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PROFESSOR: So, the first thing
that we did is look at a simple

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00:00:23 --> 00:00:26
polymer model not too different
from polymer models that

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we've seen before.
 

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That is, models for
molecular configurations.

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00:00:29 --> 00:00:32
The difference in the case that
we treated last time, though,

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00:00:32 --> 00:00:36
was that instead of having just
a handful of possible

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configurations with their just,
designated energies, we had

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essentially an infinite
sequence of levels.

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All of evenly spaced energy.
 

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00:00:45 --> 00:00:49
And we went through that and
saw what the thermodynamics

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00:00:49 --> 00:00:50
worked out to be.
 

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00:00:50 --> 00:00:53
And also with the high and
low temperature limits of

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the thermodynamics
turned out to be.

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00:00:55 --> 00:00:59
And the interesting consequence
of that is that that particular

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choice of configurations and
energies associated with them

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00:01:03 --> 00:01:08
maps exactly onto the
vibrational levels of

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an ordinary molecule.
 

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00:01:10 --> 00:01:14
So, of course, when we do the
statistical mechanics in the

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end, everything depends on
the partition function.

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00:01:17 --> 00:01:17
Right?
 

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00:01:17 --> 00:01:22
So we're always writing some
molecular partition function.

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00:01:22 --> 00:01:30
And we've got a sum over our
energies for all the states.

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00:01:30 --> 00:01:34
And so the only input
information we need to know

32
00:01:34 --> 00:01:37
is, what are the levels and
what are their energies.

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So, although we started
yesterday with a model of

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polymer confirmations, once we
decided what the energies were,

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00:01:44 --> 00:01:48
which was this sequence of
energies starting at zero and

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going with E, and 2E, and 3E, I
think we labeled them E0, and

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00:01:55 --> 00:01:59
so on, that's all that mattered
as far as the statistical

38
00:01:59 --> 00:02:00
mechanics was concerned.
 

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00:02:00 --> 00:02:05
And so, since we started, We
formulated this by imagining

40
00:02:05 --> 00:02:08
different confirmations for a
polymer, but in fact these are

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the vibrational energies
of any molecule too.

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Of any vibrational mode.
 

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00:02:13 --> 00:02:16
So the results are very
generally applicable.

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00:02:16 --> 00:02:19
And so we saw what
the thermodynamics

45
00:02:19 --> 00:02:20
worked out to be.
 

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00:02:20 --> 00:02:22
And what the limiting
cases were.

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00:02:22 --> 00:02:25
And a couple of them, that I'll
just review, are, one that's

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very important in lots of
settings, is that the low

49
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temperature limit when you're
down it basically zero degrees

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Kelvin, that's where you've
got kT way down here.

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00:02:44 --> 00:02:46
So there's not nearly enough
thermal energy for anything

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to get out of even
the lowest level.

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And so in that case, everything
is in the ground state.

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And then the heat capacity,
which is this often a very

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simple thing to measure, you
change the temperature.

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And you see that there's
no change in the total

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vibrational energy.
 

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In other words, the limiting
low temperature heat

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capacity is zero.
 

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00:03:06 --> 00:03:15
So we saw that for
Cv for vibrations.

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And the limit that T
goes to zero was zero.

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And that's because we were in
this limit, where everything

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is in the ground state.
 

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Which is to say that u
vibrational energy in

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that limit is zero.
 

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Unlike the case we treated
earlier, where we had a

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limited number of levels.
 

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Because we treated a simple
four-unit polymer with only

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four possible confirmations, in
this case we have essentially

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an infinite number.
 

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So even in the high temperature
limit, we don't have them,

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there's not a maximum
total energy.

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If the temperature keeps
going up, you'll get

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more and more energy.
 

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Because you can keep populating
higher and higher levels.

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But the way that happens
stops changing.

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Because the levels are
all evenly spaced.

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So if the temperature, instead
of being down here, is

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somewhere up here, now for
sure if we change the

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temperature, the energy
will continue to change.

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But what we've found is that in
the high temperature limit, we

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00:04:22 --> 00:04:26
had the equipartition
of energy result.

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And that is, u vibrational
in the high temperature

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limit, was just, for N
molecules, it was NkT.

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1/2 kT for each kinetic
energy degree of freedom.

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1/2 kT for each potential
energy degree of freedom.

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And since vibrations have each
of those, potential and kinetic

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energy, it's kT for each
molecule, for each

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degree of freedom.
 

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That's an incredibly
simple, useful thing.

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That means, if I've got a
molecule in the gas phase and

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it's in the high temperature
limit for translation, I know

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00:05:05 --> 00:05:09
each translational degree
of freedom will contribute

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for each molecule 1/2 kT.
 

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00:05:12 --> 00:05:15
Or 3/2 kT, in all
three dimensions.

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00:05:15 --> 00:05:19
So I know the translational
energy of a mole of molecules

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00:05:19 --> 00:05:22
in the gas phase at room
temperature without

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00:05:22 --> 00:05:23
doing anything all.
 

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00:05:23 --> 00:05:24
It's 3/2 NkT.
 

100
00:05:24 --> 00:05:28
Rotations, if it's a diatomic
molecule, will be two

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00:05:28 --> 00:05:30
different degrees of freedom.
 

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For rotation.
 

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In two orthogonal planes.
 

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00:05:32 --> 00:05:35
And I'll have 1/2 kT for each.
 

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1/2 NkT for N molecules.
 

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For vibrations, if I'm in a
high temperature limit, then

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00:05:40 --> 00:05:42
it'll be kT for each
vibrational mode.

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For molecules, the vibrational
frequencies are usually high,

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so that you're not in the
high temperature limit.

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00:05:47 --> 00:05:52
But for materials where you
have vibrations, acoustic

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00:05:52 --> 00:05:54
vibrations, those
are low frequency.

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You can be in the high
temperature limit.

113
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And easily are.
 

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So the result for energy for
vibrational energy in the high

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00:06:01 --> 00:06:02
temperature limit, was NkT.
 

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00:06:02 --> 00:06:08
And so that means that the heat
capacity in that same limit at

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00:06:08 --> 00:06:13
high temperature was just Nk,
derivative with respect

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to temperature.
 

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00:06:14 --> 00:06:17
Very simple thing
to measure, again.

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00:06:17 --> 00:06:20
So that's what we saw last
time in the case of both the

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00:06:20 --> 00:06:24
conformational model that we
treated and the vibrational

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energies of molecules onto
which that same model maps.

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Now what I want to do, is
look a little further.

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00:06:32 --> 00:06:35
And let me actually write the
partition functions too.

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00:06:35 --> 00:06:38
Also an important result was
that the partition function

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itself, q vibrational, in the
high temperature limit,

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00:06:46 --> 00:06:50
was just kT over E0.
 

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00:06:50 --> 00:06:57
Also very simple result.
 

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E0 is h nu 0 for vibrational
mode with frequency nu 0.

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And of course, given the
partition function you can

131
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calculate everything.
 

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00:07:13 --> 00:07:15
So in the high temperature
limit we can easily

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00:07:15 --> 00:07:16
calculate things.
 

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00:07:16 --> 00:07:18
In the low temperature limit,
it's just one, right?

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There's only one allowed
possible state in the

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00:07:21 --> 00:07:22
low temperature limit.
 

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00:07:22 --> 00:07:26
Everything is in the ground
state when it's cold enough.

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00:07:26 --> 00:07:30
So the sum of our states just
gives us exactly one term

139
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that's of any reasonable value.
 

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00:07:32 --> 00:07:35
That is, the term with this
being the lowest energy.

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00:07:35 --> 00:07:38
Everything else, the energy
is much bigger than kT.

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00:07:38 --> 00:07:45
This is a vanishingly
small number then.

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So just write q, vibrational,
in the low temperature

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00:07:50 --> 00:07:53
limit, is just one.
 

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00:07:53 --> 00:07:57
OK, that was the
case for vibration.

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00:07:57 --> 00:08:02
And remember, these low
temperature limiting cases,

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that's a common case for
nearly any degree of freedom.

148
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As long as it's quantized.
 

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At some point, you'll
get this situation.

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00:08:08 --> 00:08:11
Where the temperature is
low compared to even the

151
00:08:11 --> 00:08:12
lowest excited level.
 

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Of course, what'll happen in
the high temperature limit

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00:08:15 --> 00:08:19
might vary, depending on the
structure of the energy levels.

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00:08:19 --> 00:08:22
In the case of vibrations, it's
like this, it'll turn out

155
00:08:22 --> 00:08:23
rotations do the same thing.
 

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00:08:23 --> 00:08:28
But it's not always
as simple as this.

157
00:08:28 --> 00:08:34
OK, now what I want to do is
just go through the next step

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of what we can treat,
given the statistical

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00:08:37 --> 00:08:38
mechanics that we know.
 

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Which is chemical equilibrium.
 

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00:08:40 --> 00:08:43
Why can't we just calculate
equilibrium constants based

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on what we've seen so far.
 

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00:08:45 --> 00:08:48
If we can calculate the
partition functions for

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00:08:48 --> 00:08:51
molecules, and they undergo
reactions and they're in

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equilibrium, we should be
able to calculate the

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00:08:54 --> 00:08:55
equilibrium constants.
 

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From first principles, just
based on the statistical

168
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math we've seen so far.
 

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So let's try to do that.
 

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So it just means that, again,
just as always, if we know all

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the energy levels and what all
the possible states are,

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there's no reason we shouldn't
be able to set our sights on a

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00:09:24 --> 00:09:28
calculation of that sort.
 

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So let's just sketch out what
the levels are going to

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00:09:31 --> 00:09:34
look like for simple
chemical events.

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So I just want to draw an
energy diagram, here's energy.

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And, like we often do with
chemical equilibria, I'm going

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to set the zero of energy at
the separated atoms, and

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I'm imagining I've got
reactants and products.

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00:10:14 --> 00:10:20
So let's make this
our products.

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00:10:20 --> 00:10:23
Then over here we'll have
our reactants and make

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00:10:23 --> 00:10:32
the energies different.
 

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00:10:32 --> 00:10:33
Alright.
 

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00:10:33 --> 00:10:36
So there's some amount of
binding energy, right?

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There's a bond dissociation
energy going from the lowest

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available level in each
molecule to the dissociation

187
00:10:44 --> 00:10:47
limit where we pulled
the atoms apart.

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So now I'm going to draw
vibrational energy levels

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00:10:50 --> 00:10:51
inside the molecule.
 

190
00:10:51 --> 00:10:54
Let's imagine, it wouldn't need
to be this, but let's imagine

191
00:10:54 --> 00:10:56
it's just diatomic molecules.
 

192
00:10:56 --> 00:10:59
So there's one vibrational
mode in each.

193
00:10:59 --> 00:11:01
Just the stretching mode.
 

194
00:11:01 --> 00:11:06
And we've already seen the
levels are evenly spaced.

195
00:11:06 --> 00:11:09
So there's going to be a bunch
of evenly spaced levels.

196
00:11:09 --> 00:11:12
Actually, it stopped being
quite evenly spaced once this

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00:11:12 --> 00:11:16
stops being a simple harmonic
oscillator, but for all the

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00:11:16 --> 00:11:18
low lying levels it's
pretty close to that.

199
00:11:18 --> 00:11:22
So those are the
available levels.

200
00:11:22 --> 00:11:25
And there's a
dissociation energy.

201
00:11:25 --> 00:11:36
So minus E, and I'll call
this for the products, Ep.

202
00:11:36 --> 00:11:40
 


203
00:11:40 --> 00:11:45
And it's just minus
D0 for the products.

204
00:11:45 --> 00:11:48
Right, in language, terminology
that we've seen before.

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00:11:48 --> 00:11:56
And it's the same thing
for the reactants.

206
00:11:56 --> 00:12:00
And really, if this were more
than a diatomic molecule,

207
00:12:00 --> 00:12:02
maybe there would be a
bunch of vibrational modes.

208
00:12:02 --> 00:12:03
But it wouldn't matter.
 

209
00:12:03 --> 00:12:06
This would just represent all
the vibrational energies.

210
00:12:06 --> 00:12:14
There's some lowest
state available.

211
00:12:14 --> 00:12:18
So now we've got the
dissociation energy

212
00:12:18 --> 00:12:23
for the reactants.
 

213
00:12:23 --> 00:12:26
So those are the energies it
takes to separate the molecule.

214
00:12:26 --> 00:12:29
But those aren't the only
states available to molecules.

215
00:12:29 --> 00:12:32
Of course, they could have
extra vibrational energy.

216
00:12:32 --> 00:12:35
They're not always in
their ground states.

217
00:12:35 --> 00:12:37
And we're going to calculate
partition functions.

218
00:12:37 --> 00:12:41
In general we would sum over
all the available states.

219
00:12:41 --> 00:12:43
And then just calculate
the probability of

220
00:12:43 --> 00:12:46
being in the state.
 

221
00:12:46 --> 00:12:50
One way to look at this, since
if there's chemical equilibrium

222
00:12:50 --> 00:12:54
between the species, it means
the molecules can interconvert.

223
00:12:54 --> 00:12:57
What that really means is that
any of the molecule has access

224
00:12:57 --> 00:13:00
to any of these states.
 

225
00:13:00 --> 00:13:03
Of course, we like to group
all these states over here.

226
00:13:03 --> 00:13:06
Because they correspond to a
particular chemical structure.

227
00:13:06 --> 00:13:08
And we like to group these
states over here, because they

228
00:13:08 --> 00:13:12
correspond to this different
chemical structure.

229
00:13:12 --> 00:13:15
From a statistical mechanics
point of view, it's

230
00:13:15 --> 00:13:18
just states and levels.
 

231
00:13:18 --> 00:13:22
And we could just calculate the
probability of any molecule

232
00:13:22 --> 00:13:25
being in any one of the states.
 

233
00:13:25 --> 00:13:28
But it's useful to keep
them separated like this.

234
00:13:28 --> 00:13:31
So let's do that.
 

235
00:13:31 --> 00:13:37
And let's look at the
difference here, the difference

236
00:13:37 --> 00:13:39
in dissociation energy.
 

237
00:13:39 --> 00:13:41
Delta D0.
 

238
00:13:41 --> 00:13:44
Let's not put a double arrow.
 

239
00:13:44 --> 00:13:49
Let's define the sign of it.
 

240
00:13:49 --> 00:13:54
This way, going from
products to reactants.

241
00:13:54 --> 00:13:58
OK, now let's just look at
how equilibrium should work.

242
00:13:58 --> 00:14:01
So now, let's just take
a generic reaction.

243
00:14:01 --> 00:14:06
Little a, our number, our
stoichiometric number, A plus

244
00:14:06 --> 00:14:12
b, and B goes to c C, d D.
 

245
00:14:12 --> 00:14:20
And we know that delta G0
is minus RT log of Kp.

246
00:14:20 --> 00:14:24
 


247
00:14:24 --> 00:14:28
And that delta G0 is just the
free energy of the products

248
00:14:28 --> 00:14:30
minus the free energy
of the reactants.

249
00:14:30 --> 00:14:43
So it's c times G C0 plus
d G D0 minus a G A0

250
00:14:43 --> 00:14:47
minus b G equals B0.
 

251
00:14:47 --> 00:14:50
So we need to know the
free energy of each

252
00:14:50 --> 00:14:53
one of the species.
 

253
00:14:53 --> 00:14:56
And now we know how to
calculate that from first

254
00:14:56 --> 00:14:58
principles, through
statistical mechanics.

255
00:14:58 --> 00:15:05
So we know that G is A plus pV.
 

256
00:15:05 --> 00:15:14
A is minus kT log of capital Q.
 

257
00:15:14 --> 00:15:18
And I'm going to assume that
we're in the gas phase.

258
00:15:18 --> 00:15:21
And it's an ideal gas, so I'm
going to replace pV by NkT.

259
00:15:21 --> 00:15:31
 


260
00:15:31 --> 00:15:33
And what's Q?
 

261
00:15:33 --> 00:15:37
Well, we know what Q is.
 

262
00:15:37 --> 00:15:45
It's q, little q, to the
N power translational

263
00:15:45 --> 00:15:47
over N factorial.
 

264
00:15:47 --> 00:15:50
For atoms, this is
all it would be.

265
00:15:50 --> 00:15:51
But we have molecules.
 

266
00:15:51 --> 00:15:54
So we have individual
other degrees of freedom

267
00:15:54 --> 00:15:57
besides translation.
 

268
00:15:57 --> 00:16:04
So let me just label those
internal, q internal.

269
00:16:04 --> 00:16:06
And that's also to the N power.
 

270
00:16:06 --> 00:16:09
And q internal, if you say,
what are those degrees of

271
00:16:09 --> 00:16:12
freedom, well it's the
electronic energy. it's

272
00:16:12 --> 00:16:13
the vibrational energy.
 

273
00:16:13 --> 00:16:14
It's the rotational energy.
 

274
00:16:14 --> 00:16:17
And we'll deal with those
shortly, but let's just

275
00:16:17 --> 00:16:19
separate it this way for now.
 

276
00:16:19 --> 00:16:26
Just we can keep track of where
the N factorial belongs.

277
00:16:26 --> 00:16:28
And now we're going to use
Stirling's approximation

278
00:16:28 --> 00:16:32
for the N factorial.
 

279
00:16:32 --> 00:16:46
So our log of Q, which we need
up here, is just N log of q

280
00:16:46 --> 00:17:05
trans q internal minus
log of N factorial.

281
00:17:05 --> 00:17:14
And that's just equal to N
log q trans q internal,

282
00:17:14 --> 00:17:19
minus N log N.
 

283
00:17:19 --> 00:17:22
Plus N.
 

284
00:17:22 --> 00:17:26
And of course, this I'm going
to put down here momentarily.

285
00:17:26 --> 00:17:31
And now our expression
for G is up there.

286
00:17:31 --> 00:17:36
So it's minus kT
log of Q plus NkT.

287
00:17:36 --> 00:17:41
 


288
00:17:41 --> 00:17:47
So these factors of N
are going to cancel.

289
00:17:47 --> 00:17:50
So when I take minus kT
log of Q, that's going to

290
00:17:50 --> 00:17:52
be minus NkT over here.
 

291
00:17:52 --> 00:17:54
That's going to cancel
the plus NkT here.

292
00:17:54 --> 00:17:57
So all I'm going to have left
is this term and this term.

293
00:17:57 --> 00:18:01
Which I can combine.
 

294
00:18:01 --> 00:18:12
So it's just minus NkT
log of q translational

295
00:18:12 --> 00:18:15
q internal over N.
 

296
00:18:15 --> 00:18:19
Now these, really, are just the
same as q, it's just minus

297
00:18:19 --> 00:18:24
NkT log of little q over N.
 

298
00:18:24 --> 00:18:27
I didn't need to separate
that into this product.

299
00:18:27 --> 00:18:29
I only wanted to do it to be
clear where we were getting

300
00:18:29 --> 00:18:33
this factor of N
factorial from.

301
00:18:33 --> 00:18:37
So now we have an
expression for G.

302
00:18:37 --> 00:18:40
And if we know G for all the
species involved in a chemical

303
00:18:40 --> 00:18:43
reaction, we should be able to
calculate the chemical

304
00:18:43 --> 00:18:44
equilibrium.
 

305
00:18:44 --> 00:18:46
So let's do it.
 

306
00:18:46 --> 00:18:50
So now to do it, let's look a
little more closely at what

307
00:18:50 --> 00:18:52
these internal degrees
of freedom are.

308
00:18:52 --> 00:18:55
Because that's where these
important details are

309
00:18:55 --> 00:18:56
going to come in.
 

310
00:18:56 --> 00:18:58
Obviously the equilibrium
is going to depend

311
00:18:58 --> 00:18:59
on the energetics.
 

312
00:18:59 --> 00:19:02
How much different are the
bonding energies or the

313
00:19:02 --> 00:19:05
dissociation energies and
the molecules involved.

314
00:19:05 --> 00:19:08
And also, how much different
or the other molecular

315
00:19:08 --> 00:19:09
energy levels.
 

316
00:19:09 --> 00:19:12
The vibrations,
rotations, and so forth.

317
00:19:12 --> 00:19:27
So, q internal is the product
of rotational, vibrational, and

318
00:19:27 --> 00:19:30
electronic partition functions.
 

319
00:19:30 --> 00:19:35
Remember how we saw that if you
can write the energy as a sum

320
00:19:35 --> 00:19:38
of energies, then the partition
functions are multiplied.

321
00:19:38 --> 00:19:44
Because, of course, if this is
the sum of a rotational plus

322
00:19:44 --> 00:19:47
vibrational plus electronic
energy, then of course I can

323
00:19:47 --> 00:19:49
just separate out these things.
 

324
00:19:49 --> 00:19:50
These are in the exponent.
 

325
00:19:50 --> 00:19:55
I can write it as a product.
 

326
00:19:55 --> 00:19:56
Same with translations.
 

327
00:19:56 --> 00:20:01
I already have separated that.
 

328
00:20:01 --> 00:20:05
So now let's look at how
these things behave.

329
00:20:05 --> 00:20:13
Well, for the electronic case,
there's really only one

330
00:20:13 --> 00:20:15
electronic state of
interest in general.

331
00:20:15 --> 00:20:18
And that's the lowest state.
 

332
00:20:18 --> 00:20:20
In cases you're extremely
familiar with, if it were the

333
00:20:20 --> 00:20:23
hydrogen atom, it would be
down in the 1s orbital.

334
00:20:23 --> 00:20:26
And it would take a huge amount
of energy to get up into

335
00:20:26 --> 00:20:28
the 2s or 2p orbital.
 

336
00:20:28 --> 00:20:31
At ordinary temperatures
you never have that.

337
00:20:31 --> 00:20:34
All the atoms would be
in the ground state.

338
00:20:34 --> 00:20:37
Now, for most molecules, it
don't take as much energy

339
00:20:37 --> 00:20:39
as for the hydrogen atom.
 

340
00:20:39 --> 00:20:43
If you have benzene, for
example, the ground electronic

341
00:20:43 --> 00:20:47
state, the lowest electronic
state, is quite far below

342
00:20:47 --> 00:20:48
the first excited state.
 

343
00:20:48 --> 00:20:52
Not as much as the hydrogen
atom going from 1s to 2s to 2p,

344
00:20:52 --> 00:20:55
but still by much more than
ordinary thermal energies

345
00:20:55 --> 00:20:57
at room temperature.
 

346
00:20:57 --> 00:21:00
And what that means is,
again, only the ground state

347
00:21:00 --> 00:21:04
term is going to matter.
 

348
00:21:04 --> 00:21:07
Now, sometimes we would
define that is this as

349
00:21:07 --> 00:21:08
the zero of energy.
 

350
00:21:08 --> 00:21:09
But since we're doing chemical
equilibria, the zero

351
00:21:09 --> 00:21:12
of energy is up here.
 

352
00:21:12 --> 00:21:18
So that epsilon, that energy
term, is this amount.

353
00:21:18 --> 00:21:20
The dissociation energy.
 

354
00:21:20 --> 00:21:22
This is minus
dissociation energy.

355
00:21:22 --> 00:21:24
So we're going to have a
positive number there.

356
00:21:24 --> 00:21:37
So q electronic is just
e to the D0 over kT.

357
00:21:37 --> 00:22:13
So that's easy enough.
 

358
00:22:13 --> 00:22:18
We also know what the
vibrational partition

359
00:22:18 --> 00:22:19
function is.
 

360
00:22:19 --> 00:22:21
We saw it last time.
 

361
00:22:21 --> 00:22:24
It's this thing we could
write as a simple form.

362
00:22:24 --> 00:22:27
One over one minus e to
the minus E, I'll call

363
00:22:27 --> 00:22:31
it E vibe, over kT.
 

364
00:22:31 --> 00:22:41
It's h nu 0, where nu is
the vibrational frequency.

365
00:22:41 --> 00:22:45
Again, in the case of many
molecules, the vibrational

366
00:22:45 --> 00:22:47
energy is pretty high
compared to kT.

367
00:22:47 --> 00:22:50
So even this often
simplifies to be just one.

368
00:22:50 --> 00:22:54
In other words, all the
molecules, in many cases, are

369
00:22:54 --> 00:22:57
basically all of them are
in the ground vibrational

370
00:22:57 --> 00:23:00
state at room temperature.
 

371
00:23:00 --> 00:23:03
Certainly if you look at
molecules of high frequencies,

372
00:23:03 --> 00:23:07
if you look at the hydrogen
molecule, H2, the vibrational

373
00:23:07 --> 00:23:10
frequency is about
4,000 wave numbers.

374
00:23:10 --> 00:23:13
Remember I mentioned last
time, kT at room temperature,

375
00:23:13 --> 00:23:15
T is 300 Kelvin.
 

376
00:23:15 --> 00:23:18
Then kT corresponds
to 200 wave numbers.

377
00:23:18 --> 00:23:20
It's a factor of 2/3.
 

378
00:23:20 --> 00:23:23
Much less than 4
wave numbers.

379
00:23:23 --> 00:23:28
In other words, the vibrational
energy is much lower than kT.

380
00:23:28 --> 00:23:31
So, again, everything would
be in the lowest level.

381
00:23:31 --> 00:23:34
And this just
simplifies to one.

382
00:23:34 --> 00:23:40
In many cases, it's
reasonably close to one.

383
00:23:40 --> 00:23:54
For high vibrational
frequencies, nu 0.

384
00:23:54 --> 00:24:03
Now, we haven't talked
about rotation.

385
00:24:03 --> 00:24:08
But you probably have just an
intuitive feeling that at

386
00:24:08 --> 00:24:12
ordinary temperatures, if I do
this, if I wave my hand in the

387
00:24:12 --> 00:24:15
air, molecules that I happen to
intersect are going to

388
00:24:15 --> 00:24:17
start spinning faster.
 

389
00:24:17 --> 00:24:20
In other words, ordinary
thermal energies do populate

390
00:24:20 --> 00:24:26
some number of rotational
levels of molecules.

391
00:24:26 --> 00:24:28
Molecules probably aren't going
to start getting squished

392
00:24:28 --> 00:24:33
together and vibrate harder
when I do something like this.

393
00:24:33 --> 00:24:36
So molecules might generally
still be in the ground

394
00:24:36 --> 00:24:39
vibrational levels, thermal
energy isn't enough to

395
00:24:39 --> 00:24:41
raise them in many cases.
 

396
00:24:41 --> 00:24:42
But rotation, for sure.
 

397
00:24:42 --> 00:24:45
They're not all in
the lowest level.

398
00:24:45 --> 00:24:51
Turns out that, in fact, the
energy separation between

399
00:24:51 --> 00:24:54
rotational levels is very
small compared to kT

400
00:24:54 --> 00:24:55
at room temperature.
 

401
00:24:55 --> 00:25:00
So just like we saw for the
high temperature limit for

402
00:25:00 --> 00:25:10
vibrations, it turns out that
for q rotation in the high

403
00:25:10 --> 00:25:16
temperature limit, we have the
same situation where we have

404
00:25:16 --> 00:25:23
kT over, now it's a
rotational energy.

405
00:25:23 --> 00:25:26
And it's typically on the order
of about one to ten wave

406
00:25:26 --> 00:25:32
numbers for small to
medium sized molecules.

407
00:25:32 --> 00:25:35
Remember, too, last time when
we talked about in the

408
00:25:35 --> 00:25:38
vibrational levels and said,
well, if you're at kT and it's

409
00:25:38 --> 00:25:42
this big, it's basically
telling you roughly how many

410
00:25:42 --> 00:25:46
levels do you have
thermal access to.

411
00:25:46 --> 00:25:49
Because we saw the same
results for vibrations.

412
00:25:49 --> 00:25:52
It's not so different
for rotation.

413
00:25:52 --> 00:25:54
And it turns out that at
ordinary temperatures, you

414
00:25:54 --> 00:25:58
might have access to a few tens
to a few hundreds of rotational

415
00:25:58 --> 00:26:02
levels, depending on whether
they're closer to one or ten

416
00:26:02 --> 00:26:03
wave numbers or a
little higher.

417
00:26:03 --> 00:26:08
Again, room temperature
is 200 wave numbers.

418
00:26:08 --> 00:26:13
So, that means that, remember
q is a unitless number.

419
00:26:13 --> 00:26:15
You're counting the states,
weighted by the energy.

420
00:26:15 --> 00:26:18
Weighted by the
Boltzmann factor.

421
00:26:18 --> 00:26:21
And for rotations, it's a
number that might be on

422
00:26:21 --> 00:26:34
the order of 100 or so.
 

423
00:26:34 --> 00:26:38
Order of magnitude
estimate, that's all.

424
00:26:38 --> 00:26:44
And of course we've seen the
translational partition

425
00:26:44 --> 00:26:54
function is on the order
of 10 to the 30th, right?

426
00:26:54 --> 00:26:55
Enormous number.
 

427
00:26:55 --> 00:26:59
In other words, in the simple
lattice model that we've use to

428
00:26:59 --> 00:27:02
describe translation, just
breaking up the available

429
00:27:02 --> 00:27:04
volume into little pieces.
 

430
00:27:04 --> 00:27:04
Counting.
 

431
00:27:04 --> 00:27:07
Well, OK, you have something
on the order of 10 to the

432
00:27:07 --> 00:27:09
30th possible locations.
 

433
00:27:09 --> 00:27:13
And if you treat this properly,
quantum mechanically, for the

434
00:27:13 --> 00:27:15
translations, there's actually
a magnitude of the

435
00:27:15 --> 00:27:19
number is similar.
 

436
00:27:19 --> 00:27:23
Now I just want to go through
a very simple example.

437
00:27:23 --> 00:27:28
Just treating a particular
generic reaction.

438
00:27:28 --> 00:27:30
And look at what the
equilibrium constant is.

439
00:27:30 --> 00:27:33
Working it through, given
what we've seen so far.

440
00:27:33 --> 00:27:37
So I just want to simplify
it by having all the

441
00:27:37 --> 00:27:41
stoichiometric coefficients
equal to one.

442
00:27:41 --> 00:27:45
It's not a great complication,
if we don't do that.

443
00:27:45 --> 00:27:50
But it's a little bit simpler.
 

444
00:27:50 --> 00:27:56
So let's take a molecule
that's A-B plus C-D.

445
00:27:56 --> 00:27:59
And now let's break
bonds and reform them.

446
00:27:59 --> 00:28:05
So we get A-C plus B-D, right?
 

447
00:28:05 --> 00:28:08
So we're going to do
something simple.

448
00:28:08 --> 00:28:11
And since I've got all the
stoichiometric coefficients

449
00:28:11 --> 00:28:13
equal to one, we've got the
same number of molecules as

450
00:28:13 --> 00:28:16
reactants and products,
that'll make things

451
00:28:16 --> 00:28:18
a little bit easier.
 

452
00:28:18 --> 00:28:21
Just in the sense that if we
look at the contribution of

453
00:28:21 --> 00:28:24
the translational energies
at room temperature, they're

454
00:28:24 --> 00:28:25
going to be the same.
 

455
00:28:25 --> 00:28:27
For the reactants
and the products.

456
00:28:27 --> 00:28:30
Any of the molecules is going
to have a translational

457
00:28:30 --> 00:28:33
partition function on the order
of 10 to the 30th, at room

458
00:28:33 --> 00:28:35
temperature and an
ordinary volume.

459
00:28:35 --> 00:28:40
And nothing's going to change
from reactants to products.

460
00:28:40 --> 00:28:44
And for vibrations, let's
assume, as is often the

461
00:28:44 --> 00:28:48
case, that the vibrational
frequencies are fairly high.

462
00:28:48 --> 00:28:51
So all the partition
functions for the vibrations

463
00:28:51 --> 00:28:53
are equal to one.
 

464
00:28:53 --> 00:28:54
Like we've got
written up there.

465
00:28:54 --> 00:28:56
For all four of the molecules.
 

466
00:28:56 --> 00:28:59
So then nothing is going to
change in the vibrations.

467
00:28:59 --> 00:29:03
Going from reactants
to products.

468
00:29:03 --> 00:29:05
Finally, the rotations.
 

469
00:29:05 --> 00:29:08
So, for the rotations our
partition function is something

470
00:29:08 --> 00:29:13
on the order of, I think I
meant to write 100,

471
00:29:13 --> 00:29:15
not ten here.
 

472
00:29:15 --> 00:29:17
As our order of magnitude.
 

473
00:29:17 --> 00:29:17
That's what it's going to be.
 

474
00:29:17 --> 00:29:21
It's a number on that order.
 

475
00:29:21 --> 00:29:25
And if we assume that the
masses of the atoms involved

476
00:29:25 --> 00:29:28
are comparable, then we can
cheat a little bit and say

477
00:29:28 --> 00:29:32
that's also, those numbers are
also, going to be comparable

478
00:29:32 --> 00:29:35
for the reactants
and the products.

479
00:29:35 --> 00:29:36
We don't have to do that.
 

480
00:29:36 --> 00:29:39
We could put it in, and
rather easily calculate it.

481
00:29:39 --> 00:29:42
But I'm going to make life
simple in this way and just

482
00:29:42 --> 00:29:46
work through how to carry
through the calculation.

483
00:29:46 --> 00:29:51
And I think it'll be clear how
to put in different partition

484
00:29:51 --> 00:29:53
functions for those quantities
if they are different

485
00:29:53 --> 00:29:55
from each other.
 

486
00:29:55 --> 00:30:03
OK, so let's try it.
 

487
00:30:03 --> 00:30:16
Then, our delta G0, let's
go back over here.

488
00:30:16 --> 00:30:23
Here's delta G, let me just
put the relationship up.

489
00:30:23 --> 00:30:26
So we've got an
expression for G.

490
00:30:26 --> 00:30:28
Which is, let's start
from back there.

491
00:30:28 --> 00:30:35
It's minus.
 

492
00:30:35 --> 00:30:38
And now for each one of
the substances, it's

493
00:30:38 --> 00:30:47
Ni kT log qi over Ni.
 

494
00:30:47 --> 00:30:49
I'm just going to put
this in molar terms.

495
00:30:49 --> 00:30:55
So it's minus little ni RT.
 

496
00:30:55 --> 00:31:00
Log of qi over capital Ni.
 

497
00:31:00 --> 00:31:02
And I only want to do that
because I want to express

498
00:31:02 --> 00:31:04
this in free energy.
 

499
00:31:04 --> 00:31:05
This is for substance i.
 

500
00:31:05 --> 00:31:08
Free energy per mole.
 

501
00:31:08 --> 00:31:15
So it's just minus RT
log of qi over Ni.

502
00:31:15 --> 00:31:17
 


503
00:31:17 --> 00:31:19
That's G per mole.
 

504
00:31:19 --> 00:31:25
That's Gi naught per mole.
 

505
00:31:25 --> 00:31:30
Plus RT log pi over p0.
 

506
00:31:30 --> 00:31:33
Where p0 is an
atmosphere, basically.

507
00:31:33 --> 00:31:39
One bar.
 

508
00:31:39 --> 00:31:45
So that means Gi0
bar is minus RT.

509
00:31:45 --> 00:31:49
And I'm going to take this
minus this and combine them.

510
00:31:49 --> 00:31:57
So it's log of qi over
Ni, that's this part.

511
00:31:57 --> 00:32:00
And now I've got log of pi
over p, and I'm just going

512
00:32:00 --> 00:32:01
to use the ideal gas law.
 

513
00:32:01 --> 00:32:03
So I'm going to use pV is nRT.
 

514
00:32:03 --> 00:32:05
 


515
00:32:05 --> 00:32:23
So then I've got Ni kT over my
factor of p0, times the volume.

516
00:32:23 --> 00:32:30
So that's Gi0 for each
one of the substances.

517
00:32:30 --> 00:32:33
The Ni's cancel.
 

518
00:32:33 --> 00:32:47
And I've got minus RT log
of qi kT over p0 times V.

519
00:32:47 --> 00:32:51
Now, qi is just the total
molecular partition function.

520
00:32:51 --> 00:32:55
It's this product of q trans
times q rotational times q

521
00:32:55 --> 00:33:00
vibrational times q electronic.
 

522
00:33:00 --> 00:33:08
And now I need delta G.
 

523
00:33:08 --> 00:33:16
So delta G0, is just minus RT.
 

524
00:33:16 --> 00:33:18
And now it's just, I'm going to
take this for each one of the

525
00:33:18 --> 00:33:22
substances, for the products,
minus the reactants.

526
00:33:22 --> 00:33:24
And I'm going to
combine the log terms.

527
00:33:24 --> 00:33:26
This is the same, and I'm going
to have this in every term.

528
00:33:26 --> 00:33:27
So what's going to happen?
 

529
00:33:27 --> 00:33:32
Well, all this stuff
is going to cancel.

530
00:33:32 --> 00:33:34
Now, that doesn't
necessarily happen.

531
00:33:34 --> 00:33:36
That happens because of the
stoichiometric numbers

532
00:33:36 --> 00:33:37
being the same.
 

533
00:33:37 --> 00:33:40
Otherwise I'd have to take
them to that the power of

534
00:33:40 --> 00:33:42
the stoichiometric number.
 

535
00:33:42 --> 00:33:45
But in this case they're just
going to simply cancel.

536
00:33:45 --> 00:33:53
So I'm going to have left is
the ratio of my partition

537
00:33:53 --> 00:33:56
function for molecule AC.
 

538
00:33:56 --> 00:34:00
Partition function
for molecule BD.

539
00:34:00 --> 00:34:07
Partition function for molecule
AB, and for molecule CD.

540
00:34:07 --> 00:34:11
Products, reactants.
 

541
00:34:11 --> 00:34:13
Pretty simple, because I
know how to calculate

542
00:34:13 --> 00:34:15
all this stuff.
 

543
00:34:15 --> 00:34:19
And again I'd suggested some
simplifying assumptions for

544
00:34:19 --> 00:34:20
what those partition
functions are.

545
00:34:20 --> 00:34:23
But it wouldn't be hard to
calculate each one of them

546
00:34:23 --> 00:34:27
if I just had all the
relevant energy levels.

547
00:34:27 --> 00:34:29
To make this simple, we're
going to assume that the

548
00:34:29 --> 00:34:33
rotational energies are the
same for all the molecules.

549
00:34:33 --> 00:34:35
It wouldn't need to be, it
wouldn't be hard, to plug

550
00:34:35 --> 00:34:36
in different values.
 

551
00:34:36 --> 00:34:39
We're going to assume we're
in the low temperature

552
00:34:39 --> 00:34:40
limit for vibrations.
 

553
00:34:40 --> 00:34:42
So q vibrational for each
one of these things

554
00:34:42 --> 00:34:46
is equal to one.
 

555
00:34:46 --> 00:34:49
The translational
contributions, are

556
00:34:49 --> 00:34:51
equal for all of them.
 

557
00:34:51 --> 00:34:53
So the only thing that's
different is the

558
00:34:53 --> 00:34:56
electronic contribution.
 

559
00:34:56 --> 00:34:57
And that's different.
 

560
00:34:57 --> 00:35:03
Because, of course, you have
different electronic energies.

561
00:35:03 --> 00:35:06
The binding energies for the
products and the reactants

562
00:35:06 --> 00:35:08
aren't in general
going to be equal.

563
00:35:08 --> 00:35:13
And in most cases, it is the
energetics that dominate

564
00:35:13 --> 00:35:15
the equilibrium constants.
 

565
00:35:15 --> 00:35:18
Of course, that is, it's
the electronic energies

566
00:35:18 --> 00:35:19
that usually dominate.
 

567
00:35:19 --> 00:35:22
Of course, the other
things do matter.

568
00:35:22 --> 00:35:23
But it's not unusual
for the electronic

569
00:35:23 --> 00:35:26
energies to dominate.
 

570
00:35:26 --> 00:35:27
So here's delta G0.
 

571
00:35:27 --> 00:35:33
Of course, this is
minus RT log of Kp.

572
00:35:33 --> 00:35:38
So there's our expression for
our equilibrium constants.

573
00:35:38 --> 00:35:41
So it's just this product.
 

574
00:35:41 --> 00:35:44
So let's just put in the
partition functions for

575
00:35:44 --> 00:35:45
the electronic part.
 

576
00:35:45 --> 00:35:53
It's e to the D0 over
kT for molecule AC.

577
00:35:53 --> 00:36:11
Times e to the D0 for
molecule BD over kT.

578
00:36:11 --> 00:36:21
Divided by e to the D0 for AB
over kT, times e to the D0

579
00:36:21 --> 00:36:25
for molecule CD over kT.
 

580
00:36:25 --> 00:36:36
So the whole thing is just e to
the D0 AC, plus D0 BD minus

581
00:36:36 --> 00:36:49
D0 AB minus D0 CD over kT.
 

582
00:36:49 --> 00:36:57
Or in other words, it's e
to the delta D0 over kT.

583
00:36:57 --> 00:37:00
That's the whole story.
 

584
00:37:00 --> 00:37:06
So it's a super-simple
calculation to execute.

585
00:37:06 --> 00:37:10
And for lots of molecules, we
know the dissociation energies.

586
00:37:10 --> 00:37:13
These are measured, determined,
either thermochemically or

587
00:37:13 --> 00:37:16
spectroscopically for a
great many molecules.

588
00:37:16 --> 00:37:19
So in fact, we have the
information we need to

589
00:37:19 --> 00:37:21
do that calculation.
 

590
00:37:21 --> 00:37:24
And if we want to worry about
the vibrational and rotational

591
00:37:24 --> 00:37:26
levels, typically we have that
information, too,

592
00:37:26 --> 00:37:28
spectroscopically.
 

593
00:37:28 --> 00:37:32
So we know what the relevant
energy levels are to do

594
00:37:32 --> 00:37:33
the whole calculation.
 

595
00:37:33 --> 00:37:37
So what it means is, we can
calculate equilibrium constants

596
00:37:37 --> 00:37:42
for ordinary chemical reactions
just from first principles.

597
00:37:42 --> 00:37:45
Knowing the energy levels
of the available states

598
00:37:45 --> 00:37:49
of the molecules.
 

599
00:37:49 --> 00:37:58
Any questions?
 

600
00:37:58 --> 00:38:03
In your notes I've
included another example.

601
00:38:03 --> 00:38:06
It's a little bit of
an extra example.

602
00:38:06 --> 00:38:08
You can go through
it if you'd like.

603
00:38:08 --> 00:38:12
It turns out to have
a kind of attractive

604
00:38:12 --> 00:38:13
closed form solution.
 

605
00:38:13 --> 00:38:18
Which is the case of a chemical
reaction like an isomerization

606
00:38:18 --> 00:38:21
inside a crystalline solid.
 

607
00:38:21 --> 00:38:23
So you can imagine you've got
a crystal of some molecule,

608
00:38:23 --> 00:38:27
many crystals, thermally over
time, they might decompose.

609
00:38:27 --> 00:38:29
They might have reaction
processes that can take place.

610
00:38:29 --> 00:38:31
So I've imagined you've got
uni-molecular reactions

611
00:38:31 --> 00:38:32
that can occur.
 

612
00:38:32 --> 00:38:36
And in that case you start with
a pure crystal of one species.

613
00:38:36 --> 00:38:37
And you end up with a mixture
of some number of the

614
00:38:37 --> 00:38:40
original species, and some
number of new species.

615
00:38:40 --> 00:38:42
And there's an equilibrium
constant that'll say

616
00:38:42 --> 00:38:44
where that should be.
 

617
00:38:44 --> 00:38:48
So again, it'll depend
on the energies.

618
00:38:48 --> 00:38:52
On the dissociation energies
of the different species.

619
00:38:52 --> 00:38:55
And there's also a kind of
mixing term, because

620
00:38:55 --> 00:38:59
effectively now you've got the
different species located at

621
00:38:59 --> 00:39:00
different places
in the crystal.

622
00:39:00 --> 00:39:04
Those are distinguishable
states, in the solid.

623
00:39:04 --> 00:39:07
And you need to count those.
 

624
00:39:07 --> 00:39:13
But the one other example I
want to work through is a

625
00:39:13 --> 00:39:15
little bit different from
chemical equilibria.

626
00:39:15 --> 00:39:17
It's phase equilibria.
 

627
00:39:17 --> 00:39:22
Why shouldn't we be able to
calculate phase diagrams?

628
00:39:22 --> 00:39:24
So in the earlier part of the
course, you went through

629
00:39:24 --> 00:39:28
and saw the macroscopic
thermodynamic treatment.

630
00:39:28 --> 00:39:31
The equilibrium constants
and chemical equilibria.

631
00:39:31 --> 00:39:34
So of course you saw the
whole delta G0 as minus

632
00:39:34 --> 00:39:37
RT log Kp and so forth.
 

633
00:39:37 --> 00:39:39
And you've seen now we can
actually calculate all that

634
00:39:39 --> 00:39:42
from first principles.
 

635
00:39:42 --> 00:39:43
What about phase equilibria?
 

636
00:39:43 --> 00:39:47
What you saw before, we
just drew phase diagrams.

637
00:39:47 --> 00:39:51
And they had boiling points
or lines of boiling.

638
00:39:51 --> 00:39:54
Or in other words, liquid-solid
and liquid-gas and

639
00:39:54 --> 00:39:56
solid-gas equilibria.
 

640
00:39:56 --> 00:40:02
And what we did is, given
where those lines fell,

641
00:40:02 --> 00:40:05
you can calculate things.
 

642
00:40:05 --> 00:40:07
But where do the lines fall?
 

643
00:40:07 --> 00:40:10
What's the boiling point or
the sublimation temperature

644
00:40:10 --> 00:40:13
of some material at a
particular pressure?

645
00:40:13 --> 00:40:15
Well, we didn't offer
any prescription for

646
00:40:15 --> 00:40:16
calculating that.
 

647
00:40:16 --> 00:40:19
You had to take that from
measurement and then given

648
00:40:19 --> 00:40:22
that, you could use the
Clausius-Clapeyron equation and

649
00:40:22 --> 00:40:26
so forth and look at the way
things behaved if you moved

650
00:40:26 --> 00:40:28
along a line in a
phase diagram.

651
00:40:28 --> 00:40:31
But there was no prescription
offered to calculate where

652
00:40:31 --> 00:40:35
exactly would that line
be on the phase diagram.

653
00:40:35 --> 00:40:38
But again, if you know all the
energies of the possible

654
00:40:38 --> 00:40:43
states, in the solid, in the
liquid and the gas, statistical

655
00:40:43 --> 00:40:46
mechanics shows us that we can
calculate the equilibrium

656
00:40:46 --> 00:40:46
between those.
 

657
00:40:46 --> 00:40:49
Which is to say we know
we can calculate where

658
00:40:49 --> 00:40:51
those lines belong.
 

659
00:40:51 --> 00:40:56
So I want to just go through
maybe one of a couple of

660
00:40:56 --> 00:40:58
relatively simple cases.
 

661
00:40:58 --> 00:41:02
I want to go through the case
of a solid-solid equilibrium.

662
00:41:02 --> 00:41:05
Let's imagine we have two solid
phases and an equilibrium

663
00:41:05 --> 00:41:06
between them.
 

664
00:41:06 --> 00:41:08
And this problem was
on the problem set.

665
00:41:08 --> 00:41:13
How many of you did it?
 

666
00:41:13 --> 00:41:15
Everybody did it.
 

667
00:41:15 --> 00:41:19
Did you all get to
the end of it?

668
00:41:19 --> 00:41:23
Who got to the end of it?
 

669
00:41:23 --> 00:41:25
Some of you got to
the end of it.

670
00:41:25 --> 00:41:29
If you did, congratulations,
it's not so trivial to work

671
00:41:29 --> 00:41:31
on for the first time.
 

672
00:41:31 --> 00:41:36
So I'll just briefly go
through that problem.

673
00:41:36 --> 00:41:37
And show how it works.
 

674
00:41:37 --> 00:41:43
So, it's similar, of course, to
the case that we're doing now.

675
00:41:43 --> 00:41:46
To the case we just did,
of chemical equilibria.

676
00:41:46 --> 00:41:49
The difference, though, is
that it's cooperative.

677
00:41:49 --> 00:41:56
The solid, at least in the way
I formulated that problem, you

678
00:41:56 --> 00:41:59
either have the crystal in
phase one or phase alpha,

679
00:41:59 --> 00:41:59
or in phase beta.
 

680
00:41:59 --> 00:42:02
And there's nothing in between.
 

681
00:42:02 --> 00:42:03
There are lots of
systems like that.

682
00:42:03 --> 00:42:09
Not just crystals, you could go
from different forms of DNA.

683
00:42:09 --> 00:42:11
Act like that.
 

684
00:42:11 --> 00:42:15
Where you have super coiled
DNA and regular DNA.

685
00:42:15 --> 00:42:19
And you actually can have
some of the DNA in each.

686
00:42:19 --> 00:42:22
But it's actually
very cooperative.

687
00:42:22 --> 00:42:26
So there's a huge tendency
to either be all in one,

688
00:42:26 --> 00:42:26
or be all in the other.
 

689
00:42:26 --> 00:42:30
Or at least very very, large
pieces of it like that.

690
00:42:30 --> 00:42:32
Lots of other systems
act like that.

691
00:42:32 --> 00:42:36
In other words, unlike the case
where we're thinking about

692
00:42:36 --> 00:42:39
chemical equilibrium among
molecules in the gas phase,

693
00:42:39 --> 00:42:42
these two molecules over here
crash into each other

694
00:42:42 --> 00:42:44
and they react.
 

695
00:42:44 --> 00:42:45
Nothing else cares.
 

696
00:42:45 --> 00:42:47
The other whole mole of
molecules does whatever

697
00:42:47 --> 00:42:48
they were doing.
 

698
00:42:48 --> 00:42:51
And a little later some other
pair of molecules happen to

699
00:42:51 --> 00:42:53
crash into each other and
maybe react or maybe don't.

700
00:42:53 --> 00:42:56
In other words, there's
no cooperativity at all.

701
00:42:56 --> 00:43:00
What happens to some pair of
reactants and products,

702
00:43:00 --> 00:43:03
everything else is completely
independent of it.

703
00:43:03 --> 00:43:05
That's not the case when
you have, in many cases,

704
00:43:05 --> 00:43:06
a phase transition.
 

705
00:43:06 --> 00:43:08
You can see it by eye, often.
 

706
00:43:08 --> 00:43:13
If you do something
like super cool water.

707
00:43:13 --> 00:43:14
This is a kind of
fun experiment.

708
00:43:14 --> 00:43:18
If you take dry ice, and you've
got a little bit of water.

709
00:43:18 --> 00:43:21
Right or a little pot of water.
 

710
00:43:21 --> 00:43:23
And you put some
dry ice into it.

711
00:43:23 --> 00:43:26
So you can actually get
the water to be below the

712
00:43:26 --> 00:43:28
freezing temperature and
it won't freeze yet.

713
00:43:28 --> 00:43:33
Then you drop a crystal in
there and boom, it all freezes.

714
00:43:33 --> 00:43:35
Very cooperatively.
 

715
00:43:35 --> 00:43:38
Lots of phase transitions
behave like that.

716
00:43:38 --> 00:43:52
So let's just see how it works.
 

717
00:43:52 --> 00:44:04
In this case, it's really
similar to what we just did.

718
00:44:04 --> 00:44:09
We can treat it,
here's phase alpha.

719
00:44:09 --> 00:44:14
Here's phase beta.
 

720
00:44:14 --> 00:44:18
And the crystal, the
interactions between molecules

721
00:44:18 --> 00:44:21
or the atoms in the crystal are
different in the two phases.

722
00:44:21 --> 00:44:23
So effectively, the binding
energy is different.

723
00:44:23 --> 00:44:26
In other words, if you say, now
let's think of the energy it

724
00:44:26 --> 00:44:29
would take to evaporate all the
atoms or molecules and let

725
00:44:29 --> 00:44:31
them loose in the gas phase.
 

726
00:44:31 --> 00:44:34
That's the analog of
dissociation for the molecule.

727
00:44:34 --> 00:44:36
You're pulling everything
apart from the crystal

728
00:44:36 --> 00:44:38
and separating them all.
 

729
00:44:38 --> 00:44:41
So that's what we'll
call these energies.

730
00:44:41 --> 00:44:48
This'll be minus E alpha.
 

731
00:44:48 --> 00:44:55
And this will be minus E beta.
 

732
00:44:55 --> 00:44:58
And then there are
vibrational energies.

733
00:44:58 --> 00:45:02
So the lattice has a bunch
of vibrational energies.

734
00:45:02 --> 00:45:05
We can assume they're
evenly spaced.

735
00:45:05 --> 00:45:07
And they're going
to be different.

736
00:45:07 --> 00:45:10
They're not the same in the two
different crystalline forms.

737
00:45:10 --> 00:45:11
That's typically the case.
 

738
00:45:11 --> 00:45:14
You have a phase transition
from one crystal to another.

739
00:45:14 --> 00:45:16
The lattice vibrational
frequencies aren't

740
00:45:16 --> 00:45:17
the same any more.
 

741
00:45:17 --> 00:45:18
Speed of sound is different.
 

742
00:45:18 --> 00:45:20
Things are different.
 

743
00:45:20 --> 00:45:21
And that's all.
 

744
00:45:21 --> 00:45:24
That's the only thing
that's different.

745
00:45:24 --> 00:45:40
So these are binding
energies per atom.

746
00:45:40 --> 00:45:45
So now, let's just
try to calculate Q.

747
00:45:45 --> 00:45:51
And what we are supposed to get
out of this is, what's the

748
00:45:51 --> 00:45:52
phase transition temperature?
 

749
00:45:52 --> 00:45:55
That's the thing that we
didn't have any prescription

750
00:45:55 --> 00:45:57
for calculating before.
 

751
00:45:57 --> 00:45:58
We just said, here it is.
 

752
00:45:58 --> 00:46:00
Here's the boiling point.
 

753
00:46:00 --> 00:46:01
Here's the melting point,
or whatever the phase

754
00:46:01 --> 00:46:02
transition was.
 

755
00:46:02 --> 00:46:05
Or if we drew a phase
diagram, here's the line.

756
00:46:05 --> 00:46:07
But now we're going to try
to calculate where that is.

757
00:46:07 --> 00:46:09
Or what those temperatures are.
 

758
00:46:09 --> 00:46:16
So, Q for either phase it's
just e to the E over kT.

759
00:46:16 --> 00:46:18
Just like we saw before
for the dissociation

760
00:46:18 --> 00:46:20
energies of molecules.
 

761
00:46:20 --> 00:46:22
To the Nth power.
 

762
00:46:22 --> 00:46:23
No N factorial any more.
 

763
00:46:23 --> 00:46:26
There's no translation.
 

764
00:46:26 --> 00:46:28
Times the vibrational part.
 

765
00:46:28 --> 00:46:32
This is the electronic
partition function.

766
00:46:32 --> 00:46:34
And then there's the
vibrational part.

767
00:46:34 --> 00:46:40
And that's one over one minus
e to the minus h nu, what's

768
00:46:40 --> 00:46:44
called E over kT, for the
lattice I'm assuming it's

769
00:46:44 --> 00:46:47
just one frequency.
 

770
00:46:47 --> 00:46:52
So this energy is h nu E.
 

771
00:46:52 --> 00:46:57
And of course, it's different
for the alpha and beta phases.

772
00:46:57 --> 00:46:59
And I'm going to assume we're
in the high temperature limit.

773
00:46:59 --> 00:47:02
Which is often the case
for lattice vibrations.

774
00:47:02 --> 00:47:06
So it's e to the,
little E over kT.

775
00:47:06 --> 00:47:09
For our electronic part,
to the Nth power.

776
00:47:09 --> 00:47:11
And we've seen what the
high temperature limit is.

777
00:47:11 --> 00:47:14
It's kT over h nu E.
 

778
00:47:14 --> 00:47:17
And that's also, that
to the 3N power.

779
00:47:17 --> 00:47:19
I should have
written this here.

780
00:47:19 --> 00:47:24
In other words if you say, how
many vibrations are there in

781
00:47:24 --> 00:47:28
the lattice, well if there are
N atoms, each atom in the gas

782
00:47:28 --> 00:47:30
phase would have three
degrees of freedom.

783
00:47:30 --> 00:47:32
Translational
degrees of freedom.

784
00:47:32 --> 00:47:35
In the crystal, it can't
freely translate.

785
00:47:35 --> 00:47:36
But those degrees of
freedom are still there.

786
00:47:36 --> 00:47:38
The atoms can all move.
 

787
00:47:38 --> 00:47:41
So now, those are the
lattice vibrations.

788
00:47:41 --> 00:47:42
When the atoms try to
move, they vibrate

789
00:47:42 --> 00:47:44
against each other.
 

790
00:47:44 --> 00:47:47
So how many different
modes are there?

791
00:47:47 --> 00:47:48
They may be degenerate.
 

792
00:47:48 --> 00:47:50
They may all have
the same energy.

793
00:47:50 --> 00:47:52
But those modes are
all still there.

794
00:47:52 --> 00:47:55
Actually, they're the acoustic
modes of the crystal.

795
00:47:55 --> 00:47:58
And there are 3N of them.
 

796
00:47:58 --> 00:48:01
One for each translational
degree or freedom

797
00:48:01 --> 00:48:02
that each atom had.
 

798
00:48:02 --> 00:48:06
OK, so that's it.
 

799
00:48:06 --> 00:48:14
A, minus kT log of Q,
so it's just minus NE,

800
00:48:14 --> 00:48:16
and the kT cancels.
 

801
00:48:16 --> 00:48:28
Minus 3NkT log of
kT over h nu E.

802
00:48:28 --> 00:48:29
Second part.
 

803
00:48:29 --> 00:48:33
And then we can calculate
the chemical potential.

804
00:48:33 --> 00:48:40
It's just the dA/dN,
d constant T and V.

805
00:48:40 --> 00:48:41
So this goes away.
 

806
00:48:41 --> 00:48:43
It's just minus little E.
 

807
00:48:43 --> 00:48:52
This goes away, it's minus
3kT log of kT over h nu E.

808
00:48:52 --> 00:48:55
Well, basically we
just finished.

809
00:48:55 --> 00:48:58
At the phase transition
temperature for the two phases,

810
00:48:58 --> 00:49:02
those things have to be equal.
 

811
00:49:02 --> 00:49:03
And that's all there is to it.
 

812
00:49:03 --> 00:49:18
So, at, we'll call it Tc, the
phase transition temperature, I

813
00:49:18 --> 00:49:26
guess I called it T1 here, mu
alpha is equal to mu beta.

814
00:49:26 --> 00:49:30
So this stuff for alpha is
equal to this for beta.

815
00:49:30 --> 00:49:32
That's it.
 

816
00:49:32 --> 00:49:33
So what does it say?
 

817
00:49:33 --> 00:49:44
It says E beta minus E alpha,
these terms, equals 3kT1 log

818
00:49:44 --> 00:49:50
of nu E beta over nu E alpha.
 

819
00:49:50 --> 00:49:52
Solve for T1.
 

820
00:49:52 --> 00:49:54
That's it.
 

821
00:49:54 --> 00:50:03
It's E beta minus E alpha
over 3k log nu E beta

822
00:50:03 --> 00:50:06
over nu E alpha.
 

823
00:50:06 --> 00:50:09
There is our phase
transition temperature.

824
00:50:09 --> 00:50:11
That's the whole story.
 

825
00:50:11 --> 00:50:15
So if we know the electronic
energy that binds the crystal

826
00:50:15 --> 00:50:18
together, usually something
that can be measured easily, so

827
00:50:18 --> 00:50:21
it's known for many materials.
 

828
00:50:21 --> 00:50:23
And if we know the lattice
vibrational frequency.

829
00:50:23 --> 00:50:29
Also something that's pretty
routinely measured, we're done.

830
00:50:29 --> 00:50:34
So again, from a very simple
first principles approach, we

831
00:50:34 --> 00:50:38
can calculate that phase
transition temperature.

832
00:50:38 --> 00:50:42
We could do it for a solid-gas
equilibria too, if we said OK,

833
00:50:42 --> 00:50:45
let's think of sublimation
of the solid.

834
00:50:45 --> 00:50:48
And now we know how to
calculate the chemical

835
00:50:48 --> 00:50:50
potential in the gas phase.
 

836
00:50:50 --> 00:50:51
We can have those be equal.
 

837
00:50:51 --> 00:50:53
That's actually worked
through in your notes.

838
00:50:53 --> 00:50:55
Again, it's an extra problem.
 

839
00:50:55 --> 00:50:57
If you would like to take
a look at it, it's a

840
00:50:57 --> 00:50:59
straightforward calculation.
 

841
00:50:59 --> 00:51:02
The point is, we can calculate
those phase equilibria.

842
00:51:02 --> 00:51:03
Liquids, I will
say, are harder.

843
00:51:03 --> 00:51:05
Just because it's harder to
define and know all the

844
00:51:05 --> 00:51:07
energies available.
 

845
00:51:07 --> 00:51:10
Doable, approximately.
 

846
00:51:10 --> 00:51:13
But the point is, we can
actually now locate all

847
00:51:13 --> 00:51:16
those lines that we sort
of arbitrarily drew on

848
00:51:16 --> 00:51:17
the phase diagrams.
 

849
00:51:17 --> 00:51:21
And make sense of what the
temperatures and what the

850
00:51:21 --> 00:51:26
pressures are, for that
matter, where they occur.

851
00:51:26 --> 00:51:27
Any questions?
 

852
00:51:27 --> 00:51:30