Transcript Document

Chapter 7
Quantum Theory of the Atom
John A. Schreifels
Chemistry 211
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Overview
• Light Waves, Photons, and the Bohr Theory
– Wave Nature of Light
– Quantum Effects and Photons
– Bohr Theory: Hydrogen and Hydrogen-like atoms
• Quantum Mechanics and Quantum Numbers
– Quantum Mechanics
– Quantum Numbers and Atomic Orbitals
John A. Schreifels
Chemistry 211
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The Wave Nature of the Light
• Atomic structure elucidated by interaction of matter with light.
• Light properties: characterized by wavelength, , and
frequency,.
• Light = electromagnetic radiation, a wave of oscillating electric
and magnetic influences called fields.
• Frequency and wavelength inversely proportional to each other.
c = 
where c = the speed of light = 3.00x108 m/s; units  = s1,  = m
E.g. calculate the frequency of light with a wavelength of 500 nm.
E.g.2 calculate the frequency of light if the wavelength is 400 nm.
John A. Schreifels
Chemistry 211
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Electromagnetic Radiation and
Atomic Spectra - 2
• Line spectra result from the emission of radiation
from an excited atom.
• Spectrum: characteristic pattern of wavelengths
absorbed (or emitted) by a substance.
• Emission Spectrum: spontaneous emission of
radiation from an excited atom or molecule.
• Line Spectrum: spectrum containing only certain
wavelengths.
• Balmer : hydrogen has a line spectrum in the visible
region with wavelengths of 656.3 nm, 486.1 nm,
434.0 nm, 410.1 nm. 
1  R  1  1 
• Balmer equation: 
where n = 3.

4 2

John A. Schreifels
Chemistry 211
n

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Quantized Energy and Photons
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Light = wave arriving as stream of particles called "photons".
Each photon = quantum of energy
E  hν  hc
λ
where h (Planck's constant) = 6.63x1034J*s.
An increase in the frequency = an increase in the energy
An increase in the wavelength gives an decrease in the energy of the
photon.
E.g. determine the energies of photons with
wavelengths of 650 nm, 700 nm and
frequencies 4.50x1014 s1, 6.50x1014 s1
Photoelectric effect: E = h   where  = constant
the energy of the electron is directly related to the energy of the photon.
the threshold of energy must be exceeded for electron emission.
The total energy of a stream of particles (photons) of that energy will
be:
E
 n  h where n = 1, 2, …(only discrete energies).
total
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Chemistry 211
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Bohr’s Model of the Hydrogen
Atom
• Line spectra in other spectral regions also were observed:
• Lyman series
ultraviolet
• Paschen, Brackett, Pfund
infrared
• Balmer-Rydberg equation predicted the wavelengths of
 1
emission.
1
1 


 R
2
m

n2 
where m = 1, 2, 3,… and n = 2, 3, …(always at least m + 1
• Longest wavelength observed when n = m + 1.
• Shortest wavelength observed when n = .
E.g. Determine the wavelength of emission for the first line in the
Paschen series (m = 3, n = 4).
E.g. Determine the shortest wavelength in the Paschen series (m =
3 and n = ).
John A. Schreifels
Chemistry 211
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Bohr model of the Hydrogen
Atom II
• Duality of matter led to the hypothesis that electrons behave as
waves.
• Bohr model assumed
– Only circular orbits around the nucleus and that the angular
momentum around the atom must be quantized.
– Stable orbital where constructive interference occurs.
• Assumption led to the conclusions:
– Radius of an orbital: rn = n2r1.
– Energy of an orbital: En = E1/n2 = 21.93x1019J/n2 where E1 =
energy of the most stable hydrogen orbital. E1<E2<E3.
– Most stable state E1,r1 = ground state.
– Higher energy states = excited states.
• A photon is emitted when the electron moves from a higher
energy state to a lower one.
• Photon energy equals the difference in energy of the two states.
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Chemistry 211
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Bohr model of the Hydrogen
Atom III
E E
• If Ei = the initial state energy
and Ef = final state energy,
then the energy of the
transition would be: E = Ei 
Ef.
R = Rydberg constant =
1.097x107m1.
• Theory and experiment agree
for hydrogen and hydrogenlike particles.
John A. Schreifels
Chemistry 211
E 
1
ni2
n2f
1






hc  E  1  1
1 n2 n2

i
f




1  E1   1  1 
 hc  n2 ni2 
f









2 
i 
 R 1  1
n2f n
8
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Wave Nature of Matter
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Light behaves like matter since it can only have certain energies.
Light had both wave- and particle-like properties  matter did too.
Einstein equation helps describe the duality of light:
E = mc2
Particle behavior
E = h
Wave behavior
Wave and particle behavior
h  mc 2
hc
 mc 2

h
m
c
Duality of matter expressed by replacing the speed of light with the
speed of the particle to get:
h
  mv
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where  called the de Broglie wavelength of any moving particle.
E.g. determine the de Broglie wavelength of a person with a mass
of 90 kg who is running 10 m/s.
John A. Schreifels
Chemistry 211
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Quantum Mechanics:
Hydrogen
Bohr model did not work with multielectron atoms,
i.e. line spectra not predicted.
Quantum mechanics provides universal
description of the electron distribution in atoms.
Heisenberg uncertainty principle = impossible to
determine the position and momentum with
absolute precision or (position
uncertainty)(momentum uncertainty) 
x  (mv)  h
4
Schroedinger used wave concepts to derive the
wave equation.Electrons allowed to be in
anywhere.
Solution of the Schroedinger three dimensional
wave equation, , led to the discrete energy levels
of the hydrogen atom.
Lowest level is spherical.
Predicts distribution of electrons in other elements.
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Chemistry 211
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Quantum Mechanics and
Atomic Orbitals
The first orbital of all elements is spherical.
Other orbitals have a characteristic shape and position as described by
4 quantum numbers: n,l,ml,ms. All are integers except ms
Principal Quantum Number (n): an integer from 1... Total # e in a
shell = n2.
Angular quantum number (l). (permitted values l = 0 to n1): the
subshell shape.
– Common usage for l = 0, 1, 2, 3, 4, and use s, p, d, f, g,... respectively.
– Subshell described as 1s, 2s, 2p, etc.
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Magnetic quantum number,ml, (allowed l to +l ) directionality of an l
subshell orbital.
– Total number of possible orbitals is 2l+1.
– E.g. s and p subshells have 1 & 3 orbitals, respectively.
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Spin quantum number,ms (allowed values 1/2). Due to induced
magnetic fields from rotating electrons.
Pauli exclusion principle: no two electrons in an atom can have the
same four quantum numbers.
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Chemistry 211
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Permissible Quantum States
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Chemistry 211
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Figure 7.23: Orbital energies of the
hydrogen atom.
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Chemistry 211
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Orbital Energies of Multielectron
Atoms
• All elements have the same number of
orbitals (s,p, d, and etc.).
• In hydrogen these orbitals all have the same
energy.
• In other elements there are slight orbital
energy differences as a result of the presence
of other electrons in the atom.
• The presence of more than one electron
changes the energy of the electron orbitals
(click here)
John A. Schreifels
Chemistry 211
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Shape of 1s Orbital
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Chemistry 211
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Shape of 2p Orbital
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Chemistry 211
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Shape of 3d Orbitals
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Chemistry 211
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