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PROFESSOR: Alright.
00:00:21.660 --> 00:00:24.690
Well, we've been looking in the
last couple lectures at a
00:00:24.690 --> 00:00:27.690
really important topic
in thermodynamics.
00:00:27.690 --> 00:00:31.500
Which is, how do you tell
what's going to happen.
00:00:31.500 --> 00:00:33.800
Which way does a process
want to go?
00:00:33.800 --> 00:00:36.040
Which way will it go
spontaneously?
00:00:36.040 --> 00:00:38.470
And if it goes in one direction
or another, where
00:00:38.470 --> 00:00:39.370
does is lead?
00:00:39.370 --> 00:00:42.660
In other words, what is
the equilibrium state?
00:00:42.660 --> 00:00:44.660
And this is just an incredibly
important area that
00:00:44.660 --> 00:00:48.330
thermodynamics allows
us to speak to.
00:00:48.330 --> 00:00:52.180
So we started to see this.
00:00:52.180 --> 00:01:09.060
Sort of direction of
spontaneous change.
00:01:09.060 --> 00:01:10.860
And where the equilibrium
lies.
00:01:10.860 --> 00:01:18.520
So what we did is, remember we
started with the second law.
00:01:18.520 --> 00:01:20.200
Right?
00:01:20.200 --> 00:01:27.490
That dS is greater than
dq over T. And for the
00:01:27.490 --> 00:01:33.950
spontaneous change which happens
irreversibly That
00:01:33.950 --> 00:01:37.180
means that'll be dq
irreversible.
00:01:37.180 --> 00:01:39.690
It would be equal for
the reversible case.
00:01:39.690 --> 00:01:49.790
And we combine this with first
law, which for the case of
00:01:49.790 --> 00:01:57.610
pressure volume changes
we write as this.
00:01:57.610 --> 00:02:01.590
And so what this gave us was
a very, very useful general
00:02:01.590 --> 00:02:04.310
criterion for determining
whether something happened
00:02:04.310 --> 00:02:06.220
spontaneously.
00:02:06.220 --> 00:02:18.620
Namely, du plus p external dV
minus T for the surroundings
00:02:18.620 --> 00:02:26.780
dS, is greater than zero.
00:02:26.780 --> 00:02:31.950
Sorry.
00:02:31.950 --> 00:02:34.730
It's less than zero.
00:02:34.730 --> 00:02:43.360
And this is for any spontaneous
change.
00:02:43.360 --> 00:02:50.390
If it equals zero, then
we're at equilibrium.
00:02:50.390 --> 00:02:59.890
And if it's greater than
zero, then the process
00:02:59.890 --> 00:03:01.830
goes the other way.
00:03:01.830 --> 00:03:04.565
We would write the process in
the reverse to have it be less
00:03:04.565 --> 00:03:09.150
than 0 and it would
go spontaneously.
00:03:09.150 --> 00:03:15.720
And based on this one result,
we then looked under various
00:03:15.720 --> 00:03:19.720
constraints and said OK, what
about looking at our
00:03:19.720 --> 00:03:23.540
variables, volume, pressure,
temperature, other things,
00:03:23.540 --> 00:03:27.190
entropy, if we constrain those,
what's the condition
00:03:27.190 --> 00:03:28.660
for equilibrium?
00:03:28.660 --> 00:03:32.240
And that's what led us to a
number of results to determine
00:03:32.240 --> 00:03:35.980
what quantities we even
need to be looking at.
00:03:35.980 --> 00:03:37.540
To figure out equilibrium.
00:03:37.540 --> 00:03:38.730
And what the conditions were.
00:03:38.730 --> 00:03:41.890
And so what we discovered
were the following.
00:03:41.890 --> 00:03:49.670
This one, which we already
had seen, which is dS, is
00:03:49.670 --> 00:03:50.760
greater than zero.
00:03:50.760 --> 00:03:53.770
Change in entropy is
greater than zero,
00:03:53.770 --> 00:04:02.180
for an isolated system.
00:04:02.180 --> 00:04:08.130
We also saw that dS for constant
H and p was greater
00:04:08.130 --> 00:04:20.110
than zero. du, regular energy,
at constant entropy and volume
00:04:20.110 --> 00:04:41.370
is less then zero.
00:04:41.370 --> 00:04:43.300
And u is minimized
at equilibrium.
00:04:43.300 --> 00:04:46.500
And this is the familiar
result from ordinary
00:04:46.500 --> 00:04:48.560
mechanics, where you're not
worrying about something like
00:04:48.560 --> 00:04:51.420
entropy for a whole collection
of particles.
00:04:51.420 --> 00:04:54.850
That is, you minimize potential
energy and you see
00:04:54.850 --> 00:04:57.930
things falling under the force
of gravity and so forth, going
00:04:57.930 --> 00:05:05.560
to potential energy minima in
conformance with this result.
00:05:05.560 --> 00:05:10.970
dH, S and p is less than zero.
00:05:10.970 --> 00:05:16.140
So our H is u plus
pV, as you know.
00:05:16.140 --> 00:05:27.870
And H is minimized
at equilibrium.
00:05:27.870 --> 00:05:33.290
And this is, of course, with
constant S V. This is
00:05:33.290 --> 00:05:37.560
constant S and p.
00:05:37.560 --> 00:05:41.770
But of course, the need to have
entropy constrained is
00:05:41.770 --> 00:05:44.250
never going to be the most
convenient one experimental.
00:05:44.250 --> 00:05:47.040
There may be circumstances under
which it's the case, but
00:05:47.040 --> 00:05:49.740
it's often difficult
to control.
00:05:49.740 --> 00:05:53.100
On the other hand, temperature,
volume and
00:05:53.100 --> 00:05:56.480
pressure are variables that are
much easier in the lab to
00:05:56.480 --> 00:05:57.170
keep constant.
00:05:57.170 --> 00:05:58.530
To keep control over.
00:05:58.530 --> 00:06:02.160
And so that led us to the
definitions of other energy
00:06:02.160 --> 00:06:05.860
quantities, the Helmholtz
and Gibbs free energy.
00:06:05.860 --> 00:06:11.480
We discovered that the quantity
dA, under conditions
00:06:11.480 --> 00:06:15.770
of constant volume
and temperature,
00:06:15.770 --> 00:06:16.720
is less than zero.
00:06:16.720 --> 00:06:20.440
And A is u minus TS.
00:06:28.370 --> 00:06:34.830
And A is minimized at
equilibrium, under conditions
00:06:34.830 --> 00:06:43.210
of constant T and V. And
finally, and in many cases the
00:06:43.210 --> 00:06:45.670
most important of the results,
because of the conditions it
00:06:45.670 --> 00:06:52.540
applies to, we saw that this
Gibbs free energy is less than
00:06:52.540 --> 00:06:57.310
zero, that's our condition
for spontaneous change.
00:06:57.310 --> 00:07:01.850
Where the Gibbs free energy,
u plus pV minus
00:07:01.850 --> 00:07:05.550
TS is H minus TS.
00:07:08.380 --> 00:07:22.330
Also A plus pV and G is
minimized at equilibrium with
00:07:22.330 --> 00:07:26.090
constant temperature
and pressure.
00:07:26.090 --> 00:07:27.760
And that's why the Gibbs
free energy is just
00:07:27.760 --> 00:07:28.980
so enormously important.
00:07:28.980 --> 00:07:31.560
Because so much of what we do
in chemistry does take place
00:07:31.560 --> 00:07:34.290
with constant temperature
and pressure.
00:07:34.290 --> 00:07:37.880
So we have this condition
that's established in a
00:07:37.880 --> 00:07:40.390
quantity that we know
how to calculate.
00:07:40.390 --> 00:07:43.930
That tells us the direction
of spontaneous change for
00:07:43.930 --> 00:07:47.140
ordinary processes, chemical
processes, mixing and you name
00:07:47.140 --> 00:07:54.180
it, under conditions that are
easy to achieve in the lab.
00:07:54.180 --> 00:08:01.030
OK, now what we'd like to do is
be able to calculate any of
00:08:01.030 --> 00:08:04.940
these quantities in terms of
temperature, pressure, volume
00:08:04.940 --> 00:08:05.370
properties.
00:08:05.370 --> 00:08:08.370
That is, in terms of
equations of state.
00:08:08.370 --> 00:08:09.230
For any material.
00:08:09.230 --> 00:08:10.460
Then we would really be able to
00:08:10.460 --> 00:08:12.530
essentially calculate anything.
00:08:12.530 --> 00:08:14.250
Anything thermodynamic.
00:08:14.250 --> 00:08:15.990
About a material.
00:08:15.990 --> 00:08:18.760
Of course, that's assuming we
know the equation of state.
00:08:18.760 --> 00:08:20.640
We may or may not.
00:08:20.640 --> 00:08:25.190
But because in many cases we can
reasonably either model or
00:08:25.190 --> 00:08:28.470
measure equations of state,
collect data for a material
00:08:28.470 --> 00:08:32.590
for its temperature, pressure,
volume relations, then in fact
00:08:32.590 --> 00:08:35.290
if we can relate all these
quantities to those, then in
00:08:35.290 --> 00:08:37.460
fact we really can calculate
essentially all of the
00:08:37.460 --> 00:08:38.500
thermodynamics.
00:08:38.500 --> 00:08:42.830
For the material.
00:08:42.830 --> 00:08:58.450
So let's relate the
thermodynamic quantities to
00:08:58.450 --> 00:09:07.620
equation of state
p, V, T data.
00:09:07.620 --> 00:09:12.380
And we can do that by going
through and deriving what
00:09:12.380 --> 00:09:15.800
we'll call the fundamental
equations of thermodynamics.
00:09:15.800 --> 00:09:17.330
that'll provide these
relations.
00:09:17.330 --> 00:09:19.250
And at this point we know
enough to do this in a
00:09:19.250 --> 00:09:21.020
straightforward way.
00:09:21.020 --> 00:09:38.630
So if we start with a relation
for energy, T dS minus p dV.
00:09:38.630 --> 00:09:40.270
Where u is written
as a function
00:09:40.270 --> 00:09:43.870
of entropy and volume.
00:09:43.870 --> 00:09:47.790
And we've seen that that's
generally the case.
00:09:47.790 --> 00:09:59.450
It comes from the fact that dq
reversible is T dS, and dw
00:09:59.450 --> 00:10:04.760
reversible is minus p dV.
00:10:04.760 --> 00:10:08.310
And of course du is
the some of those.
00:10:08.310 --> 00:10:14.180
So, this is generally true.
00:10:14.180 --> 00:10:15.930
Since these are all
state functions.
00:10:15.930 --> 00:10:19.720
That is, this is derived in the
case for reversible paths.
00:10:19.720 --> 00:10:22.590
But since these are all simply
state functions and
00:10:22.590 --> 00:10:25.440
quantities, this is
generally true.
00:10:25.440 --> 00:10:28.710
Now we can use it to derive
differential relations for all
00:10:28.710 --> 00:10:29.960
of the thermodynamics
quantities.
00:10:29.960 --> 00:10:32.010
So let's just go through
and do that.
00:10:32.010 --> 00:10:38.590
So H is u plus pV.
00:10:41.180 --> 00:10:49.590
So dH is just du plus
p dV plus V dp.
00:10:49.590 --> 00:10:53.940
And now we're just going to
substitute du in here.
00:10:53.940 --> 00:10:58.380
And the p dV terms are
going to cancel.
00:10:58.380 --> 00:11:08.970
So we have the result that
dH is T dS plus V dp.
00:11:08.970 --> 00:11:11.570
Right?
00:11:11.570 --> 00:11:16.740
And that shows us that H is
written naturally as a
00:11:16.740 --> 00:11:23.580
function of entropy
and pressure.
00:11:23.580 --> 00:11:26.130
And now let's keep going.
00:11:26.130 --> 00:11:29.370
A is u minus TS.
00:11:31.880 --> 00:11:41.160
dA is du minus T
dS minus S dT.
00:11:41.160 --> 00:11:42.360
We're going to do
the same thing.
00:11:42.360 --> 00:11:45.660
Substitute this for du.
00:11:45.660 --> 00:11:51.570
This time, the T dS terms
are going to cancel.
00:11:51.570 --> 00:11:58.900
So we have dA is minus
S dT minus T dS.
00:12:02.570 --> 00:12:03.510
That can't be right.
00:12:03.510 --> 00:12:06.760
And it isn't.
00:12:06.760 --> 00:12:12.040
Minus S dT, that's the p dV term
that's left, minus p dV.
00:12:12.040 --> 00:12:15.520
And it shows us that A is
written naturally as a
00:12:15.520 --> 00:12:26.140
function of T and V. G,
we can write in any
00:12:26.140 --> 00:12:27.100
of a number of ways.
00:12:27.100 --> 00:12:31.230
Let's write it as H minus TS.
00:12:31.230 --> 00:12:40.720
So dG is dH minus
T dS minus S dT.
00:12:40.720 --> 00:12:43.000
Here's dH.
00:12:43.000 --> 00:12:46.420
We'll substitute that in,
and the T dS terms
00:12:46.420 --> 00:12:48.990
are going to cancel.
00:12:48.990 --> 00:12:56.150
So dG is minus S dT plus V dp.
00:12:59.840 --> 00:13:02.650
And this shows that G is
written naturally as a
00:13:02.650 --> 00:13:10.510
function of T and p.
00:13:10.510 --> 00:13:22.000
So these, which we will exalt
and celebrate by our
00:13:22.000 --> 00:13:30.210
sparingly-used colored chalk,
are our fundamental equations
00:13:30.210 --> 00:13:54.790
of thermodynamics.
00:13:54.790 --> 00:13:58.300
So what they do is, they're
describing how these
00:13:58.300 --> 00:14:03.500
thermodynamic properties change,
in terms of only state
00:14:03.500 --> 00:14:07.180
functions and state variables.
00:14:07.180 --> 00:14:08.520
Very, very useful.
00:14:08.520 --> 00:14:11.010
And that's what it means, when
we say well, it's natural
00:14:11.010 --> 00:14:14.180
then, to express say, G as a
function of T and p, that's
00:14:14.180 --> 00:14:14.830
what we're saying.
00:14:14.830 --> 00:14:16.870
Is that we can express
its changes in
00:14:16.870 --> 00:14:19.530
terms of these variables.
00:14:19.530 --> 00:14:23.300
Related only through
quantities that are
00:14:23.300 --> 00:14:25.740
functions of state.
00:14:25.740 --> 00:14:29.360
I don't need to know about
a specific path here.
00:14:29.360 --> 00:14:34.030
If I know about the states
involved, I just need to know
00:14:34.030 --> 00:14:40.940
what the volume was
in each of them.
00:14:40.940 --> 00:14:45.130
Now, before, of course, in the
first part of the class we
00:14:45.130 --> 00:14:49.470
started out looking at u and
then looking at H not as
00:14:49.470 --> 00:14:53.810
functions of S and V or S and
p, but as functions of
00:14:53.810 --> 00:14:55.640
temperature, mostly.
00:14:55.640 --> 00:14:58.900
In general, temperature and
volume or pressure.
00:14:58.900 --> 00:15:00.740
And it doesn't mean
that something was
00:15:00.740 --> 00:15:04.100
somehow wrong with that.
00:15:04.100 --> 00:15:07.960
It certainly is, it still is
going to be useful to do
00:15:07.960 --> 00:15:08.470
thermochemistry.
00:15:08.470 --> 00:15:11.890
To ask questions like how much
heat is released in a chemical
00:15:11.890 --> 00:15:15.640
reaction that takes place
at constant temperature.
00:15:15.640 --> 00:15:17.130
Not one of these variables.
00:15:17.130 --> 00:15:19.740
And we can calculate that.
00:15:19.740 --> 00:15:22.870
So it's not that we're somehow
throwing away our
00:15:22.870 --> 00:15:23.940
ability to do that.
00:15:23.940 --> 00:15:28.710
However, the thing to remember
is, when you look at heats of
00:15:28.710 --> 00:15:30.510
reaction under those
conditions it's
00:15:30.510 --> 00:15:32.440
all well and good.
00:15:32.440 --> 00:15:35.940
But it doesn't tell you, this
is the direction that the
00:15:35.940 --> 00:15:37.930
reaction is going to go.
00:15:37.930 --> 00:15:41.380
It doesn't tell you, this is the
equilibrium concentration
00:15:41.380 --> 00:15:43.950
that you'll end up with.
00:15:43.950 --> 00:15:46.950
That doesn't come out of what
we calculated before in
00:15:46.950 --> 00:15:47.650
thermochemistry.
00:15:47.650 --> 00:15:51.470
What does come out, which is
very useful is, if you do run
00:15:51.470 --> 00:15:54.360
the reaction, here's how
much heat comes out.
00:15:54.360 --> 00:15:57.950
And if you want to run a furnace
and provide energy,
00:15:57.950 --> 00:16:01.840
that's an extremely important
thing to be able to calculate.
00:16:01.840 --> 00:16:03.940
Because you're going to run it
and you'll probably find
00:16:03.940 --> 00:16:05.940
conditions under which you can
run it more or less to
00:16:05.940 --> 00:16:07.170
completion.
00:16:07.170 --> 00:16:10.100
But it doesn't tell you,
by itself, which
00:16:10.100 --> 00:16:14.060
direction things run in.
00:16:14.060 --> 00:16:16.710
Whereas under these conditions,
these quantities,
00:16:16.710 --> 00:16:19.610
if you look at free energy
change, for example, at
00:16:19.610 --> 00:16:23.180
constant temperature and
pressure, you can still
00:16:23.180 --> 00:16:25.180
calculate H. You can
still calculate
00:16:25.180 --> 00:16:26.320
the heat that's released.
00:16:26.320 --> 00:16:28.410
This is what will tell you
under some particular
00:16:28.410 --> 00:16:31.110
conditions what will
actually happen.
00:16:31.110 --> 00:16:33.040
Where will you end up.
00:16:33.040 --> 00:16:34.500
Very, very important, of
course, to be able to
00:16:34.500 --> 00:16:39.220
understand that.
00:16:39.220 --> 00:16:55.520
Now, it's also very useful to
look at some of the relations
00:16:55.520 --> 00:16:57.970
that come out of these
fundamental equations.
00:16:57.970 --> 00:16:59.990
And they're straightforward
to derive.
00:16:59.990 --> 00:17:04.200
So, all I want to do now is look
at the derivatives of the
00:17:04.200 --> 00:17:06.680
free energies with respect
to temperature
00:17:06.680 --> 00:17:10.070
and volume and pressure.
00:17:10.070 --> 00:17:14.830
So for example, if I look at A,
which we now have written
00:17:14.830 --> 00:17:18.800
as the function of T and V, of
course, in general I can
00:17:18.800 --> 00:17:23.550
always write dA as partial of
A, with respect to T at
00:17:23.550 --> 00:17:28.750
constant volume dT, plus partial
of A with respect to
00:17:28.750 --> 00:17:32.210
V, at constant temperature dV.
00:17:32.210 --> 00:17:35.330
And I know what those
turn out to be.
00:17:35.330 --> 00:17:41.320
It's minus S dT minus p dV.
00:17:41.320 --> 00:17:42.840
So what does that tell me?
00:17:42.840 --> 00:17:47.690
It tells me that the partial
of A with respect to T at
00:17:47.690 --> 00:17:52.260
constant V is minus S. Right?
00:17:52.260 --> 00:17:55.760
In other words, now I know how
to tell how the Helmholtz free
00:17:55.760 --> 00:17:59.300
energy changes as a function
of temperature.
00:17:59.300 --> 00:18:05.040
Or as a function of volume.
dA/dV, at constant T, must be
00:18:05.040 --> 00:18:08.230
negative p.
00:18:08.230 --> 00:18:12.770
Things that I can measure.
00:18:12.770 --> 00:18:15.150
So I can in a very
straightforward way say, OK,
00:18:15.150 --> 00:18:19.560
well, here is my Helmholtz
free energy.
00:18:19.560 --> 00:18:21.870
If I'm working under conditions
of constant
00:18:21.870 --> 00:18:23.370
temperature and volume,
that's very useful.
00:18:23.370 --> 00:18:26.130
Now, if I want to change those
quantities; change the
00:18:26.130 --> 00:18:28.580
temperature, change the volume,
how will it change?
00:18:28.580 --> 00:18:32.360
Well, I can, for any given case,
measure the pressure,
00:18:32.360 --> 00:18:35.010
determine the entropy and
I'll know what the slope
00:18:35.010 --> 00:18:38.680
of change will be.
00:18:38.680 --> 00:18:45.040
Similarly for G as a function of
temperature and pressure, I
00:18:45.040 --> 00:18:47.560
can go through the
same procedure.
00:18:47.560 --> 00:18:54.210
That is, it's easy to write down
straight away that dG,
00:18:54.210 --> 00:18:59.060
with respect to temperature at
constant pressure is minus S.
00:18:59.060 --> 00:19:03.090
That is, this is, dG/dT
at constant pressure.
00:19:03.090 --> 00:19:18.020
And this is dG/dp at constant
temperature.
00:19:18.020 --> 00:19:22.450
So again with the Gibbs free
energy, now I see how to
00:19:22.450 --> 00:19:25.680
determine, if I change the
pressure, if I change the
00:19:25.680 --> 00:19:29.200
temperature by some modest
amount, how much is the Gibbs
00:19:29.200 --> 00:19:30.300
free energy going to change?
00:19:30.300 --> 00:19:39.240
Well, it's easy to see.
00:19:39.240 --> 00:19:42.990
These two relations involving
entropy are also useful
00:19:42.990 --> 00:19:46.890
because they'll let us see
how entropy depends
00:19:46.890 --> 00:19:48.680
on volume and pressure.
00:19:48.680 --> 00:20:09.410
And let me show you
how that goes.
00:20:09.410 --> 00:20:11.930
Now, you've already seen
how entropy depends on
00:20:11.930 --> 00:20:22.890
temperature.
00:20:22.890 --> 00:20:30.630
We've already seen that, going
to write dS as dq reversible
00:20:30.630 --> 00:20:42.050
over T. And it's Cv dT over
T at constant volume.
00:20:42.050 --> 00:20:51.040
It's Cp dT over T at
constant pressure.
00:20:51.040 --> 00:20:56.500
So we know that dS/dT at
constant volume is Cv over T,
00:20:56.500 --> 00:21:04.430
and dS/dT at constant pressure
is Cp, over T. And we've seen
00:21:04.430 --> 00:21:08.180
that on a number of occasions.
00:21:08.180 --> 00:21:11.420
So that tells us what to do
to know the entropy as the
00:21:11.420 --> 00:21:13.280
temperature changes.
00:21:13.280 --> 00:21:19.360
But now, what happens if,
instead we look at what
00:21:19.360 --> 00:21:21.780
happens when we go to some state
one to some other state
00:21:21.780 --> 00:21:22.840
two and it's the pressure.
00:21:22.840 --> 00:21:24.420
Or the volume, that changes.
00:21:24.420 --> 00:21:27.930
And by the way, just to be
explicit about this, let's
00:21:27.930 --> 00:21:32.910
take this example, it means that
delta S, if we undergo a
00:21:32.910 --> 00:21:35.110
change from, say, T1 to T2.
00:21:35.110 --> 00:21:42.140
There's Cp over T dT.
00:21:42.140 --> 00:21:51.370
So it's Cp log of T2 over T1,
and we saw this before.
00:21:51.370 --> 00:21:56.260
So now, instead, let's
look at some process.
00:21:56.260 --> 00:21:59.890
State one goes to state two.
00:21:59.890 --> 00:22:06.190
Let's have constant T. And
look at what happens if
00:22:06.190 --> 00:22:09.730
pressure goes from p1 to p2.
00:22:09.730 --> 00:22:14.760
Or volume goes from V1 to V2.
00:22:14.760 --> 00:22:17.650
And see what happens there.
00:22:17.650 --> 00:22:20.030
We looked at pressure change
before, actually, in
00:22:20.030 --> 00:22:23.740
discussing the third law, the
fact that the entropy goes to
00:22:23.740 --> 00:22:27.105
zero as the absolute temperature
goes to zero for a
00:22:27.105 --> 00:22:28.350
pure, perfect crystal.
00:22:28.350 --> 00:22:31.320
But, actually, we didn't do
that in a general way.
00:22:31.320 --> 00:22:34.830
We just treated the one case
of an ideal gas as the
00:22:34.830 --> 00:22:36.850
temperature is reduced.
00:22:36.850 --> 00:22:39.890
But we can do this, generally,
by using what are called
00:22:39.890 --> 00:22:41.450
Maxwell relations.
00:22:41.450 --> 00:22:45.990
And all this is, is saying that
when you take a mixed
00:22:45.990 --> 00:22:50.470
second derivative, it doesn't
matter in which order you take
00:22:50.470 --> 00:22:54.980
the two derivatives.
00:22:54.980 --> 00:23:03.300
So, let's, we're going to
use this relationship.
00:23:03.300 --> 00:23:13.830
And we're going to
use these two.
00:23:13.830 --> 00:23:18.770
So, using those, now, what
happens if we take the second
00:23:18.770 --> 00:23:22.170
derivative of A, the mixed
derivative, partial with
00:23:22.170 --> 00:23:29.900
respect to T and the partial
with respect to V. So let's
00:23:29.900 --> 00:23:35.910
leave these off for a moment,
and now let's try that.
00:23:35.910 --> 00:23:41.110
And the point is that the second
derivative of A, with
00:23:41.110 --> 00:23:47.630
respect to V and T in this
order is the same as the
00:23:47.630 --> 00:23:50.760
second derivative of
a with respect to T
00:23:50.760 --> 00:23:52.160
and V in this order.
00:23:52.160 --> 00:23:56.340
It doesn't matter which order.
00:23:56.340 --> 00:23:58.840
But that turns out
to be useful.
00:23:58.840 --> 00:24:01.230
So let's do this explicitly.
00:24:01.230 --> 00:24:03.900
Which means we're going to
take the derivative with
00:24:03.900 --> 00:24:06.620
respect to volume of dA/dT.
00:24:09.680 --> 00:24:15.590
Now, the dA/dT isn't
constant volume.
00:24:15.590 --> 00:24:18.700
The derivative we're taking with
respect to volume, when
00:24:18.700 --> 00:24:23.570
we take that it's at constant
temperature.
00:24:23.570 --> 00:24:24.770
But what is it?
00:24:24.770 --> 00:24:30.360
Well, we already know what
dA/dT at constant V is.
00:24:30.360 --> 00:24:35.520
It's negative S. So this
is negative dS/dV.
00:24:38.440 --> 00:24:43.110
At constant temperature.
00:24:43.110 --> 00:24:46.150
Now let's take it in
the other order.
00:24:46.150 --> 00:24:57.410
So d/dT of dA/dV,
just like this.
00:24:57.410 --> 00:25:01.410
The dA/dV is calculated at
constant temperature.
00:25:01.410 --> 00:25:02.290
We know it.
00:25:02.290 --> 00:25:04.460
Then we can take the derivative
of that quantity,
00:25:04.460 --> 00:25:07.820
when we vary the temperature,
holding the volume constant.
00:25:07.820 --> 00:25:11.380
But again, dA/dV dT,
there it is.
00:25:11.380 --> 00:25:14.300
It's negative p.
00:25:14.300 --> 00:25:24.100
So this is just negative dp/dT
at constant volume.
00:25:24.100 --> 00:25:31.850
These things have to be
equal to each other.
00:25:31.850 --> 00:25:33.610
Because these mixed
second derivatives
00:25:33.610 --> 00:25:36.530
are the same thing.
00:25:36.530 --> 00:25:37.640
But that's very useful.
00:25:37.640 --> 00:25:42.570
Because this is what comes
directly out of an equation of
00:25:42.570 --> 00:25:43.930
state, right?
00:25:43.930 --> 00:25:46.660
You know how pressure changes
with temperature at constant
00:25:46.660 --> 00:25:48.490
volume if you know the
equation of state.
00:25:48.490 --> 00:25:50.280
It relates the pressure,
volume,
00:25:50.280 --> 00:26:03.690
and temperature together.
00:26:03.690 --> 00:26:06.600
So from measured equation of
state data, or from a model
00:26:06.600 --> 00:26:09.380
like the ideal gas or the van
der Waal's gas or another
00:26:09.380 --> 00:26:11.910
equation of state,
you know this.
00:26:11.910 --> 00:26:14.620
Can determine how entropy
is going to behave
00:26:14.620 --> 00:26:31.510
as the volume changes.
00:26:31.510 --> 00:26:42.070
If we try that for an ideal
gas, pV is nRT.
00:26:42.070 --> 00:26:54.690
So dp/dT at constant volume,
it's just nR over V. And that,
00:26:54.690 --> 00:27:04.725
now, we know must equal dS/dV,
with a positive sign.
00:27:04.725 --> 00:27:08.280
At constant temperature.
00:27:08.280 --> 00:27:11.710
So now let's try looking
at something where
00:27:11.710 --> 00:27:13.720
are V1 goes to V2.
00:27:13.720 --> 00:27:17.790
The volume is going to change,
and we can see
00:27:17.790 --> 00:27:20.310
how the entropy changes.
00:27:20.310 --> 00:27:29.500
So, if one goes to two and V1
goes to V2, and it's constant
00:27:29.500 --> 00:27:33.770
temperature, just what we've
specified there.
00:27:33.770 --> 00:27:48.930
Delta S is S(T, V2) minus S(T,
V1), T's staying the same.
00:27:48.930 --> 00:27:57.900
So it's just the integral from
V1 to V2 of dS/dV At constant
00:27:57.900 --> 00:27:59.810
temperature dV.
00:27:59.810 --> 00:28:00.900
And now we know what that is.
00:28:00.900 --> 00:28:11.380
So it's nR integral from V1 to
V2 dV over V. So it's nR log
00:28:11.380 --> 00:28:16.690
V2 over V1.
00:28:16.690 --> 00:28:22.990
There's our delta S. So we
know how to calculate it.
00:28:22.990 --> 00:28:30.440
Make sense?
00:28:30.440 --> 00:28:33.890
Now, we can do the same
procedure for
00:28:33.890 --> 00:28:38.300
the pressure change.
00:28:38.300 --> 00:28:41.840
And all we do is, I'll just
outline this, I think.
00:28:41.840 --> 00:28:45.780
I won't write it all
on the board.
00:28:45.780 --> 00:28:48.970
But, of course, it's going to
come from the fact that these
00:28:48.970 --> 00:28:52.590
second derivatives
are also equal.
00:28:52.590 --> 00:29:05.530
So d squared G dT dp is equal
to d squared G dp dT.
00:29:05.530 --> 00:29:08.560
In other words, the order of
taking the derivatives with
00:29:08.560 --> 00:29:13.150
respect to pressure and
temperature doesn't matter.
00:29:13.150 --> 00:29:20.590
And what this will show is
that dS/dp at constant
00:29:20.590 --> 00:29:23.590
temperature, here we saw how
entropy varies with volume,
00:29:23.590 --> 00:29:27.210
this is going to show us how
it varies with pressure.
00:29:27.210 --> 00:29:34.470
Is equal to minus dV/dT
at constant pressure.
00:29:34.470 --> 00:29:37.905
And again, this is something
that comes from
00:29:37.905 --> 00:29:40.750
an equation of state.
00:29:40.750 --> 00:29:44.940
We know how the volume and
temperature vary with respect
00:29:44.940 --> 00:29:46.400
to each other at constant
pressure.
00:29:46.400 --> 00:29:53.030
That's what the equation
of state tells us.
00:29:53.030 --> 00:29:57.050
So, again, I can just
use that result.
00:29:57.050 --> 00:30:03.260
So, if we do a process where
one goes to two at constant
00:30:03.260 --> 00:30:11.590
temperature, and now the
pressure, p1, goes to p2, well
00:30:11.590 --> 00:30:19.750
then delta S is just the
integral from p1 to p2 of
00:30:19.750 --> 00:30:26.910
dS/dp times dS, so
it's just this.
00:30:26.910 --> 00:30:30.680
And so of course it's
still pV equals nRT.
00:30:30.680 --> 00:30:38.680
So now we just have
nR over p dp.
00:30:38.680 --> 00:30:39.890
Right?
00:30:39.890 --> 00:30:41.930
So we're going to see
the same story.
00:30:41.930 --> 00:30:49.590
It's nR log of p2 over p1 for
the process where there's a
00:30:49.590 --> 00:31:01.180
pressure change.
00:31:01.180 --> 00:31:04.030
Any questions about this part?
00:31:04.030 --> 00:31:07.140
So what we've done is take
one step further.
00:31:07.140 --> 00:31:16.020
We've used the fundamental
equations that are hiding down
00:31:16.020 --> 00:31:19.670
here, out of sight but
never out of mind.
00:31:19.670 --> 00:31:23.390
And what we've done is look at
the derivatives of the new
00:31:23.390 --> 00:31:28.740
free energies that we've just
recently introduced, A and G.
00:31:28.740 --> 00:31:33.330
And then, the only thing we've
done beyond that is say, OK,
00:31:33.330 --> 00:31:35.940
well now let's just take the
mixed second derivatives, they
00:31:35.940 --> 00:31:37.990
have to be equal
to each other.
00:31:37.990 --> 00:31:42.660
And what's fallen out when we do
that, because in each case,
00:31:42.660 --> 00:31:46.370
one of the first derivatives
gives us the entropy.
00:31:46.370 --> 00:31:48.630
Then the second derivative gives
the change in entropy
00:31:48.630 --> 00:31:51.500
with respect to the variable
that we're differentiating,
00:31:51.500 --> 00:31:54.790
with respect to which is either
pressure or volume.
00:31:54.790 --> 00:31:59.520
And the useful outcome of all
that is that we get to see how
00:31:59.520 --> 00:32:03.910
entropy changes with one of
those variables in terms of
00:32:03.910 --> 00:32:09.200
only V, T, and p, which come out
of some equation of state.
00:32:09.200 --> 00:32:11.660
And all we did, further, is take
that second derivative.
00:32:11.660 --> 00:32:13.060
That mixed second derivative.
00:32:13.060 --> 00:32:16.340
And, of course, see that either
way we do that we'll
00:32:16.340 --> 00:32:24.600
have an equality.
00:32:24.600 --> 00:32:34.840
Now, let's go back to our older
friends u and H. Which
00:32:34.840 --> 00:32:46.590
we've expressed now in terms
of S and V, S and p.
00:32:46.590 --> 00:32:52.500
So, so far we don't have a way
to just write off, relate them
00:32:52.500 --> 00:32:53.940
to equation of state data.
00:32:53.940 --> 00:32:57.860
Which also would
be very useful.
00:32:57.860 --> 00:33:02.110
Here, A and G, we've already
got as functions of these
00:33:02.110 --> 00:33:03.710
easily controlled, conveniently
00:33:03.710 --> 00:33:05.560
controlled state variables.
00:33:05.560 --> 00:33:13.450
Let's look at those quantities.
u and H. And look
00:33:13.450 --> 00:33:17.270
at, for example, the
V dependence of u.
00:33:17.270 --> 00:33:35.780
The volume dependence.
00:33:35.780 --> 00:33:44.810
And in particular let's look
at, for example, du/dV at
00:33:44.810 --> 00:33:46.690
constant temperature.
00:33:46.690 --> 00:33:53.540
Now, we can immediately
see what du/dV at
00:33:53.540 --> 00:33:57.210
constant entropy is.
00:33:57.210 --> 00:33:59.270
Experimentally, though, that's
not such an easy
00:33:59.270 --> 00:34:02.330
situation to arrange.
00:34:02.330 --> 00:34:07.440
Of course, this is a much
more practical one.
00:34:07.440 --> 00:34:10.570
But it doesn't just fall out
immediately from the one
00:34:10.570 --> 00:34:14.980
fundamental equation for du.
00:34:14.980 --> 00:34:16.810
But we can start there.
00:34:16.810 --> 00:34:24.190
So, du is T dS minus p dV.
00:34:24.190 --> 00:34:33.560
And I can take this derivative.
du/dV at constant
00:34:33.560 --> 00:34:35.380
T. And so, what is it?
00:34:35.380 --> 00:34:42.650
Well, it's not just p because
there's some dS/dV at constant
00:34:42.650 --> 00:34:44.270
T. This isn't zero.
00:34:44.270 --> 00:34:50.370
There's some variation, dS/dV,
at constant temperature.
00:34:50.370 --> 00:34:53.430
That's going to matter.
00:34:53.430 --> 00:35:00.050
This part, of course,
is just minus p.
00:35:00.050 --> 00:35:15.160
But we just figured out what
dS/dV at constant T is.
00:35:15.160 --> 00:35:20.310
This is dp/dT at constant
V. So that's neat.
00:35:20.310 --> 00:35:25.230
So in other words, we can write
this as T, dp/dT at
00:35:25.230 --> 00:35:29.420
constant V, minus p.
00:35:29.420 --> 00:35:35.970
Let's just check T, p, T, V, p.
00:35:35.970 --> 00:35:37.070
Right?
00:35:37.070 --> 00:35:38.920
In other words, we just
have pressure,
00:35:38.920 --> 00:35:40.420
temperature and volume.
00:35:40.420 --> 00:35:42.820
Again, if we know the equation
of state, we
00:35:42.820 --> 00:35:45.390
know all this stuff.
00:35:45.390 --> 00:35:47.670
So again, we can measure
equation of state data.
00:35:47.670 --> 00:35:50.940
Or, if we know the equation of
state from a model, ideal gas,
00:35:50.940 --> 00:35:56.260
van der Waal's gas, whatever,
now we can determine u.
00:35:56.260 --> 00:35:58.810
From equation of state data.
00:35:58.810 --> 00:36:17.960
Terrific, right?
00:36:17.960 --> 00:36:22.000
So let's take our one model that
we keep going back to.
00:36:22.000 --> 00:36:25.520
Equation of state, and just
see how it works.
00:36:25.520 --> 00:36:28.110
That is, ideal gas.
00:36:28.110 --> 00:36:30.790
And see how it works
with that.
00:36:30.790 --> 00:36:36.070
Now, we saw before, or really
I should say we accepted
00:36:36.070 --> 00:36:39.770
before, that for an ideal
gas, u was a function of
00:36:39.770 --> 00:36:40.950
temperature only.
00:36:40.950 --> 00:36:43.135
Well, now let's try it.
00:36:43.135 --> 00:36:50.640
So, dp/dT, for our ideal gas,
at constant volume,
00:36:50.640 --> 00:36:54.680
remember pV is nRT.
00:36:54.680 --> 00:37:01.460
So this nR over V. And then,
using the relation again, we
00:37:01.460 --> 00:37:06.380
can just write this as p over
T. In other words, we're
00:37:06.380 --> 00:37:07.810
taking advantage of
the fact that we
00:37:07.810 --> 00:37:10.420
now know that quantity.
00:37:10.420 --> 00:37:12.110
In the case of the ideal
gas, we just have a
00:37:12.110 --> 00:37:13.180
simple model for it.
00:37:13.180 --> 00:37:14.610
More generally, we
could measure it.
00:37:14.610 --> 00:37:16.940
We could just collect
a bunch of data.
00:37:16.940 --> 00:37:17.640
For a material.
00:37:17.640 --> 00:37:22.040
What's the volume it occupies
at some pressure and
00:37:22.040 --> 00:37:23.080
temperature?
00:37:23.080 --> 00:37:25.190
Now let's change the pressure
and temperature and sweep
00:37:25.190 --> 00:37:27.030
through a whole range of
pressures and temperatures and
00:37:27.030 --> 00:37:28.590
measure the volume in
every one of them.
00:37:28.590 --> 00:37:30.680
Well, then, we could
just use that for
00:37:30.680 --> 00:37:31.210
our equation of state.
00:37:31.210 --> 00:37:33.710
One way or another, we can
determine this quantity.
00:37:33.710 --> 00:37:35.780
For the ideal gas it's this.
00:37:35.780 --> 00:37:44.970
So now our du/dV, at constant T
is just T times dp/dT, which
00:37:44.970 --> 00:37:52.510
is just p over T minus
p, it's zero.
00:37:52.510 --> 00:37:55.560
Remember the Joule expansion.
00:37:55.560 --> 00:37:58.880
And we saw that, you saw that
the Joule coefficient for an
00:37:58.880 --> 00:38:01.390
ideal gas was zero.
00:38:01.390 --> 00:38:06.160
So that you could see that for
the ideal gas, u would not be
00:38:06.160 --> 00:38:08.860
a function of volume, but
only of temperature.
00:38:08.860 --> 00:38:13.350
But actually, when you saw that
before, you weren't given
00:38:13.350 --> 00:38:15.220
any proof of that.
00:38:15.220 --> 00:38:19.160
It was just that when the
good Mr. Joule made the
00:38:19.160 --> 00:38:23.450
measurements, to the precision
that he could measure, he
00:38:23.450 --> 00:38:26.680
discovered that for some gases
it was extremely small.
00:38:26.680 --> 00:38:29.920
At least, smaller than anything
he could detect.
00:38:29.920 --> 00:38:33.280
So it sure seemed like it was
going to zero, under ideal gas
00:38:33.280 --> 00:38:33.800
conditions.
00:38:33.800 --> 00:38:36.260
And that was the result that
we came to accept.
00:38:36.260 --> 00:38:38.340
Here, though, you can just
derive straight away.
00:38:38.340 --> 00:38:41.160
That for an ideal gas it has to
be the case that there's no
00:38:41.160 --> 00:38:43.060
volume dependence
of the energy.
00:38:43.060 --> 00:38:53.460
Only a temperature dependence.
00:38:53.460 --> 00:39:23.690
It's the same for H. Just like
u, we'd like to be able to
00:39:23.690 --> 00:39:27.190
express it in a way that allows
us to calculate what
00:39:27.190 --> 00:39:30.820
happens only from equation
of state data.
00:39:30.820 --> 00:39:36.880
But, again, our fundamental
equations show us how it
00:39:36.880 --> 00:39:45.690
changes as a function of
entropy and pressure.
00:39:45.690 --> 00:39:53.920
So, dH is T dS plus V dp.
00:39:53.920 --> 00:39:55.900
So let's look at dH/dp.
00:39:58.930 --> 00:40:00.470
We know how to get it
immediately if we
00:40:00.470 --> 00:40:01.640
keep entropy constant.
00:40:01.640 --> 00:40:04.350
But we'd like to relate it to
what happens if we keep the
00:40:04.350 --> 00:40:06.950
temperature constant.
00:40:06.950 --> 00:40:09.890
So then, just like we saw,
analogous to what saw just
00:40:09.890 --> 00:40:19.380
before, it's T dS/dp at constant
T. Plus V. But now
00:40:19.380 --> 00:40:24.830
we've seen from the Maxwell
relations that dS/dp is minus
00:40:24.830 --> 00:40:26.860
dV/dT, for constant p.
00:40:26.860 --> 00:40:29.530
Again, this is this quantity,
one of these quantities that
00:40:29.530 --> 00:40:31.850
again we can determine from
equation of state data.
00:40:31.850 --> 00:40:35.180
Only V, p and T appear.
00:40:35.180 --> 00:40:45.170
So it's minus T dV/dT at
constant p, plus V. And so,
00:40:45.170 --> 00:41:03.120
again, this can come from
equation of state data.
00:41:03.120 --> 00:41:09.550
And if you do this again for
an ideal gas, let me see.
00:41:09.550 --> 00:41:11.810
So we have pV is nRT.
00:41:11.810 --> 00:41:24.100
So dV/dT at constant pressure
is just nR over p.
00:41:24.100 --> 00:41:26.810
But we can plug that in again
just like we did before.
00:41:26.810 --> 00:41:36.480
It's just equal to V over T.
And so dH/dp under our
00:41:36.480 --> 00:41:43.170
condition of constant
temperature is just minus T
00:41:43.170 --> 00:41:47.220
times V over T plus
V, everything
00:41:47.220 --> 00:41:50.820
cancels, and that's zero.
00:41:50.820 --> 00:41:54.430
That's our Joule - Thompson
expansion.
00:41:54.430 --> 00:41:56.790
That was a constant
enthalpy change.
00:41:56.790 --> 00:42:01.700
And again there, too, you saw
an experimental result you
00:42:01.700 --> 00:42:04.060
were presented with that says,
well at least to the extent
00:42:04.060 --> 00:42:05.410
that it could be measured,
it was obviously
00:42:05.410 --> 00:42:06.950
getting very small.
00:42:06.950 --> 00:42:11.530
For gases that approach
ideal gas conditions.
00:42:11.530 --> 00:42:13.910
Well, there you can see it.
00:42:13.910 --> 00:42:16.020
Sure better have gotten
small because in fact
00:42:16.020 --> 00:42:20.790
it has to be zero.
00:42:20.790 --> 00:42:27.550
Now let's take just
one somewhat
00:42:27.550 --> 00:42:30.490
more complicated case.
00:42:30.490 --> 00:42:37.290
Let's look at a van
der Waal's gas.
00:42:37.290 --> 00:42:40.710
Let's try it with a different
equation of state, that isn't
00:42:40.710 --> 00:42:47.230
quite as simple as the
ideal gas case.
00:42:47.230 --> 00:42:59.280
So, then p plus a over molar
volume squared times V minus b
00:42:59.280 --> 00:43:07.460
molar volume V minus b is
equal to RT, remember?
00:43:07.460 --> 00:43:10.340
This was back from the
first or second
00:43:10.340 --> 00:43:14.350
lecture in the course.
00:43:14.350 --> 00:43:18.970
So, we can separate out p.
00:43:18.970 --> 00:43:25.730
It's RT over molar volume minus
b minus a over molar
00:43:25.730 --> 00:43:31.670
volume V squared.
00:43:31.670 --> 00:43:33.440
And then we can take the
derivative with respect to
00:43:33.440 --> 00:43:38.190
temperature, it's just R over
molar volume minus b.
00:43:38.190 --> 00:43:53.870
So it's dp/dT at constant V is
just R over V bar minus b.
00:43:53.870 --> 00:44:04.190
Well, let's now look, given
this, let's now look in that
00:44:04.190 --> 00:44:09.100
case, at what happens to u as a
function of V. For the ideal
00:44:09.100 --> 00:44:12.860
gas, we know that u is
volume independent.
00:44:12.860 --> 00:44:16.270
It only depends on
the temperature.
00:44:16.270 --> 00:44:20.650
But for the van der
Waal's gas, now
00:44:20.650 --> 00:44:22.080
it's going to be different.
00:44:22.080 --> 00:44:27.120
And that's because this is
different from what it is in
00:44:27.120 --> 00:44:28.760
the ideal gas case.
00:44:28.760 --> 00:44:49.770
Namely, now du/dV at constant T,
for the van der Waal's gas.
00:44:49.770 --> 00:44:57.010
So it's this.
00:44:57.010 --> 00:45:04.550
So it's RT over molar
volume minus b.
00:45:04.550 --> 00:45:09.750
Minus p, right?
00:45:09.750 --> 00:45:13.130
But in fact, if you go back to
the van der Waal's equation of
00:45:13.130 --> 00:45:17.690
state, here's RT
over v minus b.
00:45:17.690 --> 00:45:19.220
If we put it as minus
b, that's just
00:45:19.220 --> 00:45:22.100
equal to a over V squared.
00:45:22.100 --> 00:45:28.610
Equals a over molar
volume squared.
00:45:28.610 --> 00:45:33.220
But the point is, the main
point is, it's not zero.
00:45:33.220 --> 00:45:37.110
It's some number. a over the
molar volume squared. a is a
00:45:37.110 --> 00:45:40.220
positive number in the van der
Waal's equation of state.
00:45:40.220 --> 00:45:44.050
So this is greater than zero.
00:45:44.050 --> 00:45:52.990
In other words, u is a function
of T and V. If we
00:45:52.990 --> 00:45:59.150
don't have an ideal gas.
00:45:59.150 --> 00:46:01.260
By the way, just to think about
it a little bit, it's a
00:46:01.260 --> 00:46:02.620
positive number.
00:46:02.620 --> 00:46:04.490
What that means is, I've got my
00:46:04.490 --> 00:46:08.720
ideal gas in some container.
00:46:08.720 --> 00:46:12.490
There's some energy, some
internal energy.
00:46:12.490 --> 00:46:13.760
Now I make the volume bigger.
00:46:13.760 --> 00:46:17.630
I allow it to expand.
00:46:17.630 --> 00:46:22.790
And the energy changes,
it goes up.
00:46:22.790 --> 00:46:25.840
In some sense it's less
favorable energetically.
00:46:25.840 --> 00:46:31.310
What's happening there, that a
term in the van der Waal's
00:46:31.310 --> 00:46:35.730
equation of state, is describing
interactions,
00:46:35.730 --> 00:46:38.660
favorable attractions, between
gas molecules.
00:46:38.660 --> 00:46:42.290
The b is describing
repulsions.
00:46:42.290 --> 00:46:44.290
Effectively, the
volume changes.
00:46:44.290 --> 00:46:47.300
The molar volume is being
changed a little bit by the
00:46:47.300 --> 00:46:50.160
fact that if you're really
trying to make things collide
00:46:50.160 --> 00:46:54.110
with each other, they can't
occupy the same volume.
00:46:54.110 --> 00:46:56.270
And that is being
expressed here.
00:46:56.270 --> 00:47:02.800
The a, though is expressing
attraction between molecules
00:47:02.800 --> 00:47:04.740
at somewhat longer range.
00:47:04.740 --> 00:47:07.380
So now, if you make the
volume bigger, those
00:47:07.380 --> 00:47:08.670
attractions die out.
00:47:08.670 --> 00:47:11.500
Because the molecules are
farther apart from each other.
00:47:11.500 --> 00:47:13.520
So the energy goes up.
00:47:13.520 --> 00:47:17.970
Now, you might ask, well why
does it do that, right?
00:47:17.970 --> 00:47:22.160
I mean, if the energy is lower
to occupy a smaller volume,
00:47:22.160 --> 00:47:26.290
then if I have this room and a
bunch of molecules of oxygen,
00:47:26.290 --> 00:47:29.120
and nitrogen and what have you
in the air, and there are weak
00:47:29.120 --> 00:47:31.490
attractions between them, why
don't they all just sort of
00:47:31.490 --> 00:47:34.890
glum together and find whatever
volume they like.
00:47:34.890 --> 00:47:37.720
So that the attractive
forces can exert
00:47:37.720 --> 00:47:38.580
themselves a little bit.
00:47:38.580 --> 00:47:39.980
Not too close, right?
00:47:39.980 --> 00:47:42.530
Not so close that the
repulsions dominate.
00:47:42.530 --> 00:47:44.630
Why don't they do that?
00:47:44.630 --> 00:47:54.790
What else matters besides any
of those considerations?
00:47:54.790 --> 00:47:58.720
What else matters that I haven't
considered in this
00:47:58.720 --> 00:48:02.370
little discussion?
00:48:02.370 --> 00:48:04.350
Yeah.
00:48:04.350 --> 00:48:06.750
What about entropy, right?
00:48:06.750 --> 00:48:08.710
If I only worry about
minimizing the
00:48:08.710 --> 00:48:10.900
energy, it's true.
00:48:10.900 --> 00:48:13.660
They'll stick together
a little bit.
00:48:13.660 --> 00:48:15.540
But entropy also matters.
00:48:15.540 --> 00:48:18.680
And there's disorder achieved
by occupying the full
00:48:18.680 --> 00:48:19.930
available volume.
00:48:19.930 --> 00:48:21.990
Many more states possible.
00:48:21.990 --> 00:48:24.950
And that will end up winning out
at basically any realistic
00:48:24.950 --> 00:48:28.040
temperature where the stuff
really is a gas.
00:48:28.040 --> 00:48:29.000
OK.
00:48:29.000 --> 00:48:32.540
Next time, what you're going
to see is the following.
00:48:32.540 --> 00:48:36.120
It turns out we can express all
these functions in terms
00:48:36.120 --> 00:48:39.730
of G, we wouldn't need to choose
G, but it's a very
00:48:39.730 --> 00:48:40.820
useful function to choose.
00:48:40.820 --> 00:48:44.280
Because of its natural
expression in
00:48:44.280 --> 00:48:46.630
terms of T and p.
00:48:46.630 --> 00:48:49.690
So we can write any of these
functions in terms of G. Which
00:48:49.690 --> 00:48:51.970
means that we can really
calculate all the
00:48:51.970 --> 00:48:55.350
thermodynamics in
terms of only g.
00:48:55.350 --> 00:48:58.080
It's not necessary to do that,
but it can be quite
00:48:58.080 --> 00:48:59.450
convenient.
00:48:59.450 --> 00:49:03.190
And then we can say, OK, if we
have many constituents, what
00:49:03.190 --> 00:49:04.800
if we have a mixture of stuff?
00:49:04.800 --> 00:49:08.590
We can take the derivative of
G with respect to how much
00:49:08.590 --> 00:49:10.510
material there is.
00:49:10.510 --> 00:49:13.010
With respect to n, the
number of moles.
00:49:13.010 --> 00:49:15.420
And if there are one, and two,
and three constituents with
00:49:15.420 --> 00:49:18.280
respect to n1, and n2, and n3.
00:49:18.280 --> 00:49:21.160
Each individual amount
of stuff.
00:49:21.160 --> 00:49:23.750
What that's going to allow us to
do is, if we say, OK I have
00:49:23.750 --> 00:49:26.560
a mixture of stuff, how does
the free energy change?
00:49:26.560 --> 00:49:29.020
If I change the composition
of the mixture?
00:49:29.020 --> 00:49:32.320
If I take something away, or
put something else in?
00:49:32.320 --> 00:49:34.440
And we'll be able to determine
equilibrium under those
00:49:34.440 --> 00:49:35.140
conditions.
00:49:35.140 --> 00:49:38.730
That's very useful for things
like chemical reactions, where
00:49:38.730 --> 00:49:41.380
this constituent changes
to this one.
00:49:41.380 --> 00:49:43.530
And I can calculate what happens
to G under those
00:49:43.530 --> 00:49:44.360
conditions.
00:49:44.360 --> 00:49:46.390
And that's what you'll see
starting next time.
00:49:46.390 --> 00:49:47.970
And professor Blendi
will be taking over
00:49:47.970 --> 00:49:49.860
for that set of lectures.