Prof: This is the slide
from the end of last time,
we didn't quite finish it.
Equilibrium versus Resonance.
Remember, equilibrium is a case
where you have two different
species,
different structures,
and you go back and forth
between them,
maybe fast, maybe slow.
Like the hydrogen could be
attached to the top or it could
be attached to the bottom
oxygen,
and those are different,
two different,
what's called isomers;
we'll get to that later on.
Or you could imagine the same
thing if you take the hydrogen
off,
you could have a short double
bond to oxygen and a long single
bond to oxygen,
or they could exchange as to
which oxygen was long and which
was short.
So you could have an
equilibrium there.
But it turns out you don't,
that in fact it's not two
different species,
it's one species.
It's a single minimum not a
double minimum.
Now how do you know this?
How do you know it's just one
nuclear geometry with an
intermediate bond distance?
The only way you know is by
experiment or by some really
fancy calculation that you have
to believe.
A lot of people would believe
experiments before calculations;
some are the other way around.
But there's evidence from a
technique called electron
paramagnetic resonance,
or EPR, that shows that indeed
this is one species,
a single minimum.
If you have an extra electron
on it and you have a carboxylate
anion, then again it's just one
species, a single minimum.
And there's evidence of that
from infrared spectroscopy that
we'll talk about next semester.
But don't be disappointed that
you're not able to predict this.
A lot of really smart,
experienced people couldn't
predict it.
This is lore.
If you look it up in the Oxford
English Dictionary it says that
lore is, "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."
So a lot of things are lore.
You just have to learn them.
You can't predict them ahead of
time, they're way too subtle.
So don't be disappointed,
because you haven't had enough
time yet to get the lore;
you're not supposed to know it
yet.
It would be nice if Lewis
theory was so accurate and
straightforward that if you draw
two structures,
there are two
structures and it's a double
minimum.
But that's not true,
and you have to know from
something, and you don't have
time to know yet.
As time goes on maybe you'll
figure it out.
This is from a good textbook;
it might be the one that we'll
adopt next semester.
It's a quote that says,
"Empirical rules"
(it's going to give)
"Empirical rules for
assessing the relative
importance of the resonance
structures of molecules and
ions."
That is, if you have two
different pictures that you can
draw for the thing,
does it look halfway in between
them,
almost all like one,
almost all like the other,
or some fraction of the way
between them.
How far is it,
one way or the other?
These are two different
pictures you draw to try to show
different aspects of the same
thing.
But there is one real thing.
The molecule doesn't know about
resonance, it just knows what it
is.
The problem is with our
notation.
Okay, but anyhow here's what
they say.
So rules that will allow you to
use this concept more
productively by deciding which
ones are better,
which look more like the real
thing.
Okay, so it gives rules and it
numbers them.
"(1) Resonance structures
involve no change in the
positions of nuclei;
only electron distribution is
involved."
That is, when you draw these
two structures you don't move
the atoms,
you just change where you draw
a single bond or a double bond
or a dotted bond or something
like that.
And in fact that's not even
true because it's not -- the
electrons know where they want
to be, it's the way we draw them
that's uncertain.
So when you draw two different
resonance structures,
you're not changing where the
electrons are,
you're just changing the lines
you draw.
Is that clear?
It's our notation that's at
fault.
"(2) Structures in which
all first-row atoms have filled
octets are generally important;
however, resulting formal
charges"
(we talked last time about how
you get formal charges)
"and electronegativity
differences"
(and of course we have to know
what electronegativity is,
but you've heard about it at
least) "can make
appropriate nonoctet structures
comparably important."
So if you have a bad charge
distribution,
even though you have octets,
it still might not be a very
good structure.
"(3) The more important
structures are those involving a
minimum of charge
separation"
(so you don't want to have
formal charge separation)
"particularly among atoms
of comparable electronegativity.
Structures with negative
charges assigned to
electronegative atoms may also
be important."
So if you're going to get a
charge, put it where it wants to
be.
Now, look at this more
carefully.
Number two, it says 'generally
important.'
It doesn't say "always
important,"
but "generally."
That's not such a great rule
because you have to know when
there's going to be an
exception.
Or 'however;'
this doesn't sound like the Ten
Commandments graven in stone.
Or "can make" a thing;
not that "it will,"
but "it can."
Or '"particularly"
or "may also be
important."
These are all weasel words
because the rules are not rules,
and this is all lore again.
So people write these rules but
really the people who are
writing the rules know the
answers ahead of time and think
"Ah, it generally works
that way and mostly we can get
away with it."
But these are not rules like
the rules you want to learn in
physics.
You want to learn these because
it's sort of handy,
but don't believe them.
And notice at the top that it's
"empirical"
rules.
It's not a fundamental theory,
these are just ways of sort of
correlating a bunch of the lore
that's come in.
And what's important is the
experiments, not theories like
this.
And we'll get better theories
later.
Okay, so the goal of this Lewis
stuff was from the number of
valence electrons it would be
nice to be able to predict the
constitution,
that is, the valence numbers
for the different atoms,
how many atoms of one kind or
another get together to make a
molecule.
Reactivity, we've seen a little
bit of that at least,
that unshared pairs can get
together with vacant orbitals
and make a new bond.
That wasn't something that was
part of the original rules of
valence.
And maybe something about
charge distribution as well.
Now let's look at the case of
O_2 and O_3, and apply some of
this stuff.
So O_2_,
you can have -- complete
the octets by bringing two
oxygen atoms together and
forming two bonds between them.
So that looks pretty good,
and it's a double bond,
and we can draw it that way
with just lines and forget the
unshared pairs if we're not
particularly concerned with them
at a given time.
Okay, now suppose you make O_3.
You could do the same kind of
trick there and make a
three-membered ring,
or you could make it linear,
or bent instead of a
symmetrical three-membered ring.
So you could have an
equilateral triangle like that,
or you could have it like this,
have O_2_ and bring
in another oxygen on one of the
lone pairs.
Now that's not an equilateral
triangle;
it's a triangle,
or it could conceivably be a
straight line,
we're not quite sure what the
geometric implications are.
Okay, but we could call it an
open structure as compared to
the ring structure.
And we can draw it like that,
and the formal charges would be
as shown.
Why?
Because that pair that
originally belonged only to the
original top oxygen of O_2 is
now shared with the other
oxygen.
So one of them loses
half-interest in a pair,
the other gains half-interest
in the pair and it's plus,
minus.
So the trivalent oxygen is
positive, as you would expect.
Now, what is the true structure
of the molecule ozone,
O_3?
Well you have to do some
experiment to find that out or
some high-faluting calculation
that you believe;
a quantum mechanical
calculation.
So it could be a ring or it
could be an open structure.
And notice, in the open
structure you should have two
resonance structures,
because you could draw it that
way,
or you could draw it that way,
and it could be two minima --
it could be a double minima --
double minimum situation,
or it could be distorted one
way or the other and click back
and forth.
Or it could be that the true
structure is in between with
equal bond distances.
So we have to find out
something about this.
Now one way of finding it out
is to do calculations and draw a
graph that shows what the answer
is, and that's what we're going
to do here.
It's based on some fairly
recent high-level calculations.
But the problem is drawing the
graph,
because if you want to be able
to show the structure,
you have to -- how many
variables do you have to specify
to say what the structure of O_3
is?
How many numbers would you have
to have?
It's three particles, right?
So if you gave this distance
and this distance and this
distance, that would fix the
triangle.
Or you could give this distance
and this distance and the angle
here.
That would also specify the
triangle.
But any way you slice it,
you have to have three numbers
to tell the structure,
and then you have to have
another number in your graph to
say what the energy is,
when it has that structure.
So you have to plot four
variables.
And that's not trivial on a
piece of paper,
to plot a graph with four
variables in it.
So three distances plus energy,
or two distances and an angle
and energy you need to plot.
So if you're good at plotting
things and have had a lot of
experience,
great, but if you need a little
warm-up exercise you could look
at this webpage that you get by
clicking up at the top there,
and it uses these graphs to
give you a little exercise in
plotting things that are in many
dimensions.
But we're not going to do that
in class, that's just to do on
your own or with a discussion
section.
And we'll show the specific
example of how we're going to do
it in the case of O_3.
So to specify O_3 we need four
dimensions.
Now here we have a plot that
shows one distance,
a second distance and the
angle.
So that'll specify where the
three oxygens are.
But Rutenberg and his coworkers
here, in 1997,
when they did the fancy
calculation, did it a little
differently.
They constrained it.
They didn't allow everything to
vary.
They said we're going to
require the two distances to be
the same.
So now instead of three
variables, you have only two,
and you can plot those two on a
two-dimensional paper;
you can plot two things.
So here we're going to plot the
position.
If the two distances are equal,
then if you know the position
of the top right you know the
position of the top left,
because the original oxygen is
at the origin 0,0.
Everybody with me on that?
So we're going to look only at
where the top right oxygen is,
and the other one will be
someplace symmetrically related
to it.
So we're going to blow that up,
and here's the plot they give
then.
So now any point on this will
specify a precise geometry of
the three oxygens.
Everybody with me?
We choose a point that tells
where the top right one is.
The bottom one is at 0,0,
and the top left is just on the
other side, in the corresponding
mirror-image position.
So any point on that thing will
specify the structure.
But now we still -- but that's
used up our two dimensions of
the paper.
How do we show the energy?
Can you think of a
three-dimensional graph you've
ever used?
Student: Color codes.
Prof: Pardon me?
Student: Color codes.
Prof: You could color
code it.
You could make red really high
and blue really low.
That would be one way to do it.
There are earlier ways,
before printing made it -- or
computers made it easy to make
colors.
Student: Contour lines.
Prof: Did you ever go
hiking?
Yes what?
Student: Contour lines.
Prof: Contour lines,
like a geological map,
right?
Okay, so we can draw contour
lines that show what the energy
is for every one of these
geometries.
And so if that -- notice that
at that position,
that X, it turns out to be what
the red structure would be here,
which is a ring;
equilateral, okay?
And this structure is open.
And I chose those particular
ones because when you do the
fancy calculation,
those turn out to be the ones
that are at the bottom of
valleys,
low energy.
And if you distort away from
those, the energy goes up.
Now you can go up -- any
direction you go from one of the
bottoms of the valleys you go up
in energy,
but there's some particularly
interesting positions.
A particularly interesting
position is that one,
because that's the pass,
that's the lowest you can go,
the lowest energy that's
required to go from one valley
to the other.
Everybody, I think,
has probably done enough hiking
to realize that that's the way
you'd like to hike,
or that if you spilled a cup of
water up at the pass it would
run down both ways.
Now in fact we can use that
concept to make it even a
simpler graph,
where we don't need to draw
contour lines,
because we could take the
steepest descent path --
if you pour the water out at
the pass and follow how it'll
trickle down,
it'll take the steepest way to
go down.
It turns out it crosses every
contour perpendicularly.
So you follow that and it would
go down to the bottom,
and then if you kept going,
like if you dropped a marble or
something,
and it rolled down to the
bottom it would keep going.
So there is a particularly
interesting path.
Not that the true molecule
would necessarily have to follow
that path, but it's a
well-defined path that gets from
one valley to the other.
So now what you could do is
take a knife and slice this
thing along that green line and
unfold it so it's flat.
Does everybody see how that
would be?
Like I remember once when I was
young we took a family vacation
and the AAA sent you a map that
showed where you would go,
but it also showed -- along a
particular highway --
and it also showed another map
that showed how high you were
all the time as you went along
the roads.
Got the idea?
So this is exactly that kind of
thing, where as you go along
that green road there are
different altitudes.
So we could just -- we could
draw it this way.
Does everyone see how that
works?
So it's not quite as specific
about geometry anymore,
the way the previous one was,
but it shows how much energy
you need as you go along.
And a particularly important
one -- well there are two things
that are really important.
One is how high one valley is
compared to the other,
how much more stable one is,
and the other is how much
higher is the pass,
how hard is it to get from one
to the other?
Okay, so there now is,
in a two-dimensional graph that
you can draw on a piece of
paper, is something that gives
you this information.
Okay, but this required that we
choose R_12 equals R_23,
in order to make this
simplification.
The guys who did the
calculations said if we don't
make R_12 equal to R_23,
then it turns out that all the
structures are still higher in
energy.
So these are the lowest energy
structures, although they can't
plot them on a piece of paper.
So the lowest energy structure
is this open form and it does
have R_12 equal to R_23.
It's a resonance structure,
halfway between;
not one double bond and one
single bond but a symmetrical
one.
Okay, so it's a symmetrical
single minimum,
found by calculation.
And there's experimental
evidence that supports that too,
but it's more complicated.
Now how about the charge
distribution?
What would the Lewis structures
that we've drawn here predict
about charge separation?
Would there be charge anyplace,
do you think,
on the basis of this,
in the real molecule,
which is symmetrical?
What do you think?
Anybody got a suggestion?
Yes?
Student: Well,
it would be positive on the
central oxygen because even
converted,
regardless of which of the two
structures are on that side,
that --
Prof: Both structures
have it positive in the middle,
so that looks like a good
prediction.
Student: And the ends
would be negative.
Prof: Right,
and then partially negative but
equally negative on the two
ends.
Now is that true?
Well here's a picture that,
based again on -- so the
prediction is positive in the
middle, negative on the ends.
Is it true?
Well there's the structure,
the minimum energy according to
calculation, quantum-mechanical
calculation.
And we can draw what's called
the surface potential of that
structure, which again comes
from calculation.
So you put a proton -- you
define the molecular surface --
and that is a little bit of a
problem but we'll talk about
that later,
where the molecular surface is
-- but then you put a proton at
a point on it;
and we talked about this
before, for ammonium chloride.
You put a proton on and you
find -- or no,
it was the BNH_6 we talked
about the surface potential.
Here's the same thing for O_3.
And you'll notice the surface
potential is high in the middle,
a bad place to put a proton
because there's positive charge
there,
and it's low on the two ends,
at least some place on the two
ends.
So more or less the Lewis
structure predicted this right.
So that's great.
We're not confident that it
will always work but maybe the
lore will build up that way;
and it does.
Okay, so charge distribution
also.
So reactivity we saw was a
special attribute of this nice
Lewis theory,
and charge distribution,
at least qualitatively if not
in detail,
for O_3_ and also for
the BNH_6.
Now how about specific
distances and specific angles?
We'll get into that later on as
to whether you can do that from
something like a Lewis
structure.
And how about the energy
content?
Well that's not so good
actually, with just Lewis
structures, but we'll test it
later.
Okay, so the Lewis-dot
structures.
It attempts to provide a
physical basis for the valence
rules, based on completing
octets and sharing electrons for
bonds.
What it gives that's new is
reactivity due to unshared pairs
where both of the
"hooks"
in the bond come from the same
atom,
you might say.
And it's convenient for
electron bookkeeping.
It certainly tells you what the
molecular charge is.
The formal atomic charges are
qualitatively realistic,
as we've just seen,
at least in the case of O_3.
And there's this question of
stability and resonance.
But resonance actually is not
something nature knows anything
about.
It's just a correction we try
to apply to having made drawings
based on Lewis theory.
It's not something serious.
Now, but this leaves us with
some serious questions.
Why does Lewis theory work?
What's so great about octets;
why not have a different number?
Or if you have a sestet,
instead of an octet,
how bad is it?
Or how bad are structures that
have charge separation?
It said in those rules that you
don't want to have charge
separation.
Well how bad is it?
Suppose you had a choice,
you either had to have a sextet
instead of an octet,
or you had to have charge
separation.
Which one would win?
How bad is "bad"
charge separation?
Remember it said that if you
put the negative charge on an
electronegative atom,
that's not so very bad,
but how bad?
Now last year in the Wiki there
came an interesting comment.
Somebody said,
"I have a question when
drawing these structures.
Is it more important to try to
fill the octet or to have the
lowest formal charge on as many
atoms, especially carbon,
as possible?
And why?"
That's a good question,
because Lewis doesn't tell you
that at all.
And there's further the
question, "Is this at all
true?"
Are there
electrons --
this is a really important
question,
we're going to spend some time
on it --
are there electrons in between
the nuclei that hold them
together,
pairs of electrons?
It'll take us several lectures
to get to answer that question.
Are there electron pairs
between nuclei and are there
unshared ones on some atoms?
What's the nature of the force
laws?
That's what we're after,
remember, here altogether.
Now, as to whether Lewis theory
is right,
there's a very fundamental
theorem in physics that was
developed in 1839 by Samuel
Earnshaw,
who was a tutor at Cambridge.
And the statement of the
theorem is that in systems
governed by inverse-square force
laws,
things like gravity,
magnetism, electrostatic
interaction,
there can be no local minimum
or maximum of potential energy.
He proved it mathematically.
It's not our business to repeat
his proof, but that's the
statement of it.
But we want to understand what
that means.
One thing that means is that if
you have Coulombic interaction,
positive/negative,
you can't have a minimum energy
structure that has a nucleus
here and eight electrons at the
corner of a cube,
because there are
inverse-square force laws and
that can't be a minimum energy;
if it distorted,
it would keep going.
Now we can visualize Earnshaw's
theorem here in terms of for
electrostatics by the analogy
that you've all seen of magnetic
lines of force;
everybody's seen something like
this I think,
right?
And the idea,
if it's electrostatic,
rather than magnetic,
the idea is that lines of force
emerge from a positive charge
and converge on a negative
charge,
and then you'd make them
continuous by drawing these
lines of force;
which iron filings of course
did in that case.
And this was the idea of
Michael Faraday,
whom we met a little while
back, and he thought these lines
of force were real physical
things.
Most people don't think that
now, they think they're just
graphs that involve
inverse-square force laws,
but he thought they were real.
And the neat thing about them
is they not only show the
direction of the force that a
charged body would feel,
because of the other charged
body, the one that created the
lines of force,
they not only show the force,
the direction of the force,
they also show how strong the
force is.
And the strength of the force
is by the density of lines.
The denser the lines,
the stronger the force.
Everybody's familiar with this
idea?
Speak up if things are not
familiar because I'm assuming
they are, if I say so.
Okay?
So you've seen that.
Now let's just think about that
a little bit.
Suppose you look at the line
density here.
Through that little line,
there are three lines of force
that pass.
So let's say that means there
are three, the force is three at
that -- and it's obviously
pointing to the right.
Okay?
Now suppose you checked it out
at that distance.
Stronger force or weaker force?
Students: Weaker force.
Prof: Weaker.
In fact only one or one and a
half, or something like that,
lines of force are going
through that.
Now, how does it depend on
distance, the number of lines
going through this standard
linked line, the blue one?
Well you see that in flatland
-- we're just doing this in two
dimensions,
we'll get to three in just a
second --
in flatland,
in two dimensions,
the circumference through which
all these lines of force pass,
the length of the circumference
is proportional to the radius.
Right?
So if you go out twice as far
there's twice the circumference.
So the density of lines passing
through it is half as big.
Everybody got that?
Okay, so that means the force,
which is proportional to the
line density,
must be proportional to 1/r.
Okay?
So then as you move out twice,
there'll be half as many lines;
move out three times,
a third as many lines.
Okay?
But this is just in two
dimensions.
Now let's think about,
how's it going to be different
in three dimensions?
Katelyn?
Speak up a little bit,
my hearing isn't so great.
I said I went to my 50th high
school reunion,
right?
Student: Instead of a
line it would be more like a
square, some sort of an area.
Prof: Yes,
it would be a two-dimensional
area that this stuff's going to
be.
Now if you have a certain
number of lines passing through
at a certain distance,
and you go out twice as far,
what's the density of the lines
going to be?
Can you think?
Anybody help?
Yes?
Student: It'll be
inverse-square.
Prof: Inverse-square,
because now we're not talking
about the circumference of a
circle,
we're talking about the area of
a sphere as we go out.
Right?
So if we go to three dimensions
here and take your area and move
it out, the surface is going to
be proportional to r^2,
as you go out.
So the line density is going to
be 1/r^2.
So if you have an
inverse-square force law,
then you can draw lines of
force.
The lines of force won't work
if you don't have an
inverse-square force law,
because they have to drop off
this way in order to give lines
of force.
So in 3D such diagrams work
only for inverse-square forces.
You can't draw a thing like
that for Hooke's law.
Right?
Now, so here's a whole bunch of
charges, positive and negative,
and the lines of force between
them.
And notice something
interesting.
The lines of force start on
positive charges and end on
negative charges.
There's no other place in space
where all the lines go away from
it, or go toward it.
Everybody see that?
It's only at the charges that
they all go away or all go
toward.
Right?
And that has a very important
meaning, that you can't have
someplace off in free space
where all the lines of force
would converge.
Notice if you were to have
something that was positively
charged,
and it's at a minimum of
energy, then any place you
displace it,
it'll get pushed back,
if it's at a lowest point in
energy.
You push it any direction,
it'll come back.
That is, all the lines of force
must converge on that point.
Okay, everybody with me on that?
But it can't be.
The only place that all the
lines of force converge is on a
negative charge.
It can't be any place -- if you
have inverse-square laws,
you can't have someplace in
space which is the lowest energy
place for the charge to be,
except on another charge.
The same is true,
you can't have a maximum
either.
That's a visualization of
Earnshaw's theorem.
So if you have inverse-square
force laws,
or any combination of
inverse-square force laws,
like a combination of gravity
and electrostatics and
magnetics,
you can't get a minimum energy
structure,
unless everything just falls
together or blows apart.
Okay?
So Earnshaw's Theorem:
"In systems governed by
inverse-square force laws there
can be no local maximum (or
minimum) of potential energy in
free space."
Okay?
And that's why I don't just
float here, I have to be on the
floor.
Right?
Did you ever see anything truly
levitating, something that just
sits in space and doesn't move,
not touching anything else?
That's been levitating for ten
or fifteen years.
It's not plugged into anything,
it's just this thing right
here.
What do you conclude from
seeing that, about the force
laws that are involved?
<<Students speak over one
another>>
Prof: There must be some
force involved in that that's
not an inverse-square force law;
otherwise it wouldn't work.
You can read about it on the
Web, if you want to,
it's not the business of this
class.
There is a non-inverse-square
force law involved in that
little magnet sitting there.
The only stationary points
allowed by Earnshaw's theorem
are saddle points where it's
flat in energy,
for all directions,
but you go one direction and
then you go down,
go the other direction and you
go up.
That's like a potato chip or a
saddle.
So you can have saddle points,
but you can't have absolute
minima or maxima of energy.
There's the picture of that
thing.
Let me do the -- this reminds
me to get back to seeing who's
here.
<<Professor McBride takes
attendance >>
Prof: Now we'll
continue, got a few more minutes
here.
And yet it stands still.
So there must be something
that's not an inverse-square
force law there.
Okay, so J.J. Thomson,
in 1897, discovered the
electron.
So the idea is maybe electrons
have something to do with bonds;
that's what brought Lewis into
the game.
But Thomson himself came up
with what came to be called --
and I don't know by who first,
I've tried to find out and
can't --
the "plum-pudding"
atom.
Has anyone ever heard of that?
Yes, okay.
And I suspect that when people
told you about the plum-pudding
atom they sort of snickered.
Am I right?
This was sort of a naïve
idea.
It's not that naïve at all.
Let me show you why.
Here's the book by
J.J. Thomson called The
Corpuscular Theory of
Matter, and if you look in
there you'll find out what he's
talking about.
He says -- he has a model of
electron configuration --
he says: "Consider the
problem of how to arrange 1,2,
3, up to n
corpuscles"
(that's what he called
electrons,
he called them corpuscles);
"Consider the problem as
to how they would arrange
themselves if placed in a sphere
filled with positive electricity
of uniform density."
So that's the idea of the
plum-pudding.
You have this sphere of uniform
positive density --
why it should be like that
nobody knows --
but suppose you have a sphere,
and you put the electrons in
it,
like plums in a plum-pudding,
which is like a fruitcake in
England.
Okay, now notice that he said,
"placed in a sphere
filled with positive
electricity."
Why did he do that?
Why didn't he just have -- like
Rutherford ultimately did it --
a nucleus with positive charge
and electrons around and about?
Why did he put the electrons
inside the positive
charge?
Zach?
Student: Was it lowest
potential energy?
Prof: I can't hear very
well.
Student: Lowest
potential energy?
Prof: No.
Yes?
Student: Well they were
thinking
>
maybe that they were going in.
Prof: No,
no, they weren't thinking about
things moving,
they wanted them just sitting
there.
Student: Maybe because
on a macroscopic scale opposite
charges attract.
So maybe he might not --
Prof: Yes,
but that's well known,
Coulomb's law.
Yes Keith -- or Kevin, right?
Student: If you have
them all,
you have a positive sphere and
if you have the negative
corpuscle inside,
then they'll cancel each other
out and be a neutral body.
Prof: Yes,
but that could be -- it
wouldn't have to be a big sphere
of uniform density,
you could have a little
particle of positive charge,
of the same charge, right?
It'd be the same deal.
Student: They didn't
know that the inverse existed,
so they thought that --
Prof: No,
none of this is it.
It's what we've just been
talking about.
Yes?
Student: Isn't it about
accounting for the bonding
between the electrons and the
protons and containing the
electrons?
Prof: No,
bonding hasn't come up yet.
This is something very
fundamental.
Yes?
Student: He was a big
fan of plum-pudding.
>
Prof: He probably liked
plum-pudding at Christmas.
Student: Did he have
experimental evidence?
Prof: I can't hear.
Student: Did he have
experimental evidence?
Prof: No.
Yes?
Student: Because
Earnshaw said that there should
be no minimum or maximum --
Prof: Ah ha!
Earnshaw said you can't have an
energy minimum for separate
particles,
but if you put the negative
ones inside the positive ones,
then you could get a stable
structure.
It's Earnshaw's theorem that
required it to be a
plum-pudding.
Okay?
Now why a sphere,
why not a doughnut or some
other shape, a barbell or
something, right?
And he said that the positive
charge is distributed in a way
most amenable to mathematical
calculation.
He chose a sphere so it would
be simple.
You've heard about spherical
cows and so on like that,
that physicists like to
calculate.
Let's suppose there's a cow,
let it be a sphere,
right?
So that's what he did.
That's why it's a sphere.
Okay, we can solve the special
case where the corpuscles are
confined to a plane,
if you do it in two dimensions
-- it's difficult mathematically
in three dimensions --
but you can do it in two
dimensions.
And he gives a picture in this
book,
like this, which is a solenoid
magnet that attracts little
needles that are magnetized,
and those needles are stuck
into corks,
which float in the water, right?
So they have to,
the corks have to stay at the
level of the surface of the
water.
So it's a two-dimensional
problem.
The needles are parallel,
the magnets,
so they repel one another.
But the big magnet attracts
them to the center.
Okay?
So what he does is toss a
certain number of needles and
corks in there and see what
pattern they form.
Okay?
So here are some patterns.
You can get these on the Web,
at Greg Blonder's website here.
So if you have just one,
it goes to the center;
no big deal.
If you have two, you get a line;
there's no big deal about that.
Three make an equilateral
triangle.
Put in four, they make a square.
Put in five,
you make a pentagon.
No one's surprised so far I
suspect.
Except that sometimes when you
put in five and shake it up,
you get a square with one in
the middle.
Can you see where this might be
going?
Let's keep going.
Okay, if you put in six,
you get one inside a pentagon;
seven, one inside a hexagon;
eight, one inside a heptagon;
nine, two inside;
ten, two inside.
And sometimes it's like that,
and sometimes it's like that.
These are experimental, right?
And if you put in eleven,
it's three inside,
and then three inside nine;
although sometimes you get two
-- pardon me,
I'm screwing up here --
sometimes you get --
there are two different
patterns --
sometimes you get three inside
eight,
sometimes you get two inside
nine, for that number.
Okay, and then you can get four
inside nine,
four inside ten,
five inside ten,
and then after five,
if you put in more,
if you put in sixteen,
you get one inside five inside
ten.
What is this reminding you of?
Yes?
Student: Of orbitals.
Prof: The shell
structure of atoms.
Right?
As you go down the periodic
table you complete a shell.
So it's a model of shells,
and then you can get more and
more and more.
So this is what Thomson was
thinking about.
But that's just two dimensions.
Right?
Three dimensions is a bigger
problem.
But he could say,
he was able to say something
mathematically about eight.
"The equilibrium of eight
corpuscles at the corners of a
cube is unstable."
Even if you have the spherical
charge -- so it's not exactly
Earnshaw -- still you can't get
eight at the corners of a cube.
Now Lewis comes alon,g and in
1923,
as I've told you before,
he writes: "I have ever
since regarded the cubic octet
as representing essentially the
arrangement of electrons in the
atom."
This is long after Thomson had
written this about not being
able to do that.
So was Lewis ignorant of
Earnshaw's theorem?
Because by 1923 they know that
the nucleus is not a
plum-pudding -- that is the
positive sphere -- that it's a
point.
So was Lewis just naïve?
No, look what he wrote, in 1916.
"The electric forces
between particles which are very
close together do not obey the
simple laws of inverse-squares
which holds at greater
distances."
So Coulomb's law breaks down.
You don't have inverse-square.
So then you don't have
Earnshaw's theorem,
and you don't have to worry,
and you can get a structure if
it's not an inverse-square force
law.
Okay?
No trouble.
But what is the force law?
Well Thomson thought the same
thing in 1923 in his book The
Electron in Chemistry.
He wrote: "If electron
nuclear attraction were to vary
strictly as the inverse-square,
we know by Earnshaw's theorem
that no stable configuration is
possible with the electrons at
rest or oscillating about
positions of equilibrium…
I shall assume that the law of
force between a positive charge
and an electron is expressed by
this equation…
Then a number of electrons can
be in equilibrium about a
positive charge without
necessarily describing orbits
around it."
And look at the -- we're going
to end right now by looking at
this equation.
What's that bit of it,
the first bit,
the "one"
part?
Students: Coulomb's law.
Prof: That's Coulomb's
law.
But he's got a correction to
Coulomb's law,
here at the end,
c/r, where c is a distance --
you divide it by r and you get
a number --
and c is a distance that's on
the scale of atomic lengths.
Right?
And that means that as long as
c is very small -- pardon me,
as long as -- okay so have I
got this right?
Okay, so when distance r gets
smaller than c,
then the force changes sign.
Okay?
So what was attractive becomes
repulsive, and then you can have
the electrons sitting around the
nucleus.
Okay?
So we'll see what happened
three years later next time.