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PROFESSOR: So, last time we
started in on a discussion

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of a new topic, with was
statistical mechanics.

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So what we're trying to do now
is revisit the thermodynamics

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that we've spent most of the
term deriving and trying to use

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in a microscopic approach.
 

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So our hope is to be able to
start with a microscopic model

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of matter, starting with atoms
and molecules that we

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know are out there.
 

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And formulate thermodynamics
starting from that

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microscopic point of view.
 

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In contrast to the way it was
first formulated historically,

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and the way we've presented it
also, which is as an entirely

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empirical subject based on
macroscopic observation

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and deduction from that.
 

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So, we looked at the
probabilities that states with

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different energies would be
occupied and we inferred that

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there would be a simple way to
describe the distribution of

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atoms or molecules in states
of different levels.

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First of all, although I didn't
state it explicitly,

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essentially assumed there is
that if I've got a bunch of

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molecules, and I've got states
that they could be in of equal

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energy, then the probability
that they would be in one

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or another of those
states is the same.

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If the energies of the states
are equal, the probabilities

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that those states
will be occupied.

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That they'll be populated by
molecules, will be equal.

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And then we deduced what's
called the Boltzmann

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probability distribution.
 

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Which says that the probability
that a molecular state, i, will

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be occupied is proportional to
e to the minus Ei over kT,

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00:02:31 --> 00:02:40
where little Ei is the energy
of that molecular state.

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And then realizing that the
probability that some state out

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there has to be occupied, we
saw that of course if we sum

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over all these probabilities
that sum has to

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equal one, right?
 

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In other words, the sum
over all of i of P of

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00:02:57 --> 00:03:00
i is equal to one.
 

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00:03:00 --> 00:03:04
So that means that we could
write not just that this is

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proportional to this, but we
could write that Pi is equal

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to e to the minus Ei over kT
divided by the sum

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of all such terms.
 

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So now we know how to figure
out how likely it is that a

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certain state is occupied.
 

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And what it means is, let's say
we have a whole bunch of states

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whose energy is pretty low
compared to kT. kT, the

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Boltzmann constant times
temperature is energy units.

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So if it's pretty warm, maybe
it's room temperature,

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maybe it's warmer.
 

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And there are a whole lot of
states that are accessible

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because the energies
are less than kT.

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What does that mean?
 

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That means that you could put
a bunch of different i's

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here whose energies are all
quite low compared to kT.

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And they'll all have
significant probabilities.

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Now let's decrease the energy.
 

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Go to cold temperature.
 

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So kT becomes really small.
 

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So that e to the minus energy
over kT starts to, if this gets

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to be bigger than kT by a lot,
then this whole thing is

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a very small number.
 

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Suddenly, you get very
few states accessible.

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So if we just plot this
distribution, of course it's

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easy to do, it's just a
decaying exponential.

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So Pi, which is a
function of Ei.

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It just looks like this.
 

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And here are a bunch of states.
 

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And if we have a classical
mechanics picture of matter,

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then there would just
be continuous states.

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And if we have a quantum
mechanical picture of matter,

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there might be individual
states with gaps in

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between the energies.
 

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Either way, what happens, what
this is saying is, if we go out

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to higher and higher energies,
then you have a smaller and

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smaller probability to be
in a state like that.

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And if you go to low energies,
the probability gets bigger.

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And it depends on temperature.
 

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Because it's the ratio between
the energy and kT that dictates

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the size of that term.
 

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So let's say this is
moderate temperature.

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If we go to low temperature,
it might look more like this.

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Hardly any states can be
occupied because even states

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of rather moderate energies,
suddenly now those energies

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are much bigger than kT.
 

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In other words, kT is
measuring thermal energy.

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It's saying there's not enough
thermal energy to populate, to

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knock things into states
whose energy is much

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higher than that.
 

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So you have a very
precipitous decay.

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If you go to the other extreme,
of very high temperature, then

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this will tend to flatten out.
 

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Eventually it'll decay, but
it may take a long time.

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Because now kT is enormous.
 

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So the energy has to
get to be very big.

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Before it's bigger than kT.
 

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And as long as it isn't,
then this exponential

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term is not small.
 

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So there be a whole bunch of
states that may be occupied.

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In other words, a whole bunch
of states that are thermally

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accessible at equilibrium.
 

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Systems at thermal equilibrium,
molecules are getting knocked

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around with whatever thermal
energy is available.

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Crashing into each other or
into the walls of a vessel.

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And there'll be a distribution
of molecular energies.

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And that distribution is skewed
either low or high depending

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on the temperature.
 

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So that's what that
distribution, the Boltzmann

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distribution, is telling us.
 

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This is often called a
Boltzmann factor, because it's

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telling what the population of
some particular state is.

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OK.
 

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Now, this is just dealing with
individual molecule states.

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Then we said, OK, what
about the whole system?

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Well, the same kind of
relation holds there, too.

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In other words, if these are
individual molecule energies,

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now if I look at the entire
system, right well, still,

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there's no reason that same
distribution doesn't hold.

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And it does.
 

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So in other words, Pi of Ei,
that's the whole system energy,

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is e to the minus Ei over kT,
over the sum over

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i, Ei over kT.
 

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Now, this i doesn't refer to
a single molecule state.

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We're talking about
a whole system.

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It might be a mole of atoms
or molecules in the gas

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phase, or what have you.
 

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It refers to a system state
where the energy, the state

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of every one of those atoms
or molecules is specified.

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So this is a system microstate.
 

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Every molecular
state is specified.

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So for a mole of stuff, that
means there might be 10 to the

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24 or so molecular states that
this single subscript

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is indicating.
 

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But the point is,
those states exist.

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They have a total energy.
 

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What's the probability that
the whole system will

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be in such a state?
 

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00:10:00 --> 00:10:12
Still going to be proportional
to the total system energy.

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It turns out that these
summations end up taking on

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an enormous importance in
statistical mechanics.

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And the reason is, as we'll see
shortly, it turns out that

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every single macroscopic
thermodynamic function can be

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derived by knowing just that.
 

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Just these sums, what are
called partition functions.

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So of course they take
on enormous importance.

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So, we call them partition
functions because what they're

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doing is, they're indicating
how the molecules are

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partitioned among the
different available levels.

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The molecules or
a whole system.

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So the molecular partition
function is labeled little q.

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And the system partition
function is labeled big Q.

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It's called the canonical
partition function.

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And because they are going
to take on such special

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importance, let's just look
at some of their properties

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for different kinds of
systems and in general.

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OK, first of all, let's talk
about units and values.

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They are unitless.
 

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This is an exponential
function.

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Here are units of
energy and energy.

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But this is a
unitless was number.

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It's just some number, right?
 

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Could be 1.
 

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Could be 10.
 

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Could be 50.
 

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Whatever, right?
 

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Could be 10 to the 24.
 

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Its magnitude tells you about
more or less how many states

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are thermally accessible.
 

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Because, look at it.
 

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And then go back to the example
that I showed you if lots of

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these terms are big, are
significant because this is

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really a big number, right?
 

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It's really hot.
 

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So lots of states have energies
lower than kT, which means this

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is not too small, for lots and
lots of values of i, then

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this just keeps adding up.
 

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And of course, same here.
 

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So the number might
be very large.

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But if we're in the low
temperature limit, maybe we're

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in such low temperature that
only the lowest possible

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state is occupied.
 

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And everything else,
it's just too cold.

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There's not enough thermal
energy to occupy anything

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but the very lowest
state available.

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Well, in that case the lowest
state, this would be one.

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Essentially we could
label this zero.

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We could put the energy,
the zero of energy there.

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Everything else is really big
compared to kT, which means

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this exponential gets to be a
really small number for

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every state except one.
 

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And this is just equal to one.
 

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In other words, the magnitude
of these numbers tells us about

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how many states are accessible,
thermally accessible, to

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00:14:04 --> 00:14:09
molecules or to a whole system.
 

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So let's just go through a
couple of specific examples

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to try to make that a
little more concrete.

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One is, let's start in
the simplest case.

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Which it'll turn out
I just alluded to.

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Let's consider a perfect
atomic crystal at essentially

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zero degrees Kelvin.
 

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Zero Kelvin.
 

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So every atom.
 

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Or even if it's a molecular
crystal, every molecule,

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00:14:46 --> 00:14:47
they're all in the
ground state.

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There's no excess
thermal energy.

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Every molecule is in
its proper place.

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Every atom, if it's
an atomic lattice.

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In other words, it's
in the ground state.

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That's it.
 

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00:15:01 --> 00:15:06
So again, we can place
the zero where we like.

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00:15:06 --> 00:15:08
We'll place it there.
 

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00:15:08 --> 00:15:12
That one, for that one state,
this will be equal to one.

219
00:15:12 --> 00:15:25
And for everything
else it'll be zero.

220
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So Q is just the sum of e
to the minus Ei over kT.

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It's equal to e to the minus
zero over kT plus e to the

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minus E1 over kT plus e
to the minus E2 over kT.

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But remember, T is really tiny.
 

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It's almost zero
degrees Kelvin.

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So all these things are
much bigger than that.

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So this is vanishingly small.
 

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This is vanishingly small.
 

228
00:16:00 --> 00:16:04
The whole thing is
approximately equal to one.

229
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That's it.
 

230
00:16:08 --> 00:16:13
Also, if we say OK, now what's
the probability of the system

231
00:16:13 --> 00:16:15
being in a particular state.
 

232
00:16:15 --> 00:16:21
Well, we have an
expression for that.

233
00:16:21 --> 00:16:25
So let's look at P0.
 

234
00:16:25 --> 00:16:33
It's e to the minus E0
over kT over the sum.

235
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Which we've just seen.
 

236
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So it's e to the minus E0 over
to kT divided by e to the minus

237
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E0 over kT plus e to the
minus E1 over kT, and so on.

238
00:16:57 --> 00:17:00
This is the only term
that's significant.

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So it's approximately
equal to one.

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00:17:03 --> 00:17:06
Now, while I've got this
written here, let's just make

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sure we've got something clear.
 

242
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What if we hadn't
arbitrarily set the zero

243
00:17:12 --> 00:17:14
of energy equal to zero?
 

244
00:17:14 --> 00:17:15
I mean, it's arbitrary, right?
 

245
00:17:15 --> 00:17:17
We can put the zero
of energy anywhere.

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And you might think, well,
gee that's going to have a

247
00:17:19 --> 00:17:21
big effect on everything.
 

248
00:17:21 --> 00:17:24
Well, it would have an
effect on the actual

249
00:17:24 --> 00:17:27
number that we get for Q.
 

250
00:17:27 --> 00:17:30
But what you'll see is that any
measurable quantity that we

251
00:17:30 --> 00:17:34
calculate won't be affected.
 

252
00:17:34 --> 00:17:36
It's only the zero of
the energy scale that's

253
00:17:36 --> 00:17:37
going to be affected.
 

254
00:17:37 --> 00:17:41
So for example, let's look
at this probability.

255
00:17:41 --> 00:17:46
It doesn't matter where we
put the zero of energy.

256
00:17:46 --> 00:17:48
This term is still going
to be enormously bigger

257
00:17:48 --> 00:17:49
then the next one.
 

258
00:17:49 --> 00:17:50
And the next one.
 

259
00:17:50 --> 00:17:52
And the next one.
 

260
00:17:52 --> 00:17:56
So in this sum only this
term is going to matter.

261
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It's going to cancel
with this term.

262
00:17:58 --> 00:18:00
So whether or not these
individual terms are equal to

263
00:18:00 --> 00:18:04
one, which happens if we set
this to zero, or whether

264
00:18:04 --> 00:18:06
they're equal to
some other number.

265
00:18:06 --> 00:18:10
Still, the probability that
the system is in the lowest

266
00:18:10 --> 00:18:13
state is one, right?
 

267
00:18:13 --> 00:18:16
That doesn't depend on where
we arbitrarily put the

268
00:18:16 --> 00:18:18
zero of the energy scale.
 

269
00:18:18 --> 00:18:20
The state is still going to
be in the ground state, at

270
00:18:20 --> 00:18:23
essentially zero
degrees Kelvin.

271
00:18:23 --> 00:18:25
And it'll turn out to be
that way with any property

272
00:18:25 --> 00:18:26
that we can measure.
 

273
00:18:26 --> 00:18:30
Only the zero of this scale
moves if we arbitrarily

274
00:18:30 --> 00:18:32
move it somewhere.
 

275
00:18:32 --> 00:18:36
But the observable, measurable
quantities that we calculate,

276
00:18:36 --> 00:18:38
they won't change.
 

277
00:18:38 --> 00:18:41
Other than that scale.
 

278
00:18:41 --> 00:18:47
So that's one simple example.
 

279
00:18:47 --> 00:18:50
Now let's look at
another example.

280
00:18:50 --> 00:18:53
Let's consider a mole of atoms,
roaming around in the gas

281
00:18:53 --> 00:19:06
phase at room temperature.
 

282
00:19:06 --> 00:19:11
OK, so now what I want to do is
just have a simple model for

283
00:19:11 --> 00:19:14
their translational motion.
 

284
00:19:14 --> 00:19:17
And of course, we could do that
either quantum mechanically or

285
00:19:17 --> 00:19:19
classically solve for that.
 

286
00:19:19 --> 00:19:22
We're going to use an
even simpler model.

287
00:19:22 --> 00:19:24
And this model is going to be
very, very useful for a lot of

288
00:19:24 --> 00:19:26
the things that we'll treat.
 

289
00:19:26 --> 00:19:29
It's called a lattice model.
 

290
00:19:29 --> 00:19:33
All it means is, we're going to
divide up the available volume.

291
00:19:33 --> 00:19:37
This room, for example,
into zillions of tiny

292
00:19:37 --> 00:19:38
little elements.
 

293
00:19:38 --> 00:19:39
Little volume elements.
 

294
00:19:39 --> 00:19:43
Each one about the
size of an atom.

295
00:19:43 --> 00:19:46
The idea being, we're going
to specify the state of the

296
00:19:46 --> 00:19:48
atom by saying where is it.
 

297
00:19:48 --> 00:19:50
Is it in this lattice sight,
in this one, in this

298
00:19:50 --> 00:19:53
one, in this one.
 

299
00:19:53 --> 00:20:11
So it's a lattice model.
 

300
00:20:11 --> 00:20:13
Might be an atom there.
 

301
00:20:13 --> 00:20:19
Might be one there.
 

302
00:20:19 --> 00:20:22
So we're going to
divide the volume up.

303
00:20:22 --> 00:20:36
So let's call our atomic
volume little v.

304
00:20:36 --> 00:20:41
Our total volume big V.
 

305
00:20:41 --> 00:20:44
And an atomic volume, it's
going to be on the order

306
00:20:44 --> 00:20:48
of 1 angstroms cubed.
 

307
00:20:48 --> 00:20:52
Or 10 to the minus
30 meters cubed.

308
00:20:52 --> 00:20:55
And our room, our volume
macroscopic one, might be on

309
00:20:55 --> 00:20:59
the order of one meter cubed.
 

310
00:20:59 --> 00:21:03
Ordinary sort of size.
 

311
00:21:03 --> 00:21:09
So now let's figure out our
molecular partition function.

312
00:21:09 --> 00:21:13
Now, implicit in this, we're
basically saying that the

313
00:21:13 --> 00:21:17
energy, the translational
energy, is basically zero.

314
00:21:17 --> 00:21:20
In other words, all these
states have the same energy.

315
00:21:20 --> 00:21:23
They're just located in
different positions.

316
00:21:23 --> 00:21:27
At any given instant of time.
 

317
00:21:27 --> 00:21:30
And what that means is all
these terms, all these

318
00:21:30 --> 00:21:32
Boltzmann factors, we just
set them equal to one.

319
00:21:32 --> 00:21:34
We'll set the zero of energy
there, be done with it.

320
00:21:34 --> 00:21:36
It's a simple model.
 

321
00:21:36 --> 00:21:38
But it's going to give us the
right order of magnitude

322
00:21:38 --> 00:21:39
that we're after.
 

323
00:21:39 --> 00:21:45
So, how many states are
there that are accessible?

324
00:21:45 --> 00:21:50
Well, on the order of
10 to the 30th, right?

325
00:21:50 --> 00:21:57
So q, little q, we'll call
it little q translational,

326
00:21:57 --> 00:22:02
it's just discussing
where things are.

327
00:22:02 --> 00:22:05
Is on the order of
10 to the 30th.

328
00:22:05 --> 00:22:09
And if we do a more careful
treatment, if we treat the

329
00:22:09 --> 00:22:12
translational energy of atoms,
either classically or quantum

330
00:22:12 --> 00:22:13
mechanically and solve it.
 

331
00:22:13 --> 00:22:16
We'll still get, we still
do get, about the same

332
00:22:16 --> 00:22:20
order of magnitude.
 

333
00:22:20 --> 00:22:23
Now let's treat
the whole system.

334
00:22:23 --> 00:22:25
So, what happens?
 

335
00:22:25 --> 00:22:30
What are our total
possible states?

336
00:22:30 --> 00:22:33
Because we have to add this
up for every possible state.

337
00:22:33 --> 00:22:36
Well, let's start
with the first atom.

338
00:22:36 --> 00:22:37
It has to be somewhere.
 

339
00:22:37 --> 00:22:39
It has 10 to the 30th
possibilities for

340
00:22:39 --> 00:22:41
where it can be.
 

341
00:22:41 --> 00:22:43
Let's start with
the second atom.

342
00:22:43 --> 00:22:45
And put it somewhere.
 

343
00:22:45 --> 00:22:47
Well, it has 10 to the 30th
minus one, which is still

344
00:22:47 --> 00:22:50
pretty close to 10 to the 30th.
 

345
00:22:50 --> 00:22:52
Let's let's go to
the third atom.

346
00:22:52 --> 00:22:53
And the fourth.
 

347
00:22:53 --> 00:22:56
If we have about a mole of
atoms, let's say 10 to the 24th

348
00:22:56 --> 00:23:00
atoms, that's still going
to occupy a very small

349
00:23:00 --> 00:23:01
fraction of the site.
 

350
00:23:01 --> 00:23:03
Only one in a million.
 

351
00:23:03 --> 00:23:05
So we don't have to
keep careful track.

352
00:23:05 --> 00:23:07
Every one of them, we can just
say, look, there are 10 to

353
00:23:07 --> 00:23:11
the 30th available sites.
 

354
00:23:11 --> 00:23:12
Because we're not going
to worry about a change

355
00:23:12 --> 00:23:15
in one in a millionth.
 

356
00:23:15 --> 00:23:24
So what that means is, capital
Q, trans for the system, is 10

357
00:23:24 --> 00:23:29
to the 30th times
10 to the 30th.

358
00:23:29 --> 00:23:31
In other words, let's now
take the whole state.

359
00:23:31 --> 00:23:34
Well, I could put
atom one here.

360
00:23:34 --> 00:23:36
I have 10 to the
30th possibilities.

361
00:23:36 --> 00:23:37
Atom two, I can put
anywhere else.

362
00:23:37 --> 00:23:40
So the joint probability of
that particular state, for just

363
00:23:40 --> 00:23:44
the two atoms, is 10th to the
30th times 10 to the 30th.

364
00:23:44 --> 00:23:46
It's 10 to the 30th squared.
 

365
00:23:46 --> 00:23:56
So this is going to keep going.
 

366
00:23:56 --> 00:24:00
It's going to be 10 to the
30th to the Nth power.

367
00:24:00 --> 00:24:06
Where N is the number of atoms,
which is 10 of the 24th.

368
00:24:06 --> 00:24:07
Huge number, right?
 

369
00:24:07 --> 00:24:09
It is a huge number.
 

370
00:24:09 --> 00:24:12
And that, too, if we treat the
whole thing classically or

371
00:24:12 --> 00:24:15
quantum mechanically
and work it all out.

372
00:24:15 --> 00:24:16
We'll get that number.
 

373
00:24:16 --> 00:24:17
Or something on that order.
 

374
00:24:17 --> 00:24:22
Because you know there really
is a simply astronomical number

375
00:24:22 --> 00:24:26
of states accessible to the
whole bunch of atoms or

376
00:24:26 --> 00:24:29
molecules in this room.
 

377
00:24:29 --> 00:24:32
It really is that big.
 

378
00:24:32 --> 00:24:37
So in other words, capital Q is
just an astronomical number.

379
00:24:37 --> 00:24:42
And it is the case.
 

380
00:24:42 --> 00:24:42
OK.
 

381
00:24:42 --> 00:24:47
There's an important
sort of nuance that

382
00:24:47 --> 00:24:51
we need to introduce.
 

383
00:24:51 --> 00:24:53
And it's the following.
 

384
00:24:53 --> 00:24:54
Turns out, it is an
astronomical number.

385
00:24:54 --> 00:24:58
But a tiny bit less
astronomical than what I've

386
00:24:58 --> 00:25:00
treated so far would indicate.
 

387
00:25:00 --> 00:25:02
So let's look a little
more carefully.

388
00:25:02 --> 00:25:08
What I've said is that Q
translational is little q

389
00:25:08 --> 00:25:12
translational, that is, that
10 to the 30th number.

390
00:25:12 --> 00:25:15
To the Nth power.
 

391
00:25:15 --> 00:25:17
But there's one failing here.
 

392
00:25:17 --> 00:25:23
Which is, when I got that, when
I decided that if I have just

393
00:25:23 --> 00:25:26
the first two atoms I've got
10th to the 30th times 10 to

394
00:25:26 --> 00:25:30
the 30th possible states, the
trouble with that is then if I

395
00:25:30 --> 00:25:33
keep counting all the possible
atoms starting with each one,

396
00:25:33 --> 00:25:37
I'll double-count it, right?
 

397
00:25:37 --> 00:25:41
In other words, what if I
interchange those two atoms?

398
00:25:41 --> 00:25:43
Well, those states are
identical, right?

399
00:25:43 --> 00:25:45
Indistinguishable.
 

400
00:25:45 --> 00:25:51
And it's only distinguishable
states that count.

401
00:25:51 --> 00:25:56
When you specify these
things, these are

402
00:25:56 --> 00:26:00
indicating distinct states.
 

403
00:26:00 --> 00:26:02
In some way, at least in
principle, measurably

404
00:26:02 --> 00:26:04
different.
 

405
00:26:04 --> 00:26:07
If the atom's are identical, in
other words, if it's a mole of

406
00:26:07 --> 00:26:10
the same stuff, I don't have a
way of distinguishing

407
00:26:10 --> 00:26:11
between those two.
 

408
00:26:11 --> 00:26:14
I have to correct for that.
 

409
00:26:14 --> 00:26:17
And when I come to the third
atom, I have to correct for all

410
00:26:17 --> 00:26:18
the possible interchanges.
 

411
00:26:18 --> 00:26:20
Of course, it's
three factorial.

412
00:26:20 --> 00:26:23
And in general,
it's N factorial.

413
00:26:23 --> 00:26:26
So this result needs
to be modified.

414
00:26:26 --> 00:26:37
This is true for
distinguishable particles.

415
00:26:37 --> 00:26:40
In other words, if I had all
different atoms, so I could

416
00:26:40 --> 00:26:42
label them all, then I don't
have that correction.

417
00:26:42 --> 00:26:45
Because then there's really a
difference between that state

418
00:26:45 --> 00:26:48
and the state with the
two atoms interchanged.

419
00:26:48 --> 00:26:50
But if it's a mole of
identical atoms, that's

420
00:26:50 --> 00:26:51
no longer the case.
 

421
00:26:51 --> 00:26:58
So then, Q translational is
little q translational to the N

422
00:26:58 --> 00:27:04
power divided by N factorial.
 

423
00:27:04 --> 00:27:07
It's still going to be an
absolutely enormous number.

424
00:27:07 --> 00:27:09
But it's going to be a little
less absolutely enormous

425
00:27:09 --> 00:27:13
than it was a minute ago.
 

426
00:27:13 --> 00:27:16
Now finally, I just want to
introduce a handy approximation

427
00:27:16 --> 00:27:19
to N factorial that's going
to turn out to be very

428
00:27:19 --> 00:27:22
useful again and again.
 

429
00:27:22 --> 00:27:31
And that's called
Stirling's approximation.

430
00:27:31 --> 00:27:43
For the log of a big number, ln
N factorial is N log N minus N.

431
00:27:43 --> 00:27:49
If we take e to those, both
sides, then we find that N

432
00:27:49 --> 00:28:02
factorial is equal to e to the
minus N N to the nth power.

433
00:28:02 --> 00:28:08
So this, then, is approximately
equal to q translational

434
00:28:08 --> 00:28:14
to the Nth power.
 

435
00:28:14 --> 00:28:20
Over N to the N times
e to the minus N.

436
00:28:20 --> 00:28:24
Now let's put the
numbers back in.

437
00:28:24 --> 00:28:32
There's our 10 to the 30th to
the 10 to the 24th power.

438
00:28:32 --> 00:28:35
And now we're going to
diminish that just a bit.

439
00:28:35 --> 00:28:37
It's N to the Nth power.
 

440
00:28:37 --> 00:28:47
So it's 10 to the 24th to
the 10 to the 24th power.

441
00:28:47 --> 00:28:55
Times e to the minus
10 to the 24th power.

442
00:28:55 --> 00:28:58
So, we can cancel
something here.

443
00:28:58 --> 00:29:06
So we have 10 to the 6th to the
10 to the 24th power times e to

444
00:29:06 --> 00:29:17
the 10 to the 24th power. e is
about 10 to the 0.4th power.

445
00:29:17 --> 00:29:24
So this whole thing is about 10
to the 6.4th power to the 10 to

446
00:29:24 --> 00:29:31
the 24th power It's still a
pretty respectable number.

447
00:29:31 --> 00:29:33
You could still
say astronomical.

448
00:29:33 --> 00:29:35
But maybe astronomical,
but not quite to the

449
00:29:35 --> 00:29:36
edge of the universe.
 

450
00:29:36 --> 00:29:39
Whereas the one before
maybe hit the edge of the

451
00:29:39 --> 00:29:44
universe and then beyond.
 

452
00:29:44 --> 00:29:46
So that's our second example.
 

453
00:29:46 --> 00:29:48
And again, part of what I'm
trying to do here is just

454
00:29:48 --> 00:29:50
introduce some ideas.
 

455
00:29:50 --> 00:29:52
Things like this way
of modeling positions

456
00:29:52 --> 00:29:53
and so forth.
 

457
00:29:53 --> 00:29:55
But also, again,
orders of magnitude.

458
00:29:55 --> 00:29:58
How big are these numbers.
 

459
00:29:58 --> 00:30:00
For different sorts of systems.
 

460
00:30:00 --> 00:30:02
So let's do one more example.
 

461
00:30:02 --> 00:30:07
Now, let's consider a
polymer in a liquid.

462
00:30:07 --> 00:30:08
And it has different
configurations.

463
00:30:08 --> 00:30:10
And they might be a little
bit different in energy.

464
00:30:10 --> 00:30:13
For example, some
configurations might bring

465
00:30:13 --> 00:30:16
neighboring regions of the
polymer into proximity where

466
00:30:16 --> 00:30:18
they could hydrogen bond.
 

467
00:30:18 --> 00:30:22
And the point is that then the
molecular energies involved

468
00:30:22 --> 00:30:23
will change a little bit.
 

469
00:30:23 --> 00:30:26
Because of some sorts of
interactions that are possible

470
00:30:26 --> 00:30:28
in some configurations.
 

471
00:30:28 --> 00:30:33
And not in other
configurations.

472
00:30:33 --> 00:30:36
So what I'm trying to do is
introduce a very simple

473
00:30:36 --> 00:30:39
framework through which we
might be able to look at things

474
00:30:39 --> 00:30:42
like protein folding, or
DNA hydrogen bonding.

475
00:30:42 --> 00:30:43
Things like this.
 

476
00:30:43 --> 00:30:46
And just in a simple way model
how those work and what the

477
00:30:46 --> 00:31:15
forces are that drive them.
 

478
00:31:15 --> 00:31:22
So we'll think about
polymer configurations.

479
00:31:22 --> 00:31:28
So let's look at a
few configurations.

480
00:31:28 --> 00:31:33
Here is going to be one.
 

481
00:31:33 --> 00:31:37
I'm going to label
this one here.

482
00:31:37 --> 00:32:02
And then here are a few others.
 

483
00:32:02 --> 00:32:05
So I've labeled them this way
so that you can see how they

484
00:32:05 --> 00:32:07
are distinct from each other.
 

485
00:32:07 --> 00:32:12
This one would have a
possible interaction.

486
00:32:12 --> 00:32:14
So we'll label its energy.
 

487
00:32:14 --> 00:32:25
So this is molecular
state i, here's going

488
00:32:25 --> 00:32:28
to be our energy, Ei.
 

489
00:32:28 --> 00:32:33
And let's call this one
negative e int, for an

490
00:32:33 --> 00:32:36
interaction energy
that's favorable.

491
00:32:36 --> 00:32:44
And these will be zero.
 

492
00:32:44 --> 00:32:46
Let's just put that there.
 

493
00:32:46 --> 00:32:56
Zero, zero, zero.
 

494
00:32:56 --> 00:33:01
And let's also indicate
the degeneracy.

495
00:33:01 --> 00:33:04
How many states, different
states are there with

496
00:33:04 --> 00:33:07
the same energy.
 

497
00:33:07 --> 00:33:09
And that's called gi.
 

498
00:33:09 --> 00:33:11
And here it's one.
 

499
00:33:11 --> 00:33:16
And here are these three.
 

500
00:33:16 --> 00:33:20
So that's the framework
of our model.

501
00:33:20 --> 00:33:25
Well, so what's our molecular
partition function for

502
00:33:25 --> 00:33:29
this configurational
degree of freedom?

503
00:33:29 --> 00:33:37
Well, we can label it
little q configurational.

504
00:33:37 --> 00:33:43
So we're going to sum over
these states. e to the minus

505
00:33:43 --> 00:33:50
Ei for the different
configurations over kT.

506
00:33:50 --> 00:33:57
So it's e to the e int over kT.
 

507
00:33:57 --> 00:34:02
Plus three times e to the
zero over kT, we'll get all

508
00:34:02 --> 00:34:04
those other three terms.
 

509
00:34:04 --> 00:34:05
So that's it.
 

510
00:34:05 --> 00:34:12
It's e to the e int
over kT plus three.

511
00:34:12 --> 00:34:16
Remember, the interaction
energy is negative e int.

512
00:34:16 --> 00:34:22
It's a favorable interaction.
e int is a positive number.

513
00:34:22 --> 00:34:25
So that's our result.
 

514
00:34:25 --> 00:34:32
Now we've described the
probabilities in terms of the

515
00:34:32 --> 00:34:34
states that can be occupied.
 

516
00:34:34 --> 00:34:35
That is, we've added it up.
 

517
00:34:35 --> 00:34:38
But of course, even the way
I wrote it just out of

518
00:34:38 --> 00:34:41
convenience, I didn't actually
write out each term in the sum.

519
00:34:41 --> 00:34:42
In the notes I
actually did that.

520
00:34:42 --> 00:34:45
But of course it's not
necessary to write e to the

521
00:34:45 --> 00:34:48
zero over kT plus e to the zero
over kT and write that three

522
00:34:48 --> 00:34:50
times for each of these three.
 

523
00:34:50 --> 00:34:54
Rather, it's convenient
to group them together.

524
00:34:54 --> 00:34:58
And the point I'm illustrating
here is that in our expression,

525
00:34:58 --> 00:35:03
for the partition function,
we've written that in terms

526
00:35:03 --> 00:35:06
of the individual states.
 

527
00:35:06 --> 00:35:09
But we doing a sum over the
individual molecular states.

528
00:35:09 --> 00:35:14
But we could also sum
over energy levels.

529
00:35:14 --> 00:35:17
Including the degeneracy.
 

530
00:35:17 --> 00:35:21
So we could say, let's not do
the sum over every state.

531
00:35:21 --> 00:35:24
After all, what if there
are a hundred states that

532
00:35:24 --> 00:35:24
had the same energy.
 

533
00:35:24 --> 00:35:26
Rather than just three.
 

534
00:35:26 --> 00:35:28
Gets to be kind of
painful, right?

535
00:35:28 --> 00:35:31
Instead let's just sum
over energy levels.

536
00:35:31 --> 00:35:34
They're all going to be
the same factor anyway.

537
00:35:34 --> 00:35:36
And then we'll have, we'll
just explicitly write

538
00:35:36 --> 00:35:40
the degeneracy in there.
 

539
00:35:40 --> 00:35:51
So, we can write the
partition function as a

540
00:35:51 --> 00:35:57
sum over energy level.
 

541
00:35:57 --> 00:36:06
So, in other words q is
the sum over i, e to

542
00:36:06 --> 00:36:11
the minus Ei over kT.
 

543
00:36:11 --> 00:36:22
That's individual
molecular states i.

544
00:36:22 --> 00:36:27
But we also could write it as
the sum over i, where this now

545
00:36:27 --> 00:36:36
is molecular energy levels i.
 

546
00:36:36 --> 00:36:40
And then we need to incorporate
the degeneracy gi e to

547
00:36:40 --> 00:36:44
the minus Ei over kT.
 

548
00:36:44 --> 00:36:47
Same thing, of course.
 

549
00:36:47 --> 00:36:49
But again, sometimes
much more convenient to

550
00:36:49 --> 00:36:53
write things this way.
 

551
00:36:53 --> 00:36:56
And of course, we can write
the probabilities of the

552
00:36:56 --> 00:36:59
occupancies in the
same way, too.

553
00:36:59 --> 00:37:07
That is, we could talk about
the Pi in terms of individual

554
00:37:07 --> 00:37:17
states. e to the minus Ei over
kT over q, the whole sum.

555
00:37:17 --> 00:37:34
Or Pi summing over energy
levels, gi e to the minus

556
00:37:34 --> 00:37:40
Ei over kT, divided by q.
 

557
00:37:40 --> 00:37:43
Here, of course, this is bigger
if it's degenerate, right?

558
00:37:43 --> 00:37:46
It's saying that the
probability of being in this

559
00:37:46 --> 00:37:49
energy level is three times the
probability of it being in any

560
00:37:49 --> 00:37:53
individual one of the states.
 

561
00:37:53 --> 00:37:55
But sometimes it's
useful to keep track

562
00:37:55 --> 00:37:57
of things in this way.
 

563
00:37:57 --> 00:38:02
What it shows you too is that,
remember when we looked

564
00:38:02 --> 00:38:05
at, did I erase it?
 

565
00:38:05 --> 00:38:07
I guess it's gone.
 

566
00:38:07 --> 00:38:13
When we looked at this
probability distribution for

567
00:38:13 --> 00:38:18
the occupancies of the levels,
of course, what it says is at

568
00:38:18 --> 00:38:22
any temperature, the lowest
level is the most probable.

569
00:38:22 --> 00:38:24
For intermediate temperatures.
 

570
00:38:24 --> 00:38:29
For low temperatures,
for high temperatures.

571
00:38:29 --> 00:38:32
At high temperature, it might
be only a little more probable

572
00:38:32 --> 00:38:34
than the next one over, and
the next one, and so forth.

573
00:38:34 --> 00:38:37
But the very lowest
state always has the

574
00:38:37 --> 00:38:41
highest probability.
 

575
00:38:41 --> 00:38:44
But the lowest energy doesn't
always have the highest

576
00:38:44 --> 00:38:46
probability because
of degeneracies.

577
00:38:46 --> 00:38:49
There might be many states
with an energy up here.

578
00:38:49 --> 00:38:51
And the probability of any
one of them is only a little

579
00:38:51 --> 00:38:54
lower than the probability
of the lowest energy.

580
00:38:54 --> 00:38:55
That could be the case here.
 

581
00:38:55 --> 00:39:00
Let's say, the interaction
energy isn't enormously strong.

582
00:39:00 --> 00:39:07
So there is some energetic
favoring of this state.

583
00:39:07 --> 00:39:10
Maybe there are 10%
at room temperature.

584
00:39:10 --> 00:39:12
Maybe it turns out there are
10% more molecules like

585
00:39:12 --> 00:39:15
this than in any one
of these states.

586
00:39:15 --> 00:39:18
But of course, these three
altogether mean that there are

587
00:39:18 --> 00:39:21
many more molecules at this
energy than at the

588
00:39:21 --> 00:39:32
lowest energy.
 

589
00:39:32 --> 00:39:37
Now of course we could do the
same thing for the canonical

590
00:39:37 --> 00:39:37
partition function.
 

591
00:39:37 --> 00:39:41
Not just the molecular one.
 

592
00:39:41 --> 00:39:53
So in other words, capital
Q sum over i system

593
00:39:53 --> 00:40:03
microstates. e to the
minus capital Ei over kT.

594
00:40:03 --> 00:40:08
But we could also write it
as a sum over energies.

595
00:40:08 --> 00:40:17
Sum over system energies.
 

596
00:40:17 --> 00:40:19
Ei.
 

597
00:40:19 --> 00:40:22
And then we have to
include the degeneracy.

598
00:40:22 --> 00:40:35
Capital Omega i e to
the minus Ei over kT.

599
00:40:35 --> 00:40:46
Degeneracy of the
system energy Ei.

600
00:40:46 --> 00:41:03
Little g here is the degeneracy
of molecular energy Ei.

601
00:41:03 --> 00:41:05
Now, same form.
 

602
00:41:05 --> 00:41:08
Just an important
difference, though.

603
00:41:08 --> 00:41:12
This number, this little gi,
is typically a small number.

604
00:41:12 --> 00:41:16
It could be one, it
could be a few.

605
00:41:16 --> 00:41:18
This number is usually
astronomical.

606
00:41:18 --> 00:41:22
That's basically, that is,
what we calculated here.

607
00:41:22 --> 00:41:26
In other words, how many total
system states are there

608
00:41:26 --> 00:41:27
with a particular energy.
 

609
00:41:27 --> 00:41:31
Well, in many, many, many
cases the answer is some

610
00:41:31 --> 00:41:32
astronomical number.
 

611
00:41:32 --> 00:41:36
So this number might often
be between one and ten.

612
00:41:36 --> 00:41:38
This number might
be 10 to the 24th.

613
00:41:38 --> 00:41:43
It might be 10 to the
24th to a large power.

614
00:41:43 --> 00:41:46
And, of course, that has
a big effect on the way

615
00:41:46 --> 00:41:47
things end up working.
 

616
00:41:47 --> 00:41:51
In statistical mechanics
and in thermodynamics.

617
00:41:51 --> 00:41:55
A lot of thermodynamics
results are the way they are

618
00:41:55 --> 00:42:00
because you have so many
possible states with

619
00:42:00 --> 00:42:02
a particular energy.
 

620
00:42:02 --> 00:42:06
That that energy can be
strongly favored just by

621
00:42:06 --> 00:42:09
virtue of the number of
states that there are.

622
00:42:09 --> 00:42:11
Just the way, in a very small
way, this energy might be

623
00:42:11 --> 00:42:14
favored just because there are
more states with it then

624
00:42:14 --> 00:42:15
there are states here.
 

625
00:42:15 --> 00:42:17
But again, with the
system, it's not a

626
00:42:17 --> 00:42:18
factor of three to one.
 

627
00:42:18 --> 00:42:21
It might be a factor of
10 to the 24th to the

628
00:42:21 --> 00:42:23
power of something.
 

629
00:42:23 --> 00:42:28
It might be just enormously
larger than other possibilities

630
00:42:28 --> 00:42:40
that'll tend to put the
energy at a certain place.

631
00:42:40 --> 00:42:43
And just to continue, of
course, the same thing goes

632
00:42:43 --> 00:42:46
for the system probabilities.
 

633
00:42:46 --> 00:42:51
Pi, right, which is e to
the minus Ei over kT

634
00:42:51 --> 00:42:54
divided by capital Q.
 

635
00:42:54 --> 00:43:05
If this is a sum over states,
or omega i e to the minus Ei

636
00:43:05 --> 00:43:17
over kT over Q, let's write
this over, if now we're

637
00:43:17 --> 00:43:25
calculating the probability
of an energy level.

638
00:43:25 --> 00:43:27
And it's important to
make the distinction.

639
00:43:27 --> 00:43:31
Because in many cases,
that's what we care about.

640
00:43:31 --> 00:43:34
What's the energy
of the system?

641
00:43:34 --> 00:43:38
And we often don't care about
exactly where's that molecule

642
00:43:38 --> 00:43:40
and that one, and that
one, and that one, right?

643
00:43:40 --> 00:43:44
The individual states that
might be involved that would

644
00:43:44 --> 00:43:51
comprise that energy.
 

645
00:43:51 --> 00:43:57
Now, what I want to do is start
deriving thermodynamics.

646
00:43:57 --> 00:44:01
Like I promised, we're going
to be able to derive every

647
00:44:01 --> 00:44:03
thermodynamic quantity
if we just know the

648
00:44:03 --> 00:44:06
partition function.
 

649
00:44:06 --> 00:44:32
So now I just want to show that
that might really be true.

650
00:44:32 --> 00:44:39
So, the point is that from
Q we're going to get all

651
00:44:39 --> 00:44:44
of our thermodynamics.
 

652
00:44:44 --> 00:44:47
And let's start
with the energy.

653
00:44:47 --> 00:44:50
Remember u, right?
 

654
00:44:50 --> 00:44:54
That's our system energy.
 

655
00:44:54 --> 00:45:00
It's an average energy.
 

656
00:45:00 --> 00:45:04
It's an average of the energy
that we would get by looking

657
00:45:04 --> 00:45:09
at the states that the
system might be occupying.

658
00:45:09 --> 00:45:12
So we can write it
as a sum over i.

659
00:45:12 --> 00:45:17
Of Pi times Ei.
 

660
00:45:17 --> 00:45:20
In other words, it's going to
be determined by the energy

661
00:45:20 --> 00:45:23
of any system state times
the probability that the

662
00:45:23 --> 00:45:25
system is in that state.
 

663
00:45:25 --> 00:45:28
Add them all up.
 

664
00:45:28 --> 00:45:31
Well, we know what that is.
 

665
00:45:31 --> 00:45:45
It's the sum i of Ei, e to the
minus Ei over kT divided by Q.

666
00:45:45 --> 00:45:48
Now, I'm going to just make
a simple substitution.

667
00:45:48 --> 00:45:52
I'm going to use the term
beta to mean one over kT.

668
00:45:52 --> 00:45:54
I'm really just using that
so I don't need to use the

669
00:45:54 --> 00:45:57
chain rule a billion times
and doing derivatives.

670
00:45:57 --> 00:46:03
So I'm going to write this
is sum over i Ei e to the

671
00:46:03 --> 00:46:16
minus beta Ei over Q.
 

672
00:46:16 --> 00:46:20
So Q, then, in this term
is just a sum over i e

673
00:46:20 --> 00:46:23
to the minus beta Ei.
 

674
00:46:23 --> 00:46:24
So.
 

675
00:46:24 --> 00:46:25
Now I do want to take
some derivatives.

676
00:46:25 --> 00:46:37
If I take dQ / d beta, keeping
V and N constant, then I

677
00:46:37 --> 00:46:43
get derivative with
respect to beta.

678
00:46:43 --> 00:46:47
Sum over i e to the
minus beta Ei.

679
00:46:47 --> 00:46:52
So that's going to
bring out Ei, right?

680
00:46:52 --> 00:46:57
So it's going to bring out
minus Ei minus the sum over i,

681
00:46:57 --> 00:47:01
Ei e to the minus beta Ei.
 

682
00:47:01 --> 00:47:03
That obviously looks
like a handy thing.

683
00:47:03 --> 00:47:21
Because that looks
like that term.

684
00:47:21 --> 00:47:29
Then, our average energy,
remember, it's one over Q.

685
00:47:29 --> 00:47:34
Sum of i, Ei e to
the minus beta Ei.

686
00:47:34 --> 00:47:45
So now, it's just minus
one over q, dQ / d beta.

687
00:47:45 --> 00:47:53
So that's just the same as
minus d log Q / d beta.

688
00:47:53 --> 00:47:55
And now I'm going to
use the chain rule.

689
00:47:55 --> 00:48:11
So it's minus d log Q /
dT times dT / d beta.

690
00:48:11 --> 00:48:15
All at constant V and N.
 

691
00:48:15 --> 00:48:18
Now I can easily
get dT / d beta.

692
00:48:18 --> 00:48:25
Because d beta / dT is just
minus one over kT squared.

693
00:48:25 --> 00:48:30
It's just the derivative of one
over kT with respect to T.

694
00:48:30 --> 00:48:38
So finally, my average E, which
is u, is just kT squared times

695
00:48:38 --> 00:48:47
d log Q / dT constant V, N.
 

696
00:48:47 --> 00:48:51
That's terrific.
 

697
00:48:51 --> 00:48:55
In other words, if I have an
expression for Q, I know the

698
00:48:55 --> 00:48:58
partition function, and I
can calculate it at

699
00:48:58 --> 00:48:59
any temperature.
 

700
00:48:59 --> 00:49:02
I just need to take log of
it, take its derivative with

701
00:49:02 --> 00:49:04
respect to temperature.
 

702
00:49:04 --> 00:49:08
Multiply it by k T squared
and I've got my energy.

703
00:49:08 --> 00:49:16
Not very complicated, right?
 

704
00:49:16 --> 00:49:19
So in other words, macroscopic
thermodynamic properties come

705
00:49:19 --> 00:49:24
straight out of our microscopic
model of statistical mechanics.

706
00:49:24 --> 00:49:29
Statistical thermodynamics.
 

707
00:49:29 --> 00:49:31
Now I'm just going to state the
next result, just because I

708
00:49:31 --> 00:49:35
want to get there and it'll be
followed up more next time.

709
00:49:35 --> 00:49:36
But it's the following.
 

710
00:49:36 --> 00:49:42
You can see, of course, our Q
is a function of V and N and T.

711
00:49:42 --> 00:49:45
It's a function of V because in
principle the energies that

712
00:49:45 --> 00:49:50
are going into all this can
be a function of volume.

713
00:49:50 --> 00:49:53
What thermodynamic
function is naturally

714
00:49:53 --> 00:49:55
a function of N, V, T?
 

715
00:49:55 --> 00:49:58
Who remembers?
 

716
00:49:58 --> 00:49:59
Gibbs free energy?
 

717
00:49:59 --> 00:50:05
Helmholtz free energy?
 

718
00:50:05 --> 00:50:06
Enthalpy?
 

719
00:50:06 --> 00:50:09
Which one?
 

720
00:50:09 --> 00:50:13
Nobody knows.
 

721
00:50:13 --> 00:50:17
What's a function
of N, V, and T.

722
00:50:17 --> 00:50:20
Or, V and T, is what we
really formulate it as.

723
00:50:20 --> 00:50:22
N was introduced later.
 

724
00:50:22 --> 00:50:23
The Helmholtz free energy.
 

725
00:50:23 --> 00:50:26
Thank you.
 

726
00:50:26 --> 00:50:29
What that suggests is that
actually the simplest and most

727
00:50:29 --> 00:50:33
natural connection between Q
and macroscopic thermodynamics

728
00:50:33 --> 00:50:36
is to the Helmholtz
free energy.

729
00:50:36 --> 00:50:40
And the result that you'll see
derived next time is A is

730
00:50:40 --> 00:50:47
just minus kT log of Q.
 

731
00:50:47 --> 00:50:51
What a simple result.
 

732
00:50:51 --> 00:50:52
And you'll see the
derivations in your notes.

733
00:50:52 --> 00:50:55
You can see it's a
couple of lines, right?

734
00:50:55 --> 00:50:58
But of course, if you know
A, and you know E, you

735
00:50:58 --> 00:50:59
know everything, right?
 

736
00:50:59 --> 00:51:00
Because S is in there.
 

737
00:51:00 --> 00:51:06
And then other combinations
can give S and G and mu.

738
00:51:06 --> 00:51:07
And p.
 

739
00:51:07 --> 00:51:08
And anything else you want.
 

740
00:51:08 --> 00:51:11
So that's the point, is
that all of macroscopic

741
00:51:11 --> 00:51:12
thermodynamics follows.
 

742
00:51:12 --> 00:51:16
And you'll see that
elaborated more next time.

743
00:51:16 --> 00:51:20
And in addition, a very simple
natural expression for the

744
00:51:20 --> 00:51:24
entropy in terms of the states
available follows, that

745
00:51:24 --> 00:51:25
we've alluded to before.
 

746
00:51:25 --> 00:51:26
And now you'll see
it played out.

747
00:51:26 --> 00:51:28
Right.