WEBVTT

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Okay, what I did last time was
describe to you a certain heat

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engine called the Carnot engine.

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So, let me refresh your memory
on what that was.

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The whole engine consisted of a
gas at some pressure and some

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volume.
And the gas was subject to a

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cycle, and the cycle went like
this.

00:29.500 --> 00:34.500
You start at some point
A, you expand at a

00:34.496 --> 00:38.656
certain temperature
T_1,

00:38.660 --> 00:42.410
isothermally,
until you come to point

00:42.408 --> 00:45.898
B.
So, the way you should imagine

00:45.899 --> 00:49.949
that is that this gas is placed
on a reservoir whose temperature

00:49.950 --> 00:53.230
is frozen, I mean stuck at
T_1.

00:53.230 --> 00:55.600
Reservoirs never change the
temperature.

00:55.600 --> 00:59.600
And what you do is you lift one
grain after another of sand,

00:59.601 --> 01:03.671
so you let the gas expand,
but you don't let it cool down.

01:03.670 --> 01:07.300
The minute it tries to cool
some heat will go from below to

01:07.302 --> 01:11.002
maintain the temperature and
start at that volume and end at

01:10.997 --> 01:13.717
that volume.
During this process,

01:13.719 --> 01:18.019
some amount of heat
Q_1 enters the

01:18.018 --> 01:20.148
gas.
Next thing you do,

01:20.154 --> 01:25.104
you let the gas expand some
more, but now you let it expand

01:25.100 --> 01:28.970
adiabatically.
Adiabatic means you put the

01:28.971 --> 01:34.531
whole thing in a heat insulating
material, so no heat can go in

01:34.533 --> 01:37.053
or out.
Now, you can see if it expands,

01:37.047 --> 01:40.227
it's going to have to pay for
it with its own internal energy,

01:40.230 --> 01:41.900
so the temperature will drop.

01:41.900 --> 01:45.320
So, you wait till it drops to a
lower temperature

01:45.324 --> 01:48.684
T_2,
then you go backwards like

01:48.677 --> 01:51.117
this;
that's when you start

01:51.115 --> 01:55.985
compressing the gas by putting
some grains of sand back,

01:55.989 --> 02:00.099
but this time the gas is kept
in a lower temperature on top of

02:00.096 --> 02:03.256
a temperature reservoir
T_2,

02:03.260 --> 02:09.640
and you drop more sand on it to
compress it, that is this part.

02:09.639 --> 02:13.229
During this portion,
heat will actually be rejected

02:13.227 --> 02:15.767
by the gas.
Why is heat rejected?

02:15.770 --> 02:18.010
Because normally,
when you compress a gas,

02:18.010 --> 02:20.630
you're doing work on it,
the energy should go up,

02:20.633 --> 02:23.533
the energy goes up,
the temperature should go up;

02:23.530 --> 02:25.100
but you're not letting the
temperature go up,

02:25.104 --> 02:26.544
you're forcing it to have the
temperature

02:26.535 --> 02:27.425
T_2.

02:27.430 --> 02:30.880
So, it will reject heat into
this heat bath.

02:30.879 --> 02:33.939
And finally,
you come back to the starting

02:33.940 --> 02:37.150
point by further compression,
but adiabatic,

02:37.151 --> 02:40.881
and adiabatic means it's
insulated from the outside

02:40.884 --> 02:43.094
world;
no heat flow.

02:43.090 --> 02:44.890
So, there are four parts.

02:44.889 --> 02:49.629
There are two things you have
to notice about the Carnot

02:49.633 --> 02:51.443
engine.
In fact, let me draw a

02:51.436 --> 02:52.926
schematic of the Carnot engine.

02:52.930 --> 02:55.750
This is the standard way people
draw that.

02:55.750 --> 02:59.060
The engine takes some heat
Q_1 from the

02:59.056 --> 03:01.766
furnace at T_1,
it has some

03:01.770 --> 03:05.180
Q_2 rejected at
the exhaust, which is at

03:05.183 --> 03:08.533
T_2 and it
delivers a certain amount of

03:08.534 --> 03:10.684
work which,
by the Law of Conservation of

03:10.680 --> 03:12.090
Energy, has to be
Q_1 -

03:12.092 --> 03:12.942
Q_2.

03:12.940 --> 03:16.480


03:16.480 --> 03:18.310
Do you understand that?

03:18.310 --> 03:20.810
The work done by the gas equal
to that.

03:20.810 --> 03:22.270
And who paid for it?

03:22.270 --> 03:24.770
It is the reservoir,
upper reservoir furnished the

03:24.773 --> 03:27.693
heat, some heat was rejected
downstairs and the difference

03:27.685 --> 03:29.315
between the two is the work.

03:29.320 --> 03:33.830


03:33.830 --> 03:38.250
Now, we call this a cyclic
process because at the end of

03:38.251 --> 03:41.951
the day, the gases come back to
where it is.

03:41.949 --> 03:44.639
That's very important,
if you want to make an engine,

03:44.639 --> 03:47.899
it's not enough to convert;
the whole purpose of heat

03:47.901 --> 03:52.061
engines was to burn something
and get some work out of it.

03:52.060 --> 03:53.720
But you don't want it to be a
one shot thing.

03:53.720 --> 03:56.440
You want to be able to do it
over and over and over again and

03:56.437 --> 03:59.107
you can do that with this engine
because after the cycle,

03:59.110 --> 04:01.720
it's back to where it started
and you can do this any number

04:01.722 --> 04:04.842
of times.
Now, we define an efficiency

04:04.837 --> 04:09.057
for this engine,
which is what you get divided

04:09.061 --> 04:11.221
by what you pay for.

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What you get from the engine is
of course, the work W and

04:14.794 --> 04:17.634
what you pay for is the coal
that you burn which is

04:17.631 --> 04:19.051
Q_1.

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Now, writing W as
Q_1 -

04:21.618 --> 04:24.428
Q_2 you can write
this as 1 minus

04:24.431 --> 04:27.551
Q_2 over
Q_1.

04:27.550 --> 04:29.850
This is true for any engine.

04:29.850 --> 04:32.180
Any engine, even if it's not a
Carnot engine;

04:32.180 --> 04:36.730
with the Law of Conservation of
Energy you can write something

04:36.725 --> 04:39.205
like this.
Anyway, let me say with the

04:39.213 --> 04:41.813
Carnot engine I got this as the
efficiency.

04:41.810 --> 04:45.560
Now, why is the efficiency not
100%?

04:45.560 --> 04:47.690
That's because
Q_2 is not

04:47.686 --> 04:49.956
zero.
You can say why are we building

04:49.962 --> 04:52.432
a stupid engine that rejects
some heat?

04:52.430 --> 04:55.060
Why not just use all of it in
some other fashion,

04:55.059 --> 04:57.469
and that's what we're going to
talk about,

04:57.470 --> 05:00.430
but assuming that the engine
operates between two

05:00.428 --> 05:03.138
temperatures,
the furnace temperature and the

05:03.140 --> 05:06.540
ambient atmospheric temperature,
you can calculate the

05:06.535 --> 05:09.275
efficiency for this engine by
actually calculating

05:09.277 --> 05:12.017
Q_2 and
Q_1.

05:12.019 --> 05:16.039
That's a simple problem of
integrating the work done and

05:16.039 --> 05:17.939
going from here to here.

05:17.940 --> 05:19.380
Do you remember that?

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1 - nRT_1
log V_B over

05:25.407 --> 05:30.777
V_A and
downstairs you have NR

05:30.777 --> 05:35.047
T_2 log
V_C over

05:35.051 --> 05:38.011
V_D.

05:38.009 --> 05:39.999
Again, you've got to remember
why that is true.

05:40.000 --> 05:43.000
When you go from A to
B there is no change in

05:43.002 --> 05:45.592
internal energy because
temperature is the same.

05:45.589 --> 05:47.829
Therefore, the work done and
the heat import are equal

05:47.828 --> 05:49.808
numerically, so
Q_2 is the work

05:49.812 --> 05:52.222
done and Q_1 is
the work done there.

05:52.220 --> 05:53.640
So, that's what I got.

05:53.639 --> 05:56.659
Then, I did a little bit of
calculation to show you by using

05:56.661 --> 05:59.431
the fact that B and
C are on an adiabatic

05:59.427 --> 06:02.287
curve,
A and D are on an

06:02.293 --> 06:05.333
adiabatic curve,
that these logs actually

06:05.330 --> 06:06.850
cancel,
these guys of course,

06:06.845 --> 06:08.965
cancel to give the final result
1 minus T_2

06:08.967 --> 06:10.007
over T_1.

06:10.010 --> 06:11.980
That's the bottom line.

06:11.980 --> 06:15.410
I don't care what else you
remember but this is something I

06:15.407 --> 06:17.177
need for today's discussion.

06:17.180 --> 06:20.700
Efficiency of the Carnot engine
is 1 minus T_2

06:20.700 --> 06:22.290
over T_1.

06:22.290 --> 06:28.350


06:28.350 --> 06:32.440
Okay now, you can say why I'd
be interested in this very

06:32.441 --> 06:36.831
primitive engine containing a
cylinder and gas and so on.

06:36.829 --> 06:39.379
Okay, this efficiency of this
stupid engine,

06:39.380 --> 06:41.990
I'm sure people can build a
better engine.

06:41.990 --> 06:45.290
So, here is Carnot's claim.

06:45.290 --> 06:51.360
Mr.
Carnot say,s "No engine can

06:51.358 --> 06:56.858
beat my engine."
Okay that's his claim.

06:56.860 --> 07:01.000
By "beat my engine" he means,
no engine can be more efficient

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than my engine.
And he's going to demonstrate

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that.
I'm going to demonstrate that

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for you.
You don't get something for

07:08.175 --> 07:10.005
nothing.
It's based on the following

07:10.011 --> 07:12.051
postulate, which I told you last
time.

07:12.050 --> 07:14.640
I'm just now going to write it;
I will repeat it.

07:14.639 --> 07:18.989
We are going to postulate,
as the Second Law of

07:18.994 --> 07:23.254
Thermodynamics,
that it's impossible to find a

07:23.254 --> 07:26.454
process,
the sole result of which is to

07:26.445 --> 07:29.935
transfer some heat from a cold
body to a hot body.

07:29.939 --> 07:33.579
We all know you can transfer
heat from a cold body to a hot

07:33.580 --> 07:35.150
body in a refrigerator.

07:35.149 --> 07:37.329
You take heat out of the
freezer and dump it into the

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room;
no one says that's wrong.

07:40.269 --> 07:42.589
But that's not the sole effect,
because you get an electric

07:42.594 --> 07:44.884
bill at the end of the month,
because there's a compressor

07:44.878 --> 07:45.838
doing a lot of work.

07:45.839 --> 07:49.059
The claim is--not--no other
agency in the end should be

07:49.059 --> 07:50.609
affected in any fashion.

07:50.610 --> 07:53.910
If at the end of the day all
you had was heat flow up hill.

07:53.910 --> 07:56.920
That's not allowed;
that's a postulate and you have

07:56.915 --> 07:59.595
to grant him that postulate
which we accept to be

07:59.602 --> 08:01.172
phenomenologically valid.

08:01.170 --> 08:04.620
But taking that postulate,
Carnot will now show you that

08:04.615 --> 08:06.615
no engine can beat his engine.

08:06.620 --> 08:11.310
The key to the whole Carnot
engine is that it is a

08:11.314 --> 08:13.234
reversible engine.

08:13.230 --> 08:18.610
By reversible engine I mean on
every step in the Carnot cycle,

08:18.610 --> 08:23.990
let's say going from here to
here, I can also go backwards.

08:23.990 --> 08:27.020
I can put a grain of sand and
compress a piston,

08:27.019 --> 08:30.499
or I can take a grain it'll go
back to where it was.

08:30.500 --> 08:33.080
You're never far from
equilibrium and if you can go

08:33.076 --> 08:34.876
one way you can go the other
way.

08:34.879 --> 08:38.099
That also means the Carnot
engine, starting at the point

08:38.101 --> 08:41.561
A, can go backwards to
D then C then to

08:41.558 --> 08:43.548
B and then to A.

08:43.549 --> 08:47.489
If the Carnot engine went
backwards it would look like

08:47.493 --> 08:51.513
this: it will take heat
Q_1--I'm sorry

08:51.510 --> 08:54.710
it will take heat
Q_2,

08:54.710 --> 08:58.290
somebody will give it work
W and it'll reject

08:58.290 --> 09:01.310
Q_1 at this
temperature.

09:01.309 --> 09:07.039
This is the Carnot refrigerator
and that's the Carnot engine.

09:07.039 --> 09:09.279
The Carnot engine and the
refrigerator are the same

09:09.281 --> 09:10.911
machine;
you just can make it run

09:10.911 --> 09:12.341
one-way or the opposite way.

09:12.340 --> 09:15.440
That's what we're going to use.

09:15.440 --> 09:18.830
So, let me give you a
demonstration that's for--as far

09:18.831 --> 09:22.161
as I can tell is good enough for
our present purpose,

09:22.158 --> 09:24.908
that you cannot beat the Carnot
engine.

09:24.909 --> 09:29.999
So, let us say a Carnot engine
takes some heat,

09:30.003 --> 09:35.433
say 100 calories,
delivers 20 calories in work,

09:35.429 --> 09:37.869
rejects 80 calories is
operating between some two

09:37.869 --> 09:39.699
temperatures
T_1 and

09:39.699 --> 09:40.969
T_2.

09:40.970 --> 09:41.480
This is the Carnot.

09:41.480 --> 09:46.770


09:46.769 --> 09:51.139
I am taking a particular
example where the efficiency is

09:51.139 --> 09:54.409
what, one-fifth.
But don't worry about the fact

09:54.408 --> 09:56.978
that, yeah, so it's one-fifth,
so one minus

09:56.981 --> 10:00.351
T_2 over
T_1 should be

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one-fifth.
Now, if you say you have a

10:03.397 --> 10:07.077
better engine,
you have a better efficiency.

10:07.080 --> 10:11.440
What you really mean is that
your engine can take a 100

10:11.438 --> 10:15.068
calories and deliver more than
20 calories,

10:15.070 --> 10:18.520
maybe even deliver 40 calories
and reject only 60 calories,

10:18.515 --> 10:22.075
but it's also operating between
the same two temperatures.

10:22.080 --> 10:23.890
So, this is your engine.

10:23.889 --> 10:28.449
U for--no,
Y for your engine,

10:28.450 --> 10:30.810
okay;
your engine is better than

10:30.812 --> 10:33.342
Carnot's engine,
okay that's your claim.

10:33.340 --> 10:35.810
You're now going to get shot
down.

10:35.809 --> 10:38.769
So, how am I going to shoot
this down?

10:38.769 --> 10:43.459
I am going to run the Carnot
engine backwards first thing,

10:43.462 --> 10:48.572
then I am going to get a Carnot
engine which is twice as big as

10:48.566 --> 10:51.966
this Carnot engine,
it's not more efficient,

10:51.969 --> 10:54.169
it has just got twice as much
gas.

10:54.169 --> 10:57.569
If you take a Carnot engine and
do the following things,

10:57.570 --> 11:00.290
run it backwards and make it
twice as big.

11:00.290 --> 11:02.620
What will that Carnot engine do?

11:02.620 --> 11:07.280
It will look like this;
it will take up 160 calories

11:07.275 --> 11:11.835
from downstairs,
it will want 40 calories input

11:11.835 --> 11:15.795
and it'll dump 200 calories
upstairs.

11:15.800 --> 11:20.330
So, this is a two-times Carnot;
let me put a star saying it's

11:20.331 --> 11:21.911
Carnot run backwards.

11:21.909 --> 11:27.389
See, it doesn't take any
special genius to make an engine

11:27.389 --> 11:29.369
bigger.
It won't be more efficient;

11:29.370 --> 11:30.820
it'll have the same efficiency.

11:30.820 --> 11:33.080
But if you can build something
with one piston,

11:33.083 --> 11:35.893
I can make the same design with
a bigger piston or smaller

11:35.888 --> 11:38.348
piston.
I'm just saying I want it to be

11:38.346 --> 11:41.186
twice as big for the following
simple reason.

11:41.190 --> 11:45.280
Your engine's giving out 40
units of work;

11:45.280 --> 11:47.890
my refrigerator needs 40 to run.

11:47.889 --> 11:52.079
So, what we do is we directly
take the output from your engine

11:52.082 --> 11:54.352
and feed it to my refrigerator.

11:54.350 --> 11:59.860
Okay, your heat engine makes
work, my refrigerator needs work

11:59.861 --> 12:05.741
and I've scaled mine so that its
appetite matches your output;

12:05.740 --> 12:07.230
that's all I've done.

12:07.230 --> 12:12.060
Now, let's draw a box around
these guys, don't look under the

12:12.062 --> 12:14.562
hood and see what you've got.

12:14.559 --> 12:17.259
At the end of a full cycle,
when everything is done,

12:17.256 --> 12:18.836
all the gases,
all the pistons,

12:18.842 --> 12:21.382
everything has come back to
where it starts.

12:21.379 --> 12:27.069
But, no need to plug this
gadget into the wall.

12:27.070 --> 12:30.150
You don't have to plug this
into anything because this

12:30.148 --> 12:33.338
refrigerator is getting the
power from this heat engine,

12:33.342 --> 12:35.552
so it doesn't need external
power.

12:35.550 --> 12:39.090
I look at the lower reservoir;
I see 100 calories leaving.

12:39.090 --> 12:41.620
You see that 60 coming down and
160 going up?

12:41.620 --> 12:47.660
I look at the upper reservoir;
I see 200 calories out,

12:47.655 --> 12:49.975
in and 100 out,
so basically,

12:49.975 --> 12:53.945
the combined gadgets,
yours and mine are equal to

12:53.951 --> 12:57.741
this gadget.
It simply transfers heat from a

12:57.744 --> 13:02.204
cold body to hot body with no
other changes anywhere in the

13:02.204 --> 13:05.054
universe, and that is not
allowed.

13:05.049 --> 13:09.469
Therefore, you cannot have an
engine more efficient that the

13:09.471 --> 13:12.211
Carnot engine.
The logic is pretty simple.

13:12.210 --> 13:14.220
These numbers picked are
representative,

13:14.218 --> 13:17.358
but you can take any number as
long as your engine does better

13:17.359 --> 13:20.269
than mine,
instead of 40 calories,

13:20.274 --> 13:23.714
would have made 30 calories
needed.

13:23.710 --> 13:26.010
If it produced 30,
I'll get an engine which is

13:26.006 --> 13:28.656
one-half times as bigger than
the standard engine;

13:28.659 --> 13:31.109
run it backwards,
your 30 will feed my engine and

13:31.105 --> 13:34.315
you will find once again heat is
flowing from a cold body to hot

13:34.316 --> 13:37.166
body with no other changes
anywhere in the universe,

13:37.169 --> 13:40.239
that is forbidden by the
postulate.

13:40.240 --> 13:44.780
So, this is how Carnot's
engine, even though it's a very

13:44.778 --> 13:48.078
primitive engine,
is the standard for all

13:48.079 --> 13:50.259
engines.
No engine can be better than

13:50.261 --> 13:51.071
the Carnot engine.

13:51.070 --> 13:53.760
Now, the key to the Carnot
engine is that it's a reversible

13:53.762 --> 13:55.642
engine.
If I had more time I could show

13:55.641 --> 13:58.091
you that all reversible engines
operating between two

13:58.091 --> 14:00.261
temperatures will have the same
efficiency,

14:00.259 --> 14:02.519
namely that of the Carnot
engine.

14:02.519 --> 14:10.609
Okay, now what has the Carnot
engine got to do with what I

14:10.611 --> 14:14.161
started saying earlier?

14:14.159 --> 14:17.129
You remember I started saying
earlier there are certain things

14:17.127 --> 14:19.947
in our world that seem allowed
but don't seem to happen.

14:19.950 --> 14:22.090
Like you know,
drop an egg it doesn't come

14:22.087 --> 14:23.127
back in your hand.

14:23.129 --> 14:25.529
You let the weight go down and
turn the paddle.

14:25.529 --> 14:27.889
The paddle doesn't turn
backwards and lift the weight.

14:27.889 --> 14:30.589
I said many things are
forbidden and there is a law

14:30.585 --> 14:33.655
that's going to forbid all of
them, so I am coming to that

14:33.658 --> 14:35.628
law.
But it all started with a

14:35.631 --> 14:39.331
person asking a very practical
question, "How good can I make

14:39.330 --> 14:42.170
engines?"
And what Carnot is telling you

14:42.166 --> 14:45.796
is the best efficiency any
engine can have is this.

14:45.799 --> 14:50.279
Therefore, even today hundreds
of years after Carnot,

14:50.284 --> 14:54.814
no company in America or Japan,
China, anywhere can build an

14:54.809 --> 14:58.229
internal combustion engine for
example, whose efficiency's

14:58.230 --> 14:59.670
better than this one.

14:59.670 --> 15:01.740
It's a theoretical maximum.

15:01.740 --> 15:05.620
In reality, efficiency will be
less than this because most

15:05.621 --> 15:08.091
engines have a loss;
there is heat leaking,

15:08.090 --> 15:10.520
there is friction that's good
for nothing, so in the end

15:10.515 --> 15:12.495
efficiency will be always lower
than this.

15:12.500 --> 15:17.820
Okay now, let me leave the
practical domain and go to more

15:17.818 --> 15:21.828
theoretical issues that follow
from this.

15:21.830 --> 15:24.530
What I want you to notice if
the following:

15:24.529 --> 15:27.359
for a Carnot engine
Q_1 over

15:27.358 --> 15:31.338
Q_2 turned out
to be T_1 over

15:31.344 --> 15:33.084
T_2.

15:33.080 --> 15:35.240
Right, I did that for you here.

15:35.240 --> 15:38.370
Therefore, I am going to write
it as follows:

15:38.369 --> 15:41.999
Q_1 over
T_1 minus

15:41.996 --> 15:45.836
Q_2 over
T_2 equal to

15:45.837 --> 15:49.567
zero.
That's just a simple rewrite.

15:49.570 --> 15:52.750
But I am going to write this in
another rotation.

15:52.750 --> 15:55.860
My cycle had four parts;
remember, I did this,

15:55.860 --> 15:57.570
then I did this,
then I did this,

15:57.566 --> 16:00.716
then I did that.
In this segment I had

16:00.719 --> 16:05.609
Q_1 over
T_1.

16:05.610 --> 16:08.320
In this segment there was no
Q_1 or

16:08.321 --> 16:10.611
Q_2,
there was no Q,

16:10.607 --> 16:12.837
because this is the adiabatic
process.

16:12.840 --> 16:15.120
In this segment,
I had minus

16:15.120 --> 16:19.600
Q_2 over
T_2 and the

16:19.597 --> 16:23.987
last segment I had zero and
whole thing is zero.

16:23.990 --> 16:27.780
In other words,
what I'm telling you is the

16:27.777 --> 16:30.817
following.
At every stage,

16:30.815 --> 16:37.065
if you looked at the heat
absorbed by the system,

16:37.072 --> 16:41.922
in stage i,
divided by the temperature of

16:41.922 --> 16:45.932
stage i and you added all
of them you get zero.

16:45.930 --> 16:48.950
My process had four stages.

16:48.950 --> 16:51.920
Stage 1 is here,
stage 2 is adiabatic,

16:51.923 --> 16:55.543
state 3 is this,
and stage 4 is zero again.

16:55.539 --> 17:01.279
In this summation
Q_i is defined

17:01.277 --> 17:08.927
to be ∆Q_i is
the heat input in stage i

17:08.927 --> 17:13.227
in the cycle,
or in the process.

17:13.230 --> 17:16.180
So, we define
Q_2 to be

17:16.178 --> 17:20.258
positive, even though it was
rejected by the engine,

17:20.259 --> 17:22.559
but in this summation,
the agreement I make is

17:22.557 --> 17:25.617
∆Q is positive if the
heat comes into the system,

17:25.619 --> 17:28.469
∆Q is negative if heat
leaves the system.

17:28.470 --> 17:32.510
So, why is that important?

17:32.509 --> 17:36.249
Now, this is the heart of the
whole entropy concept.

17:36.250 --> 17:41.790
Remember I told you there is
nothing called the heat in a

17:41.789 --> 17:44.119
system.
You cannot look at a gas and

17:44.122 --> 17:46.442
say that's the amount of heat in
the system.

17:46.440 --> 17:49.620
Why?
Because if you take a point

17:49.618 --> 17:54.068
there and say there's some
amount of heat in the system,

17:54.069 --> 17:57.589
if you go through some kind of
cycle and come back to the same

17:57.585 --> 18:00.865
point, then since you come back
to the same point internal

18:00.871 --> 18:04.191
energy doesn't change,
this is the work done,

18:04.190 --> 18:08.990
therefore, the work done is
equal to the heat and it is not

18:08.988 --> 18:11.228
zero.
So, in this example what

18:11.226 --> 18:15.296
would've happened is you'd added
heat Q to the system.

18:15.299 --> 18:19.109
So, if there is some notion of
how much heat is there in the

18:19.112 --> 18:23.182
beginning, you got that plus the
Q that you added and yet

18:23.183 --> 18:25.383
you are back to where you are.

18:25.380 --> 18:28.230
Therefore, you cannot define
something and say that's the

18:28.228 --> 18:31.178
amount of Q in the system
because you're able to add

18:31.178 --> 18:34.228
Q to the system and bring
it back to where it is.

18:34.230 --> 18:35.970
What happened,
of course, is you added

18:35.969 --> 18:37.519
Q and you did some work.

18:37.519 --> 18:40.689
But the point is you cannot say
the Q here is so and so.

18:40.690 --> 18:42.460
But you can say the energy here
is so and so.

18:42.460 --> 18:44.910
Because if you come back to the
same point P times

18:44.913 --> 18:47.543
V is equal to 3 is equal
to RT and the internal

18:47.541 --> 18:49.821
energy of a gas is proportional
to the T,

18:49.820 --> 18:51.650
energy returns to the old value.

18:51.650 --> 18:55.660
But now I'm going to tell you
so listen very carefully.

18:55.660 --> 19:00.120
Energy is a state variable
because it comes to the starting

19:00.116 --> 19:03.416
value, if you come to the
starting point.

19:03.420 --> 19:05.800
It doesn't matter where you
wander off in the PV

19:05.799 --> 19:07.879
diagram.
Heat is not a state variable,

19:07.881 --> 19:10.321
there's nothing called a heat
at that point,

19:10.315 --> 19:12.405
because I am able to go on a
loop,

19:12.410 --> 19:14.960
change the value of Q or
add some Q to it,

19:14.955 --> 19:16.385
and I come to the same point.

19:16.390 --> 19:19.530
So, there's no unique Q
associated with that point.

19:19.529 --> 19:23.049
But there is a new unique
quantity S,

19:23.049 --> 19:26.569
called entropy,
which has a fixed value at a

19:26.568 --> 19:31.398
given point and if you go for a
little loop in the PV

19:31.397 --> 19:35.077
diagram and you come to the same
point,

19:35.079 --> 19:37.879
S will return to the
starting value.

19:37.880 --> 19:39.840
So, who is this S?

19:39.839 --> 19:45.749
That S is defined by
saying the change in the entropy

19:45.750 --> 19:50.660
is the heat you add divided by
the temperature.

19:50.660 --> 19:53.920
In a tiny little process,
if you are at some temperature

19:53.922 --> 19:56.772
T, you add a little
amount of Q,

19:56.769 --> 20:03.169
keep track of all the changes,
and that change will be zero,

20:03.169 --> 20:07.399
as I showed you in this Carnot
cycle.

20:07.400 --> 20:11.250
You can show more generally
that if you take any path,

20:11.252 --> 20:14.892
not just the Carnot engine
bounded by adiabatic and

20:14.886 --> 20:17.776
isothermals,
but any path you take,

20:17.781 --> 20:21.561
you can show that the
∆Q added up is not

20:21.560 --> 20:23.400
zero,
but ∆Q/T,

20:23.404 --> 20:27.394
namely give a weighting factor
of 1/T to the heat you

20:27.387 --> 20:29.427
add,
then the positives and

20:29.426 --> 20:32.556
negatives cancel and give you
exactly zero.

20:32.559 --> 20:34.119
In other words,
Q_1 was not

20:34.120 --> 20:35.970
Q_2,
Q_1 is bigger

20:35.971 --> 20:38.131
than Q_2,
but Q_1/T

20:38.125 --> 20:40.685
_1 precisely
balances Q_2/T

20:40.691 --> 20:41.661
_2.

20:41.660 --> 20:44.530
The heat absorbed at higher
temperature if divided by

20:44.531 --> 20:47.021
T_1,
to compute the change in

20:47.017 --> 20:49.507
entropy,
then in the upper part of the

20:49.513 --> 20:52.233
Carnot cycle here,
and the lower part of the

20:52.231 --> 20:55.511
Carnot here;
the two cancel as far as the

20:55.513 --> 20:57.933
entropy change is concerned.

20:57.930 --> 21:01.050
So, entropy is a new variable;
it's a mathematically

21:01.045 --> 21:02.275
discovered variable.

21:02.279 --> 21:05.289
What we find is that if I
postulate there's a variable

21:05.291 --> 21:07.451
called entropy,
the change in which,

21:07.450 --> 21:11.510
in any process,
is the heat transfer divided by

21:11.505 --> 21:16.965
temperature, then that has the
property that when you go around

21:16.971 --> 21:20.851
on a loop you come back net zero
change.

21:20.849 --> 21:23.839
That means every point can be
associated with a number you can

21:23.843 --> 21:25.763
call entropy.
Entropy is like a height.

21:25.759 --> 21:29.259
Suppose you are walking around
on some landscape,

21:29.258 --> 21:31.078
each point is a height.

21:31.079 --> 21:34.689
You can walk here and there and
come back at every stage;

21:34.690 --> 21:37.300
you keep track of the change in
height if you come back to where

21:37.303 --> 21:39.463
you are, the change in height
will add up to zero.

21:39.460 --> 21:42.850
That's what I told you in the
case of a potential energy

21:42.851 --> 21:45.101
function.
It is just like a potential

21:45.100 --> 21:48.170
energy function.
Internal energy and the entropy

21:48.171 --> 21:49.851
are now state variables.

21:49.849 --> 21:53.459
At every point the gas has a
certain internal energy and an

21:53.457 --> 21:55.427
entropy.
They deserve to be called state

21:55.432 --> 21:58.232
variables because when you go on
a loop and come back they return

21:58.225 --> 21:59.355
to the starting values.

21:59.360 --> 22:02.550


22:02.549 --> 22:06.199
Now, you have no idea what this
quantity stands for;

22:06.200 --> 22:07.460
you agree it's bizarre.

22:07.460 --> 22:09.460
Work done, we understand.

22:09.460 --> 22:12.550
Heat absorbed and heat rejected
we understand.

22:12.549 --> 22:14.839
What is ∆Q over
T?

22:14.839 --> 22:18.089
Why is dividing by T
make such a big difference?

22:18.089 --> 22:19.519
Why does it produce a new
variable?

22:19.520 --> 22:22.370
We can see it is true;
at least in the Carnot cycle

22:22.368 --> 22:25.708
you can verify in detail that
the change in the total of all

22:25.714 --> 22:28.554
the ∆Qs over T
is in fact zero.

22:28.549 --> 22:31.109
So, we'll develop up a feeling
for what it means,

22:31.111 --> 22:33.301
but historically this is what
happened.

22:33.299 --> 22:36.379
People realized,
hey there's another state

22:36.375 --> 22:38.555
variable.
We introduce this new variable,

22:38.562 --> 22:41.112
we have no idea what it means
but it is a state variable so

22:41.110 --> 22:42.780
we'd better take it very
seriously.

22:42.779 --> 22:45.719
So, I'm going to tell you what
it means.

22:45.720 --> 22:49.070
But first I want you to get
some practice calculating the

22:49.074 --> 22:51.654
entropy change for a couple of
processes.

22:51.650 --> 22:55.630
So, let us take for example,
one gram of water,

22:55.633 --> 22:59.533
let's say m grams of
some substance.

22:59.529 --> 23:03.769
It's got some specific heat
C.

23:03.769 --> 23:07.909
And I'll change the temperature
say from some T initial

23:07.906 --> 23:09.666
for some T final.

23:09.670 --> 23:14.130


23:14.130 --> 23:17.910
I do that by keeping the
system--starting the system at

23:17.905 --> 23:21.395
T_i and I
should never be far from

23:21.401 --> 23:23.651
equilibrium.
That's one of the conditions on

23:23.651 --> 23:26.221
this.
In fact, you can say it should

23:26.223 --> 23:28.893
be near, always near
equilibrium.

23:28.890 --> 23:31.730
So, the system starts at
temperature

23:31.732 --> 23:35.632
T_i,
I put it on a heat bath,

23:35.630 --> 23:39.560
which is infinitesimally warmer
than this and I let it heat up

23:39.559 --> 23:41.929
at that point.
Then, I put it on another

23:41.933 --> 23:44.193
reservoir, slightly hotter than
this one.

23:44.190 --> 23:48.610
At all stages I want the system
to be almost near equilibrium.

23:48.609 --> 23:51.779
In the sense of calculus you
can make the difference as small

23:51.783 --> 23:54.593
as you like provided you do
enough number of times and

23:54.586 --> 23:57.016
slowly I raise this guy from
here to here.

23:57.019 --> 23:59.839
At every stage the system has a
well-defined temperature,

23:59.838 --> 24:02.958
has a well-defined heat input
and I wanted to add it all up.

24:02.960 --> 24:04.800
So, what's the change in
entropy?

24:04.799 --> 24:09.119
S final minus S
initial is equal to dQ

24:09.124 --> 24:12.334
over T,
summed over all the parts,

24:12.329 --> 24:18.739
I should write this as an
integral, mC ∆T over T.

24:18.740 --> 24:21.400
Do you see that, dQ is mC ∆T?

24:21.400 --> 24:24.980
But I got to divide by T
and that integral is done

24:24.980 --> 24:28.820
between some initial and final
temperature and we can do this

24:28.815 --> 24:32.455
integral rather trivially in an
mC log T final over

24:32.459 --> 24:33.929
T initial.

24:33.930 --> 24:37.940


24:37.940 --> 24:40.800
That is the increase in entropy
of some m grams of some

24:40.802 --> 24:43.292
substance of specific heat
C that's heated from

24:43.289 --> 24:45.119
T initial to T
final.

24:45.119 --> 24:49.049
We're just getting practice
calculating this.

24:49.049 --> 24:50.909
We still don't know what this
guy means.

24:50.910 --> 24:52.750
So, don't worry about that
right now.

24:52.750 --> 24:55.940
If it did not divide by
T, what are you

24:55.938 --> 24:58.668
calculating?
mC ∆T integrated is

24:58.668 --> 25:01.398
just mC times change in
temperature.

25:01.400 --> 25:03.920
That's the old calorimetric
problem you did.

25:03.920 --> 25:07.100
How many calories does it take
to raise the substance from

25:07.100 --> 25:08.830
initial to final temperature?

25:08.830 --> 25:10.820
That you understand.

25:10.819 --> 25:13.479
When you divide by T,
something you don't understand,

25:13.479 --> 25:15.959
but anyway, let's make sure we
know how to calculate the

25:15.958 --> 25:18.618
increase in entropy when water
or something is heated from a

25:18.617 --> 25:20.147
lower to higher temperature.

25:20.150 --> 25:23.000
If you cooled it from a higher
to lower temperature,

25:23.004 --> 25:25.974
you can use the same formula
T_f over

25:25.970 --> 25:28.350
T_i,
but T_f will

25:28.350 --> 25:29.770
now be smaller than
T_i;

25:29.769 --> 25:31.629
if you cooled it,
if the log of a number less

25:31.630 --> 25:33.830
than one is negative and the
entropy changes it'll be

25:33.830 --> 25:36.810
negative.
So, let me do one more entropy

25:36.811 --> 25:41.351
calculation that's going to be
pretty important for us.

25:41.349 --> 25:43.629
That entropy calculation is
this.

25:43.630 --> 25:49.040
Take a gas and watch it expand
isothermally from some starting

25:49.038 --> 25:53.918
point, at some fixed temperature
T that goes from

25:53.915 --> 25:58.255
V_1 to
V_2.

25:58.259 --> 26:01.839
What is the change in entropy
when it goes from here to here?

26:01.840 --> 26:06.180


26:06.180 --> 26:09.630
Again, the system must always
be in equilibrium or near

26:09.634 --> 26:13.154
equilibrium so I can plot it as
a point in the PV

26:13.152 --> 26:15.532
diagram,
so I slowly take grain after

26:15.530 --> 26:19.210
grain, do the whole thing I told
you, and find S_2 -

26:19.212 --> 26:22.722
S_1,
this is 2 and this is 1.

26:22.720 --> 26:27.070
That is equal to the sum of all
the heat transfers divided by

26:27.072 --> 26:30.412
temperature at every little step
of the way.

26:30.410 --> 26:38.000
But remember this is also the
same as P∆V over

26:38.003 --> 26:40.483
T.
Why?

26:40.480 --> 26:43.640
Because on an isothermal
∆U is zero,

26:43.642 --> 26:47.382
I repeated it many times but
you've got to know this,

26:47.380 --> 26:49.680
∆Q is P∆V.

26:49.680 --> 26:55.530
But P over T is
nRT over V,

26:55.532 --> 26:59.272
∆V.
You also want to divide it by a

26:59.265 --> 27:02.485
T.
Do you understand P over

27:02.487 --> 27:05.477
T is nR over
V?

27:05.480 --> 27:09.970
So, P = nRT over
V, I'm just saying,

27:09.965 --> 27:14.725
PV is nRT,
so P over T is

27:14.726 --> 27:17.286
NR over V.

27:17.289 --> 27:20.819
Now, dV over V
when you sum or integrate will

27:20.821 --> 27:23.261
give you nR log
V_2 over

27:23.257 --> 27:24.777
V_1.

27:24.780 --> 27:29.380


27:29.380 --> 27:32.390
That is the change in entropy
of this gas when it went from

27:32.387 --> 27:35.237
volume V and went to
volume V_2,

27:35.239 --> 27:36.639
at a given temperature.

27:36.640 --> 27:40.070
You don't have to use calculus
to do this.

27:40.069 --> 27:42.919
Does everybody understand why
this could have been done a lot

27:42.922 --> 27:45.052
easier?
You don't need to do the

27:45.054 --> 27:46.124
integral here.

27:46.120 --> 27:49.420


27:49.420 --> 27:52.900
Because I showed you in
studying the heat engine that

27:52.899 --> 27:56.709
the heat transfer Q is
nRT log V_2

27:56.712 --> 27:58.722
over V_1.

27:58.720 --> 28:01.640
Since the whole process takes
place at fixed temperature,

28:01.643 --> 28:03.993
instead of dividing by T
at every step,

28:03.992 --> 28:06.292
you can just divide by T
overall.

28:06.290 --> 28:11.430
Just bring the T here;
that's why this is a change in

28:11.428 --> 28:13.108
entropy.
In other words,

28:13.114 --> 28:16.454
during the whole process you
feel at one temperature then

28:16.454 --> 28:20.034
integral of ∆Q over
T is just 1 over T

28:20.032 --> 28:23.842
times integral of ∆Q,
which is the total heat

28:23.841 --> 28:26.171
transfer.
Anyway, this is the heat,

28:26.165 --> 28:29.435
this is the change in entropy,
S_2 -

28:29.443 --> 28:32.913
S_1 is nR
log V_2 over

28:32.911 --> 28:34.551
V_1.

28:34.550 --> 28:43.680


28:43.680 --> 28:46.760
Okay, so what have I done so
far?

28:46.760 --> 28:50.220
Let's collect our thoughts here.

28:50.220 --> 28:54.110
I went to the Carnot engine
today and reminded you what the

28:54.110 --> 28:56.660
efficiency of the Carnot engine
was.

28:56.660 --> 28:59.300
Then, I showed you no engine
can be better than the Carnot

28:59.302 --> 29:01.382
engine.
That's a separate story.

29:01.380 --> 29:04.930
That has to do with how
efficient things can be.

29:04.930 --> 29:08.590
Then, a byproduct of the Carnot
engine was this great

29:08.594 --> 29:12.894
realization that if you go on a
closed loop and at every stage

29:12.894 --> 29:15.404
you add,
you compute the heat absorbed,

29:15.400 --> 29:17.890
but divide by the temperature
at that point,

29:17.890 --> 29:20.020
the sum of all those is zero.

29:20.019 --> 29:23.979
What that means is that,
there is a quantity S

29:23.980 --> 29:26.650
whose change,
if it is defined to be

29:26.645 --> 29:30.355
∆Q over T,
has a property the total change

29:30.355 --> 29:32.465
in S is zero and you go
around a loop.

29:32.470 --> 29:35.840
That means at every point you
can associate an S.

29:35.839 --> 29:38.849
S is a property of the
point.

29:38.849 --> 29:42.119
Another example--so then,
I said let's get used to

29:42.118 --> 29:44.118
computing change in entropy.

29:44.119 --> 29:47.549
I took one example of heating a
substance of some mass and

29:47.546 --> 29:50.906
specific heat C by a
temperature dT and the

29:50.913 --> 29:54.403
change in entropy was mC log
TF over TI.

29:54.400 --> 29:57.710
You got the log because of
integral of dT over

29:57.709 --> 29:59.299
T was logarithm.

29:59.299 --> 30:03.239
Then, I took an ideal gas,
I let it expand isothermally,

30:03.240 --> 30:06.680
I found the entropy change,
I got this answer.

30:06.680 --> 30:08.550
By the way, here's an
interesting exercise,

30:08.552 --> 30:11.052
I don't have time to do it,
but you can ask the following

30:11.050 --> 30:13.750
question.
If you tell me that every point

30:13.752 --> 30:17.382
there is a unique entropy,
then the entropy difference

30:17.380 --> 30:21.150
between 2 and 1 should be
independent of how I go from 1

30:21.145 --> 30:22.825
to 2.
Do you understand?

30:22.829 --> 30:24.839
If you're walking on a
mountain, you take two points,

30:24.835 --> 30:27.415
they have a height difference,
and I can find the height

30:27.417 --> 30:30.757
difference by going on this path
keeping track of the change in

30:30.760 --> 30:32.270
height or any other path.

30:32.269 --> 30:34.439
In the end, the height
difference between two points is

30:34.443 --> 30:35.413
the height difference.

30:35.410 --> 30:37.900
So, I could find the entropy
another way.

30:37.900 --> 30:40.850
Let me show you another way
that's easy for you guys to work

30:40.849 --> 30:43.399
out.
Come down like this and go to

30:43.400 --> 30:45.030
the right like this.

30:45.030 --> 30:47.640
And find the entropy change.

30:47.640 --> 30:51.300
You'll get the same answer as I
got on this thing.

30:51.299 --> 30:54.789
Okay, it's too tempting for me
to just leave it here.

30:54.790 --> 30:56.100
Let me tell you how it works.

30:56.099 --> 31:00.309
Call this intermediate point,
give a subscript zero for all

31:00.306 --> 31:02.966
its parameters.
Then in this step,

31:02.974 --> 31:07.364
when you come from here to
here, the entropy change will

31:07.358 --> 31:12.458
be--let's take one mole of a gas
when I go from here to here.

31:12.460 --> 31:16.420
Then for one mole of a gas the
heat transfer dQ is

31:16.421 --> 31:20.601
C_VdT and I
divide by T and I do the

31:20.595 --> 31:23.495
integral from
T_1 to this

31:23.495 --> 31:25.825
point T_0.

31:25.830 --> 31:28.170
That's the entropy change here.

31:28.170 --> 31:32.080
Then in the horizontal part,
since I'm going at constant

31:32.078 --> 31:34.988
pressure, I go
C_PdT over

31:34.991 --> 31:38.121
T from
T_0 up to back

31:38.118 --> 31:40.728
to T_1,
because this

31:40.731 --> 31:42.821
T_1 and
T_2 are the

31:42.817 --> 31:43.617
same temperature.

31:43.619 --> 31:48.159
I have to add all these to get
the entropy change.

31:48.160 --> 31:51.400
Now it's a two-step process,
you come down because you're at

31:51.402 --> 31:54.482
constant volume and you're doing
something to the gas,

31:54.480 --> 31:56.740
dQ is
C_VdT.

31:56.740 --> 31:58.820
Horizontally,
you're at constant pressure,

31:58.815 --> 32:00.735
dQ is
C_PdT.

32:00.740 --> 32:03.030
That's the definition of
specific heat at constant

32:03.030 --> 32:04.340
volume, constant pressure.

32:04.339 --> 32:07.939
But notice the following:
C_P is equal

32:07.936 --> 32:10.396
to C_V +
R.

32:10.400 --> 32:14.230
So, put C_V +
R here and look what you

32:14.232 --> 32:17.282
get, then you get
C_V times

32:17.284 --> 32:21.124
log T_0 over
T_1 plus

32:21.117 --> 32:25.657
log T_1
over T_0,

32:25.660 --> 32:29.270
which we can write as the
product here,

32:29.268 --> 32:33.258
plus R log
T_1 over

32:33.256 --> 32:35.626
T_0.

32:35.630 --> 32:38.970


32:38.970 --> 32:39.960
In other words,
this log is log of

32:39.955 --> 32:41.415
T_0 over
T_1,

32:41.420 --> 32:44.260
the next log is log of
T_1 over

32:44.263 --> 32:47.353
T_0,
log A plus log B

32:47.348 --> 32:48.798
is log of AB.

32:48.799 --> 32:50.789
But when you do this look what
happens here.

32:50.789 --> 32:55.219
This all cancels,
log of 1 is zero;

32:55.220 --> 32:57.730
the total entropy change is
R log T_1 over

32:57.726 --> 32:58.716
T_0.

32:58.720 --> 33:02.610


33:02.609 --> 33:07.449
But for a gas at constant
pressure, using PV = RT,

33:07.448 --> 33:12.718
this ratio of temperature is
also the ratio of the volumes.

33:12.720 --> 33:18.030


33:18.029 --> 33:21.859
So this one--I'm sorry I should
write it more carefully.

33:21.859 --> 33:25.859
When I say T_1
I really meant the temperature

33:25.861 --> 33:27.831
at this point,
second point.

33:27.829 --> 33:30.409
So, that temperature,
or this temperature is also

33:30.413 --> 33:32.623
that volume divided by the
initial volume,

33:32.620 --> 33:34.450
which is V_1.

33:34.450 --> 33:36.060
See, this is a confusing
problem because

33:36.062 --> 33:38.462
T_1 happens to
be T_0.

33:38.460 --> 33:40.670
The correct way to do this for
me would be to write

33:40.667 --> 33:42.917
T_0 over
T_1 times

33:42.918 --> 33:45.168
T_2 over
T_0,

33:45.170 --> 33:48.200
then realize the fact that
T_2 and

33:48.201 --> 33:51.901
T_1 are equal
because I am on an isotherm.

33:51.900 --> 33:53.250
That's what I should have done.

33:53.250 --> 33:55.550
So, I should cancel this factor
because T_2 is

33:55.554 --> 33:57.204
T_1,
but here T_2

33:57.197 --> 33:58.637
over T_0 is
V_2 over

33:58.637 --> 34:00.167
V_0,
which is V_2

34:00.170 --> 34:00.990
over V_1.

34:00.990 --> 34:04.360
That's, of course,
the entropy change I got in one

34:04.355 --> 34:06.875
shot here.
So, you can find entropy change

34:06.878 --> 34:09.278
anyway you like.
You usually pick the easiest

34:09.284 --> 34:09.604
path.

34:09.600 --> 34:13.460


34:13.460 --> 34:18.890
Alright, so we have now learned
how to find entropy change.

34:18.890 --> 34:21.520
We have no idea what it means.

34:21.519 --> 34:24.599
It seems as if when you heat
something entropy goes up;

34:24.599 --> 34:27.139
when you cool something entropy
goes down.

34:27.139 --> 34:30.079
That's certainly correct,
because ∆Q is positive

34:30.084 --> 34:32.244
for heating and you divide by
T,

34:32.239 --> 34:34.359
and you make it a logarithm,
but we know at every

34:34.357 --> 34:36.737
infinitesimal portion we're
adding positive numbers.

34:36.739 --> 34:38.949
And when I cooled things it
goes down.

34:38.949 --> 34:40.749
Why not just call it
temperature?

34:40.750 --> 34:43.070
Why do you need this new
concept called entropy?

34:43.070 --> 34:45.320
So, that's what the rest of the
lecture is about.

34:45.320 --> 34:47.500
It's a very,
very powerful and beautiful

34:47.501 --> 34:50.971
concept so I wanted to explain
it, make sure I get it right.

34:50.969 --> 34:56.189
Here is now the result for all
this hard work.

34:56.190 --> 35:00.960
Remember I told you there are
many, many, many phenomena that

35:00.956 --> 35:05.266
seem forbidden in our world,
and we're not quite sure what

35:05.270 --> 35:08.800
law to invoke to prevent all of
them from happening.

35:08.800 --> 35:11.950
Do we want a law for each one
that says if you drop a bottle

35:11.946 --> 35:13.916
and it shatters it won't come
back?

35:13.920 --> 35:15.420
If you drop an egg it won't
come back.

35:15.420 --> 35:19.880
If you let hot and cold mix,
they're not going to unmix.

35:19.880 --> 35:23.890
Or if you let a gas trapped in
half a room, escape to the whole

35:23.894 --> 35:27.524
room, it'll never untrap and go
back to half the room.

35:27.519 --> 35:30.209
A lot of things happen one way
but not the other and I said I

35:30.211 --> 35:32.861
am looking for a mega law that
will prevent all these things

35:32.857 --> 35:35.997
from happening.
Now, I'm ready to state that

35:36.000 --> 35:38.660
law.
This is the third law--I mean,

35:38.659 --> 35:41.209
the Second Law of
Thermodynamics.

35:41.210 --> 35:43.670
Carnot said it one way,
but I'm going to say it in a

35:43.673 --> 35:45.223
way that's very,
very general.

35:45.219 --> 35:50.799
The Second Law of
Thermodynamics says ∆S

35:50.796 --> 35:56.136
for the universe is either zero
or positive.

35:56.139 --> 35:58.729
There you have it,
that's the great law.

35:58.730 --> 36:01.740
The law says,
take any process,

36:01.743 --> 36:07.573
if at the end of the process
the entropy of the universe is

36:07.568 --> 36:12.388
bigger than it was before,
that will happen.

36:12.389 --> 36:16.029
If the entropy of the universe
is smaller than it was before,

36:16.034 --> 36:17.314
it will not happen.

36:17.309 --> 36:20.999
Now, we have to make sure that
this law has anything whatsoever

36:20.997 --> 36:24.677
to do with all the other things
we have studied and I will show

36:24.684 --> 36:26.334
you;
this will forbid everything

36:26.326 --> 36:28.866
that should be forbidden and
allow everything that should be

36:28.874 --> 36:32.564
allowed.
So, let me start with the most

36:32.555 --> 36:34.795
obvious formulation.

36:34.800 --> 36:36.840
Mr.
Carnot's version of the Second

36:36.844 --> 36:40.314
Law was that you cannot have a
process in which some heat

36:40.314 --> 36:43.354
Q goes from a hot body to
a cold body.

36:43.349 --> 36:48.449
I'm sorry this allowed,
right, according to Carnot,

36:48.447 --> 36:50.687
this is not allowed.

36:50.690 --> 36:54.900


36:54.900 --> 36:59.060
Right, heat can flow downhill,
cannot flow up hill.

36:59.059 --> 37:03.049
Let's see the entropy change
the universe with the two cases.

37:03.050 --> 37:05.510
In this case,
with this guy,

37:05.505 --> 37:10.775
the change in entropy is the
following: the upper reservoir

37:10.781 --> 37:15.181
lost some amount Q,
so ∆Q is a negative

37:15.175 --> 37:18.205
number, add some temperature
T_1,

37:18.210 --> 37:22.590
the lower reservoir gained
Q at temperature

37:22.592 --> 37:24.652
T_2.

37:24.650 --> 37:26.630
Now, what's the overall sign of
this?

37:26.630 --> 37:29.140
Think about it.
T_1 is bigger

37:29.138 --> 37:31.348
than T_2,
so this negative number is

37:31.353 --> 37:32.933
smaller than this positive
number,

37:32.930 --> 37:34.400
so the whole thing is positive.

37:34.400 --> 37:39.080
That means it's allowed;
it's okay.

37:39.079 --> 37:41.789
By the same token,
if you take this process when

37:41.790 --> 37:44.620
heat flows up hill,
then this reservoir loses some

37:44.615 --> 37:47.725
heat Q at temperature
T_2,

37:47.730 --> 37:50.520
the other one gains heat
Q at temperature

37:50.517 --> 37:53.477
T_1,
but this is clearly less than

37:53.482 --> 37:56.632
zero because this positive
number is smaller than this

37:56.625 --> 37:59.475
negative number.
So, there you have a very

37:59.476 --> 38:02.446
simple example where if heat
flows the wrong way,

38:02.446 --> 38:04.856
the entropy of the universe
goes up.

38:04.860 --> 38:08.800


38:08.800 --> 38:11.610
And it's very simple,
it's only--it's because,

38:11.607 --> 38:15.287
it's true that the heat loss of
this guy is the heat gain of

38:15.288 --> 38:16.928
that guy,
that's the Law of the

38:16.925 --> 38:17.865
Conservation of Energy.

38:17.869 --> 38:20.289
So, energy doesn't change,
but entropy changes,

38:20.294 --> 38:22.884
because for entropy it's not
the heat transferred,

38:22.877 --> 38:25.457
it's heat transferred divided
by temperature.

38:25.460 --> 38:28.800
Therefore, a heat loss at one
temperature and a [equal]

38:28.797 --> 38:32.007
heat gain at a lower
temperature don't cancel when it

38:32.011 --> 38:33.311
comes to entropy.

38:33.309 --> 38:35.449
In one case the entropy goes up
and is allowed;

38:35.449 --> 38:37.269
other case entropy goes down,
that's not allowed.

38:37.270 --> 38:40.730


38:40.730 --> 38:43.740
Okay so let's try,
that's certainly one process

38:43.735 --> 38:47.455
that you know should not be
allowed, but it's forbidden by

38:47.459 --> 38:50.529
the Third Law,
by a computation of entropy.

38:50.530 --> 38:53.920
So, the reason when you put a
hot and cold body together,

38:53.924 --> 38:57.624
heat flows from the cold to the
hot--I'm sorry from the hot to

38:57.621 --> 38:59.361
the cold,
is because that's the way the

38:59.362 --> 39:00.022
entropy will go up.

39:00.020 --> 39:05.970


39:05.970 --> 39:10.100
So, let's take one more example.

39:10.099 --> 39:14.229
Suppose I have some mass of
water at some temperature

39:14.233 --> 39:18.133
T_1 and an
equal mass of water at

39:18.128 --> 39:21.068
temperature
T_2.

39:21.070 --> 39:24.900
One is hot and one is cold.

39:24.900 --> 39:29.920
I put them together,
what will happen?

39:29.920 --> 39:32.650
We believe it will then get
this big mass,

39:32.648 --> 39:36.108
all at some common temperature
T star which is

39:36.108 --> 39:38.968
T_1 +
T_2/2;

39:38.970 --> 39:40.600
this is just by symmetry.

39:40.599 --> 39:42.589
It's equal mass,
equal specific heat,

39:42.589 --> 39:43.859
where will they meet?

39:43.860 --> 39:45.380
They will meet half way.

39:45.380 --> 39:48.080
They will meet here.

39:48.079 --> 39:49.779
Energies, of course,
is conserved,

39:49.776 --> 39:52.806
and that's how we determine in
fact where they will meet.

39:52.810 --> 39:54.310
But look at the entropy.

39:54.309 --> 39:56.869
What happens with entropy
change?

39:56.869 --> 40:00.269
With entropy change,
you should imagine this water

40:00.267 --> 40:04.357
being steadily heated by putting
it in contact with a lot of

40:04.359 --> 40:06.639
reservoir,
so it's never far from

40:06.640 --> 40:09.620
equilibrium and slowly bringing
it to this point.

40:09.619 --> 40:15.109
Then, the ∆S total
will be M specific heat

40:15.111 --> 40:19.831
is 1, then ∆T over
T starting from

40:19.831 --> 40:26.001
T_1 to T
star for one thing and dT

40:25.996 --> 40:30.806
over T from
T_2 to T

40:30.813 --> 40:33.803
star for the other one.

40:33.800 --> 40:34.740
Do you understand that?

40:34.739 --> 40:36.629
They both meet at T
star, upper limit is T

40:36.634 --> 40:38.364
star, lower one is
T_1 for one and

40:38.356 --> 40:39.696
T_2 for the
other,

40:39.699 --> 40:44.159
so the change in entropy
becomes M log T

40:44.155 --> 40:47.005
star squared over
T_1T

40:47.014 --> 40:48.784
_2.

40:48.780 --> 40:54.550


40:54.550 --> 40:56.090
Now, we have to ask,
okay you got an entropy change

40:56.087 --> 40:57.467
from start to finish,
but how do you know it's

40:57.470 --> 40:57.870
positive?

40:57.870 --> 41:02.240


41:02.239 --> 41:04.799
Can anybody give some reason
why this has to be a positive

41:04.802 --> 41:07.412
without looking into properties
of logarithms and so on?

41:07.410 --> 41:11.700


41:11.700 --> 41:12.800
Any of these in, what--yes?

41:12.800 --> 41:17.920
Student: [inaudible]
Professor Ramamurti

41:17.917 --> 41:23.137
Shankar: Yeah,
that's one way to prove that.

41:23.139 --> 41:24.999
But I am saying,
I'm going to prove it to you

41:24.996 --> 41:27.236
that way.
But can you think of a reason

41:27.242 --> 41:30.442
why at every step of the
process, when I brought this

41:30.439 --> 41:34.189
down and when I pushed that up,
the entropy change is always

41:34.188 --> 41:35.548
positive in each step.

41:35.550 --> 41:37.060
So, let me explain why.

41:37.059 --> 41:40.379
Imagine cooling this down a
little bit, by putting,

41:40.384 --> 41:43.514
taking some heat ∆Q
out of this guy.

41:43.510 --> 41:46.500
That's happening at some
temperature here,

41:46.501 --> 41:50.221
this one is gaining some
∆Q, but at a lower

41:50.222 --> 41:53.402
temperature.
Okay.

41:53.400 --> 41:56.760
Therefore, the gain has got the
same ∆Q,

41:56.764 --> 42:00.484
but at a lower temperature,
this is always at an upper

42:00.478 --> 42:02.938
temperature.
Throughout the process,

42:02.935 --> 42:05.365
at every stage,
this guy's hotter than this

42:05.373 --> 42:07.303
guy.
So, every calorie of heat it

42:07.303 --> 42:10.293
gains at a higher temperature,
every calorie it loses is

42:10.294 --> 42:13.224
divided by higher temperature,
every calorie this gained is

42:13.219 --> 42:14.429
divided by lower temperature.

42:14.429 --> 42:17.639
At every increment,
at every infinitesimal step you

42:17.636 --> 42:20.646
can see the total entropy change
is positive.

42:20.650 --> 42:24.600
Now, to prove it,
we will do what he just said.

42:24.599 --> 42:27.409
If you want to prove this is
positive, you want to show that

42:27.410 --> 42:29.840
the fellow inside the logarithm
is bigger than 1,

42:29.840 --> 42:32.810
so I'm asking is
T_1 plus

42:32.814 --> 42:37.174
T_2^(2) over 4
bigger than T_1T

42:37.166 --> 42:38.686
_2?

42:38.690 --> 42:40.230
Or I'm saying is
T_1 plus

42:40.231 --> 42:42.261
T_2^(2) bigger
than 4T_1T

42:42.260 --> 42:43.050
_2?

42:43.050 --> 42:45.580
If you rearrange this,
you can show the left-hand side

42:45.577 --> 42:48.007
becomes T_1 -
T_2^(2);

42:48.010 --> 42:50.520
that's of course,
a positive number because this

42:50.524 --> 42:53.364
will be T_1^(2) +
T_2^(2) +

42:53.360 --> 42:55.500
2T_1T
_2.

42:55.500 --> 42:58.160
When I bring this guy to the
other side, it'll become minus

42:58.163 --> 43:00.283
2T_1T
_2 and so it's

43:00.275 --> 43:01.235
clearly positive.

43:01.239 --> 43:05.879
So, when hot and cold meet and
create lukewarm,

43:05.878 --> 43:09.708
entropy of the universe has
gone up.

43:09.710 --> 43:12.680
It follows, therefore,
if lukewarm spontaneously

43:12.675 --> 43:16.265
separated into hot and cold,
the entropy would go down and

43:16.271 --> 43:18.481
that's why that doesn't happen.

43:18.480 --> 43:21.200
That's why if you leave a jar
of water at one temperature,

43:21.202 --> 43:23.642
it doesn't spontaneously
separate into a part on top

43:23.637 --> 43:25.497
which is cold and a part on
bottom,

43:25.500 --> 43:27.870
which is hot.
Such a separation would not

43:27.867 --> 43:28.777
violate anything.

43:28.780 --> 43:32.260
It would not violate the Law of
Conservation of Energy,

43:32.255 --> 43:35.855
but it will violate the law
that entropy has to always go

43:35.859 --> 43:37.829
up.
So, one way is allowed another

43:37.827 --> 43:38.837
way is not allowed.

43:38.840 --> 43:41.800
So, you see over and over
again, that anything that's

43:41.800 --> 43:44.760
allowed is in the direction of
increasing entropy;

43:44.760 --> 43:47.970
anything that is forbidden is
in the direction of decreasing

43:47.966 --> 43:51.506
entropy.
But who is this entropy?

43:51.510 --> 43:52.740
What does it mean?

43:52.739 --> 43:55.209
Well, that we have not
understood at all from any of

43:55.212 --> 43:56.232
these calculations.

43:56.230 --> 43:59.370
Because dQ over T
doesn't speak to us the way

43:59.374 --> 44:01.764
dU or dQ or
PdV does.

44:01.760 --> 44:07.490
So, that last part of what I'm
going to do is going to explain

44:07.491 --> 44:12.191
to you what is the microscopic
basis of entropy.

44:12.190 --> 44:14.130
Why is the entropy going up?

44:14.130 --> 44:19.130
Why is--why do we understand
the tendency of things to go the

44:19.134 --> 44:21.974
direction of increasing entropy?

44:21.969 --> 44:25.759
All I've shown you now is that
it's a mystical quantity called

44:25.759 --> 44:29.299
entropy, circumstantially you
find every time something is

44:29.300 --> 44:31.880
forbidden,
it's because had it taken place

44:31.882 --> 44:33.542
entropy would have gone down.

44:33.540 --> 44:34.930
Do you remember the examples?

44:34.929 --> 44:38.089
Heat flowing from hot to cold
entropy goes up,

44:38.088 --> 44:40.398
allowed.
Flowing from cold to hot,

44:40.395 --> 44:42.445
entropy goes down,
not allowed.

44:42.449 --> 44:45.189
Hot and cold mix and become
lukewarm;

44:45.190 --> 44:47.590
allowed, entropy goes up.

44:47.590 --> 44:49.480
Luke warm separates into hot
and cold;

44:49.480 --> 44:51.630
not allowed, entropy goes down.

44:51.630 --> 44:53.490
So, there's definitely a
correlation.

44:53.489 --> 44:55.729
And yet we don't know what it
means.

44:55.730 --> 44:57.800
So, we're going to talk about
what it means.

44:57.800 --> 45:01.390
For that, I'm going to consider
the following process.

45:01.389 --> 45:08.699
I take a gas and I put it in a
room in a box where there's a

45:08.703 --> 45:11.433
partition, half way.

45:11.429 --> 45:15.099
The molecules are stuck on the
left side, gas has reached

45:15.096 --> 45:19.026
equilibrium and it's done what
it can, which is to spread out

45:19.025 --> 45:20.395
over this volume.

45:20.400 --> 45:23.370
Now, I suddenly remove the
partition, rip it out.

45:23.370 --> 45:26.870


45:26.870 --> 45:30.950
So, let's follow this gas.

45:30.950 --> 45:32.640
Initially, it is a point here.

45:32.640 --> 45:37.170


45:37.170 --> 45:40.870
Then, there is a period when it
goes off the radar because I

45:40.867 --> 45:43.497
told you when you suddenly
remove the wall,

45:43.499 --> 45:46.319
the gas is not in a state of
equilibrium.

45:46.320 --> 45:48.100
It doesn't even have a
well-defined pressure.

45:48.099 --> 45:50.659
The minute you remove the wall,
pressure is something here,

45:50.655 --> 45:52.105
pressure is zero in the vacuum.

45:52.110 --> 45:56.990
So, you got to wait a bit,
so gas has gone off the radar.

45:56.989 --> 45:58.839
We cannot talk about what's
happening.

45:58.840 --> 46:02.670
Then, if you wait long enough,
I get a gas that looks like

46:02.673 --> 46:06.173
this after some seconds,
or whatever milliseconds.

46:06.170 --> 46:08.460
That is again gas in
equilibrium.

46:08.460 --> 46:11.710
It's got a certain state;
I can call it 2.

46:11.710 --> 46:16.820
What is the entropy change now?

46:16.820 --> 46:19.260
That's what I want to ask.

46:19.260 --> 46:21.210
What's the change in entropy
here?

46:21.210 --> 46:26.510
Because I know that this is
allowed, that is forbidden,

46:26.513 --> 46:30.543
so I want to ask did the
entropy go up?

46:30.539 --> 46:33.649
Now, how do we do the entropy
calculation?

46:33.650 --> 46:35.620
Here's the wrong way to do the
calculation.

46:35.619 --> 46:39.399
You go back and say ∆S
at ∆Q over T,

46:39.396 --> 46:42.476
but this whole box I forgot to
mention--I'm sorry,

46:42.479 --> 46:45.059
this whole box is thermally
isolated.

46:45.059 --> 46:47.419
It's not in contact with
anything.

46:47.420 --> 46:51.230
I just ripped out the partition
in the middle;

46:51.230 --> 46:53.480
that's it.
So, you might say,

46:53.482 --> 46:57.072
well, if it's thermally
isolated, ∆Q = 0 so you

46:57.066 --> 47:00.016
can divide by T,
you can do what you want,

47:00.018 --> 47:01.628
so this entropy change is zero.

47:01.630 --> 47:03.090
But that is a wrong argument.

47:03.090 --> 47:06.100


47:06.099 --> 47:08.899
Can you think about why that's
not the way to do the entropy

47:08.902 --> 47:11.042
change?
Why that's the wrong analysis?

47:11.040 --> 47:15.460


47:15.460 --> 47:17.560
First of all,
could that be the right answer?

47:17.560 --> 47:21.250
Yes.
Student: It can't

47:21.246 --> 47:25.346
because then that would mean
that going backwards is not a

47:25.347 --> 47:27.267
valid process.
Professor Ramamurti

47:27.265 --> 47:28.275
Shankar: No,
that is correct,

47:28.276 --> 47:29.726
but I'm saying this computation
of entropy.

47:29.730 --> 47:33.380
The one line calculation I did,
namely there is no heat inflow,

47:33.382 --> 47:36.212
the ∆Q = 0,
so ∆Q over T

47:36.209 --> 47:37.799
summed up is also zero.

47:37.800 --> 47:39.190
That's not how you do the
entropy change.

47:39.190 --> 47:42.790


47:42.789 --> 47:45.029
Is there any condition I made
on computing?

47:45.030 --> 47:47.140
Yes.
Student: Well,

47:47.144 --> 47:49.144
uhm… there's a gas
[inaudible]

47:49.142 --> 47:52.242
Professor Ramamurti
Shankar: Wait in fact,

47:52.237 --> 47:54.427
here's the interesting thing.

47:54.429 --> 47:57.449
What's going to be the
temperature difference between

47:57.447 --> 47:58.547
before and after?

47:58.550 --> 48:00.410
Which is going to be hotter or
cooler?

48:00.410 --> 48:04.350


48:04.350 --> 48:06.450
Any views on this?

48:06.450 --> 48:11.000
Yep.
Student: [inaudible]

48:10.995 --> 48:13.715
Professor Ramamurti
Shankar: Ah,

48:13.719 --> 48:17.879
well, you can say it is cooler
because it was insulated and

48:17.876 --> 48:20.936
expanded,
but you cool down because you

48:20.937 --> 48:24.057
expand against an external
pressure, right?

48:24.059 --> 48:26.429
If, when--If I were to move
this piston here,

48:26.433 --> 48:28.863
there is no pressure pushing
this gas back.

48:28.860 --> 48:30.150
Do you understand that?

48:30.150 --> 48:33.920
Doesn't do any work,
doesn't take any heat.

48:33.920 --> 48:35.720
So, what does that mean?

48:35.719 --> 48:39.329
In the end when you settle
down, what can you conclude?

48:39.329 --> 48:41.459
No heat input,
no work done,

48:41.458 --> 48:45.318
so what does that mean in terms
of temperature?

48:45.320 --> 48:46.450
Student: Same.

48:46.449 --> 48:48.449
Professor Ramamurti
Shankar: Same.

48:48.449 --> 48:50.549
Because the internal energy
cannot change because

48:50.545 --> 48:52.155
∆Q = 0,
work done is zero.

48:52.160 --> 48:54.490
So, in fact, this is a surprise.

48:54.489 --> 48:57.759
This gas when it expands will
be at the same temperature.

48:57.760 --> 48:59.860
You guys are thinking of the
computer air,

48:59.861 --> 49:02.681
where if you let it expand it
cools down because that is

49:02.680 --> 49:05.090
expanding against the
atmospheric pressure.

49:05.090 --> 49:07.940
This gas is expanding into a
complete vacuum.

49:07.940 --> 49:10.630
It's got no piston to push
against, nothing from outside

49:10.625 --> 49:13.355
pushing, so these two points are
the same temperature.

49:13.360 --> 49:16.650


49:16.650 --> 49:19.150
So, the correct answer,
maybe you didn't catch on,

49:19.152 --> 49:22.272
but I will tell you the correct
answer is: There is an entropy

49:22.268 --> 49:25.268
increase in this problem,
because the final state

49:25.265 --> 49:29.015
definitely has a well-defined
entropy you can calculate,

49:29.019 --> 49:30.639
because it's an equilibrium
state.

49:30.639 --> 49:34.039
Initial one has a well-defined
entropy, in between stages here,

49:34.036 --> 49:36.936
are not on the PV
diagram because they were not

49:36.940 --> 49:38.200
equilibrium states.

49:38.199 --> 49:40.889
So in this experiment,
the way to calculate the

49:40.888 --> 49:44.278
entropy change is not to do
∆Q over T as it

49:44.279 --> 49:46.149
happened in this experiment.

49:46.150 --> 49:49.410
What you really want to say is
my gas was here in the

49:49.405 --> 49:51.905
beginning, my gas is there in
the end,

49:51.909 --> 49:54.449
both are states of equilibrium,
both have well-defined

49:54.447 --> 49:57.217
temperature and I can find the
entropy change in going from

49:57.224 --> 49:59.144
here to here for any process I
want.

49:59.139 --> 50:02.989
As long as that process keeps
the system near equilibrium,

50:02.993 --> 50:06.643
so I can follow dot by dot
where I'm moving and add the

50:06.644 --> 50:08.744
∆Q over Ts.

50:08.739 --> 50:12.539
The correct rule for entropy
change is ∆Q/T computed

50:12.541 --> 50:16.471
on a path in which the system
never strays from equilibrium.

50:16.469 --> 50:19.879
Now, your system strayed from
equilibrium all the time.

50:19.880 --> 50:22.390
But we're not interested in how
it got from here to here.

50:22.389 --> 50:25.109
We are saying what's the
entropy here and what's the

50:25.110 --> 50:27.080
entropy here.
Since this at the same

50:27.082 --> 50:30.182
temperature, I know one way to
go from there to there is to

50:30.175 --> 50:33.205
follow the isotherm because
that'll connect the two points

50:33.213 --> 50:34.923
in this particular problem.

50:34.920 --> 50:38.400
Then I know the entropy change
is S_2 -

50:38.402 --> 50:41.952
S_1 is nR log
V_2 over

50:41.952 --> 50:43.662
V_1.

50:43.660 --> 50:54.750


50:54.750 --> 50:57.960
So, here is the subtle point I
want to explain to you.

50:57.960 --> 50:59.070
This is worth understanding.

50:59.070 --> 51:01.450
A lot of people do all the
problems and get everything

51:01.445 --> 51:03.905
right, but don't appreciate this
particular question.

51:03.909 --> 51:08.179
The free expansion of a gas
into a vacuum takes it from one

51:08.178 --> 51:12.668
equilibrium state called 1 to
another equilibrium state called

51:12.668 --> 51:14.568
2.
In the actual expansion,

51:14.565 --> 51:18.195
the system went through a stage
which cannot be even shown in

51:18.203 --> 51:20.813
the PV diagram,
they were not equilibrium

51:20.805 --> 51:22.795
states, they didn't have well
defined pressure,

51:22.803 --> 51:24.543
didn't have well defined
temperature.

51:24.539 --> 51:27.219
How do you find a temperature
when half the box is empty?

51:27.219 --> 51:28.989
Doesn't have it,
you cannot it T

51:28.987 --> 51:30.147
temperature of the gas.

51:30.150 --> 51:32.280
So, what do you mean by
∆Q over T,

51:32.280 --> 51:33.280
there is no T.

51:33.280 --> 51:35.520
Only when it settles down,
there is a T.

51:35.519 --> 51:39.279
Every settled down equilibrium
state is a well-defined entropy.

51:39.280 --> 51:41.590
What we're trying to do is to
compute that difference.

51:41.590 --> 51:45.380
The way to compute the
difference is to forget about

51:45.383 --> 51:49.403
what actually happened and
instead do the following.

51:49.400 --> 51:55.280
Take the gas like that and turn
it this way, [90 degree

51:55.284 --> 51:57.904
rotation]
if you like.

51:57.900 --> 52:01.260
Just for convenience,
put a piston there at the

52:01.261 --> 52:04.481
halfway point,
put some weights on it so that

52:04.476 --> 52:09.076
the pressure equals the pressure
inside and slowly remove little

52:09.080 --> 52:12.420
by little,
and keeping it on a reservoir a

52:12.415 --> 52:14.785
temperature
T_1.

52:14.789 --> 52:17.849
Which is the temperature at
which all of this happens,

52:17.848 --> 52:19.808
or at some temperature
T.

52:19.809 --> 52:24.429
As you remove the grain of
sand, what happens is the gas

52:24.429 --> 52:28.119
gets some energy,
that's the ∆Q over

52:28.124 --> 52:31.824
T, the reservoir loses
some heat;

52:31.820 --> 52:35.430
in fact, the reservoir loses
the same amount of heat

52:35.426 --> 52:38.746
∆Q at the same
temperature T.

52:38.750 --> 52:42.580
So in fact, what happens in
this process is entropy of the

52:42.583 --> 52:44.873
universe does not change at all.

52:44.870 --> 52:48.930
The entropy gain of the gas;
this is an entropy loss of the

52:48.932 --> 52:51.702
reservoir.
But I used this as a device for

52:51.699 --> 52:55.439
finding the entropy gain of the
gas in the other process.

52:55.440 --> 53:00.680


53:00.679 --> 53:06.219
So, this is the part that is
absolutely central to this whole

53:06.222 --> 53:09.642
argument how entropy is
calculated.

53:09.639 --> 53:13.169
In a real spontaneous expansion
of a gas, where there was no

53:13.165 --> 53:16.315
reservoir, no nothing,
we just broke the partition and

53:16.320 --> 53:18.540
expanded, there is a change in
entropy.

53:18.540 --> 53:21.220
That's an irreversible process.

53:21.219 --> 53:24.569
But to find the change in
entropy in this problem,

53:24.570 --> 53:27.580
since 1 and 2 have well-defined
entropies,

53:27.579 --> 53:31.909
we can find the change between
1 and 2 by going to a different

53:31.911 --> 53:35.391
experiment in which it was
brought from 1 to 2,

53:35.389 --> 53:37.879
not in this crazy irreversible
way, but by putting it in

53:37.882 --> 53:40.512
contact with the heat bath at
that temperature T,

53:40.510 --> 53:43.290
removing grain after grain,
never straying far from

53:43.293 --> 53:45.913
equilibrium, that's when we find
this change.

53:45.909 --> 53:49.699
So, you replace the dotted
line, where you cannot compute

53:49.704 --> 53:52.554
anything, by a solid line where
you can.

53:52.550 --> 53:55.820
But since the change in entropy
between 1 and 2 doesn't depend

53:55.819 --> 53:58.069
on how you got there,
that's the answer.

53:58.070 --> 54:00.920
But this process,
where you can actually see it

54:00.921 --> 54:04.701
is the one in which ∆S
of the universe is actually at

54:04.702 --> 54:06.202
zero.
Why?

54:06.199 --> 54:09.109
Because you are putting in
contact with the reservoir at

54:09.105 --> 54:11.895
the same temperature,
so ∆Q over T

54:11.896 --> 54:15.156
for the gas and ∆Q over
T for the reservoir

54:15.162 --> 54:18.372
exactly cancel because they're
at the same temperature and

54:18.371 --> 54:21.751
∆Q for one is minus
∆Q for the other.

54:21.750 --> 54:26.390
So, don't confuse the two
processes.

54:26.389 --> 54:29.879
One is a reversible process in
which a gas is slowly allowed to

54:29.878 --> 54:33.198
expand from the initial state of
the final stage by removing

54:33.198 --> 54:36.408
grain after grain of some
externally applied pressure in a

54:36.405 --> 54:39.945
process where the entropy of the
universe doesn't change,

54:39.949 --> 54:42.629
gas and reservoir change by
equal amounts.

54:42.630 --> 54:45.510
But the punch line is I don't
care what happens in reservoir,

54:45.511 --> 54:47.481
I want to know how much the gas
gained,

54:47.480 --> 54:50.520
because that is the gain I need
to find S_2 -

54:50.517 --> 54:52.487
S_1 for this
process.

54:52.489 --> 54:56.089
So, this is how we always find
entropy change.

54:56.090 --> 54:59.730
In real life processes happen
in a way in which entropy of the

54:59.734 --> 55:00.874
universe goes up.

55:00.869 --> 55:03.859
Because take this box,
in the real experiment,

55:03.864 --> 55:07.994
remember there is no reservoir,
there's only the box of gas.

55:07.989 --> 55:09.829
It expanded,
its entropy went up,

55:09.827 --> 55:12.467
so entropy of the universe
actually went up.

55:12.470 --> 55:16.540
That's why it's allowed.

55:16.539 --> 55:20.379
The entropy of the universe
went up by an amount equal to

55:20.379 --> 55:24.629
the increase in entropy of the
gas when it goes from 1 to 2.

55:24.630 --> 55:28.400
To find the increase in entropy
of the gas when it goes from 1

55:28.399 --> 55:31.919
to 2, I create a second process
in which I can compute the

55:31.921 --> 55:35.511
change by letting it go in a
sequence of connected dots and

55:35.505 --> 55:37.045
finding the change.

55:37.050 --> 55:40.840
So, the way you use to find the
change in entropy is not the way

55:40.840 --> 55:44.090
the entropy increase took place
in the real process.

55:44.090 --> 55:46.250
So, I don't think I want to
repeat it anymore,

55:46.245 --> 55:49.115
but it's an important point and
you should really think about

55:49.118 --> 55:53.028
that.
Alright, now the last point is,

55:53.034 --> 55:59.784
what does--Let's analyze this
final formula I got for entropy

55:59.778 --> 56:04.048
change for free expansion of a
gas.

56:04.050 --> 56:07.730


56:07.730 --> 56:12.610
Let's take a simple case where
the gas expanded from some

56:12.614 --> 56:16.544
volume V to volume 2
times V.

56:16.539 --> 56:19.529
What is S_2 -
S_1?

56:19.530 --> 56:25.040
S_2 -
S_1 = nR

56:25.041 --> 56:27.601
log 2.
2 happens to be

56:27.601 --> 56:30.121
V_2/V
_1;

56:30.120 --> 56:31.760
the volume is doubled.

56:31.760 --> 56:38.120
Let me re-write this as Nk
log 2, because the number of

56:38.116 --> 56:43.536
moles times R is the
number of molecules times

56:43.535 --> 56:47.565
k.
Let me re-write that as log of

56:47.571 --> 56:50.611
2 to the N times
k.

56:50.610 --> 56:55.680


56:55.679 --> 57:02.399
Now, I will tell you what all
these factors mean and that

57:02.400 --> 57:05.640
comes from the following.

57:05.639 --> 57:09.289
Thermodynamics only told you
the formula for the change in

57:09.289 --> 57:12.169
the entropy of a gas,
didn't tell you what the

57:12.171 --> 57:13.581
entropy itself is.

57:13.579 --> 57:17.519
So, the formula for the entropy
of a gas, without talking about

57:17.524 --> 57:21.534
the change and also this formula
∆S is ∆Q over

57:21.532 --> 57:23.702
T,
doesn't even believe in atoms.

57:23.699 --> 57:25.359
You don't need atoms,
you don't need molecules,

57:25.363 --> 57:27.283
you don't need to know what
anything is made up of.

57:27.280 --> 57:30.760
In fact, it was discovered long
before atoms were proven to

57:30.760 --> 57:33.950
exist.
Boltzmann is the guy who gave

57:33.948 --> 57:39.918
the formula for what the entropy
of a gas is in terms of what the

57:39.924 --> 57:43.104
individual molecules are doing.

57:43.099 --> 57:46.659
That's the punch line of what's
called statistical mechanics,

57:46.659 --> 57:50.159
which gives you the microscopic
origin of thermodynamics.

57:50.159 --> 57:53.549
Thermodynamics to this stage
talk about fluids and solids and

57:53.553 --> 57:57.003
gases and nothing there requires
that everything be made up of

57:57.004 --> 57:58.684
atoms,
you don't have to know what

57:58.676 --> 58:00.986
it's made of.
But if you tell me it's made up

58:00.991 --> 58:04.481
of atoms, you have a better idea
of what entropy means and here

58:04.475 --> 58:06.605
is Boltzmann's formula for
entropy.

58:06.610 --> 58:08.610
I am going to write it down.

58:08.610 --> 58:14.740
Boltzmann constant times log
Ω.

58:14.739 --> 58:16.769
I'll tell you what omega is in
a minute.

58:16.769 --> 58:20.399
This is such an important
formula it is written in

58:20.401 --> 58:22.181
Boltzmann's tombstone.

58:22.179 --> 58:24.939
So, when physicists go to
Vienna, we skip the orchestras

58:24.944 --> 58:27.984
and everything else,
we go to Boltzmann's tomb and

58:27.977 --> 58:31.297
we read this formula once more
with some passion.

58:31.300 --> 58:33.720
That's a summary of a lifetime
of work.

58:33.719 --> 58:36.299
But it's a great formula and
you will understand once you

58:36.295 --> 58:38.085
know the formula what all this
means.

58:38.090 --> 58:41.880
So, let me tell you what omega
means.

58:41.880 --> 58:43.940
I am going to say it in words
and then I am going to apply it

58:43.941 --> 58:46.081
to this problem.
If you took any gas,

58:46.077 --> 58:50.167
let it be under certain
conditions that macroscopically

58:50.174 --> 58:53.744
has some pressure and some
volume and so on.

58:53.739 --> 58:57.549
So, here is the gas at one
instant;

58:57.550 --> 59:00.430
now we believe in atoms,
here are these atoms scattered

59:00.429 --> 59:01.549
all over the place.

59:01.550 --> 59:05.260


59:05.260 --> 59:11.420
Suppose I wait a little bit,
what did the gas look like?

59:11.420 --> 59:12.590
It looks like that.

59:12.590 --> 59:17.530
If you can see every atom,
the atomic configuration has

59:17.534 --> 59:20.994
changed.
But macroscopically nothing has

59:20.992 --> 59:23.262
changed.
Instead of atom A being

59:23.261 --> 59:26.121
here, maybe A has gone
there and B has come

59:26.121 --> 59:27.271
here.
In particular,

59:27.274 --> 59:30.524
if you divide the volume into
some number of little cells that

59:30.516 --> 59:33.366
you can count,
on the average the gas' uniform

59:33.368 --> 59:37.198
density, there are some number
of atoms, per every little cube

59:37.204 --> 59:40.274
you can form,
a little later I move on and

59:40.266 --> 59:42.026
somebody else moves in.

59:42.030 --> 59:44.130
Macroscopically,
it looks the same to me.

59:44.130 --> 59:46.010
Microscopically,
stuff is going on.

59:46.010 --> 59:51.100
Omega is the number of
different microscopic

59:51.099 --> 59:58.199
arrangements that agree with
what is seen macroscopically.

59:58.199 --> 1:00:00.239
When I say microscopic
arrangements,

1:00:00.238 --> 1:00:03.438
I really mean the following:
give every molecule a name,

1:00:03.442 --> 1:00:05.662
A, B,
C, D,

1:00:05.656 --> 1:00:07.776
whatever.
Then put A here and

1:00:07.777 --> 1:00:09.697
B here,
C here and D

1:00:09.697 --> 1:00:11.687
there;
that's one arrangement.

1:00:11.690 --> 1:00:13.350
Then permute them,
put D here and B

1:00:13.349 --> 1:00:14.769
here and C here and
A there,

1:00:14.767 --> 1:00:15.837
that's another arrangement.

1:00:15.840 --> 1:00:17.960
To my naked eye they,
all look the same,

1:00:17.959 --> 1:00:20.569
but they all produce the same
macroscopic effect,

1:00:20.568 --> 1:00:23.338
but they're different
microscopic arrangements.

1:00:23.340 --> 1:00:27.560
And you're supposed to take the
log of the number of these

1:00:27.556 --> 1:00:30.586
arrangements multiply by this
constant,

1:00:30.590 --> 1:00:33.670
called Boltzmann's constant,
to get the entropy of the

1:00:33.665 --> 1:00:36.415
state.
So, now let's apply it to this

1:00:36.417 --> 1:00:37.177
gas here.

1:00:37.180 --> 1:00:40.640


1:00:40.639 --> 1:00:43.609
Now, the number of microscopic
arrangements,

1:00:43.608 --> 1:00:46.298
if you really want to get down
to it,

1:00:46.300 --> 1:00:49.080
requires dividing the box into
tiny little cells,

1:00:49.079 --> 1:00:52.319
maybe 1 cubic millimeter and
saying how many atoms are in

1:00:52.323 --> 1:00:54.773
each region.
But we're going to do a crude

1:00:54.769 --> 1:00:57.339
calculation in which we'll only
say the following:

1:00:57.336 --> 1:01:00.476
that either a molecule is on
the left or is on the right.

1:01:00.480 --> 1:01:04.620
We are saying these are the
only two positions open to the

1:01:04.623 --> 1:01:06.283
gas;
left or right.

1:01:06.280 --> 1:01:08.720
Of course, in the left,
there is left corner top and

1:01:08.715 --> 1:01:10.955
left corner bottom,
but don't worry about it.

1:01:10.960 --> 1:01:13.680
Just say that only two things
it can do, it can be on the left

1:01:13.681 --> 1:01:14.931
or it can be on the right.

1:01:14.929 --> 1:01:20.739
Now, let's ask ourselves if the
gas looks like this.

1:01:20.740 --> 1:01:25.170


1:01:25.170 --> 1:01:29.750
How many ways can it be in this
arrangement?

1:01:29.750 --> 1:01:34.620
As compared to how many ways it
can be in this arrangement?

1:01:34.620 --> 1:01:40.530


1:01:40.530 --> 1:01:44.350
It is like the following:
molecules are moving randomly,

1:01:44.352 --> 1:01:48.522
a given molecule has got 50/50
chance of being on the left or

1:01:48.521 --> 1:01:50.121
being on the right.

1:01:50.119 --> 1:01:53.889
You are demanding that all the
N molecules on their own,

1:01:53.888 --> 1:01:56.748
there's no partition now,
on their own be on the

1:01:56.745 --> 1:02:00.825
left-hand side.
You can see it's like tossing a

1:02:00.827 --> 1:02:05.547
coin N times and wanting
heads every time.

1:02:05.550 --> 1:02:09.900
Because you want every guy in a
random process to end up picking

1:02:09.896 --> 1:02:12.146
left.
So, if you want them all to be

1:02:12.153 --> 1:02:15.823
left, there's only one way to do
that, and that's like saying I

1:02:15.820 --> 1:02:18.540
tossed a million coins,
I wanted all heads.

1:02:18.540 --> 1:02:20.530
There's only one way to do that.

1:02:20.530 --> 1:02:23.150
So, that arrangement,
if you like,

1:02:23.148 --> 1:02:26.718
we will say looks like this,
left, left, left,

1:02:26.719 --> 1:02:28.969
left, left;
for everybody,

1:02:28.968 --> 1:02:31.158
that's the arrangement here.

1:02:31.159 --> 1:02:36.219
Take another one where one
molecule is over here,

1:02:36.222 --> 1:02:41.182
and all the others are here on
the left side.

1:02:41.179 --> 1:02:44.579
That looks like left,
left, left, right,

1:02:44.582 --> 1:02:46.242
left, left, left.

1:02:46.240 --> 1:02:48.820
But that's one arrangement.

1:02:48.820 --> 1:02:51.420
But have I done my complete
homework here?

1:02:51.420 --> 1:02:52.740
Are there more arrangements?

1:02:52.740 --> 1:02:54.170
I want you to think about it.

1:02:54.170 --> 1:02:56.490
Student: [inaudible]
arrangements Professor

1:02:56.489 --> 1:02:58.049
Ramamurti Shankar: Pardon
me.

1:02:58.050 --> 1:02:59.890
Student: [inaudible]
arrangements.

1:02:59.889 --> 1:03:01.219
Professor Ramamurti
Shankar: This is only one of

1:03:01.221 --> 1:03:02.431
the arrangements for the
molecule on the right.

1:03:02.429 --> 1:03:06.529
There's another arrangement in
which I can have the second

1:03:06.526 --> 1:03:10.476
molecule, molecule Moe or Joe or
Poe, whoever's here.

1:03:10.480 --> 1:03:12.300
You can pick them in N
ways.

1:03:12.300 --> 1:03:15.150
Already when one is on the left
and N - 1 on the right,

1:03:15.146 --> 1:03:17.336
there are more ways in which
that can happen.

1:03:17.340 --> 1:03:20.350
So, your eye sees one on the
right, N - 1 on the left.

1:03:20.349 --> 1:03:22.559
There are N ways in
which that can happen.

1:03:22.559 --> 1:03:27.859
It's like saying if I toss
coins, I want 1 head and 99

1:03:27.860 --> 1:03:30.930
tails.
Well, it turns out there are a

1:03:30.933 --> 1:03:35.333
100 ways in which it can happen,
because the one head I got

1:03:35.327 --> 1:03:39.947
could have occurred in one of
the hundred different turns.

1:03:39.949 --> 1:03:45.709
Now, suppose I want 2 heads and
98 tails, then as you know the

1:03:45.710 --> 1:03:50.810
way to do that is 100 times 99
divided by 1 times 2.

1:03:50.809 --> 1:03:53.289
These are some combinatorics
you learn in high school.

1:03:53.289 --> 1:03:59.659
So, you find that if you plot
number heads over the total

1:03:59.655 --> 1:04:06.585
number, and ask for how many
ways in which that can happen,

1:04:06.590 --> 1:04:11.350
all heads can occur in only
way, every coin has to be head,

1:04:11.350 --> 1:04:14.720
1 head less,
1 tail and 99 heads can occur

1:04:14.716 --> 1:04:17.926
in N ways,
then it'll become more and more

1:04:17.929 --> 1:04:20.129
and more.
And you will find the 50/50

1:04:20.134 --> 1:04:23.714
chance of half the number of
heads and half the number of

1:04:23.711 --> 1:04:26.331
tails has the biggest occurrence
rate,

1:04:26.330 --> 1:04:30.280


1:04:30.280 --> 1:04:33.320
because that's the most number
of ways to get 50/50 than

1:04:33.316 --> 1:04:36.916
anything else.
And 100 heads is as bad as 100

1:04:36.916 --> 1:04:38.656
tails.
Again, there's only one way in

1:04:38.658 --> 1:04:40.838
which you can get all heads in
one way and which you can get

1:04:40.839 --> 1:04:42.639
all tails.
But as you start scrambling

1:04:42.639 --> 1:04:44.349
them there are more and more
ways.

1:04:44.349 --> 1:04:48.959
That means for the molecule,
if you have a 100 molecules,

1:04:48.960 --> 1:04:53.650
the number of states omega is
largest when they're equally

1:04:53.652 --> 1:04:56.492
distributed.
In fact, as the number

1:04:56.492 --> 1:05:00.302
molecules goes from 100 to 1000
to million to 10^(23),

1:05:00.300 --> 1:05:04.250
this function is so sharply
peaked that the approximate

1:05:04.246 --> 1:05:07.896
formula for omega,
when it's at the 50/50 thing,

1:05:07.900 --> 1:05:09.590
is in fact 2 to the N.

1:05:09.590 --> 1:05:13.420
Namely, all the arrangements
more or less belong to this

1:05:13.416 --> 1:05:16.586
configuration.
Therefore, if you go back to

1:05:16.586 --> 1:05:20.546
the Boltzmann formula and,
wait let me keep the formula

1:05:20.552 --> 1:05:23.312
here,
and look at S = K log

1:05:23.314 --> 1:05:27.844
Ω, then you find
S_1 is when

1:05:27.843 --> 1:05:32.863
everybody is on the left,
that is k log 1,

1:05:32.862 --> 1:05:39.122
and S_2 is
50/50 mix, half on the left and

1:05:39.123 --> 1:05:44.563
half on the right,
that entropy is k log

1:05:44.561 --> 1:05:48.341
2^(N),.
Therefore, you can see that's

1:05:48.340 --> 1:05:52.000
what I wrote down here,
K log 2^(N).

1:05:52.000 --> 1:05:57.200


1:05:57.199 --> 1:06:01.689
So, the way to understand
entropy is the following.

1:06:01.690 --> 1:06:07.950
If I give you a gas and ask,
"Will they ever like to be in

1:06:07.954 --> 1:06:12.464
the left half of the box on
their own?"

1:06:12.460 --> 1:06:13.450
Of course they won't.

1:06:13.449 --> 1:06:16.189
But if you start them out that
way, you force them to be on the

1:06:16.186 --> 1:06:17.506
left they'll be on the left.

1:06:17.510 --> 1:06:20.330
The question is if you remove
the partition,

1:06:20.326 --> 1:06:22.156
what would they likely do?

1:06:22.159 --> 1:06:24.079
Unlike coins,
which once they land on the

1:06:24.084 --> 1:06:26.974
head, they have to stay on the
head, these molecules can move

1:06:26.970 --> 1:06:29.120
around.
The question is if they were

1:06:29.120 --> 1:06:31.960
all left to begin with,
how long can they possibly

1:06:31.956 --> 1:06:34.306
last?
If they start moving randomly,

1:06:34.310 --> 1:06:37.900
they will always end up doing
this because there are many,

1:06:37.901 --> 1:06:40.801
many more ways to do this,
than to do this.

1:06:40.800 --> 1:06:44.050


1:06:44.050 --> 1:06:47.230
So, things go from one
arrangement, macroscopic to

1:06:47.226 --> 1:06:49.816
another one, left to their own
device,

1:06:49.820 --> 1:06:53.650
because the final state can be
realized in many,

1:06:53.650 --> 1:06:56.910
many more ways than the initial
state.

1:06:56.909 --> 1:07:02.089
Similarly, if you combine a hot
gas and cold gas--This is hot

1:07:02.089 --> 1:07:05.359
and this is cold,
with a thermally insulating

1:07:05.355 --> 1:07:09.025
partition, then all the hots are
on the left and all the colds

1:07:09.026 --> 1:07:10.226
are on the right.

1:07:10.230 --> 1:07:13.040
If you remove the thermal
insulation, but not even remove

1:07:13.037 --> 1:07:15.337
it, just replace it by a
heat-conducting thing,

1:07:15.343 --> 1:07:17.703
eventually the temperatures
would equalize.

1:07:17.699 --> 1:07:20.679
So, what you find every time
is, there's many,

1:07:20.676 --> 1:07:24.706
many ways in which you can pack
your atoms--if you put all the

1:07:24.712 --> 1:07:28.022
hot on the left and all the cold
on the right,

1:07:28.019 --> 1:07:30.329
that can be achieved in fewer
ways than in which you let them

1:07:30.329 --> 1:07:31.179
go wherever they like.

1:07:31.176 --> 1:07:33.176
[He should have said,
"If you put all hot ones on one

1:07:33.178 --> 1:07:34.818
side and cold on the other,
there are fewer ways of doing

1:07:34.819 --> 1:07:35.529
that, than if you let them go
wherever they liked"]

1:07:35.525 --> 1:07:37.625
So,
what you find is entropy is the

1:07:37.632 --> 1:07:40.832
direct measure of how disordered
your system is.

1:07:40.829 --> 1:07:44.499
One technical measure of
disorder is to ask how many

1:07:44.498 --> 1:07:47.878
microscopic arrangements can
lead to what I see,

1:07:47.880 --> 1:07:50.470
and take the log of that
number.

1:07:50.469 --> 1:07:53.919
And what you find is this has a
very low entropy;

1:07:53.920 --> 1:07:55.870
this is a very high entropy.

1:07:55.869 --> 1:07:59.119
Because this configuration--In
other words, if you took a gas

1:07:59.122 --> 1:08:01.722
in a whole room and asked will
it ever go to this

1:08:01.724 --> 1:08:05.454
configuration,
there is a slight chance it

1:08:05.449 --> 1:08:07.679
will.
That chance is one part and

1:08:07.677 --> 1:08:10.007
2^(N),
where N is 10^(23),

1:08:10.014 --> 1:08:12.474
so it's not something you
should wait for,

1:08:12.471 --> 1:08:13.851
but it can happen.

1:08:13.849 --> 1:08:17.069
So, the Second Law of
Thermodynamics is a statistical

1:08:17.068 --> 1:08:18.498
law.
Microscopically,

1:08:18.496 --> 1:08:22.456
it's perfectly allowed for a
gas, for suddenly all the gas in

1:08:22.458 --> 1:08:25.098
the whole room to come to where
I am.

1:08:25.100 --> 1:08:28.120
It's allowed but you don't hold
your breath, because that's not

1:08:28.123 --> 1:08:29.053
likely to happen.

1:08:29.050 --> 1:08:32.340
The odds for that is again one
part in 1 over 2 to some huge

1:08:32.338 --> 1:08:33.688
number.
On the other hand,

1:08:33.685 --> 1:08:35.885
if in this part of the room we
release some gas,

1:08:35.894 --> 1:08:38.714
it'll very quickly spread out,
because there are more ways to

1:08:38.713 --> 1:08:41.683
do it.
So, we understand completely

1:08:41.684 --> 1:08:45.984
now why certain things occur and
why they don't.

1:08:45.979 --> 1:08:48.909
Because if you took hot water
and cold water,

1:08:48.911 --> 1:08:53.111
you are separating them to fast
molecules and slow molecules,

1:08:53.109 --> 1:08:55.429
as long as they're in different
insulated containers,

1:08:55.430 --> 1:08:56.770
that's the best they can do.

1:08:56.770 --> 1:08:59.710
But if you put them in contact,
so that they no longer have to

1:08:59.708 --> 1:09:01.538
be separated,
the question is will they

1:09:01.538 --> 1:09:02.548
remain separated.

1:09:02.550 --> 1:09:05.590
It's like saying I have bunch
of coins which are all left and

1:09:05.590 --> 1:09:08.630
all right, but they're allowed
to flip as a function time,

1:09:08.630 --> 1:09:10.440
but they'll never remain that
way.

1:09:10.439 --> 1:09:13.409
Okay, so what'll happen is the
cold molecules and hot will

1:09:13.412 --> 1:09:16.492
mingle, and soon the container
will have cold everywhere and

1:09:16.488 --> 1:09:19.928
hot everywhere and you will get
some intermediate temperature.

1:09:19.930 --> 1:09:22.830
Or if you took two different
dyes, you know water here,

1:09:22.827 --> 1:09:24.917
on top of it you pour some red
paint,

1:09:24.920 --> 1:09:27.890
they'll be initially separated
after a while they will mix,

1:09:27.894 --> 1:09:30.924
because there's no reason for
the hot--for the red molecules

1:09:30.920 --> 1:09:33.890
to stay on the top forever;
they would like to occupy the

1:09:33.892 --> 1:09:35.852
whole box and so would the
colorless ones,

1:09:35.847 --> 1:09:37.657
and eventually it will become
pink.

1:09:37.659 --> 1:09:41.169
Because there's more ways to
remain pink than to remain

1:09:41.170 --> 1:09:44.050
separated.
Likewise, if pink spontaneously

1:09:44.050 --> 1:09:47.170
separated into red and
colorless, that is very,

1:09:47.167 --> 1:09:50.277
very improbable,
because entropy for that state

1:09:50.284 --> 1:09:53.494
would be lower.
So, you have to understand the

1:09:53.485 --> 1:09:57.045
Second Law of Thermodynamics is
saying certain things occur

1:09:57.052 --> 1:10:00.432
because if you go in that
direction you can realize that

1:10:00.434 --> 1:10:02.284
arrangement in more ways.

1:10:02.279 --> 1:10:06.809
And since microscopic motion is
random, it's like tossing coins

1:10:06.811 --> 1:10:10.321
to ask for N heads many
thousand times;

1:10:10.319 --> 1:10:12.779
it's like asking all the
molecules to be on the left side

1:10:12.782 --> 1:10:14.852
of the room, to ask for 50/50
head and tails,

1:10:14.850 --> 1:10:18.520
is like asking for molecules to
populate the room equally.

1:10:18.520 --> 1:10:21.520
Okay, so that's called the
arrow of time.

1:10:21.520 --> 1:10:23.540
But you've got to be careful
about one thing;

1:10:23.539 --> 1:10:26.479
the entropy of a part of the
universe can go down.

1:10:26.479 --> 1:10:29.859
It's just the entropy of the
whole universe will not go down.

1:10:29.859 --> 1:10:33.679
So, all of life is an example
of lowering entropy because,

1:10:33.675 --> 1:10:37.755
you know, the creation of life
or the creation of tomatoes out

1:10:37.757 --> 1:10:41.767
of mud is a highly organizing
process so the entropy there is

1:10:41.773 --> 1:10:43.383
really going down.

1:10:43.380 --> 1:10:45.620
But if you kept track of the
rest of the world,

1:10:45.616 --> 1:10:48.096
you will find there's some
corresponding increase of

1:10:48.097 --> 1:10:49.407
entropy somewhere else.

1:10:49.409 --> 1:10:51.979
Or take your freezer,
you know, your refrigerator

1:10:51.975 --> 1:10:55.075
sucks heat out of your freezer,
∆Q over T for

1:10:55.082 --> 1:10:57.172
that is negative,
but somewhere there's bigger

1:10:57.165 --> 1:10:59.985
heat emitted outside from the
exhaust of the refrigerator.

1:10:59.989 --> 1:11:01.959
If you took care of all of
that, entropy of the universe

1:11:01.964 --> 1:11:51.004
will go up or remain same;
it really doesn't go down.