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PROFESSOR: Last few lectures,
and we started to see some of

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the consequences and looked at
simple changes, transformations

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of expansion and mixtures
of liquids and gases.

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And saw how in the framework of
statistical mechanics, we could

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derive the thermodynamic
results that you saw before,

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based on an empirical
framework.

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On the thermodynamic
framework we've been

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working with all term.
 

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But you saw them derived from
a microscopic perspective.

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Now what I want to do is move
to examples that are more

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commonly encountered
in chemistry.

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One that we did last time
actually was very common,

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or is at least a prototype
for something common.

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That is we looked at what would
be a simple model for a polymer

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with different configurations
available to it.

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And just tried to understand
what the basic thermodynamics

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is that it would exhibit.
 

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What the heat capacities would
be in various limiting cases.

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High and low temperature.
 

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I want to continue that today.
 

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But for an even
more common case.

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So the model that I want to
lay out is indicated on

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your notes for today.
 

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And essentially, you could look
at it as double stranded DNA.

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Any kind of polymer with
a whole variety of

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configurations.
 

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Ultimately what's going
to matter is what is the

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set of energy levels
available to the system.

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And so here's what
it looks like.

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The way I've tried
to depict it.

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I'm imagining something like,
covalently bonded chains that

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have links between them.
 

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So of course, it's reminiscent
of DNA with hydrogen bonding

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between the strands.
 

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And the idea in this simple
model is that you can

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have various numbers of
these hydrogen bonds.

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So this is the case
that would be the most

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energetically stable.
 

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That is, all the available
hydrogen bonds are formed

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between neighboring pairs.
 

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So this would be
our lowest energy.

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Now let's starts breaking
some hydrogen bonds.

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And what I'm imagining is they
break starting at one end

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and working their way down.
 

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So the next highest energy
state would be like this.

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So now these are still
hydrogen bonded.

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This is not.
 

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And there's a cost in
energy for breaking

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that hydrogen bond.
 

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We'll call it epsilon zero.
 

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It's a positive number.
 

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So that this is higher
energy than this.

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And now we'll go to the
next one, which could

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have two broken bonds.
 

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Broken hydrogen bonds.
 

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And so on.
 

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And if we imagine that this is
very long, there may be many

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many, members of the chain.
 

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Then of course this sequence of
structures with corresponding

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energies might go on
for a very long way.

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So this is a simple
construction that gives

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us a set of states with
distinct energies.

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And so we have a series of
configurational energy levels.

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They're evenly spaced
in this model.

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So there's zero, E
zero, two E zero.

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And so on.
 

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So those are available
energy states.

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And now, just based on that
simple model, we should be

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able to figure out all
the thermodynamics.

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We should should be able
to figure out what the

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equilibrium energy is at
a particular temperature.

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What the average energy is.
 

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And so forth.
 

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And we can do it in a
straightforward way.

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So we'll start with our
molecular partition

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function, as always.
 

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So it's q configurational.
 

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And it's just a sum over our
available states with their

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associated Boltzmann factors.
 

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So sum over n of e to
the minus En over kT.

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Now the way I've modeled this,
there are no degeneracies.

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Each energy level has just one
microscopic state of the

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molecule corresponding to it.
 

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So there are no
molecular degeneracies.

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Turns out we'll be able to
put this into a simple form.

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So the first thing is, let's
approximate that we could take

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the sum -- not to some finite
level, which would be the case

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if there's a finite number
of elements in the chain.

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But all the way
through infinity.

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And the idea here
is the following.

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We'll assume that the chain at
least has a great many members.

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Even if it's finite.
 

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And then at some point, this
becomes a really small number.

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Even if epsilon naught,
the energy of one of the

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bonds, is less than kT.
 

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Once we multiply it by a very
large number n, it becomes

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very much greater than kT.
 

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So the underlying assumption
here is, well there will not

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be a significant number of
molecules in the highest

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energy state anyway.
 

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In other words, the terms
higher than a finite but large

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value of n between there and
infinity are going to be so

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small that those terms will
contribute negligibly

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to the sum anyway.
 

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00:06:40 --> 00:06:43
And on that basis, we could
make this approximation.

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And the incentive to make this
approximation and carry the sum

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00:06:45 --> 00:06:49
to infinity is that then we can
put this in what turns out to

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be a very simple closed form.
 

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So now if we just write out a
few of the individual terms in

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the sum, it's one plus e to the
minus epsilon zero over kT.

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Plus e to the minus
epsilon zero over kT.

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I'll write this as squared.
e to the minus two

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00:07:10 --> 00:07:12
epsilon zero over k t.
 

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And so on.
 

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So this has the form of a sum
that looks like one plus x,

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plus x squared, and so on.
 

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And because the sum goes to
infinity, we can simply write

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it as one over one minus x.
 

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Sort of a very simple result.
 

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And now putting this back into
x, it's one over one minus e to

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the minus epsilon zero over kT.
 

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That's our q zero.
 

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Our molecular
partition function.

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So this model yields a
particularly simple

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result for q zero.
 

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And then everything else
follows from there.

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So then we can write the
canonical partition

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function capital Qn.
 

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It's just q configurational to
the capital Nth power, where

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capital N is the total
number of these molecules.

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00:08:25 --> 00:08:29
So it's just one over one
minus e to the minus epsilon

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00:08:29 --> 00:08:35
over kT to the Nth power.
 

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00:08:35 --> 00:08:38
And then we can start writing
out the results for the various

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thermodynamic properties.
 

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So A configurational is minus
NkT, log q configurational.

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That is, remember, A is
minus kT log capital Q.

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00:09:02 --> 00:09:05
And this is just going
to come straight out.

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So it's minus NkT
log of little q.

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Minus NkT, log of one
over one minus e to the

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00:09:16 --> 00:09:19
minus epsilon over kT.
 

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Or positive NkT, log
of one minus e to the

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00:09:26 --> 00:09:30
minus epsilon over kT.
 

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So a pretty simple form for
the Helmholtz free energy.

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00:09:35 --> 00:09:37
And I'm not going to write
out all of the individuals

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thermodynamic terms, but
I'll write a few of them.

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So mu, the chemical potential
for these configurations,

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is just dA/dN.
 

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00:09:46 --> 00:09:53
 


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With T and V constant.
 

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And so the only capital N
dependence, the only dependence

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00:09:57 --> 00:10:00
on the number of molecules
is multiplicative.

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00:10:00 --> 00:10:03
Right here.
 

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00:10:03 --> 00:10:08
So this just gives us kT log
of one minus e to the minus

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00:10:08 --> 00:10:10
epsilon naught over kT.
 

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00:10:10 --> 00:10:12
And the important point to
realize here that we mentioned

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last time too, in the examples
we considered then, especially

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00:10:15 --> 00:10:21
the last one, is that because
the only place that N

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00:10:21 --> 00:10:26
figures in is as a
multiplicative factor.

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What this is telling us is
that we just have a chemical

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potential, of Helmholtz
free energy per molecule.

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The molecules are
all independent.

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The total energy doesn't depend
on any interaction between this

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molecule and its neighbor
somewhere else.

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So terms like this
are simply additive.

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So if we work out the chemical
potential, it's just

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00:10:48 --> 00:10:56
one over N times A.
 

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And I'll just write the result
for u configurational.

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00:11:04 --> 00:11:08
Because I do want to look
at the heat capacity.

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00:11:08 --> 00:11:22
So it's Nk T squared, d log of
little q configurational / dT.

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00:11:22 --> 00:11:25
With N and V held constant.
 

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And it turns out to just
be N epsilon zero.

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00:11:31 --> 00:11:40
One over E zero
over kT minus one.

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So now we can look at
the heat capacity.

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00:11:43 --> 00:11:49
Cv configurational.
 

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00:11:49 --> 00:11:54
So it's du
configurational / dT.

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00:11:54 --> 00:11:57
With constant N and V.
 

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And taking this derivative
with respect to temperature.

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Gives us an expression
of the following form.

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It's Nk epsilon zero over k T
squared. e to the E zero over

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kT, over e to the E zero
over kT minus one squared.

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Just taking the derivative
in the usual way.

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Now, this looks kind
of complicated for

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00:12:31 --> 00:12:33
the heat capacity.
 

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00:12:33 --> 00:12:36
But let's take a look, just
like we did last time for the

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simpler case that we treated
then, let's take a look at the

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00:12:39 --> 00:12:42
high and low temperature limits
of what happens to

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the heat capacity.
 

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00:12:44 --> 00:12:59
So at high temperature,
well we can start by just

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looking at the energy.
 

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00:13:01 --> 00:13:03
That has a form that's
fairly simple.

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So when temperature is large,
this is small, and we

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00:13:07 --> 00:13:09
can Taylor expand it.
 

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00:13:09 --> 00:13:11
So then we're just going
to have one plus epsilon

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00:13:11 --> 00:13:14
zero over kT minus one.
 

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00:13:14 --> 00:13:16
Which means the
ones will cancel.

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00:13:16 --> 00:13:18
And we end up with a
fairly simple result.

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So u configurational, in
that case, is just NkT.

206
00:13:24 --> 00:13:28
 


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The epsilon zeroes are
going to cancel also.

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00:13:31 --> 00:13:34
And what that says is that the
heat capacity -- and of course

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00:13:34 --> 00:13:36
we could take the limit of
this, but we can just take the

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00:13:36 --> 00:13:40
derivative of this with respect
to temperature more easily.

211
00:13:40 --> 00:13:45
So Cv configurational.
 

212
00:13:45 --> 00:13:47
It's just N times k.
 

213
00:13:47 --> 00:13:50
In other words, the heat
capacity in the high

214
00:13:50 --> 00:13:53
temperature limit
is a constant.

215
00:13:53 --> 00:13:56
Last time we treated a simpler
case, where there were only

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four configurations altogether
available, to this sort of

217
00:14:01 --> 00:14:04
simple polymer model
that we drew.

218
00:14:04 --> 00:14:08
Unlike this present model where
we're saying there are

219
00:14:08 --> 00:14:12
essentially infinite number of
configurations and different

220
00:14:12 --> 00:14:14
energies available.
 

221
00:14:14 --> 00:14:17
So many that the highest ones
we're never even going to

222
00:14:17 --> 00:14:21
access, because they'll
be much higher than kT.

223
00:14:21 --> 00:14:25
Here it's different, so now we
have -- You know before, when

224
00:14:25 --> 00:14:29
we had a limited number of
total states, remember what

225
00:14:29 --> 00:14:32
happened in the heat capacity
at high temperature.

226
00:14:32 --> 00:14:34
What was the limiting case?
 

227
00:14:34 --> 00:14:36
Finite number of states.
 

228
00:14:36 --> 00:14:41
What's the heat capacity
at high temperature?

229
00:14:41 --> 00:14:45
This is going to
be on the exam.

230
00:14:45 --> 00:14:47
What's the heat capacity at
high temperature, if there's

231
00:14:47 --> 00:14:49
a finite number of
states available?

232
00:14:49 --> 00:14:50
STUDENT: Zero.
 

233
00:14:50 --> 00:14:52
PROFESSOR: It's zero.
 

234
00:14:52 --> 00:14:57
What was the low temperature
limit of the heat capacity?

235
00:14:57 --> 00:14:57
STUDENT: Zero.
 

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00:14:57 --> 00:14:58
PROFESSOR: It was also zero.
 

237
00:14:58 --> 00:14:59
Right.
 

238
00:14:59 --> 00:15:11
And the idea was for a system
with a finite number of states.

239
00:15:11 --> 00:15:15
The idea was -- so let's say,
the way we had it before, there

240
00:15:15 --> 00:15:16
was one state with energy zero.
 

241
00:15:16 --> 00:15:20
And we had three states with
some energy epsilon zero.

242
00:15:20 --> 00:15:21
And that was it.
 

243
00:15:21 --> 00:15:24
Those were all the molecular
states available.

244
00:15:24 --> 00:15:27
So we looked at the two cases.
 

245
00:15:27 --> 00:15:30
In one case, when the high
temperature limit where kT

246
00:15:30 --> 00:15:33
is much bigger than
any of this stuff.

247
00:15:33 --> 00:15:38
In that limit, the molecules
are just equally likely to

248
00:15:38 --> 00:15:40
be in any of these states.
 

249
00:15:40 --> 00:15:43
Because there's much, much
more thermal energy than

250
00:15:43 --> 00:15:44
this energy difference.
 

251
00:15:44 --> 00:15:45
And the molecules are
constantly getting kicked

252
00:15:45 --> 00:15:48
around by the available
thermal energy.

253
00:15:48 --> 00:15:52
So if you raise the temperature
a little bit more, it

254
00:15:52 --> 00:15:53
doesn't make any difference.
 

255
00:15:53 --> 00:15:55
The molecules already
are evenly distributed

256
00:15:55 --> 00:15:57
among the states.
 

257
00:15:57 --> 00:16:00
There's no additional
configurational energy.

258
00:16:00 --> 00:16:03
So du/dT is zero.
 

259
00:16:03 --> 00:16:06
It can't increase any more.
 

260
00:16:06 --> 00:16:08
So in that case, the high
temperature limiting

261
00:16:08 --> 00:16:11
heat capacity is zero.
 

262
00:16:11 --> 00:16:21
And in the other case -- In the
low temperature limit, now

263
00:16:21 --> 00:16:22
let's say kT is really low.
 

264
00:16:22 --> 00:16:24
It's much less than
epsilon zero.

265
00:16:24 --> 00:16:26
So now we'll redraw
this as zero.

266
00:16:26 --> 00:16:36
And put the epsilon
zero states up here.

267
00:16:36 --> 00:16:39
And kT is here.
 

268
00:16:39 --> 00:16:43
Well when it's like this, the
temperature is so low that

269
00:16:43 --> 00:16:46
there's not nearly enough
thermal energy to populate

270
00:16:46 --> 00:16:47
any of these states.
 

271
00:16:47 --> 00:16:48
And if you change the
temperature by a little

272
00:16:48 --> 00:16:51
bit, it's still not enough
thermal energy to populate

273
00:16:51 --> 00:16:52
any of these states.
 

274
00:16:52 --> 00:16:55
So once again, du/dT is zero.
 

275
00:16:55 --> 00:16:57
You change the temperature
and the configurational

276
00:16:57 --> 00:16:59
energy doesn't change.
 

277
00:16:59 --> 00:17:02
So the heat capacity
is zero again.

278
00:17:02 --> 00:17:06
This is why it's so informative
to measure heat capacities.

279
00:17:06 --> 00:17:10
Because you can learn an awful
lot about the intrinsic

280
00:17:10 --> 00:17:12
structure of the material.
 

281
00:17:12 --> 00:17:14
What are the energy
levels available to it?

282
00:17:14 --> 00:17:15
What do they do?
 

283
00:17:15 --> 00:17:18
You can learn a tremendous
amount about that by

284
00:17:18 --> 00:17:20
making measurements
of the heat capacity.

285
00:17:20 --> 00:17:24
Well, now we're in
a different case.

286
00:17:24 --> 00:17:30
We have a whole set of evenly
spaced energy levels.

287
00:17:30 --> 00:17:33
Never ends.
 

288
00:17:33 --> 00:17:36
So now let's look at the
high temperature limit.

289
00:17:36 --> 00:17:38
But we're never in a limit
that's higher than the

290
00:17:38 --> 00:17:43
highest level, because we're
assuming it goes on forever.

291
00:17:43 --> 00:17:45
So it's up here somewhere.
 

292
00:17:45 --> 00:17:47
There's kT.
 

293
00:17:47 --> 00:17:48
So what happens?
 

294
00:17:48 --> 00:17:52
Well if you raise the
temperature, there still are

295
00:17:52 --> 00:17:54
higher lying levels that can
be populated, and that

296
00:17:54 --> 00:17:56
will get populated.
 

297
00:17:56 --> 00:17:59
So du/dT isn't going to be
zero in the high temperature

298
00:17:59 --> 00:18:00
limit, in this case.
 

299
00:18:00 --> 00:18:04
But it stops changing
at some point.

300
00:18:04 --> 00:18:08
Because nothing's very
different about this than

301
00:18:08 --> 00:18:12
having kT be, let's say,
up here or up here.

302
00:18:12 --> 00:18:15
So in the high temperature
limit, yes, the energy does

303
00:18:15 --> 00:18:17
change with temperature.
 

304
00:18:17 --> 00:18:22
But it changes in the same
way at any temperature.

305
00:18:22 --> 00:18:25
In other words, the energy
is just linearly varying

306
00:18:25 --> 00:18:26
with temperature.
 

307
00:18:26 --> 00:18:30
And the heat capacity is a
constant. du/dT doesn't change

308
00:18:30 --> 00:18:34
anymore, once you're in the
high temperature limit.

309
00:18:34 --> 00:18:38
Now without me writing any
expression on the board, what's

310
00:18:38 --> 00:18:40
the low temperature limit of
the heat capacity going

311
00:18:40 --> 00:18:45
to be in this case?
 

312
00:18:45 --> 00:18:46
Going to be on the exam.
 

313
00:18:46 --> 00:18:48
STUDENT: Zero?
 

314
00:18:48 --> 00:18:51
PROFESSOR: Zero, that's still
going to be the same, right?

315
00:18:51 --> 00:18:55
Put kT way down here.
 

316
00:18:55 --> 00:18:57
That's just like this, right?
 

317
00:18:57 --> 00:19:01
Not near enough energy to
populate even though lowest,

318
00:19:01 --> 00:19:04
you know, the first level
above the ground state.

319
00:19:04 --> 00:19:07
Change the temperature a little
bit, still not enough thermal

320
00:19:07 --> 00:19:08
energy to get up there.
 

321
00:19:08 --> 00:19:10
So the energy doesn't change
with temperature in that

322
00:19:10 --> 00:19:12
low temperature limit.
 

323
00:19:12 --> 00:19:15
So you can immediately see
what's going to happen

324
00:19:15 --> 00:19:33
at low temperature.
 

325
00:19:33 --> 00:19:35
Any questions?
 

326
00:19:35 --> 00:19:35
Yeah?
 

327
00:19:35 --> 00:19:42
STUDENT: [UNINTELLIGIBLE]
 

328
00:19:42 --> 00:19:43
will still be n k t, and then
it's just -- [UNINTELLIGIBLE]

329
00:19:43 --> 00:19:45
PROFESSOR: So, of course,
that's a limiting

330
00:19:45 --> 00:19:48
case here, right?
 

331
00:19:48 --> 00:19:51
That happened because in the
limit of high temperature,

332
00:19:51 --> 00:19:54
then this exponent is
really small, right?

333
00:19:54 --> 00:19:56
So then you can
Taylor expand it.

334
00:19:56 --> 00:19:59
In the limit of low
temperature, this

335
00:19:59 --> 00:20:02
is really big.
 

336
00:20:02 --> 00:20:04
This is nearly zero.
 

337
00:20:04 --> 00:20:09
So e epsilon zero is much
bigger than kT in that case.

338
00:20:09 --> 00:20:10
This is big.
 

339
00:20:10 --> 00:20:13
This is negligible, right?
 

340
00:20:13 --> 00:20:14
Oh.
 

341
00:20:14 --> 00:20:19
Wait a minute.
 

342
00:20:19 --> 00:20:24
Something's making
me real unhappy.

343
00:20:24 --> 00:20:42
I can't have this right.
 

344
00:20:42 --> 00:20:43
Ah.
 

345
00:20:43 --> 00:20:46
Yes.
 

346
00:20:46 --> 00:20:50
It needs to be zero, is
what it needs to be.

347
00:20:50 --> 00:20:51
What am I thinking?
 

348
00:20:51 --> 00:20:51
Of course.
 

349
00:20:51 --> 00:20:52
It's one over a huge number.
 

350
00:20:52 --> 00:20:54
It's zero.
 

351
00:20:54 --> 00:20:55
OK.
 

352
00:20:55 --> 00:20:58
So of course you
can't expanded it.

353
00:20:58 --> 00:21:01
It's just, this is much
bigger than this.

354
00:21:01 --> 00:21:03
This is an enormous
number at that point.

355
00:21:03 --> 00:21:07
So in other words, what it's
saying is the configurational

356
00:21:07 --> 00:21:11
energy is zero, because
everything is stuck

357
00:21:11 --> 00:21:13
in the ground state.
 

358
00:21:13 --> 00:21:15
And it stays zero if you
vary the temperature.

359
00:21:15 --> 00:21:19
So you get the zero limiting
value for the heat capacity,

360
00:21:19 --> 00:21:23
and the energy itself
is also zero.

361
00:21:23 --> 00:21:26
Other questions?
 

362
00:21:26 --> 00:21:32
OK.
 

363
00:21:32 --> 00:21:37
Let me just make a few
comments about the entropy.

364
00:21:37 --> 00:21:42
So I didn't write
it out before.

365
00:21:42 --> 00:21:47
And I'm tempted not to
do it again, but I

366
00:21:47 --> 00:21:57
suppose I'll do it.
 

367
00:21:57 --> 00:22:02
So it turns out to be Nk, minus
log of one minus e to the

368
00:22:02 --> 00:22:07
minus epsilon zero over kT.
 

369
00:22:07 --> 00:22:10
Plus epsilon zero over kT.
 

370
00:22:10 --> 00:22:19
Over e to the epsilon
zero over kT, minus one.

371
00:22:19 --> 00:22:24
And this comes from combining
the terms for A and u.

372
00:22:24 --> 00:22:29
It comes from minus A over T.
 

373
00:22:29 --> 00:22:36
Plus u over T.
 

374
00:22:36 --> 00:22:41
And what I want to do is look
at its limiting cases also.

375
00:22:41 --> 00:22:44
In particular, what happens
here in the limit of

376
00:22:44 --> 00:22:58
high temperature.
 

377
00:22:58 --> 00:23:05
And what happens turns out
to be k log kT over epsilon

378
00:23:05 --> 00:23:10
zero to the Nth power.
 

379
00:23:10 --> 00:23:16
Or put the N over here.
 

380
00:23:16 --> 00:23:19
OK.
 

381
00:23:19 --> 00:23:24
And what this is telling
us about is the number

382
00:23:24 --> 00:23:26
of available states.
 

383
00:23:26 --> 00:23:31
Roughly, how many states are
there that are accessible to

384
00:23:31 --> 00:23:34
the system at some temperature.
 

385
00:23:34 --> 00:23:38
So in other words, think of it
as -- let's put the N back

386
00:23:38 --> 00:23:43
there. k log of kT over epsilon
zero to the Nth power.

387
00:23:43 --> 00:23:47
And think of it as k
log capital omega.

388
00:23:47 --> 00:23:49
Where that would be
the degeneracy.

389
00:23:49 --> 00:23:52
Now all the states are in equal
energy, but remember for the

390
00:23:52 --> 00:23:57
whole system, remember we
discussed this before.

391
00:23:57 --> 00:24:00
How you have a very, very
narrow distribution of system

392
00:24:00 --> 00:24:05
energy states at equilibrium.
 

393
00:24:05 --> 00:24:08
So you can think of this as the
degeneracy of the system states

394
00:24:08 --> 00:24:11
that are actually going to
exist at a particular

395
00:24:11 --> 00:24:16
temperature.
 

396
00:24:16 --> 00:24:28
So if we look at the limiting
value of the partition

397
00:24:28 --> 00:24:34
function, it's just
kT over omega.

398
00:24:34 --> 00:24:46
Or the same thing for capital
Q. kT over epsilon zero, sorry.

399
00:24:46 --> 00:24:48
To the Nth power.
 

400
00:24:48 --> 00:24:51
So you have a very
simple expression.

401
00:24:51 --> 00:24:54
And so again, what this is
doing, is it's giving us a

402
00:24:54 --> 00:24:57
measure of about how many
states are available.

403
00:24:57 --> 00:24:59
And it's particularly
informative to look at

404
00:24:59 --> 00:25:03
that for the molecular
partition function.

405
00:25:03 --> 00:25:13
What it's telling us is, if
I look at kT over epsilon.

406
00:25:13 --> 00:25:15
And these levels, remember,
are evenly spaced.

407
00:25:15 --> 00:25:19
So here's epsilon zero,
two epsilon zero, three

408
00:25:19 --> 00:25:22
epsilon zero, and so on.
 

409
00:25:22 --> 00:25:27
It says, you know, if kT
is about ten times bigger

410
00:25:27 --> 00:25:30
than epsilon zero.
 

411
00:25:30 --> 00:25:34
So this is ten.
 

412
00:25:34 --> 00:25:38
It's telling us, roughly how
many states does the system

413
00:25:38 --> 00:25:40
have thermal access to.
 

414
00:25:40 --> 00:25:42
The molecular state.
 

415
00:25:42 --> 00:25:45
It has about ten states.
 

416
00:25:45 --> 00:25:50
So going over to our picture of
the set of structures, you

417
00:25:50 --> 00:25:54
could have anywhere up to
about ten bonds broken.

418
00:25:54 --> 00:25:59
Now, the individual molecules
are going to be in a

419
00:25:59 --> 00:26:01
whole range of states.
 

420
00:26:01 --> 00:26:03
Some of them will have fewer
than that, and some of them

421
00:26:03 --> 00:26:06
will have more than that
number of bonds broken.

422
00:26:06 --> 00:26:12
But on average, it's going
to be about that number.

423
00:26:12 --> 00:26:15
And then, if you look at the
whole system, the number

424
00:26:15 --> 00:26:19
of states available, of
course it's astronomical.

425
00:26:19 --> 00:26:23
Because you have to take each
molecule, and say, well it

426
00:26:23 --> 00:26:26
could be in any one
of something on the

427
00:26:26 --> 00:26:28
order of ten states.
 

428
00:26:28 --> 00:26:30
And then whole set of N other
molecules can be in the

429
00:26:30 --> 00:26:33
states they might be in.
 

430
00:26:33 --> 00:26:34
Change the first one,
and do it again.

431
00:26:34 --> 00:26:40
So you have this astronomical
number of system states.

432
00:26:40 --> 00:26:46
But remember, like we discussed
once before, it'll turn out

433
00:26:46 --> 00:26:49
that although the individual
molecule states vary

434
00:26:49 --> 00:26:54
considerably with energy, the
system states, which are

435
00:26:54 --> 00:26:57
averaging over some
astronomical number of

436
00:26:57 --> 00:27:01
molecules, where capital N is
something like ten to the 24th.

437
00:27:01 --> 00:27:06
Once you average over that many
individual molecules, you find

438
00:27:06 --> 00:27:10
that there's very, very little
fluctuation in the energy.

439
00:27:10 --> 00:27:12
In the system energy.
 

440
00:27:12 --> 00:27:16
The individual molecule
energies vary considerably.

441
00:27:16 --> 00:27:21
Realistically the variation of
the molecular energies -- that

442
00:27:21 --> 00:27:23
variation is going to be
comparable to the

443
00:27:23 --> 00:27:25
energy itself.
 

444
00:27:25 --> 00:27:27
To the average energy.
 

445
00:27:27 --> 00:27:30
So if the average energy is
roughly you know, the ten

446
00:27:30 --> 00:27:33
epsilon zero, you say okay,
how much might it vary?

447
00:27:33 --> 00:27:35
Well, there are going to be
some molecules that have only a

448
00:27:35 --> 00:27:37
couple of bonds broken, and
some that might have

449
00:27:37 --> 00:27:39
20 bonds broken.
 

450
00:27:39 --> 00:27:41
The variation will be on the
same order of magnitude

451
00:27:41 --> 00:27:44
as the average itself.
 

452
00:27:44 --> 00:27:49
But then you average
over capital N of them.

453
00:27:49 --> 00:27:51
And then you immediately
discover that there's very,

454
00:27:51 --> 00:27:54
very, very little variation.
 

455
00:27:54 --> 00:28:04
And in particular, what happens
then, is, you know, for

456
00:28:04 --> 00:28:14
molecular average energy,
epsilon zero, and variance, or

457
00:28:14 --> 00:28:22
standard deviation about
the same magnitude.

458
00:28:22 --> 00:28:29
System average energy is
u, and it's going to

459
00:28:29 --> 00:28:31
be capital N times.
 

460
00:28:31 --> 00:28:37
Well let's say that -- average
-- Oh sorry, let me not

461
00:28:37 --> 00:28:38
put the zero There.
 

462
00:28:38 --> 00:28:41
It's just average energy.
 

463
00:28:41 --> 00:28:44
And the system average
energy is just N times

464
00:28:44 --> 00:28:48
the molecular energy.
 

465
00:28:48 --> 00:28:56
But the system variance is
going to be on the order of the

466
00:28:56 --> 00:29:02
square root of N times epsilon.
 

467
00:29:02 --> 00:29:03
If you've done statistics,
then you've seen

468
00:29:03 --> 00:29:05
that sort of result.
 

469
00:29:05 --> 00:29:09
You do a bunch of samplings a
capital N number of samplings,

470
00:29:09 --> 00:29:15
and the variation looks like
the square root of that number.

471
00:29:15 --> 00:29:18
So what ends up happening then,
if you look at the relative

472
00:29:18 --> 00:29:20
variation, you might look and
then say, well it's pretty big.

473
00:29:20 --> 00:29:21
This is ten to the 12, right?
 

474
00:29:21 --> 00:29:23
It's still a huge variance.
 

475
00:29:23 --> 00:29:47
But let's compare
it to the average.

476
00:29:47 --> 00:29:50
So this is ten to the 24th.
 

477
00:29:50 --> 00:29:51
This is ten to the 12th.
 

478
00:29:51 --> 00:29:54
It's on the order of
ten to the minus 12.

479
00:29:54 --> 00:30:00
An incredibly tiny fractional
variation in the system energy.

480
00:30:00 --> 00:30:03
You could never -- There
would be no practical

481
00:30:03 --> 00:30:04
way to measure it.
 

482
00:30:04 --> 00:30:06
And of course that is
consistent with what

483
00:30:06 --> 00:30:07
you would expect.
 

484
00:30:07 --> 00:30:12
If you say let's measure the
configurational energy of a

485
00:30:12 --> 00:30:15
bunch of molecules in a liquid
solution, or molecules in

486
00:30:15 --> 00:30:16
a gas floating around.
 

487
00:30:16 --> 00:30:21
And it's a mole of them, that
total average energy is not

488
00:30:21 --> 00:30:24
going to fluctuate
significantly, even though

489
00:30:24 --> 00:30:27
individual molecules that you
pick out of that whole system

490
00:30:27 --> 00:30:30
might have quite widely
varying energies.

491
00:30:30 --> 00:30:34
And that's the point.
 

492
00:30:34 --> 00:30:37
So it's an incredibly
small variation.

493
00:30:37 --> 00:30:39
Now you can derive this.
 

494
00:30:39 --> 00:30:40
I didn't derive
this, of course.

495
00:30:40 --> 00:30:43
I asserted that
this is the case.

496
00:30:43 --> 00:30:45
And it's probably familiar
to some of you if you've

497
00:30:45 --> 00:30:47
seen some statistics.
 

498
00:30:47 --> 00:30:49
But the way you would derive it
is, you know in addition to

499
00:30:49 --> 00:30:52
calculating the average energy,
the average of E, you can

500
00:30:52 --> 00:30:55
also calculate the average
of the energy squared.

501
00:30:55 --> 00:30:58
And you can calculate the
standard deviation that way.

502
00:30:58 --> 00:31:01
You calculate the
average of E squared.

503
00:31:01 --> 00:31:05
And then you minus the average
of E, the quantity squared.

504
00:31:05 --> 00:31:06
Take the square root.
 

505
00:31:06 --> 00:31:09
That's your root mean square.
 

506
00:31:09 --> 00:31:13
And that's what leads
to this result.

507
00:31:13 --> 00:31:17
So it's pretty straight
forward to calculate it also.

508
00:31:17 --> 00:31:17
Alright.
 

509
00:31:17 --> 00:31:21
So any questions about just the
extent of variation of the

510
00:31:21 --> 00:31:27
individual molecule energies
or the system energies?

511
00:31:27 --> 00:31:28
Okay.
 

512
00:31:28 --> 00:31:42
Then, now what I'd like to do
is look at another kind of

513
00:31:42 --> 00:31:46
energy that's going to turn out
to have exactly the same set of

514
00:31:46 --> 00:31:50
levels that we just derived
from this simple model.

515
00:31:50 --> 00:31:52
And some of you may
have seen this before.

516
00:31:52 --> 00:31:55
Many of you may not have.
 

517
00:31:55 --> 00:31:57
And for right now,
I'll just assert it.

518
00:31:57 --> 00:32:00
But it will illustrate
why this is so useful.

519
00:32:00 --> 00:32:06
It turns out that if I've got
molecular vibrations -- you

520
00:32:06 --> 00:32:08
know, there's nitrogen
and oxygen in the air.

521
00:32:08 --> 00:32:11
If I look at those nitrogen
vibrational energy levels.

522
00:32:11 --> 00:32:14
Or oxygen energy levels.
 

523
00:32:14 --> 00:32:17
They look like this.
 

524
00:32:17 --> 00:32:23
Evenly spaced, non
degenerate energy levels.

525
00:32:23 --> 00:32:26
So the model that we've
constructed here, based on this

526
00:32:26 --> 00:32:34
simple two-chain polymer,
actually gives us a set of

527
00:32:34 --> 00:32:38
energy levels that maps
directly onto the vibrational

528
00:32:38 --> 00:32:40
energy levels of a molecule.
 

529
00:32:40 --> 00:32:50
So all the results that we've
just seen apply, not just for

530
00:32:50 --> 00:32:59
conformations of a polymer, but
for vibrations of a molecule.

531
00:32:59 --> 00:33:03
So everywhere where it says
configurational, you can

532
00:33:03 --> 00:33:08
just write in vibrational.
 

533
00:33:08 --> 00:33:11
And you'll still be right.
because of course, what matters

534
00:33:11 --> 00:33:15
is what are the states that are
available to the system, and

535
00:33:15 --> 00:33:17
what are their energies?
 

536
00:33:17 --> 00:33:19
After that, the formalism
of statistical mechanics

537
00:33:19 --> 00:33:22
takes over, and calculates
partition functions and

538
00:33:22 --> 00:33:24
thermodynamic functions.
 

539
00:33:24 --> 00:33:28
The input into that is states.
 

540
00:33:28 --> 00:33:29
And their energies.
 

541
00:33:29 --> 00:33:36
Well, we have the exact same
set of states and energies.

542
00:33:36 --> 00:33:41
So we immediately arrive at a
super important case and its

543
00:33:41 --> 00:33:46
results for molecular
vibrations.

544
00:33:46 --> 00:33:51
And not only molecular
vibrations, but vibrations

545
00:33:51 --> 00:33:57
of a crystal lattice.
 

546
00:33:57 --> 00:34:00
Acoustic vibrations of a glass.
 

547
00:34:00 --> 00:34:02
Or even a liquid.
 

548
00:34:02 --> 00:34:07
So there's an enormously wide
ranging set of results that

549
00:34:07 --> 00:34:13
we've derived by starting
at this simple picture.

550
00:34:13 --> 00:34:15
And when you take quantum
mechanics, you'll see that

551
00:34:15 --> 00:34:19
indeed you do get this set of
evenly spaced energy levels.

552
00:34:19 --> 00:34:21
You may well have seen the
results before, and you'll

553
00:34:21 --> 00:34:24
see it derived there.
 

554
00:34:24 --> 00:34:25
OK.
 

555
00:34:25 --> 00:34:31
Given that, then I want to
talk a little bit further

556
00:34:31 --> 00:34:33
about the heat capacity.
 

557
00:34:33 --> 00:34:39
So we've seen the limiting
cases for the heat capacity.

558
00:34:39 --> 00:34:42
Namely, the low temperature
limit is zero.

559
00:34:42 --> 00:34:45
That's the case whenever
you have quantized levels.

560
00:34:45 --> 00:34:49
You can always get kT
lower by far than the

561
00:34:49 --> 00:34:52
lowest excited state.
 

562
00:34:52 --> 00:34:54
So that everything is stuck
in the ground state.

563
00:34:54 --> 00:34:56
How low that is depends
on how far apart the

564
00:34:56 --> 00:34:57
states are spaced.
 

565
00:34:57 --> 00:34:59
But there's got to be some
temperature somewhere

566
00:34:59 --> 00:35:02
that's down there.
 

567
00:35:02 --> 00:35:16
So we've seen the heat
capacity limiting cases.

568
00:35:16 --> 00:35:36
Let me just sort of
draw them again.

569
00:35:36 --> 00:35:41
So here's kT much less
than epsilon zero.

570
00:35:41 --> 00:35:46
And up here, kT is much
bigger than epsilon zero.

571
00:35:46 --> 00:35:50
Now I'm just going to sketch
the full temperature

572
00:35:50 --> 00:35:50
dependence.
 

573
00:35:50 --> 00:35:52
In other words, I'm going
to connect those two

574
00:35:52 --> 00:35:56
limits that we've seen.
 

575
00:35:56 --> 00:36:14
So here's what that looks like.
 

576
00:36:14 --> 00:36:20
And I'll write it
as vibrational.

577
00:36:20 --> 00:36:25
So we know it's got to be zero
at the low temperature limit.

578
00:36:25 --> 00:36:30
And we know that it's this
constant value in the

579
00:36:30 --> 00:36:31
high temperature limit.
 

580
00:36:31 --> 00:36:33
It's Nk.
 

581
00:36:33 --> 00:36:40
And in some way, it
smoothly connects.

582
00:36:40 --> 00:36:43
And if you look at the actual
scale, here -- so here's

583
00:36:43 --> 00:36:47
kT over epsilon zero.
 

584
00:36:47 --> 00:36:48
Here's the limit of
low temperature.

585
00:36:48 --> 00:36:53
Actually, you don't have to be
very high above epsilon zero to

586
00:36:53 --> 00:36:55
be in in this high
temperature limit.

587
00:36:55 --> 00:37:00
So if you look at the plot
that's on your notes, this is

588
00:37:00 --> 00:37:02
already -- it's not quite
leveling off precisely.

589
00:37:02 --> 00:37:04
But it's already pretty close.
 

590
00:37:04 --> 00:37:10
When kT is just two
times epsilon zero.

591
00:37:10 --> 00:37:22
It's already almost at
the limiting case.

592
00:37:22 --> 00:37:24
Now this has tremendous
significance.

593
00:37:24 --> 00:37:30
So a long time before quantum
mechanics was developed, people

594
00:37:30 --> 00:37:34
made measurements of the heat
capacities of materials.

595
00:37:34 --> 00:37:37
And they were familiar with the
fact that when you got to

596
00:37:37 --> 00:37:41
ordinary temperature -- room
temperature, the heat capacity

597
00:37:41 --> 00:37:45
was temperature independent.
 

598
00:37:45 --> 00:37:49
And that was understandable for
reasons we'll discuss shortly.

599
00:37:49 --> 00:37:55
Basically, in that case, the
heat capacity -- it's not just

600
00:37:55 --> 00:37:58
Nk for one vibrational mode.
 

601
00:37:58 --> 00:38:01
Of course, that's what I
describe for a single

602
00:38:01 --> 00:38:03
vibrational mode of,
say, a molecule.

603
00:38:03 --> 00:38:06
If you have a crystal
lattice with N atoms in it.

604
00:38:06 --> 00:38:08
Let's say it's an
atomic crystal.

605
00:38:08 --> 00:38:10
Each atom has three
degrees of freedom.

606
00:38:10 --> 00:38:14
It can move in each of three
independent directions.

607
00:38:14 --> 00:38:15
And there are N of those atoms.
 

608
00:38:15 --> 00:38:18
There are 3N total
degrees of freedom.

609
00:38:18 --> 00:38:23
That's how many vibrations
the lattice has.

610
00:38:23 --> 00:38:29
So in fact, you have to
multiply this by 3N.

611
00:38:29 --> 00:38:36
So what would be seen is, you'd
have 3R, the gas constant,

612
00:38:36 --> 00:38:38
per mole, in other words.
 

613
00:38:38 --> 00:38:44
And if you looked at
Cv over 3R, then this

614
00:38:44 --> 00:38:50
value would be one.
 

615
00:38:50 --> 00:38:53
Very useful to see that.
 

616
00:38:53 --> 00:38:55
To figure that out.
 

617
00:38:55 --> 00:38:56
And people understood it.
 

618
00:38:56 --> 00:39:03
What people didn't
understand is this stuff.

619
00:39:03 --> 00:39:06
Why did it do that?
 

620
00:39:06 --> 00:39:08
That was a great mystery.
 

621
00:39:08 --> 00:39:13
Why did the heat capacity go
to zero at low temperature?

622
00:39:13 --> 00:39:15
And the reason they didn't
understand it is because

623
00:39:15 --> 00:39:18
the model for vibration
was really simple.

624
00:39:18 --> 00:39:21
The lattice is a bunch
of masses and springs.

625
00:39:21 --> 00:39:25
I know the vibrational energy.
 

626
00:39:25 --> 00:39:26
And I can calculate it.
 

627
00:39:26 --> 00:39:31
That is, I know the classical
vibrational energy.

628
00:39:31 --> 00:39:35
Remember, the reason it
goes to zero is this.

629
00:39:35 --> 00:39:39
It's all because of the fact
that the energy levels are

630
00:39:39 --> 00:39:43
discrete, quantized levels,
with gaps in between them.

631
00:39:43 --> 00:39:47
If I've got classical
mechanics, then that means

632
00:39:47 --> 00:39:48
it's vibrational energy.
 

633
00:39:48 --> 00:39:51
The energy just gets bigger
if the amplitude get bigger.

634
00:39:51 --> 00:39:54
It's not discrete,
it's continuous.

635
00:39:54 --> 00:39:56
There are always
energies in there.

636
00:39:56 --> 00:39:59
In that case, there's never
a situation like this.

637
00:39:59 --> 00:40:03
Where kT is lower than the
first available excited

638
00:40:03 --> 00:40:04
level, and everything's
in the ground state.

639
00:40:04 --> 00:40:06
That never happens in classical
mechanics, because there

640
00:40:06 --> 00:40:09
are always levels there.
 

641
00:40:09 --> 00:40:13
But it does happen in
quantum mechanics.

642
00:40:13 --> 00:40:17
And Einstein recognized that
this was a way to explain

643
00:40:17 --> 00:40:20
this low temperature
limiting heat capacity.

644
00:40:20 --> 00:40:24
So actually if you look at
early development of quantum

645
00:40:24 --> 00:40:26
mechanics, really it
was all predicated on

646
00:40:26 --> 00:40:28
statistical mechanics.
 

647
00:40:28 --> 00:40:31
And the idea that, well,
that you could then do the

648
00:40:31 --> 00:40:35
statistical mechanics with
quantized levels, just

649
00:40:35 --> 00:40:36
the way we've done it.
 

650
00:40:36 --> 00:40:38
And what you immediately
discover is, gee,

651
00:40:38 --> 00:40:39
it all makes sense.
 

652
00:40:39 --> 00:40:43
You go to low temperature,
everything's stunk down here.

653
00:40:43 --> 00:40:45
And suddenly you've got zero
heat capacity, because you

654
00:40:45 --> 00:40:46
changed the temperature a
little bit, and everything

655
00:40:46 --> 00:40:51
is still stuck back here.
 

656
00:40:51 --> 00:40:56
So that's the vibrational
heat capacity of a solid.

657
00:40:56 --> 00:40:57
That's what it looks like.
 

658
00:40:57 --> 00:41:01
And at moderate temperature,
this doesn't have to be very

659
00:41:01 --> 00:41:03
much bigger than epsilon zero.
 

660
00:41:03 --> 00:41:05
You're already in,
essentially, the limit.

661
00:41:05 --> 00:41:06
The high temperature limit.
 

662
00:41:06 --> 00:41:12
And now for molecules, that
limit isn't usually reached

663
00:41:12 --> 00:41:13
at room temperature.
 

664
00:41:13 --> 00:41:20
Here's a kind of calibration.
kT at room temperature is about

665
00:41:20 --> 00:41:27
equal to 200 wave numbers.
 

666
00:41:27 --> 00:41:32
So molecular vibrations, you
know, you've taken IR spectra

667
00:41:32 --> 00:41:36
they're typically on the order
of 1,000 wave numbers or so. kT

668
00:41:36 --> 00:41:40
isn't bigger than the
vibrational energy.

669
00:41:40 --> 00:41:43
But a crystal lattice, you
know the vibrations are

670
00:41:43 --> 00:41:46
the acoustics vibrations.
 

671
00:41:46 --> 00:41:48
And those are much
lower in frequency.

672
00:41:48 --> 00:41:52
If you have an atomic crystal,
it just has the sound

673
00:41:52 --> 00:41:56
vibrations at all the different
wavelengths that are available.

674
00:41:56 --> 00:41:58
They're never very high.
 

675
00:41:58 --> 00:42:01
So it's easy to get into
the high temperature

676
00:42:01 --> 00:42:04
limit, in that case.
 

677
00:42:04 --> 00:42:07
Where you basically see a
temperature independent

678
00:42:07 --> 00:42:14
heat capacity.
 

679
00:42:14 --> 00:42:15
OK.
 

680
00:42:15 --> 00:42:18
By the way, this was actually
used commonly to determine the

681
00:42:18 --> 00:42:20
molecular weights of
molecular crystals.

682
00:42:20 --> 00:42:25
Because you've got a factor of
the number of moles in there.

683
00:42:25 --> 00:42:26
If you ask how big is
the heat capacity?

684
00:42:26 --> 00:42:29
Well it does depend on how many
moles of material you have.

685
00:42:29 --> 00:42:31
Because it depends
on how many atoms.

686
00:42:31 --> 00:42:34
That control how many modes
there are in the crystal.

687
00:42:34 --> 00:42:35
How many vibrational modes.
 

688
00:42:35 --> 00:42:39
Because each atom has
3N degrees of freedom.

689
00:42:39 --> 00:42:41
So if you just weigh the whole
crystal, and then you measure

690
00:42:41 --> 00:42:44
the heat capacity, you know how
many moles there are, and now

691
00:42:44 --> 00:42:45
you know the weight, you
can figure out the

692
00:42:45 --> 00:42:49
molecular weight.
 

693
00:42:49 --> 00:42:53
OK.
 

694
00:42:53 --> 00:42:54
So that's the low
temperature limit.

695
00:42:54 --> 00:42:57
And now I want to talk a
little bit further about

696
00:42:57 --> 00:42:59
the high temperature limit.
 

697
00:42:59 --> 00:43:04
And in particular, I want to
talk about both the heat

698
00:43:04 --> 00:43:07
capacity and the energy
that we've seen.

699
00:43:07 --> 00:43:11
Namely, this thing.
 

700
00:43:11 --> 00:43:13
Of course it's really the
same result for the energy

701
00:43:13 --> 00:43:14
and the heat capacity.
 

702
00:43:14 --> 00:43:18
They're obviously
intimately connected.

703
00:43:18 --> 00:43:22
So these are the high
temperature limits.

704
00:43:22 --> 00:43:32
Now it turns out that that high
temperature limit doesn't

705
00:43:32 --> 00:43:35
just obtain for vibrations.
 

706
00:43:35 --> 00:43:39
But it's also the case
for molecular rotations,

707
00:43:39 --> 00:43:41
translations.
 

708
00:43:41 --> 00:43:46
All these low energy
degrees of freedom.

709
00:43:46 --> 00:43:50
So let's see why.
 

710
00:43:50 --> 00:43:51
It has a name, that result.
 

711
00:43:51 --> 00:43:59
It's called the
equipartition of energy.

712
00:43:59 --> 00:44:06
Sometimes it's called the
classical equipartition

713
00:44:06 --> 00:44:10
of energy theorem.
 

714
00:44:10 --> 00:44:23
And what it says is one half kT
per degree of freedom equals

715
00:44:23 --> 00:44:31
energy in the high T limit.
 

716
00:44:31 --> 00:44:38
And in particular, for
translation, so E

717
00:44:38 --> 00:44:44
translational is 3/2.
 

718
00:44:44 --> 00:44:49
Well, NkT for N atoms.
 

719
00:44:49 --> 00:44:55
Or molecules.
 

720
00:44:55 --> 00:45:01
E rotational is, now it depends
how many rotational degrees

721
00:45:01 --> 00:45:02
of freedom there are.
 

722
00:45:02 --> 00:45:05
If I've got a linear molecule,
there are only two.

723
00:45:05 --> 00:45:08
It can rotate this way, and
it can rotate this way.

724
00:45:08 --> 00:45:10
If I've got a non linear
molecule, parts all

725
00:45:10 --> 00:45:14
over the place, it has
three unique axes.

726
00:45:14 --> 00:45:15
It can also spin this way.
 

727
00:45:15 --> 00:45:17
And that's a rotational
degree of freedom.

728
00:45:17 --> 00:45:19
Of course the linear
molecule wasn't like that.

729
00:45:19 --> 00:45:23
Nothing is moving
when you do that.

730
00:45:23 --> 00:45:30
So it's either NkT, linear.
 

731
00:45:30 --> 00:45:34
Or 3/2 NkT.
 

732
00:45:34 --> 00:45:34
Non linear.
 

733
00:45:34 --> 00:45:38
Great.
 

734
00:45:38 --> 00:45:49
And then E vibrational is
NkT per vibrational mode.

735
00:45:49 --> 00:45:52
That's because vibrational
energy is potential and kinetic

736
00:45:52 --> 00:45:56
energy, and it's 1/2 kT each.
 

737
00:45:56 --> 00:45:59
Why does all that stuff happen?
 

738
00:45:59 --> 00:46:10
Turns out, you can see
why pretty easily.

739
00:46:10 --> 00:46:12
All those degrees of freedom.
 

740
00:46:12 --> 00:46:20
Those classical
degrees of freedom.

741
00:46:20 --> 00:46:31
If you say, let's write out an
expression for the energy.

742
00:46:31 --> 00:46:35
Classical.
 

743
00:46:35 --> 00:46:39
You know it's 1/2 m v squared.
 

744
00:46:39 --> 00:46:42
Or it's 1/2 k x squared.
 

745
00:46:42 --> 00:46:51
Or for rotational energy
it's 1/2 I omega squared.

746
00:46:51 --> 00:46:52
You see a functional
form emerging here?

747
00:46:52 --> 00:46:56
That looks similar
in all these cases?

748
00:46:56 --> 00:46:59
You know, 1/2 times some
constant, times some

749
00:46:59 --> 00:47:03
variable squared.
 

750
00:47:03 --> 00:47:11
Well, now let's look at our
expressions for the energy.

751
00:47:11 --> 00:47:16
Average energy.
 

752
00:47:16 --> 00:47:21
We're going to sum over all
the energies of epsilon i,

753
00:47:21 --> 00:47:25
e to the minus Ei over kT.
 

754
00:47:25 --> 00:47:30
Over sum over i, e to
the minus Ei over kT.

755
00:47:30 --> 00:47:34
That's just how we originally
derived the average of energy.

756
00:47:34 --> 00:47:38
In other words, it's the
probability of each state times

757
00:47:38 --> 00:47:39
the energy of that state.
 

758
00:47:39 --> 00:47:44
Summed up over all the states.
 

759
00:47:44 --> 00:47:47
Well, now we have an
expression for the energy.

760
00:47:47 --> 00:47:48
It's one of these things.
 

761
00:47:48 --> 00:47:53
It's something times
a variable squared.

762
00:47:53 --> 00:48:01
So, we'll use some general
functional form, a y squared.

763
00:48:01 --> 00:48:04
And now, let's assume we're in
the high temperature limit.

764
00:48:04 --> 00:48:06
And we've seen what that means.
 

765
00:48:06 --> 00:48:11
It means kT is big,
compared to the separation

766
00:48:11 --> 00:48:13
between the energies.
 

767
00:48:13 --> 00:48:16
When that's the case, there are
lots of these terms in the sum.

768
00:48:16 --> 00:48:18
We can convert them
to integrals.

769
00:48:18 --> 00:48:21
We can forget about the fact
that the energies are discrete.

770
00:48:21 --> 00:48:26
We can say look, they're so
close together, compared to kT.

771
00:48:26 --> 00:48:29
That we can turn the
sums into integrals.

772
00:48:29 --> 00:48:36
So then we have integrals
instead of sums.

773
00:48:36 --> 00:48:37
And here's our energy.
 

774
00:48:37 --> 00:48:39
It's a y squared.
 

775
00:48:39 --> 00:48:39
Whatever that is.
 

776
00:48:39 --> 00:48:41
It could be this, it could
be this, it could be this.

777
00:48:41 --> 00:48:47
And it's not going to matter. e
to the minus a y squared over

778
00:48:47 --> 00:48:53
kT over, integral over e to the
minus a y squared

779
00:48:53 --> 00:48:56
over kT. dy, dy.
 

780
00:48:56 --> 00:49:00
 


781
00:49:00 --> 00:49:03
OK.
 

782
00:49:03 --> 00:49:06
And here's what's
going to happen.

783
00:49:06 --> 00:49:09
If you do this
integral by parts.

784
00:49:09 --> 00:49:11
This one.
 

785
00:49:11 --> 00:49:16
What ends up happening is
it gives you this integral

786
00:49:16 --> 00:49:19
times a certain number.
 

787
00:49:19 --> 00:49:23
This is going to come
out of the integral.

788
00:49:23 --> 00:49:26
And so it will turn out.
 

789
00:49:26 --> 00:49:27
And it's straightforward
to do it.

790
00:49:27 --> 00:49:28
It's in the notes.
 

791
00:49:28 --> 00:49:29
It goes through.
 

792
00:49:29 --> 00:49:31
But I'll just write
the result here.

793
00:49:31 --> 00:49:35
The result is that you get, in
this high temperature limit

794
00:49:35 --> 00:49:42
where you've gone to the
integral form, you get 1/2 kT.

795
00:49:42 --> 00:49:43
And it will always be the case.
 

796
00:49:43 --> 00:49:47
All you need to know is
the form of the energy.

797
00:49:47 --> 00:49:51
As long as that's the case --
Remember y is just a variable

798
00:49:51 --> 00:49:52
of integration here.
 

799
00:49:52 --> 00:49:56
That's not going
to be preserved.

800
00:49:56 --> 00:50:00
This comes out because
of what happens here.

801
00:50:00 --> 00:50:03
So what that's telling you is
whenever you have an energy of

802
00:50:03 --> 00:50:07
this form, and you're in the
high temperature limit, then

803
00:50:07 --> 00:50:14
you get this classical
equipartition of energy result.

804
00:50:14 --> 00:50:16
So, for translation, of course
there are three separate

805
00:50:16 --> 00:50:17
degrees of freedom.
 

806
00:50:17 --> 00:50:21
For velocity in the x,
y, or z direction.

807
00:50:21 --> 00:50:23
For rotation, if it's a linear
molecule, let's say there are

808
00:50:23 --> 00:50:24
two separate degrees
of freedom.

809
00:50:24 --> 00:50:26
You have to keep track
of how many degrees

810
00:50:26 --> 00:50:27
of freedom there are.
 

811
00:50:27 --> 00:50:29
But that's all you have to do.
 

812
00:50:29 --> 00:50:31
That's enormously powerful.
 

813
00:50:31 --> 00:50:35
It means that without doing
anything, I know the average

814
00:50:35 --> 00:50:39
translational energy of the
molecules in this room.

815
00:50:39 --> 00:50:41
Because of course I'm certainly
in the high temperature limit.

816
00:50:41 --> 00:50:43
With respect to translational
energy levels, they're

817
00:50:43 --> 00:50:45
really closely spaced.
 

818
00:50:45 --> 00:50:48
Same with rotations.
 

819
00:50:48 --> 00:50:51
Now at room temperature
vibrations, forget it.

820
00:50:51 --> 00:50:53
I have essentially no
vibrational energy.

821
00:50:53 --> 00:50:57
I'm at the low temperature
limit for the molecular

822
00:50:57 --> 00:50:59
vibrations of
nitrogen or oxygen.

823
00:50:59 --> 00:51:01
Those are high in frequency.
 

824
00:51:01 --> 00:51:04
So much higher than
200 wavenumbers.

825
00:51:04 --> 00:51:09
But for each molecule, I have
3/2 kT of translational energy.

826
00:51:09 --> 00:51:12
Linear molecules, I have
kT of rotational energy.

827
00:51:12 --> 00:51:14
Without doing any work at all.
 

828
00:51:14 --> 00:51:18
And since that applies to most
molecules at room temperature,

829
00:51:18 --> 00:51:22
it's an incredibly useful,
very, very general result.

830
00:51:22 --> 00:51:22
OK.
 

831
00:51:22 --> 00:51:25
Next time we'll do a little
bit of chemistry, and look

832
00:51:25 --> 00:51:27
at phase transformations.
 

833
00:51:27 --> 00:51:28