chemistry 125: lecture 3 double minima, earnshaw’s theorem, and plum puddings the double-well...
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Chemistry 125: Lecture 3
Double Minima, Earnshaw’s Theorem, and Plum Puddings
The double-well potential of the ozone molecule and its structural equilibrium continues the discussion of
Lewis structures and chemical forces. The inability of inverse-square force laws to account for stable
arrangements of charged particles is established by Earnshaw's Theorm, which may be visualized by
means of lines of force. J.J. Thomson circumvented Earnshaw’s prohibition on Coulombic structure by
postulating a “plum-pudding” atom, with negative corpuscles embedded in a sphere of diffuse positive
charge. When Rutherford showed that the positive nucleus was not diffuse, Thomson and G.N. Lewis
concluded incorrectly that Coulomb’s law was invalid at small distances.
Synchronize when the speaker finishes saying
“…from the slides at the end last time…” Synchrony can be adjusted by using the pause(||) and run(>) controls.
For copyright notice see final page of this file
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Equilibrium vs. Resonance
H CO
O
HH C
O
O H••
••••
•• ••
••
••
••
••
••
H CO
O
H C
O
O
••••
•• •• ••
•••
•
Two Species
Two Species?
••
H C
O
O
H C
O
O
••••
•• ••
••
••
•••
•
One Nuclear Geometry!
•
•One Species!
(Evidence: Infrared Spectroscopy)
LORE
(Evidence: Electron Paramagnetic Resonance)
LORE: That which is learned; learning, scholarship, erudition.
Also, in recent use, applied to the body of traditional facts, anecdotes, or beliefs
relating to some particular subject (Oxford English Dictionary)
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2. Structures in which all first-row atoms have filled octets are generally important; however, resulting formal charges and electronegativity differences can make appropriate nonoctet structures comparably important.
From a good Text
“empirical rules for assessing the relative importance of the resonance structures of molecules and ions.
1. Resonance structures involve no change in the positions of nuclei; only electron distribution is involved.
3. The more important structures are those involving a minimum of charge separation, particularly among atoms of comparable electronegativity. Structures with negative charges assigned to electronegative atoms may also be important.”
(our depiction of) ^
LORE
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From Number of Valence Electronswe would like to predict:
Constitution (valence numbers for different atoms)
Reactivity
Charge Distribution
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O2 O3
O•
• •• •
•
• •O ••
••••O
• ••
•
EquilateralTriangle
O•
• •• •
•O•
• •• •
• O•
• •• •
•O•
• •• •
•
OO
••
O• •
• •
OO O
OO
O+
DoubleBond
Open
Trivalent O is positive.
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What is Ozone’s Structure?
OO
O
+
_
O O
O
Ring
OO
O
+
_
Open
A Problem in4 Dimensions!(3 distances + energy)
symmetrical single minimum?
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Graph Helphttps://webspace.yale.edu/chem125/125/xray/DensityMaps/3din2d.htm
Be sure you can do the problems,but you don't have to hand them in.
(Click for an answer key)
US
GS
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Ivanic, Atchity, Ruedenberg 1997
Ring
Open
4-DimensionalStructure-
EnergyPlot
3 /Constrained by assuming symmetry
RR12 = R23
••••
Requires
e.g. R12, R23, Energy
Energy Contours
“Steepest-Descent” Path
Pass Between Valleys
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O3
More Constrained4-Dimensional
Structure-Energy
Plot
Distance along Steepest-Descent Curve
Ene
rgy
(kca
l/m
ol)
Ring
Open
0
8
2 /
R12 ≠ R23
gives higher E
symmetrical"resonant”structure
Pass
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Ozone
What of charge distribution is “predicted” by
Lewis bookkeeping?
+ in middle
- on ends?
OO
O
+
_
OO
O
+
_
Open
symmetrical single minimum?
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*) Energy of a proton on the “molecular surface”
Suface Potential* of Open Ozone (from Quantum-Mechanics)
HIGH(+ 25 kcal/mole)
(-16 kcal/mole)
LOW
+ in middle
- on ends?
YES!
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From Number of Valence Electronswe would like to predict:
Constitution (valence numbers for different atoms)
Structure (distances & angles) (we’ll test this later)
Energy Content (we’ll test this later)
Reactivity (at least for H3N: BH3)
Charge Distribution (at least qualitatively for O3, H3N-BH3)
~
~
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Lewis Dot Structure
Attempts to provide a “physical” basis for valence rules.
New: Reactivity from unshared pairs(both “hooks” from the same atom)
Convenient for electron bookkeeping(molecular charge; “formal” atomic charges;
qualitatively realistic, at least in the case of O3)
Stability and “Resonance”?
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What’s so great about octets?How bad are sestets?
How bad are structures with formal charge separation?
How bad is “bad” charge separation?from Wiki: “I have a question when drawing these structures. Is it more ‘important’ to try to fill the octet or to have lowest formal charge on
as many atoms, especially C, as possible? and WHY?”
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Is it at all True?
Force Laws?
Are there e-pairsbetween nuclei and
unshared on some atoms?
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In systems governed byinverse-square force laws
there can be no local minimum (or maximum)
of potential energy.
Earnshaw's Theorem(1839)
by p
erm
issi
on S
heff
ield
Uni
vers
ity
Samuel Earnshaw (1805-1888)
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Visualizing Earnshaw - Coulomb's Electrostatics
“Lines of Force”MagneticElectrostatic
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Faraday/Davy/Phillipsyoung
Michael Faraday
by p
erm
issi
on A
lfre
d B
ader
Col
lect
ion
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Can show magnitude (as well as direction) of Force
2-D (Flatland)
force magnitude
line density
Circumference r2
Force
line density
1/r
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Can show magnitude (as well as direction) of Force
3-Dimensions
Surface r2
Force
line density
1/r2
force magnitude
line density
In 3D such Diagrams Work only for Inverse Square Forces!
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A positive particle has a local maximum or minimum of energy only at the location of
another charged particle, never in free space.
A positive particle has a local maximum or minimum of energy (peak or valley) only at the location of another charged particle,
never in free space.
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In systems governed byinverse-square force laws
there can be no local minimum (or maximum)
of potential energy in free space.
Earnshaw's Theorem
(The only “stationary” points are saddle points.)
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Levitator by Martin Simon (UCLA)
Eppur sta fermo“and yet it stands still”
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J.J. Thomson (1856-1940)
Electron (1897) Plum-Pudding Atom
© C
ave
ndi
sh L
abo
ratp
ry,
Cam
brid
ge U
nive
rsity
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"[We can] solve the special case where the corpuscles are
confined to a plane."
Thomson's Model of Electron Configuration
"consider the problem as to how 1…2…3…n corpuscles would arrange themselves if placed in a sphere filled with positive electricity of uniform density…"
“distributed in the way most amenable to
mathematical calculation”
in
Thomson, Corpuscular Theory of Matter (1907)
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Vortex Lattice Models (Greg Blonder www.genuineideas.com)
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"[We can] solve the special case where the corpuscles are
confined to a plane."
Thomson's Model of Electron Configuration
"consider the problem as to how 1…2…3…n corpuscles would arrange themselves if placed in a sphere filled with positive electricity of uniform density…"
"the equilibrium of eight corpuscles at the corners of a cube is unstable."
in
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“I have ever since regarded [the cubic octet]as representing essentially the arrangement
of electrons in the atom” G. N. Lewis (1923)
Was Lewis ignorant of Earnshaw's Theorem?
“Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances.”
G. N. Lewis (1916)
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The Electron in Chemistry
J. J. Thomson (1923)
“… if [electron-nuclear attraction] were to vary strictly as the inverse square of the distance we know by Earnshaw's theorem than no stable configuration in which the electrons are at rest or oscillating about positions of equilibrium is possible ...
Couloumbr
cr
c
… then a number of electrons can be in equilibrium about a positive charge without necessarily describing orbits around it.”
I shall assume that the law of force between a positive charge and an electron is expressed by the equation
F =Ee
r2 1
atomic length scale(for distances r smaller than c,
the force changes sign.)
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End of Lecture 3Sept 8, 2008
Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
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The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0