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

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)

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

From Number of Valence Electronswe would like to predict:

Constitution (valence numbers for different atoms)

Reactivity

Charge Distribution

O2 O3

O•

• •• •

•

• •O ••

••••O

• ••

•

EquilateralTriangle

O•

• •• •

•O•

• •• •

• O•

• •• •

•O•

• •• •

•

OO

••

O• •

• •

OO O

OO

O+

DoubleBond

Open

Trivalent O is positive.

What is Ozone’s Structure?

OO

O

+

_

O O

O

Ring

OO

O

+

_

Open

A Problem in4 Dimensions!(3 distances + energy)

symmetrical single minimum?

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

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

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

Ozone

What of charge distribution is “predicted” by

Lewis bookkeeping?

+ in middle

- on ends?

OO

O

+

_

OO

O

+

_

Open

symmetrical single minimum?

*) 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!

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)

~

~

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”?

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?”

Is it at all True?

Force Laws?

Are there e-pairsbetween nuclei and

unshared on some atoms?

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)

Visualizing Earnshaw - Coulomb's Electrostatics

“Lines of Force”MagneticElectrostatic

Faraday/Davy/Phillipsyoung

Michael Faraday

by p

erm

issi

on A

lfre

d B

ader

Col

lect

ion

Can show magnitude (as well as direction) of Force

2-D (Flatland)

force magnitude

line density

Circumference r2

Force

line density

1/r

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!

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.

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.)

Levitator by Martin Simon (UCLA)

Eppur sta fermo“and yet it stands still”

J.J. Thomson (1856-1940)

Electron (1897) Plum-Pudding Atom

© C

ave

ndi

sh L

abo

ratp

ry,

Cam

brid

ge U

nive

rsity

"[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)

Vortex Lattice Models (Greg Blonder www.genuineideas.com)

"[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

“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)

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.)

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).

Use of this content constitutes your acceptance of the noted license and the terms and conditions of use.

Materials from Wikimedia Commons are denoted by the symbol .

Third party materials may be subject to additional intellectual property notices, information, or restrictions. 

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