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Quantum numbers
B Visegrády
The Photoelectric Effect
•  it was observed that many metals emit electrons when a
light shines on their surface
 this is called the Photoelectric Effect
•  classic wave theory attributed this effect to the light
energy being transferred to the electron
•  according to this theory, if the wavelength of light is
made shorter, or the light waves intensity made
brighter, more electrons should be ejected
 remember: the energy of a wave is directly proportional to its
amplitude and its frequency
 if a dim light was used there would be a lag time before
electrons were emitted
 to give the electrons time to absorb enough energy
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The Photoelectric Effect
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The Photoelectric Effect
The Problem
•  in experiments with the photoelectric effect, it
was observed that there was a maximum
wavelength for electrons to be emitted
 called the threshold frequency
 regardless of the intensity
•  it was also observed that high frequency light
with a dim source caused electron emission
without any lag time
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Einstein’s Explanation
•  Einstein proposed that the light energy was
delivered to the atoms in packets, called quanta
or photons
•  the energy of a photon of light was directly
proportional to its frequency
 inversely proportional to its wavelength
 the proportionality constant is called Planck’s
Constant, (h) and has the value 6.626 x 10-34 J·s
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Ejected Electrons
•  1 photon at the threshold frequency has just
enough energy for an electron to escape the atom
 binding energy, φ
•  for higher frequencies, the electron absorbs more
energy than is necessary to escape
•  this excess energy becomes kinetic energy of the
ejected electron
Kinetic Energy = Ephoton – Ebinding
KE = hν - φ
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Spectra
•  when atoms or molecules absorb energy, that energy is
often released as light energy
 fireworks, neon lights, etc.
•  when that light is passed through a prism, a pattern is
seen that is unique to that type of atom or molecule –
the pattern is called an emission spectrum
 non-continuous
 can be used to identify the material
 flame tests
•  Rydberg analyzed the spectrum of hydrogen and found
that it could be described with an equation that
involved an inverse square of integers
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Emission Spectra
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Exciting Gas Atoms to Emit Light
with Electrical Energy
Hg
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He
H
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Examples of Spectra
Oxygen spectrum
Neon spectrum
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Identifying Elements with
Flame Tests
Na
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K
Li
Ba
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Emission vs. Absorption Spectra
Spectra of Mercury
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Bohr’s Model
•  Neils Bohr proposed that the electrons could only have
very specific amounts of energy
 fixed amounts = quantized
•  the electrons traveled in orbits that were a fixed
distance from the nucleus
 stationary states
 therefore the energy of the electron was proportional the
distance the orbital was from the nucleus
•  electrons emitted radiation when they “jumped” from
an orbit with higher energy down to an orbit with lower
energy
 the distance between the orbits determined the energy of the
photon of light produced
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Bohr Model of H Atoms
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Wave Behavior of Electrons
•  de Broglie proposed that particles could have wave-like
character
•  because it is so small, the wave character of electrons is
significant
•  electron beams shot at slits show an interference
pattern
 the electron interferes with its own wave
•  de Broglie predicted that the wavelength of a particle
was inversely proportional to its momentum
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Electron Diffraction
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if electrons behave like
particles,
should
however,there
electrons
actually
only
be two
bright spots
present
an interference
onpattern,
the target
demonstrating the
behave like waves
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Complimentary Properties
•  when you try to observe the wave nature of the
electron, you cannot observe its particle nature –
and visa versa
 wave nature = interference pattern
 particle nature = position, which slit it is passing
through
•  the wave and particle nature of nature of the
electron are complimentary properties
 as you know more about one you know less about
the other
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Uncertainty Principle
•  Heisenberg stated that the product of the uncertainties
in both the position and speed of a particle was
inversely proportional to its mass
 x = position, Δx = uncertainty in position
 v = velocity, Δv = uncertainty in velocity
 m = mass
•  the means that the more accurately you know the
position of a small particle, like an electron, the less
you know about its speed
 and vice versa
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The Franck-Hertz experiment
•  The electrons were accelerated by an electric field (Figure 4.) between the
cathode (C) and the grid (G) in a tube filled with Hg vapour. After
passing through the grid, an opposite field slowed down the electrons and
prevented them from reaching the anode (A), unless they gained enough
kinetic energy in the previous acceleration. The electrons reaching the
anode formed a measurable current, I.
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The Franck-Hertz experiment
•  The Hg atoms can not absorb any energy, just well defined values, namely
4.9 eV. This energy excites the ground state electron to the first excited state,
e.g. it equals exactly to the energy difference between the E1 and E2 states of
the Hg atom. Thus, this experiment gives an excellent proof of the validity of
Bohr’s theory.
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Wave Function, ψ
•  calculations show that the size, shape and
orientation in space of an orbital are determined
by three integer terms in the wave function
 added to quantize the energy of the electron
•  these integers are called quantum numbers
 principal quantum number, n
 angular momentum quantum number, l
 magnetic quantum number, ml
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orbital
requires
n
l
3 quantum numbers
ml
magnetic
-l, …, l
orientation
angular momentum
0, 1, 2, …, (n - 1)
shape
Principal
1, 2, 3, …
size and energy
“address”
Principal Quantum Number, n
•  characterizes the energy of the electron in a particular
orbital
 corresponds to Bohr’s energy level
•  n can be any integer ≥ 1
•  the larger the value of n, the more energy the orbital has
•  energies are defined as being negative
 an electron would have E = 0 when it just escapes the atom
•  the larger the value of n, the larger the orbital
•  as n gets larger, the amount of energy between orbitals
gets smaller
for an electron in H
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Principal Energy Levels in Hydrogen
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Probability & Radial Distribution
Functions
•  ψ2 is the probability density
 the probability of finding an electron at a particular point in
space
 for s orbital maximum at the nucleus?
 decreases as you move away from the nucleus
•  the Radial Distribution function represents the total
probability at a certain distance from the nucleus
 maximum at most probable radius
•  nodes in the functions are where the probability drops to 0
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Probability Density Function
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Radial Distribution Function
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The Shapes of Atomic Orbitals
•  the l quantum number primarily determines the
shape of the orbital
•  l can have integer values from 0 to (n – 1)
•  each value of l is called by a particular letter that
designates the shape of the orbital
 s orbitals are spherical
 p orbitals are like two balloons tied at the knots
 d orbitals are mainly like 4 balloons tied at the knot
 f orbitals are mainly like 8 balloons tied at the knot
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l = 0, the s orbital
•  each principal energy state
has 1 s orbital
•  lowest energy orbital in a
principal energy state
•  spherical
•  number of nodes = (n – 1)
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2s and 3s
2s
n = 2,
l=0
3s
n = 3,
l=0
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l = 1, p orbitals
•  each principal energy state above n = 1 has 3 p orbitals
 ml = -1, 0, +1
•  each of the 3 orbitals point along a different axis
 px, py, pz
•  2nd lowest energy orbitals in a principal energy state
•  two-lobed
•  node at the nucleus, total of n nodes
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p orbitals
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l = 2, d orbitals
•  each principal energy state above n = 2 has 5 d orbitals
 ml = -2, -1, 0, +1, +2
•  4 of the 5 orbitals are aligned in a different plane
 the fifth is aligned with the z axis, dz squared
 dxy, dyz, dxz, dx squared – y squared
•  3rd lowest energy orbitals in a principal energy state
•  mainly 4-lobed
 one is two-lobed with a toroid
•  planar nodes
 higher principal levels also have spherical nodes
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d orbitals
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l = 3, f orbitals
•  each principal energy state above n = 3 has 7 f orbitals
 ml = -3, -2, -1, 0, +1, +2, +3
•  4th lowest energy orbitals in a principal energy state
•  mainly 8-lobed
 some 2-lobed with a toroid
•  planar nodes
 higher principal levels also have spherical nodes
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f orbitals
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Stern-Gerlach experiment
Particles possess intrinsic angular momentum.
Spin angular momentum is quantized (it can only take on
discrete values)
For the completely filled shells,
subshell (4d10) the orbital magnetic
momentum is zero; for the 5s orbital
ML is also zero.
Hypothesis: the argent atom possesses
no magnetic momentum >> they move
in an inhomogeneous magnetic field
along the same path.
The electron possesses an intrinsic angular momentum and an
intrinsic magnetic momentum: spin
S — intrinsic angular momentum
Ms — intrinsic magnetic momentum