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PROFESSOR: OK.
OK.
Settle down.
Settle down.
Settle down.
So the weekend doesn't begin
until after the 3.091 lecture.
OK.
A couple of announcements.
Tuesday, Quiz 3 based
on Homework 3.
The final exam has
been scheduled.
In fact, all the finals are
scheduled now, so please
consult the Registrar's
listing.
The celebration of celebrations,
the 3.091 final
exam, will be Tuesday, 15th of
December in the morning, 9:00
am to 12:00 noon over in the
Johnson Athletic Center.
So I urge you to go through the
final exam schedule and
then make your travel plans when
you know what your last
obligation is.
And book them soon, because
we've got about a quarter of a
million students in the Boston
area and the semesters all
come to a close within a very
narrow time window, and
everybody's trying to get
through a wormhole called
Logan Airport.
So you want to be ready.
Do not try to leave town before
you've met all of your
obligations.
You can't leave before you've
finished all of your finals.
But you know what they are now,
so call your travel agent
or make your internet booking.
And an announcement for
Professor Paul's section.
Owing to the holiday on Monday,
he's going to have
office hours from noon
to 1:30 today.
So if you're in Professor Paul's
section, you'd like to
catch up with him, noon
till 1:30 today.
All right.
Let's get to the lesson.
The last day we started looking
at octet stability,
and we looked at octet stability
and what it means in
terms of shell filling and some
sweet spot in energy with
respect to reactivity.
And a filled shell leaves
us with this electron
configuration, ns2np6 for
n greater than 1.
In the case of n equals 1,
there's just ns2 for helium.
Otherwise, ns2np6.
2 plus 6 is 8.
There's the octet stability.
It's observed trivially
in noble gases.
And then we saw that
it can also be
observed in certain ions.
And how do we get to
octet stability?
Electron transfer.
And here's the prototypical
reaction.
I like this.
This is very iconic.
It puts it in broad terms. The
marriage of an electron donor
with an electron acceptor leads
to the formation of a
cation and an anion, thanks to
electron transfer from the
donor to the acceptor.
It doesn't end there, though.
You have cations and anions in
the gas phase and they're
attracted to one another
by coulombic forces, or
electrostatic forces, and that
leads to ionic bonding.
And today I want to go deeper
into it and study the
energetics of ion-pair
formation.
So let's do it.
And what I'm going to do is
study this with reference to
an example of sodium chloride.
So we will go through the
energetics of sodium chloride,
which we know is going
to exist as
Na plus and Cl minus--
sodium being the electron donor
and chlorine being the
electron acceptor.
So what I'm going to do is show
you a plot of energy as a
function of separation.
So you're going to have
to be pluralistic
here in your ideas.
So, for example, if I wrote this
word by itself you don't
know if I'm saying lead or I'm
talking about the metal lead.
The only way you know
is in context.
And so you have to know
in context here.
So r is not the radius
of the atom.
In this case it's the symbol.
And I'm using this symbol
not to confuse you.
This is what professionals
use.
So if you go to the text and the
literature you'll see r.
This is the interionic
separation, and it's
determined nucleus to nucleus.
So I'm going to put a sodium
ion at the origin.
This is the origin.
Energy is on the ordinate.
So I'm going to model this
as a hard sphere.
So this is sodium ion,
Na plus, and it's
got some finite radius.
And its radius, I'm going
to call it r-plus.
That's the radius
of the cation.
And next to sodium I'm going
to put the chloride anion.
And it's bigger.
It's bigger.
It's not to scale,
but it's bigger.
So this is chloride, the
Cl minus, and it has
its radius, rCL minus.
And then the interionic
separation is measured from
the sodium nucleus to the
chloride nucleus, and it's
given the symbol r0.
That's the interionic
separation.
And we could we write that r0,
in fact, is strictly r-plus
plus r-minus because we're using
a hard sphere model.
So there's no shrinkage
when the two touch.
So that makes sense.
Now what I want to do is
calculate the energy here.
The only energy that we have
here is electrostatic.
So we're going to start and say,
imagine if we had these
two ions separated by infinite
distance and we brought them
to a separation of r0.
How much electrostatic energy
would be stored there?
So I can write that as E,
and I'm going to call it
attractive.
There's a force that attracts
these two ions together, and
that's given by Coulomb's Law.
So that's the product
q1 q2 over 4 pi
epsilon0r, in general.
I'm making this as a function
of r and then we're going to
figure out how to get to r0.
Let's put a little
ledge in here.
So q1, I'm going to make
that the sodium just
for grins and chuckles.
So q1 is equal to the z, the
valence on the sodium, times
the charge, the elementary
charge.
So the charge on the sodium is
plus 1-- so it's plus 1e--
and q2 is going to equal
the charge on the
chloride times e.
And in this case the chloride
ion is negative 1, so q2 is
minus e, q1 is plus e.
Now if instead I were doing
magnesium oxide, magnesium
would be plus 2, oxide
would be minus 2.
So that's where the charge on
the ion comes into play.
So we can put that in here and
will make this z-plus times e,
z-minus times e.
So there's q1 q2 over
4 pi epsilon0 r.
And in this case, for sodium
chloride, this is plus 1 times
minus 1 is minus 1 e squared
over 4 pi epsilon0 r.
So this is a function of r.
This e is 1/r.
That's the hyperbole, and
I can draw that here.
So it's going to look
something like this.
All right?
It's going to be a
right like this.
Good.
So that's the attractive
force.
Now how do we avoid the ions
just blending together?
Why do they stop at r0?
What puts the brakes on
the coulombic force?
Ah!
For that we have to look at the
fine structure of sodium.
So let's look at the fine
structure of sodium.
So sodium is net
charge plus 1.
But sodium more properly, if
I go to the next level of
structure, is really 11 protons
in the nucleus and
it's got electrons around it--
in total, 10 electrons.
So it's net charge is
plus 1, all right?
Now if I'm way over here and
I look at sodium I just see
something of charge plus 1.
And, in fact, I could model
sodium as just a point charge,
a plus 1, and do all the
electrostatics and I would be
perfectly accurate
in my estimation.
But as I get closer I see
there's fine structure.
It's like so many other
things in life.
You know, from a distance it
looks really good, and you get
up close you go, eww.
[LAUGHTER]
PROFESSOR: I'm not going to
mention any names, all right?
But it's Friday.
Be careful.
So as I get up closer I realize
that it's plus 1, but
the plus 1 is plus 11 mediated
by minus 10.
So when things start getting
close together this exterior
of negative charge becomes
manifest, palpable.
So let's look at the chlorine.
The chlorine's coming,
sees plus 1, plus 1.
It starts getting closer
and closer.
And now what happens?
The negative electronic
configuration on the outside
of sodium is interacting with
the negative electronic
configuration on chlorine, and
the result is you have
electron-electron repulsion.
And we've got an equation for
that, and that's given by E
repulsion is equal to some
constant b over r to the n.
And this n is called
the Born exponent.
It's not the quantum number.
So r today means interionic
separation,
n means Born exponent.
And you have to determine it
by experiment and the value
lies between 6 and 12.
And we have to determine
b by experiment.
So let's say this thing's
got a-- pick a
number in the middle.
Say it's 8.
So let's plot r to the eighth.
So that's going to hug the
abscissa, and it's going to
come in really, really close.
And then somewhere around this
distance here, where the
electron-electron repulsion
starts to be felt,
this thing takes off.
But 1 over r to the 8 goes way,
way up fast. So instead
of being a gentle curve
it's more like a
hockey stick shape here.
And so now the net energy is
the sum of the negative
attractive energy and the
positive repulsive energy.
And if we sum the two what
are we going to get?
Well, way out here 1 over r
to the 8 is negligible.
And the net value--
so E net, in red--
is essentially equal
to E attractive.
And at very, very low values of
r, r to the 8 dominates r.
So we have this as
the net value.
And the two sum--
and I can't quite do it because
of the way this is
drawn, but I think
you can see.
These two eventually
equilibrate and
go through a minimum.
So I'm going to cheat a little
bit here and I'm going to add
these two in such a way as to go
through a minimum value of
energy at r equals r0, which is
the sum of E attractive and
E repulsive.
So we can sum those two and
let's see what we get.
At r equals r0, E net is equal
to its minimum value.
So how do you find a minimum
in a function?
You take the derivative.
I'm going to put some
math to work here.
So we'll take dE by dr and
set it equal to 0.
And the value of r at which
it equals 0 is termed r0.
OK?
So we'll just go through
and take the
derivative of that thing.
And I'm not going to go through
all the math but just
show you the set-up here.
What's the other thing
that we know?
dE by dr represents what?
That's force.
So I could say that at r equals
r0 the net force is 0,
which is what you'd expect.
Because if the net force isn't
0 it's going to either push
the ions farther apart or draw
them closer together.
So this is mathematics
imitating reality.
What a concept.
Math working for you instead
of you working for math.
So what do I have?
I have everything in these
equations except
the value of b.
I'm assuming we know the value
of n from experiment.
We don't know the value of b.
And so you can solve to get
the value of b, and once
you've got that you can put
everything together and give
an expression for the energy of
the system at r equals r0.
When you plug everything in you
get this: z-plus, which is
the net charge on the cation,
times z-minus, which is the
net charge on anion, divided
by 4 pi epsilon0 r0.
1 minus 1/n, where n is
the Born exponent.
And this is valid at
r equals r0 only.
If you're not at r equals r0
then you can get the value of
b and then put it into this
expression, where E net will
equal E attractive
plus E repulsive.
So there it is.
And so this represents--
this is the energy of a single
ionic bond, because that's all
the energy that's there.
It's the single ionic bond.
And the second thing that we
realize is plus times minus is
net minus, so this means that
it's negative quantity,
as it should be.
It has to be a negative
quantity.
All right.
So what do we have here?
What we've seen by going through
this derivation is the
recognition that the ionic
bond is electrostatic
attraction mediated by
electronic repulsion.
It's the balance of the two.
And those words sound so good
to me that I'm going
to write them down.
Electrostatic attraction
mediated--
another lovely word--
mediated by electronic
repulsion.
So that's how you get to the
final setting here of the
interionic separation.
So what are the characteristics?
What does this lead
to in terms of
characteristics of this bond?
Characteristics of
the ionic bond.
First of all, it's
omnidirectional.
This is a concept based on the
fact that the electric field
radiates in all directions
uniformly.
So the negative field coming
from the chloride ion is
uniform in all directions.
There's no preferred
direction.
Omnidirectional.
OK?
E field--
oh, I better not say that.
Electric field, not the energy
field, radiates in all
directions uniformly.
And that's going to
have consequences.
I'm not just telling
you this because we
like cataloguing things.
This isn't a bookkeeping
class.
So we're going to come
back, use this fact.
And the second thing is that the
bond is unsaturated, which
is a chemical way of saying that
a given ion can bond to
more than one other ion.
In other types of bonds
that's not the case.
A given atom can only bond
once and then it's done.
Whereas in this case the ion
can bond to a plurality of
other ions.
So ions bond to more than one.
OK?
Plurality of bonds is formed.
They're polygamous,
if you like.
So what does that mean?
That means that here
is what happens.
We've got the blues as the
sodiums, and for any given
sodium it forms bonds without
limit until the number of
bonds is stopped by physical
limitations--
not because the E field
was saturated.
It's unsaturated.
You just can't jam any more
chlorides physically around
the sodium.
That's why the sodium is only
bonding to the number of
chlorides that it bonds to.
There's no intrinsic
limitation.
So what happens when you get
to this situation where you
have omnidirectional forces,
unsaturated bonds, and ions
that you can model as hard
spheres of constant radius?
All the sodiums have the same
radius, all the chlorides have
the same radius.
You make a 3-dimensional
ordered array.
So you can make an infinite
atomic ordered array, which we
use the simple Anglo-Saxon
word last day
to describe: crystal.
You form a crystal.
And as a result ionics have to
be solid at room temperature,
because if you've got thousands
and thousands of
atoms together in one
aggregate they're
not going to float.
They're going to settle.
Put another way, the strength
of the bond, the amount of
energy in here, is so great that
the thermal energy of the
room isn't great enough
to disrupt this bond.
It's a combination of
unsaturated, omnidirectional,
and high energy.
So we form solids at
room temperature.
OK.
Now I want to show the
energetics of that one because
this is good.
You know, I promised you I
wouldn't do derivations, so
I'm not going in
detail on this.
I'm giving you just enough so
that I can introduce the
characters here.
You know, how else
am I going to
introduce the Born exponent?
Am I'm just going to say,
there's this exponent n, the
Born exponent.
We're going to introduce
it in context.
So now you know what
the energetics are.
So now I want to prove to you
energetically along this line
that crystals will form.
So let's imagine--
we're going to do this
thought experiment.
We're going to take three
ion-pairs of sodium chloride.
So here's three ion-pairs
of sodium chloride.
And I want to compare these
three ion-pairs.
So this is an ion gas.
And the distance between
ion-pairs is great enough that
one pair doesn't affect
the other.
The electrostatics are only
strong within the pair.
So we'll just label this
infinity with quotation
marks around it.
They're very far apart.
When a physicist says they're
very far apart, very is code
for infinity.
So this one doesn't interact
with this one, which doesn't
interact with this one.
And I want to compare the
energetic state of the ion
dispersion to what would happen
if I were to put all of
those in a single line.
Plus, minus, plus, minus,
plus, minus.
So then this has a certain
energy state, it's the energy
of the ion line.
And I want to show you that
there's an energy decrease in
collecting all of these and
ordering them into a line.
So there's more energy
in a line dance
than in ballroom dancing.
That's what we're going
to say ultimately.
OK?
So let's compare the energies.
That's all we're going to do.
So what's the energy of this?
Well, we're not going to do 3
versus 3 Let's think big.
It's Friday.
So let's take Avogadro's number
of pairs, shall we?
So the energy of the ion
dispersion would then equal--
that's the energy of one pair,
and they're infinite distance
apart, so there's nothing to be
gained by putting them in
the same chamber.
So it's just going to be N
Avogadro, if that's the
number, times the energy
evaluated at r equals r0.
So we're done.
And we know what that is.
I don't have to rewrite
it for you.
OK.
So now what I have to do is get
an estimate of the energy
of the line and show you
that the line is at a
lower energy state.
Well, let's see.
Jocelyn, take the top one and
the middle one, please, but
not the bottom one.
Thanks.
OK.
So now let's look at the
energy of a line.
So we're going to do--
here's my line.
I'm going to get the
colored chalk.
Green and blue.
You know, some people think
being a professor is so cool
because you get to travel, you
get to research, and so on.
It's the colored chalk.
[LAUGHTER]
PROFESSOR: It's the
colored chalk.
All right.
So there's a sodium.
And on each side of the sodium
we'll put a chloride.
All right?
And I'm going to just
keep going this way.
Here's a sodium and
here's a chloride.
All right.
And now what I'm going to do,
I'm going to start here at
this sodium and I'm going
to count the energy,
electrostatic energy, that's
in this system.
So we're going to count to the
left, we're going to count to
the right, and then we're going
to multiply it by the
total number of ion-pairs.
And I know that the ones on the
end aren't the same, but
the number of ends over N
Avogadro, that's peanuts.
The edge effects are negligible
because there's
such a giant middle.
That's how you model
this stuff.
You don't obsess over the fact
that the last ion doesn't see
anything on that side.
You just do it and forget about
it, because you know you
wouldn't do it for 5.
This would be a big problem.
But if you have Avogadro's
number, who cares?
It's called risk assessment.
All right.
So let's look at the
energetics here.
So what we've got is the
energy of the ion line.
OK?
So let's start with
this central one.
And separated by distance r0 is
the chloride, and there's
an attractive energy here.
So that's going to equal
minus e squared over
4 pi epsilon0 r0.
Now I'm going to keep going,
because the field is
unsaturated and goes
in all directions.
So this sodium, that's
a distance 2r0 away.
It's got a repulsive force
exerted on this sodium.
So let's add that.
So that's going to be plus.
A repulsive force raises
the energy of a system.
That's e squared over 4pi
epsilon0 times 2r0.
And let's keep going.
So now let's go 3r0 away.
So 3 times r0, that gets me out
to the chloride over here.
Let's put 3r0.
Now that will take me from the
center of the sodium to the
center of the next chloride.
And that's going to
be attractive.
So that'll be minus e squared
over 4pi epsilon0 times 3r0
plus, et cetera.
So you see how this goes.
So you go all the way out, you
add them all up, and you go
the other way.
And so on and so forth.
So that's how the
derivation goes.
You might say, hey
wait a minute.
What happened to the Born
exponent and the repulsive
energy term?
Well, where's the repulsive
energy term going to be felt?
It's called electron-electron
repulsion.
Axiomatically, the electrons
here are
nowhere near these electrons.
See, you only have to count it
for the nearest neighbor.
So we can patch that
in at the end.
And we do.
I haven't forgotten.
But I'm not going to spend a
whole day trying to derive
this thing.
I'll show you how it
starts to evolve.
At some point you end
up with something
that looks like this.
e squared over--
in fact, I'll put the
minus sign up here.
Minus e squared over
4 pi epsilon0 r0.
And you're going to double it,
because you're going to go one
side and the other side, and
you're going to get a series
that looks like this: 1 minus
1/2 plus 1/3 minus 1/4 plus
blah, blah, blah.
Yeah.
OK.
So what does this look like?
I've really broken this
into two pieces.
So this coefficient out in front
here, you should now be
able to repeat this in your
sleep: e squared over 4 pi
epsilon0 r.
This is electrostatics,
isn't it?
Electrostatics.
This is the consequence
of Coulomb's law.
What's this second term here?
What's this all about?
Geometry.
This is dictated by atomic
arrangement.
So I could calculate this if I
took, instead of a line, what
if I put them in a sheet subject
to the constraints of
those sizes and plus
1 and minus 1?
So what would be?
I'd start at the sodium and
count how many chlorides?
If I'm on a plane there'd be
one, two, three, four.
And then how many sodiums?
Well, they'd be on the backside
of each of the chlorides.
And I'd add them all up in
2-space and I'd end up with
another coefficient here.
All right?
And we compress all of this into
a coefficient which we
call the Madelung constant.
And it's a function of the
atomic arrangements.
So different crystal structures
have different
Madelung constants.
It's named after a German
professor, Madelung.
In 1910, he published
calculations for the energy of
a system of point charges--
just abstract theoretical
paper.
And then about 10 years later
another German professor by
the name of Paul Ewald--
he did his PhD for
Sommerfeld--
he published a paper in which
he actually made the
calculation for ion crystals,
and he came
up with this constant.
And to show you the class of the
guy, instead of naming the
constant after himself he
named it after Madelung.
Now that's class.
So Madelung did the first
calculation so he gets named.
So now what we're going to do is
we're going to multiply by
the N Avogadro, because I've
got N Avogadro of these
things, and we're going to
put in the Born exponent
patch and so on.
And here's what the final
expression looks
like for the line.
There are a few algebraic tricks
that I'm not willing to
do in class because I don't
think that's a profitable use
of our time in a chemistry
class.
But if you want to try the
derivation I have it in full
and we can compare notes.
So once we get the patch in it's
going to look like this.
It'll be minus.
There'll be the Madelung
constant times
N Avogadro e squared--
and this is already assuming
it's plus 1, minus 1.
If this were magnesium oxide
there'd be a 4 in here.
4 pi epsilon0 r0 times
1 minus 1/n, where
n is the Born exponent.
So compare this to
this one here.
What's the only difference?
The only difference is the
Madelung constant, right?
It's the only difference.
So E of the pair dispersion is
really equal to E of the line
divided by the Madelung
constant.
So what I'm trying to prove to
you is that E line is more
negative than E of
the dispersion.
So it all hinges on the
magnitude of M.
If M is greater than 1 we win.
If M is less than 1 I've just
proved to you that water runs
uphill, so that's a bad day.
All right?
So let's calculate
the value of M.
And you can go to your
algebra books.
And you've got this series
natural log-- and engineers
write natural log "ln." I know
the mathematicians write
"log." Uh-uh.
Engineers--
uh.
That's 1 plus x.
OK?
Natural log, 1 plus x.
You can expand this as x minus
x squared over 2 plus x cubed
over 3, dah, dah, dah, dah.
You know, look that one up.
Set x equal to 1, which is
what we've got, right?
Because we've got 1 minus 1/2,
1/3, dah, dah, dah, dah.
Go through it and you'll get the
value that M, according to
this, will give you 2 times the
natural logarithm of 2,
which is 1.386--
which is greater than 1.
And so we're golden.
That means that the energy of
the line is lower, more
negative, than the energy
of the dispersion.
So I'm going to do
this pictorially.
Let's make an energy
level diagram.
And the energy level diagram
will look like this.
So up here the energy is 0 and
everything is negative.
So if I put this as
minus 1 unit.
All right?
These are all negative values
increasing in this direction.
So this is the dispersion
of ion-pairs.
So all we've done is
take a cation and
put it to the anion.
We've just seen that if we do
the calculation for the line
it's 1.386 times whatever
this is.
So that gives us-- this is for
the ion line, which I'm going
to take the liberty of calling
a 1-dimensional crystal.
It's a 1-dimensional
ordered array.
And what I can do is go to the
3-dimensional array, start at
the lower right-hand corner
with that chloride.
Calculate the distance to each
of the nearest neighbor
sodiums. Go through
the geometry.
The next nearest neighbor
chlorides, the next nearest
neighbor sodiums. And you'll
build an infinite series and
you'll evaluate it.
And in three dimensions
it's even lower.
It's 1.7476.
So this is for the 3-dimensional
crystal.
This is for the ionic crystal.
All right?
3-D crystal.
So what this is showing is that
the system keeps making
more and more bonds.
Why does it make bonds?
Because the more nearest
neighbors it has the lower the
energy goes.
So making a 3-dimensional
crystal is
energetically favored.
Now there are different
Madelung constants for
different crystal structures.
You say, well wait a minute.
How do you get different
crystal structures?
Suppose instead of sodium
it's potassium.
What's the only difference?
Potassium is plus 1.
What's the difference?
Size.
Potassium is larger.
They're not going to pack
quite the same.
And so depending on the
relative ion sizes--
I mean what if I have something
like silver iodide?
Iodide is huge.
Silver is so small it'll fit
into the interstices between
touching iodines.
So it's going to have a
different crystal structure,
and the different crystal
structure will give us a
different Madelung constant.
And there it is.
So we've come a long way with
that little assumption of
octet stability.
So now let's take a look at
what the properties are of
these things.
They're solid at room
temperature because we've got
strong bonds.
High melting points.
Bonding is related
to melting point.
Now think point is dictated by
bonding, because now you're
comparing thermal energy versus
the cohesive energy of
the crystal.
So tightly bonded substances
melt at high temperatures,
weakly bonded substances melt
at low temperatures.
Transparent to visible light.
How do I know that?
Because I'm the professor.
No.
How do I know that?
How do we think about it when
someone says to you is
something transparent, in this
case to visible light.
Well, what I do is I say,
here's the solid.
This is the ionic solid and here
is visible light, h nu.
And I'm going to write 2 to 3
electron volts per photon.
And what happens when I want
to decide whether this is
transparent to visible light?
Now look at the modeling here.
Photon is a squiggle.
I don't know if that's
Cartesian space.
This is like a crystal, right?
I'm speaking California.
It's like a crystal.
So now if I go inside I want
to make the energy diagram.
So what's the energy
diagram look like?
All right?
So now the question
is how does--
if this is the energy diagram of
the crystal and I make this
the energy of the photon of
visible light, I'm going to
compare how much energy the
photon has versus how much
energy it takes to excite
electrons all the
way to a new state.
Because if you don't
excite them all the
way to a new state--
they can't go part way,
so nothing happens.
And if nothing happens the
photon goes through and that's
transparent.
Now what do I know about the
binding energy and the energy
level diagram of Na plus?
Well, it's isoelectronic
with neon.
And I know that neon has an
average valence electron
energy of about 20
electron volts.
So my guess is the
visible light is
not going to do anything.
And so it just passes
right on through.
See, we got all that.
Electrical insulator.
How do I know that?
Well, all that glitters
is not gold, but it
must have free electrons.
And these electrons
are all bound, and
they're tightly bound.
Hard and brittle.
If it's going to be ductile the
atoms need to be able to
slide over one another.
Well, there's no way these can
slide over one another because
to slide over one another
requires that at some point
the two sodiums are going to be
nearest neighbors, and the
repulsive forces are so high
the crystal fractures.
So if you try to deform an ionic
solid you will get it
moving in accordance
to the elasticity.
So force will be proportional
to the extension--
Hooke's Law-- but if you try
to plastically deform
it, you shear it.
Soluble in water.
We'll come back to that later.
Melt to form ionic liquids.
And good for electrolytic
extraction of metals.
I showed you magnesium
last day.
Today I'll talk a little
bit about aluminum.
But that comes later.
OK.
So where do we find elements
that are going
to form ionic solids?
Well, you go back to the
beginning of the lecture.
What do you look for?
You've got to find the box of
really good electron donors
and the box of really good
electron acceptors.
That's where you go.
So the good electron donors are
at the left side and the
good electron acceptors
are at the right side.
So if you take sodium
plus chlorine
you get sodium chloride.
If you get calcium plus
fluorine, calcium fluoride.
Magnesium plus oxygen,
and so on.
OK?
Yeah, this shows.
Yeah.
Aluminum plus oxygen, yeah.
There's Max Born.
Max Born, he got
a Nobel Prize.
And so did Fritz Haber,
but we're going to
come to him in a minute.
All right.
So now I want to do this
energetic calculation one more
way, because right now we've
been operating with ion gas
but sodium isn't found
normally in the
form of an ion gas.
So let's do something
that starts with
elements found in nature.
So we want to form an ionic
crystal from elements in their
natural state.
And what's going to happen is
en route we're going to be
able to define a few more terms.
So that's going to make
everybody happy because we
get more definitions.
And this is called the
Born-Haber Cycle, named after
Born and Haber.
And what we're going to
do is we're going to--
pardon me, there's a C in
there-- we're going to invoke
Hess's Law.
And Hess's Law is sort of
like Kirchhoff's law
for electric circuits.
Hess's law says that the energy
of a chemical change is
path independent.
So energy change in a chemical
reaction is path independent.
It's sort of like potential
energy in Newtonian mechanics.
It doesn't matter if you take
the elevator to the top of the
Hancock Tower or if you walk up
the stairs, the change in
potential energy is the same
when you express it from the
top of the Hancock Tower,
although you might be somewhat
more exhausted having walked
the steps instead of taking
the elevator.
But you don't get any
credit in terms of
the potential energy.
So let's use Hess's Law in
order to describe the
formation of sodium chloride.
So I'm going to start with
sodium as it's found in
nature, and I'm going to talk
about room temperature.
So sodium is a solid at room
temperature, and I'm going to
react it with chlorine gas.
Chlorine's a gas at room
temperature and it's a
diatomic molecule.
And we're going to react it to
form sodium chloride, which is
a solid and a crystal.
Now you might say, well isn't
that kind of redundant?
No, because later on I'm going
to teach you about a form of
solid matter that does
not consist of atoms
in a regular array--
disordered solids.
So we're specifying, I want to
form crystal and solid sodium
chloride, because that's the
reaction that would occur if
we were to do it in the lab.
And what have we calculated
so far?
What we've calculated
so far is this.
We've calculated chloride ion
in the gas phase plus sodium
ion in the gas phase reacting
to form the crystal.
And we've called this the energy
of crystallization.
That's this Madelung stuff.
This is the Madelung
energy here.
All right?
I'll even put M here.
Madelung energy.
And why am I using H?
Because that's what
the books use.
H is enthalpy, and for
condensed matter the
difference between enthalpy
and energy
doesn't amount to much.
So H, just for the record,
is enthalpy.
And we've been operating
with E as energy.
It's almost equal to
E, which is energy
for condensed matter.
For the gas phase
it gets hairy.
OK.
So I want to get us from sodium
solid, chloride solid
over to here.
So how am I going to do that?
Well, first of all, I know how
to make sodium ion gas.
I start with sodium gas and then
by ionization I make the
electron plus sodium ion.
So this is called the ionization
energy, isn't it?
This is the ionization energy.
Sodium gas goes to that.
And now how do I get sodium
gas from sodium?
Well that's just called
sublimation.
So this I'm going to need
delta H of sublimation.
Sublimation is the conversion
of solid to vapor.
And you can look that up
on the Periodic Table.
I'll show you how to get
that in a second.
And now how do I get
chloride gas?
Well chloride gas is going to
start with atomic chlorine
gas, but instead of losing an
electron I've got to acquire
an electron.
And this action of adding an
electron is sort of an inverse
ionization, and this is called
electron affinity.
And there are tables of
electron affinity.
So each element has the ability
to lose an electron,
it has the ability to
gain an electron.
Losing an electron is ionization
energy, acquiring
an electron is electron
affinity.
And just as with ionization
energies, if you have multiple
electrons you have a first
electron affinity, second
electron affinity, and so on.
And now how do I get
to atomic chlorine?
I've got to dissociate
diatomic chlorine.
So this is called
dissociation.
So that's the whole thing.
And I'm going to now put
some numbers on here.
Let's see.
I'm going to call sublimation
step one, dissociation step
two, ionization I've got here
is step three, electron
affinity is step four, and
crystallization or
Madelung is step five.
And so working off of Hess's law
we can say that the total
energy required for the
formation of the crystal--
delta H for the reaction.
What's the reaction?
The reaction of sodium
plus chlorine
to make sodium chloride--
is going to be the sum of all of
the constituent components.
The sum of all the delta's H.
Not delta H's.
delta's H, like attorneys
general.
All right?
So now let's add these up.
So we go number one.
Number one I can look up
on the Periodic Table.
Where is it?
There's Fritz Haber.
So that's given here.
If you look on the Periodic
Table that'll
give you the number.
And for sodium it's 108
kilojoules per mole.
Number two, get that
from tables.
It's 122.
Number three is just the first
ionization energy.
You look it up on the
Periodic Table.
First ionization energy of
sodium is about 5.3 electron
volts, which turns out to be
496 kilojoules per mole.
But look, these are all positive
energies and we're
trying to make a net
negative energy.
So these three steps
are all raising the
energy of the system.
Finally, acquiring an electron
by chlorine is going to
decrease the energy of the
system, because chlorine is a
good electron acceptor.
So that's minus 349.
But watch this people.
The energy in forming the
crystal from the discrete
ion-pairs is 787 kilojoules per
mole, which gives us a net
value of minus 410 kilojoules
per mole.
So what this is showing you
is what the relative
contributions are
of the different
components of that thing.
So here it is in
graphical form.
There's the vaporization
of sodium.
This is the dissociation
of chlorine.
This is the ionization
of sodium.
All positive energies.
And now electron affinity.
And look at this contribution
from the Madelung energy.
So when things crystallize a
lot of heat's given off.
In fact, we can use that in
cooling and moderating climate
if we're clever about it.
All right.
Now just to show you what the
different values are here that
you're going to go in and get
the lattice energies, you need
to know the various
r values, you see.
The r0 is simply going to be
the r-plus and the r-minus.
So lithium fluoride,
there it is.
There's the lattice energy and
it's based on the combination
of lithium cation and
fluoride anion.
So they've gone through and
calculated these values.
So there's sodium
chloride is 787.
And, you know, you even get
things like the boiling
points, melting points.
So sodium chloride, for example,
melts at about 800
degrees Celsius.
Now if you take magnesium oxide,
magnesium oxide is,
look, 3,700 versus 700.
And the melting point of
magnesium oxide is
2,800 degrees C.
Look at aluminum.
Aluminum plus oxygen, look at
the end binding energy there.
It's phenomenal, which means
it might be useful for--
I'm standing underneath
the shuttle here.
This is the tiles underneath the
shuttle, and they're made
of aluminum oxide because the
Madelung energy is so high so
it's got the thermal
shock resistance.
So now you know how to go and
design things for thermal
ablation resistance.
All you need is this table.
That's all.
No, you need a little more than
that, but this is a good
place to start.
If you don't understand this
table I don't want you working
on the project.
OK?
What else?
All this shows you-- yeah,
we're going skip that.
All right.
So now we've got a
few minutes here.
What I want to do is last day
I talked about magnesium,
today I want to talk
about aluminum.
And it is also made in an
electrochemical process.
In this case the electrodes
are horizontal.
We feed aluminum oxide in
and we pass current--
and huge currents.
This thing typically runs at
300,000, 400,000 amperes and
about 4 volts.
So with the cathode we are
running-- remember, we're
running nature in reverse.
Instead of the electron donor
that aluminum is, we're
shoving electrons onto aluminum
ion and converting it
back to aluminum.
Unfortunately, on the anode side
we have to use carbon,
and the carbon itself
is consumed.
So we consume about a half
a ton of carbon to
make a ton of aluminum.
So aluminum smelters generate
a lot of greenhouse gases.
So you can see this is like
a drafting pencil: it's
constantly being fed.
And to give you a sense of scale
this is probably about
10 feet across, and this gap is
about 2 inches, and this is
about, I don't know, a
foot and a half deep.
It's going to get about 1,000
degrees Centigrade--
liquid aluminum and
liquid salt.
So this is what a smelter
looks like.
What's the sound of
electric current?
Yeah.
The only sound you hear is the
fans on the top that keep the
place clean.
There's the busbars that are
bringing in the current.
And all of these various
posts are these things.
So all of the magic
is occurring
below the floor here.
It was invented simultaneously
in the United States by
Charles Martin Hall and in
France by Paul Heroult.
In the same year they filed
patents independently and
eventually crossed license
when they collided at the
World Court.
So this is what happens.
We dissolve aluminum oxide in
a molten fluoride called
cryolite, which originally
came from Greenland, make
liquid aluminum carbon
dioxide.
Now if we wanted to make this
truly green, we want to
eliminate greenhouse gas
emissions, you need to find an
inert anode.
So on an inert anode aluminum
oxide would be converted into
aluminum and oxygen.
So not only would you not
produce greenhouse gases but
you'd produce tonnage oxygen,
which is marketable.
So some of the work that goes
on in my lab is in advanced
materials with a view to trying
to find an inert anode
that would then make this
process very, very clean and
justify substituting aluminum
for steel in cars.
By the way, when you have the
field, even though it's a DC
current it's a divergent
field.
And this is me in a magnesium
smelter just in Utah.
There is the busbar for
the magnesium cell.
So I'm about, oh, two
meters away from
the edge of the busbars.
The magnetic field is so high
I've got one, two, three,
four, five paper clips standing
against gravity.
And I asked them for paper
clips and there
was no office there.
They managed to find a few.
I wanted to see how
many would go.
I'm willing to bet I could
probably have about seven or
eight paper clips up before the
gravity would cause them
to collapse.
That's the intensity
of magnetic
fields in these smelters.
So when you drive up to the
smelter for the tour you park
a fair distance away, you leave
your wallet with the
credit cards in it, because
this is the biggest bulk
demagnetizer you can imagine.
If you've got a watch that's
got hands that move they're
going to be going like this.
[GESTURING]
PROFESSOR: Yeah.
This is the shuttle.
This is forged aluminum
wheels.
The shuttle lands on Centerline
racing wheels.
They're forged at a place
in California.
They're made of aluminum
alloy.
And I don't know if you can read
on the side here, it's a
little bit light, but these
are special tires.
They're made by Michelin
So, anyways, the whole message
here is learn the lessons here
in 3.091 and then we can work
together to make metal in an
environmentally acceptable
way.
And just before I send you on
your way for the weekend I
thought I'd tell a little
joke related to
the uncertainty principle.
So the joke goes like this.
Heisenberg is racing down the
Autobahn and he gets pulled
over by the state trooper who
comes to the car and says,
where's the fire buddy?
Do you know how fast
you were going?
And Heisenberg looks at
him he says, no, but I
know where I am.
[LAUGHTER]
PROFESSOR: All right.
Get out of here.
Have a good weekend.
