a given
region of space
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Wave Mechanics
E. Schrödinger Electron in an atom could be
described by equations of wave motion
Standing waves vibrations set up by plucking a
string stretched taut between two fixed pegs Ends
are fixed l of allowed oscillations satisfy
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n2 2 n1
third harmonic
n3 3n1
second harmonic
fundamental or first harmonic
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Standing waves - example of quantization only
certain discrete states allowed Electrons are
wave-like and exist in stable standing waves,
called stationary states, about the nucleus. Wave
mechanics depicts the electron as a wave-packet
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Stationary states with the circumference being
divisible into an integral number of
wavelengths From wave mechanics can show that
allowed stationary states satisfy the condition
me v r
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Wavefunctions and Energy Levels

• Since particles have wavelike properties cannot
  expect them to behave like point-like objects
  moving along precise trajectories.
• Erwin Schrödinger Replace the precise trajectory
  of particles by a wavefunction (y), a
  mathematical function that varies with position
• Max Born physical interpretation of
  wavefunctions. Probability of finding a particle
  in a region is proportional to y2.

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• y2 is the probability density. To calculate the
  probability that a particle is in a small region
  in space multiply y2 by the volume of the region.
  
• Probability y2 (x,y,z) dx dy dz

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Schrödinger Equation

• The Schrödinger equation describes the motion of
  a particle of mass m moving in a region where the
  potential energy is described by V(x).

(1-dimension)
Only certain wave functions are allowed for the
electron in an atom The solutions to the equation
defines the wavefunctions and energies of the
allowed states
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• An outcome of Schrödingers equation is that the
  particle can only possess certain values of
  energy, i.e. energy of a particle is quantized.
• For example, one of the simplest example is that
  of a particle confined between two rigid walls a
  distance of L apart - particle in a box
• Only certain wavelengths can exist in the box,
  just as a stretched guitar string can support
  only certain wavelengths.

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yn (x) (2 / L)1/2 sin (n p x / L) n 1, 2,
.. n is called the quantum number Energy of
the particle is quantized, restricted to discrete
values, called energy level. En n2 h2 / (8 m
L2) Also, a particle in the container cannot
have zero-energy - zero-point energy At the
lowest level, n 1, E1 gt 0
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The probability density for a particle at a
location is proportional to the square of the
wavefunction at that point
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• To find the energy levels of an electron in the H
  atom, solve the Schrödinger equation.
• In the H atom the potential that the electron
  feels is the electrostatic interaction between it
  and the positive nucleus
• V(r ) - e2 / (4 p eo r)
• r distance between the electron and the nucleus.
• Solution for allowed energy levels is

R (me e4) / (8 h3 eo2) 3.29 x 1015 Hz
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• Showed that quantum mechanics did indeed explain
  behavior of the electron
• Note that the energy of an electron bound in a H
  atom is always lower than that of a free electron
  (as indicated by the negative sign)

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n principle quantum number. Labels the energy
levels When n 1 gt ground state of the H atom.
Electron in its lowest energy n gt 1 excited
states energy increases as n increases E 0
when n 8 , electron has left the atom -
ionization