Carnot efficiency 2: Reversing the cycle

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.780
In the last video, I showed
you that the definition of

00:00:03.780 --> 00:00:08.189
efficiency, eta, is the work
that we do given the amount of

00:00:08.189 --> 00:00:10.450
heat we are given
to work with.

00:00:10.450 --> 00:00:13.020
And we showed that for an
engine, that could also be

00:00:13.020 --> 00:00:21.390
rewritten as 1 minus Q2 over
Q1, or essentially, 1 minus

00:00:21.390 --> 00:00:23.570
the heat we output from our
engine divided by the amount

00:00:23.570 --> 00:00:25.570
of heat we input from
our engine.

00:00:25.570 --> 00:00:28.940
Now we applied this formula to
a Carnot cycle, and we said,

00:00:28.940 --> 00:00:31.946
hey, for a Carnot engine,
we could get an

00:00:31.946 --> 00:00:34.240
efficiency of this.

00:00:34.240 --> 00:00:36.490
So let me write this here.

00:00:36.490 --> 00:00:41.310
So the efficiency for Carnot,
eta for Carnot, is 1

00:00:41.310 --> 00:00:42.800
minus T2 over T1.

00:00:42.800 --> 00:00:45.060
To get this result, we had to
use the fact that we were

00:00:45.060 --> 00:00:47.290
dealing with the Carnot cycle
to use, you know, we were

00:00:47.290 --> 00:00:49.902
moving along these isotherms,
and so I was able to take the

00:00:49.902 --> 00:00:52.500
natural log of them and do all
of that, and I was able to get

00:00:52.500 --> 00:00:55.140
this for the efficiency
of a Carnot engine.

00:00:55.140 --> 00:00:56.040
And let me be very clear.

00:00:56.040 --> 00:00:58.360
This is the efficiency that
can only be attained by a

00:00:58.360 --> 00:01:00.030
Carnot engine.

00:01:00.030 --> 00:01:02.560
The other definitions of
efficiency-- so when I just

00:01:02.560 --> 00:01:07.710
defined efficiency as equal to
the work performed divided by

00:01:07.710 --> 00:01:13.280
the heat, let me call it the
heat input-- or when I defined

00:01:13.280 --> 00:01:21.690
it as the net heat in.

00:01:21.690 --> 00:01:26.610
So Q1 minus Q2 over Q1.

00:01:26.610 --> 00:01:28.510
This applies to all
heat engines.

00:01:28.510 --> 00:01:30.370
This is true for all
heat engines,

00:01:30.370 --> 00:01:32.090
including the Carnot engine.

00:01:32.090 --> 00:01:34.940
A heat engine is an engine
that operates on heat.

00:01:34.940 --> 00:01:36.320
I probably should have said
that a while ago.

00:01:36.320 --> 00:01:39.740
And this engine that I made,
this Carnot engine, is

00:01:39.740 --> 00:01:42.930
definitely a engine that is
operating on heat, because

00:01:42.930 --> 00:01:44.810
it's taking heat here,
and later it releases

00:01:44.810 --> 00:01:46.590
the heat down here.

00:01:46.590 --> 00:01:49.180
The cycle just shows what's
happening to that engine.

00:01:49.180 --> 00:01:50.450
And I just want to make
that distinction, too.

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The engine is the actual
physical thing.

00:01:52.730 --> 00:01:55.660
The cycle just describes
what's happening to it.

00:01:55.660 --> 00:02:00.540
So with that said, I said that
this is only true for a Carnot

00:02:00.540 --> 00:02:02.150
heat engine.

00:02:02.150 --> 00:02:04.848
Now, what I'm about to embark
on-- and I don't know if I'm

00:02:04.848 --> 00:02:05.830
going to finish it
in this video.

00:02:05.830 --> 00:02:09.520
It might take into the next
video to do it properly-- is

00:02:09.520 --> 00:02:12.690
to show you that if we're
operating a heat engine

00:02:12.690 --> 00:02:16.560
between two temperature
sources-- so I have my hot

00:02:16.560 --> 00:02:24.240
temperature source, I'll call
that TH for T hot, and it's

00:02:24.240 --> 00:02:30.830
transferring some heat, Q1, and
some other heat is coming

00:02:30.830 --> 00:02:39.410
out at Q2, and I'm performing
some work, and then my other

00:02:39.410 --> 00:02:41.770
cold temperature reservoir,
I'll call that T

00:02:41.770 --> 00:02:42.820
cold, is down here.

00:02:42.820 --> 00:02:44.670
And that's where I'm releasing
the heat, too.

00:02:44.670 --> 00:02:49.770
I'm going to show over the next
few videos that the most

00:02:49.770 --> 00:02:52.960
efficient engine is this
theoretical Carnot engine.

00:02:52.960 --> 00:02:56.520
That no engine can get more
efficient than this.

00:02:56.520 --> 00:02:59.650
So if this is a Carnot
engine, this is the

00:02:59.650 --> 00:03:03.220
most efficient engine.

00:03:03.220 --> 00:03:07.380
Or this is the ideal, where
nothing is lost. Well, I'll go

00:03:07.380 --> 00:03:08.710
into that in more detail.

00:03:08.710 --> 00:03:12.510
No engine can get more efficient
than this Carnot

00:03:12.510 --> 00:03:13.670
heat engine.

00:03:13.670 --> 00:03:16.900
So to get there, to prove it to
you, I'm just going to play

00:03:16.900 --> 00:03:18.680
with the Carnot engine a little
bit, just to show you

00:03:18.680 --> 00:03:21.430
some of the tools that it
has at its disposal.

00:03:21.430 --> 00:03:23.970
So one of the things-- let me
just draw a PV diagram.

00:03:36.180 --> 00:03:39.400
In the Carnot cycle we've done
so far, we've kind of always

00:03:39.400 --> 00:03:40.330
moved in one direction.

00:03:40.330 --> 00:03:42.660
We had our isothermal
expansion.

00:03:42.660 --> 00:03:47.100
It went something like that.

00:03:47.100 --> 00:03:48.630
That was isothermal.

00:03:48.630 --> 00:03:51.670
Then we had our adiabatic
expansion-- and the whole time

00:03:51.670 --> 00:03:55.815
we were going in that
direction-- and

00:03:55.815 --> 00:03:57.990
it went like that.

00:03:57.990 --> 00:04:01.660
Then we had our isothermal
contraction.

00:04:01.660 --> 00:04:03.426
It went something like this.

00:04:06.560 --> 00:04:09.230
And then we had our adiabatic
contraction, to get to where

00:04:09.230 --> 00:04:10.500
we were to begin with.

00:04:10.500 --> 00:04:12.590
So then we went back
like that.

00:04:12.590 --> 00:04:13.960
And the whole time, we
went in this kind

00:04:13.960 --> 00:04:16.519
of clockwise direction.

00:04:16.519 --> 00:04:19.800
We went in the clockwise
direction, and we took in heat

00:04:19.800 --> 00:04:23.410
up here-- because we were doing
work-- we took in heat

00:04:23.410 --> 00:04:25.580
to keep our temperature
constant, and then we released

00:04:25.580 --> 00:04:27.650
heat here to keep our
temperature from

00:04:27.650 --> 00:04:30.530
going up from Q2.

00:04:30.530 --> 00:04:35.520
And so if I were to draw this
another way-- well, I just did

00:04:35.520 --> 00:04:37.580
one like that, but let
me draw it like this.

00:04:37.580 --> 00:04:42.370
I could also depict it like
this, where that's my engine,

00:04:42.370 --> 00:04:45.230
this is my hot reservoir--
let me put this as

00:04:45.230 --> 00:04:49.180
T1-- T1 is up here.

00:04:49.180 --> 00:04:54.760
It transferred Q1 to
my Carnot engine.

00:04:54.760 --> 00:05:00.360
My Carnot engine did some work,
and then left over, it

00:05:00.360 --> 00:05:05.130
transferred into my cold
reservoir, T2,

00:05:05.130 --> 00:05:06.580
it transferred Q2.

00:05:06.580 --> 00:05:09.490
This is another way of depicting
what went on in this

00:05:09.490 --> 00:05:10.060
Carnot cycle.

00:05:10.060 --> 00:05:13.040
And here I've actually
drawn the engine.

00:05:13.040 --> 00:05:16.630
Now, one of the tools I want to
show you is that this is a

00:05:16.630 --> 00:05:17.900
reversible reaction.

00:05:17.900 --> 00:05:20.810
Or that we can take this and
go the other way around.

00:05:20.810 --> 00:05:23.520
And it's dependent upon an
assumption that I threw out a

00:05:23.520 --> 00:05:24.420
long time ago.

00:05:24.420 --> 00:05:27.900
So when I first drew these, I
kind of introduced you to the

00:05:27.900 --> 00:05:29.940
idea of a quasistatic process.

00:05:33.690 --> 00:05:37.520
And quasistatic just means,
look, you do it really slowly,

00:05:37.520 --> 00:05:40.460
so that you can always say that
you're close enough to

00:05:40.460 --> 00:05:42.700
equilibrium that your
macro state

00:05:42.700 --> 00:05:44.300
variables are always defined.

00:05:44.300 --> 00:05:46.830
And that was the whole
justification for dealing with

00:05:46.830 --> 00:05:48.370
pebbles like this.

00:05:48.370 --> 00:05:49.890
Instead of just doing it
wholesale, instead of just

00:05:49.890 --> 00:05:52.150
moving all the pebbles, and just
getting to this state,

00:05:52.150 --> 00:05:54.930
from A to B kind of jumping, I
wanted to do it gradually, so

00:05:54.930 --> 00:05:57.570
that I would be defined at
every point in between.

00:05:57.570 --> 00:05:59.760
That's what quasistatic
did for us.

00:05:59.760 --> 00:06:03.970
And when I actually made the
video quasistatic processes, I

00:06:03.970 --> 00:06:06.570
said, you know, quasistatic
processes, for the most part,

00:06:06.570 --> 00:06:07.830
are reversible.

00:06:07.830 --> 00:06:11.820
And sometimes I used the
words interchangeably.

00:06:11.820 --> 00:06:15.480
Now, by definition, our
theoretical Carnot cycle is

00:06:15.480 --> 00:06:18.760
said to be, not only is it
quasistatic, but it is also

00:06:18.760 --> 00:06:20.100
reversible.

00:06:20.100 --> 00:06:23.990
Which means at any point in
time-- let's say we've moved a

00:06:23.990 --> 00:06:27.980
couple of pebbles, and we've
gotten right here.

00:06:27.980 --> 00:06:30.540
If we want to, if we're in the
mood, we can add some pebbles

00:06:30.540 --> 00:06:33.080
back, and just follow this right
back to where we were.

00:06:33.080 --> 00:06:36.040
That's what reversible means.

00:06:36.040 --> 00:06:37.770
It means you can reverse
something.

00:06:37.770 --> 00:06:41.180
Now, what has to be ideal about
the system in order for

00:06:41.180 --> 00:06:42.930
that to be true?

00:06:42.930 --> 00:06:48.950
Well, it means that the actual
movement of our piston, of

00:06:48.950 --> 00:06:51.720
this movable ceiling, that it
shouldn't have any friction.

00:06:51.720 --> 00:06:55.000
Because if some of the heat is
lost to friction, then when we

00:06:55.000 --> 00:06:57.610
go back, we would have lost
some of our heat.

00:06:57.610 --> 00:06:59.670
Some heat would have been
destroyed, just going from one

00:06:59.670 --> 00:07:01.360
state forward and back.

00:07:01.360 --> 00:07:07.350
So the assumption that we have
to make in order for the

00:07:07.350 --> 00:07:10.530
Carnot cycle to be reversible
is that it's frictionless.

00:07:14.640 --> 00:07:16.770
So the Carnot heat engine, this
theoretical engine, is a

00:07:16.770 --> 00:07:19.750
frictionless engine, which is
theoretically impossible.

00:07:19.750 --> 00:07:21.150
To be completely frictionless.

00:07:21.150 --> 00:07:24.150
To be-- but we'll talk more
about that in the future.

00:07:24.150 --> 00:07:27.800
So if you have a completely
frictionless engine, and it's

00:07:27.800 --> 00:07:30.890
quasistatic, it's
also reversible.

00:07:30.890 --> 00:07:31.983
So if we want to do
it reversible,

00:07:31.983 --> 00:07:33.250
what does that mean?

00:07:33.250 --> 00:07:36.770
It means I could start in this
state, my state A that I've

00:07:36.770 --> 00:07:38.680
labelled before, but instead
of going around that way, I

00:07:38.680 --> 00:07:40.520
could go around the other way.

00:07:40.520 --> 00:07:46.050
So what I could do first, is I
could adiabatically expand

00:07:46.050 --> 00:07:51.960
first-- so maybe let me redraw
it, so I do it the other way.

00:07:51.960 --> 00:07:53.810
So I could reverse
this reaction.

00:07:53.810 --> 00:07:56.120
And it would happen the
exact same way.

00:07:56.120 --> 00:07:59.060
And that's an artifact of that
I'm always in equilibrium, and

00:07:59.060 --> 00:08:02.490
that my system is frictionless,
that I don't

00:08:02.490 --> 00:08:05.050
lose energy just going
back and forth.

00:08:05.050 --> 00:08:08.120
So I could start at state
A here, and then I could

00:08:08.120 --> 00:08:11.060
adiabatically contract.

00:08:11.060 --> 00:08:13.925
Adiabatic contraction would look
something like this, and

00:08:13.925 --> 00:08:15.590
it'll get to that state.

00:08:15.590 --> 00:08:19.530
Then I can isothermically
expand.

00:08:19.530 --> 00:08:20.780
So I'm going like this.

00:08:24.650 --> 00:08:30.300
And as I isothermically expand--
so I'm going like

00:08:30.300 --> 00:08:34.010
this, I'm all along an
isotherm-- I'd doing some

00:08:34.010 --> 00:08:37.710
isothermic expansion-- so in
this case, I'm doing work, but

00:08:37.710 --> 00:08:40.440
I'm doing work isothermically,
right?

00:08:40.440 --> 00:08:44.330
At some cold isotherm.

00:08:44.330 --> 00:08:46.450
Let's call it T2, right?

00:08:46.450 --> 00:08:47.840
Just like this was T2.

00:08:47.840 --> 00:08:51.080
So in this case, if I'm
expanding, and I'm staying at

00:08:51.080 --> 00:08:53.180
T2, and I'm sitting
on top of my T2

00:08:53.180 --> 00:08:57.210
reservoir, heat is coming.

00:08:57.210 --> 00:08:59.900
This area under the curve,
the work I'm

00:08:59.900 --> 00:09:01.200
doing, is the heat added.

00:09:01.200 --> 00:09:08.080
This is Q2, and that is given
to me by my T2 reservoir.

00:09:08.080 --> 00:09:09.460
So everything is going
in reverse.

00:09:09.460 --> 00:09:10.060
That's the whole idea.

00:09:10.060 --> 00:09:17.910
Then I adiabatically contract,
like that, and then I

00:09:17.910 --> 00:09:23.800
isothermically contract,
like that, to get

00:09:23.800 --> 00:09:24.480
back where I started.

00:09:24.480 --> 00:09:27.840
When I isothermically contract,
what's happening?

00:09:27.840 --> 00:09:31.820
Work is being done to me, so now
all of this area over here

00:09:31.820 --> 00:09:33.090
will be negative.

00:09:33.090 --> 00:09:36.040
And in order to keep my
temperature constant, I have

00:09:36.040 --> 00:09:37.580
to release heat.

00:09:37.580 --> 00:09:40.740
So I'm releasing heat, but
I'm doing it at a high

00:09:40.740 --> 00:09:42.100
temperature.

00:09:42.100 --> 00:09:45.100
So I'm releasing it into
my T1 reservoir.

00:09:45.100 --> 00:09:47.330
So it's the exact same thing
as it happened before, but

00:09:47.330 --> 00:09:50.510
since when I go in a reverse
direction, some

00:09:50.510 --> 00:09:52.160
work is being applied.

00:09:52.160 --> 00:09:55.570
So now, when you look at it this
way, when you figure out

00:09:55.570 --> 00:09:57.930
all of the areas,
the area in here

00:09:57.930 --> 00:09:59.720
will actually be negative.

00:09:59.720 --> 00:10:02.530
And the reason why I'm saying
that is because the positive

00:10:02.530 --> 00:10:05.380
work values are going
to be this.

00:10:05.380 --> 00:10:07.320
This is going to be the
positive, what I'm doing in

00:10:07.320 --> 00:10:08.420
blue right here.

00:10:08.420 --> 00:10:10.200
And the negative work
values are going to

00:10:10.200 --> 00:10:13.090
be all of this stuff.

00:10:13.090 --> 00:10:15.190
So if you wanted to figure out
the total work done, it's

00:10:15.190 --> 00:10:16.070
going to be negative.

00:10:16.070 --> 00:10:20.210
So what's happening, if I run
the Carnot cycle in reverse--

00:10:20.210 --> 00:10:24.300
so I'll call it the Carnot
refrigerator.

00:10:24.300 --> 00:10:26.790
No, that's not what
I wanted to do.

00:10:26.790 --> 00:10:28.870
I'll call it Carnot reverse.

00:10:28.870 --> 00:10:33.740
But it's handy that R also
stands for refrigerator.

00:10:33.740 --> 00:10:35.820
This is the Carnot engine.

00:10:35.820 --> 00:10:42.170
It does work by using heat, by
taking advantage of the heat

00:10:42.170 --> 00:10:45.170
difference between this hot--
you could view this as the T

00:10:45.170 --> 00:10:48.040
hot and the T cold.

00:10:48.040 --> 00:10:52.640
Now, a reverse Carnot engine, or
maybe you call it a Carnot

00:10:52.640 --> 00:10:53.990
refrigerator, does
the opposite.

00:10:53.990 --> 00:10:56.830
That's exactly what I
just drew over here.

00:10:56.830 --> 00:11:02.440
What it does is, it starts with
a cold body-- I'll call

00:11:02.440 --> 00:11:09.970
that T cold, or T2-- it takes
some smaller amount of heat

00:11:09.970 --> 00:11:13.780
from the cold body.

00:11:13.780 --> 00:11:16.550
Some work has to be input
into the system

00:11:16.550 --> 00:11:18.190
in order to do this.

00:11:18.190 --> 00:11:21.630
And then it puts more heat-- you
can kind of view it as a

00:11:21.630 --> 00:11:25.250
combination of this work and
this heat taken from the cold

00:11:25.250 --> 00:11:29.060
body-- and it gives it
to the warm body.

00:11:29.060 --> 00:11:29.300
Sorry.

00:11:29.300 --> 00:11:31.320
This is Q2, and it
gives it Q1.

00:11:31.320 --> 00:11:34.090
So everything just happens
completely in reverse.

00:11:34.090 --> 00:11:36.180
And that's just a byproduct
of, this is reversible.

00:11:36.180 --> 00:11:38.990
So I can just go and I can do,
if this is the way we went

00:11:38.990 --> 00:11:41.740
before, when we're an engine,
if we want to be a

00:11:41.740 --> 00:11:44.560
refrigerator, we go the other
direction, and everything just

00:11:44.560 --> 00:11:45.540
gets reversed.

00:11:45.540 --> 00:11:46.600
And I want you to really
understand

00:11:46.600 --> 00:11:47.420
that this is doable.

00:11:47.420 --> 00:11:49.670
That there's nothing
wrong with this.

00:11:49.670 --> 00:11:52.080
You might say, doesn't this
defy the second law of

00:11:52.080 --> 00:11:53.230
thermodynamics?

00:11:53.230 --> 00:11:57.960
We're taking heat from a cold
body to a warm body?

00:11:57.960 --> 00:11:59.640
And my answer will be the
same thing I said

00:11:59.640 --> 00:12:01.030
on my entropy videos.

00:12:01.030 --> 00:12:02.020
I said, well, no.

00:12:02.020 --> 00:12:03.240
We're applying some work.

00:12:03.240 --> 00:12:04.290
This is a refrigerator.

00:12:04.290 --> 00:12:06.740
So some work has to be done
in order to do this.

00:12:06.740 --> 00:12:10.010
And whatever object that is
doing the work-- it may be

00:12:10.010 --> 00:12:12.150
some, in the case of your
refrigerator, it's a

00:12:12.150 --> 00:12:12.710
compressor.

00:12:12.710 --> 00:12:19.280
That is adding more entropy to
the universe than the entropy

00:12:19.280 --> 00:12:22.300
that's being destroyed
by our refrigerator.

00:12:22.300 --> 00:12:25.100
So this does not defy the second
law of thermodynamics.

00:12:25.100 --> 00:12:30.410
Now, I want to make another
point about the Carnot engine.

00:12:30.410 --> 00:12:33.070
Let me take the reverse
Carnot engine.

00:12:33.070 --> 00:12:35.530
Let me call it the Carnot
refrigerator.

00:12:35.530 --> 00:12:38.610
So if we take that-- and this is
really just more math than

00:12:38.610 --> 00:12:39.860
anything else.

00:12:43.220 --> 00:12:48.090
If we're taking in Q2, adding
in some work, and producing

00:12:48.090 --> 00:12:52.380
Q1, we can scale this
up arbitrarily.

00:12:52.380 --> 00:13:03.910
If we take x times Q2 in, and
we put in x times W in, then

00:13:03.910 --> 00:13:07.930
we're going to put x times Q1
into our top reservoir.

00:13:07.930 --> 00:13:09.070
And that makes sense,
because these are

00:13:09.070 --> 00:13:11.450
just arbitrary numbers.

00:13:11.450 --> 00:13:13.780
For example, if we have two
Carnot engines in parallel,

00:13:13.780 --> 00:13:17.870
you can just kind of view that
whole thing as two Carnot

00:13:17.870 --> 00:13:20.630
engines doing it together, so
all of these would be 2s.

00:13:20.630 --> 00:13:23.010
If we had three Carnot engines
doing it together, all of

00:13:23.010 --> 00:13:25.520
those would be three, you could
just view them as one

00:13:25.520 --> 00:13:27.020
collective engine.

00:13:27.020 --> 00:13:31.090
Now, with that said, I think
I've laid the framework for at

00:13:31.090 --> 00:13:36.820
least the ideas that will let us
show that the Carnot engine

00:13:36.820 --> 00:13:41.060
is the most efficient engine
that's able to be produced.

00:13:41.060 --> 00:13:46.120
And given that the Carnot
engine's efficiency is this--

00:13:46.120 --> 00:13:48.200
and we're going to prove that
it's the most efficient

00:13:48.200 --> 00:13:51.630
engine-- this becomes the upper
bound on efficiency for

00:13:51.630 --> 00:13:56.080
any engine that anybody
can or will ever make.

00:13:56.080 --> 00:13:58.350
And I'll kind of
do the crowning

00:13:58.350 --> 00:14:00.490
touch in the next video.


Carnot efficiency 3: Proving that it is the most efficient

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.280
Let's say we have
two reservoirs.

00:00:03.280 --> 00:00:08.250
Let's say up here, I have
my hot reservoir.

00:00:08.250 --> 00:00:10.720
It's at th for hot.

00:00:10.720 --> 00:00:13.210
And I have my engine.

00:00:13.210 --> 00:00:16.940
It's going to be a Carnot
engine, because we'll learn

00:00:16.940 --> 00:00:19.140
that no engine is better,
at least from an

00:00:19.140 --> 00:00:20.310
efficiency point of view.

00:00:20.310 --> 00:00:22.130
We have to be careful when
we say "better."

00:00:22.130 --> 00:00:25.870
We have our Carnot engine,
and it operates on heat

00:00:25.870 --> 00:00:26.850
differentials.

00:00:26.850 --> 00:00:30.830
So it takes some heat from
our hot heat source.

00:00:30.830 --> 00:00:33.045
It takes some heat there.

00:00:33.045 --> 00:00:34.825
Let's call that q1.

00:00:34.825 --> 00:00:38.080
And it does some work.

00:00:38.080 --> 00:00:41.220
Work is a good thing, so I'll
make that in green.

00:00:41.220 --> 00:00:48.855
It does some work, and then
the surplus energy, the

00:00:48.855 --> 00:00:53.400
surplus heat, q2, then goes
to our cold reservoir.

00:00:53.400 --> 00:00:56.970
I'll do that right there.

00:00:56.970 --> 00:00:59.020
t cold.

00:00:59.020 --> 00:01:02.350
Now, I made multiple
insinuations in the previous

00:01:02.350 --> 00:01:07.800
video that this is the most
efficient engine that can be

00:01:07.800 --> 00:01:14.230
created between two reservoirs,
th and tc.

00:01:14.230 --> 00:01:16.540
Now, you come along and say,
no, no, no, no, no.

00:01:16.540 --> 00:01:21.360
I know of a friend, he has
invented a new engine that is

00:01:21.360 --> 00:01:24.710
more efficient than this engine
between these same two

00:01:24.710 --> 00:01:25.400
reservoirs.

00:01:25.400 --> 00:01:29.350
And you go, and you proceed to
draw this same type of diagram

00:01:29.350 --> 00:01:32.140
for your friend's engine.

00:01:32.140 --> 00:01:34.730
You say, look-- let me make it
clear-- this is the same

00:01:34.730 --> 00:01:38.850
reservoir, these same reservoirs
we're dealing with.

00:01:38.850 --> 00:01:40.750
Actually, I should probably draw
this line all the way,

00:01:40.750 --> 00:01:43.250
because I'm going to have to
do multiple engines here.

00:01:43.250 --> 00:01:45.930
So the same reservoir we're
dealing with, right?

00:01:45.930 --> 00:01:49.910
This is all the hot reservoir,
th th, and this is all the

00:01:49.910 --> 00:01:50.830
cold reservoir.

00:01:50.830 --> 00:01:52.600
I need space for multiple
engines that we're

00:01:52.600 --> 00:01:53.530
going to deal with.

00:01:53.530 --> 00:01:56.210
So your friend has an engine.

00:01:56.210 --> 00:01:57.460
We'll call it the
super engine.

00:02:01.240 --> 00:02:06.430
And your friend's claim-- and
I'll show you why your

00:02:06.430 --> 00:02:09.054
friend's claim cannot be true,
if you believe the second law

00:02:09.054 --> 00:02:13.140
of thermodynamics-- so your
friend's claim, they have the

00:02:13.140 --> 00:02:14.210
super engine.

00:02:14.210 --> 00:02:17.760
And they claim that look, I
can actually take in q1.

00:02:17.760 --> 00:02:22.270
I can take in that same heat
from this heat source up here.

00:02:22.270 --> 00:02:27.300
But I can produce more work
than your Carnot engine.

00:02:27.300 --> 00:02:31.460
I could produce 1 plus x.

00:02:31.460 --> 00:02:32.520
I don't want to get
too algebraic.

00:02:32.520 --> 00:02:35.520
But let's say, you produce w.

00:02:35.520 --> 00:02:40.080
I produce w times 1
plus x of work.

00:02:40.080 --> 00:02:41.530
Where x is a positive number.

00:02:41.530 --> 00:02:43.490
So he's saying, look,
x is greater than 0.

00:02:43.490 --> 00:02:46.190
Whatever that number he
might feed you is.

00:02:46.190 --> 00:02:48.000
And then the rest of
the energy that's

00:02:48.000 --> 00:02:52.660
left over is what?

00:02:52.660 --> 00:02:55.150
It's q1 minus this.

00:02:55.150 --> 00:03:01.480
So it's q1 minus w
times 1 plus x.

00:03:01.480 --> 00:03:04.930
And just to be clear, q2 right
here, I could rewrite that as

00:03:04.930 --> 00:03:07.430
q1 minus w.

00:03:07.430 --> 00:03:08.000
Fair enough.

00:03:08.000 --> 00:03:09.030
So you look at that.

00:03:09.030 --> 00:03:11.320
You come to me with this, and
I say, no, no, no, no.

00:03:11.320 --> 00:03:12.750
This cannot be true.

00:03:12.750 --> 00:03:16.300
Because if this were, then we
would solve literally all of

00:03:16.300 --> 00:03:18.950
the world's energy problems. And
I'm about to show you why

00:03:18.950 --> 00:03:21.490
we could solve all of the
world's energy problems, and

00:03:21.490 --> 00:03:24.240
we would have a perpetual motion
machine, and be able to

00:03:24.240 --> 00:03:26.610
defy all sorts of things
if we had this.

00:03:26.610 --> 00:03:28.560
Now, this is my Carnot engine.

00:03:28.560 --> 00:03:31.600
But I could devise a reverse
Carnot engine, right?

00:03:31.600 --> 00:03:34.800
Let me make a reverse
Carnot engine.

00:03:34.800 --> 00:03:38.230
So my reverse Carnot engine
would look like this.

00:03:43.180 --> 00:03:46.930
And it's going to do the same
thing, but in reverse.

00:03:46.930 --> 00:03:55.070
So instead of producing q1 minus
w here and putting it

00:03:55.070 --> 00:04:10.420
into tc, it could take in q1
minus w from our cold source--

00:04:10.420 --> 00:04:13.520
so could take that in-- or even
better, let's scale it up

00:04:13.520 --> 00:04:14.270
a little bit.

00:04:14.270 --> 00:04:20.430
Let's say it takes in q1
minus w times 1 plus x.

00:04:20.430 --> 00:04:23.660
So I've just made a slightly
larger reversed Carnot engine.

00:04:23.660 --> 00:04:28.580
Now, if I take in that much--
in order to do this in

00:04:28.580 --> 00:04:30.180
reverse, I'm going to have to
take in, I'm going to have to

00:04:30.180 --> 00:04:32.780
scale up this Carnot engine
and reverse everything.

00:04:32.780 --> 00:04:35.860
So instead of producing work,
I'm now going to need work to

00:04:35.860 --> 00:04:39.010
go on the other direction, and
I've scaled it up by 1 plus x.

00:04:39.010 --> 00:04:43.450
So I'm going to need the
amount of work here

00:04:43.450 --> 00:04:44.820
times 1 plus x.

00:04:47.810 --> 00:04:53.360
And then I'm going to push q1,
but I've scaled it up.

00:04:53.360 --> 00:04:57.180
I'm going to push in
q1 times 1 plus x

00:04:57.180 --> 00:04:58.790
into my hot heat source.

00:04:58.790 --> 00:05:00.490
And once again, this isn't
defying the laws of

00:05:00.490 --> 00:05:01.110
thermodynamics.

00:05:01.110 --> 00:05:03.410
I'm taking up some work.

00:05:03.410 --> 00:05:06.500
There's work that needs to be
done in order to do this.

00:05:06.500 --> 00:05:10.800
But all of a sudden, you come
to me and say, look, this is

00:05:10.800 --> 00:05:11.380
an awesome deal.

00:05:11.380 --> 00:05:13.080
You have this nice engine
that works this way.

00:05:13.080 --> 00:05:14.980
My friend has a super engine.

00:05:14.980 --> 00:05:16.780
Let's just couple
them together.

00:05:16.780 --> 00:05:21.450
Let's take the work that he
produces right here-- he

00:05:21.450 --> 00:05:24.580
produces w times 1 plus x, and
that just happens to be the

00:05:24.580 --> 00:05:27.410
amount of work that you need
to operate your engine.

00:05:27.410 --> 00:05:29.310
So you just feed that
into there.

00:05:29.310 --> 00:05:31.630
So what's the net effect
of these two engines?

00:05:34.420 --> 00:05:39.750
So let me do another, scroll
a little bit more.

00:05:39.750 --> 00:05:42.190
Actually, that might be
the best way to do it.

00:05:42.190 --> 00:05:45.580
So let me make sure that we
understand that these are the

00:05:45.580 --> 00:05:49.510
same heat sinks or heat sources
that we're using the

00:05:49.510 --> 00:05:49.960
whole time.

00:05:49.960 --> 00:05:55.680
So that's my hot source, my
cold source is down here.

00:05:55.680 --> 00:05:59.400
So if I add our two engines
together-- so if I have a, you

00:05:59.400 --> 00:06:02.210
know, let me call it a-- I'm
going to pick a new color.

00:06:02.210 --> 00:06:05.320
These colors are getting
monotonous.

00:06:05.320 --> 00:06:06.160
Nope.

00:06:06.160 --> 00:06:08.760
I wanted to do the
rectangle tool.

00:06:08.760 --> 00:06:09.950
There you go.

00:06:09.950 --> 00:06:10.780
All right.

00:06:10.780 --> 00:06:12.880
So I combine these two
engines together.

00:06:12.880 --> 00:06:14.320
Essentially I just put a
big box around them.

00:06:14.320 --> 00:06:18.060
They're both operating between
these two heat sources.

00:06:18.060 --> 00:06:19.160
These two reservoirs.

00:06:19.160 --> 00:06:22.730
So I call this the, you know,
your super engine plus my

00:06:22.730 --> 00:06:25.300
reverse Carnot engine.

00:06:25.300 --> 00:06:26.990
So what's happening now?

00:06:26.990 --> 00:06:32.685
What's the net heat that's
being taken in

00:06:32.685 --> 00:06:33.570
or put out of here?

00:06:33.570 --> 00:06:38.710
So we have q1-- we have--
let me see.

00:06:38.710 --> 00:06:43.670
We have q1 minus w 1 plus x, but
in this direction, we have

00:06:43.670 --> 00:06:47.340
q1-- so in this direction,
we could rewrite this.

00:06:47.340 --> 00:06:48.620
I want to make sure you're
clear on the algebra.

00:06:48.620 --> 00:06:50.490
This could be rewritten
as what?

00:06:50.490 --> 00:06:58.360
As q1 times 1 plus x times, or
minus, w times 1 place x.

00:06:58.360 --> 00:06:59.090
Right?

00:06:59.090 --> 00:07:03.530
Now, if you compare these
terms, this is the

00:07:03.530 --> 00:07:05.340
same as this term.

00:07:05.340 --> 00:07:07.770
This term is bigger
than this term.

00:07:07.770 --> 00:07:08.120
Right?

00:07:08.120 --> 00:07:10.060
This term is clearly bigger,
because we're multiplying it

00:07:10.060 --> 00:07:11.180
by something larger than 1.

00:07:11.180 --> 00:07:12.630
It's bigger than this storm.

00:07:12.630 --> 00:07:17.640
So if we combine these two, the
upward movement, or the

00:07:17.640 --> 00:07:20.120
amount of heat I'm taking up
from my reverse Carnot, is

00:07:20.120 --> 00:07:23.530
going to be greater than the
amount of heat being put in by

00:07:23.530 --> 00:07:25.670
your friend's super engine.

00:07:25.670 --> 00:07:26.900
And we can actually calculate
the amount.

00:07:26.900 --> 00:07:29.440
We can just take this amount
minus that amount, and that's

00:07:29.440 --> 00:07:31.640
the net upward movement.

00:07:31.640 --> 00:07:36.040
So the net upward movement
from our cold

00:07:36.040 --> 00:07:38.470
reservoir is what?

00:07:38.470 --> 00:07:40.500
It's this value minus
this value.

00:07:40.500 --> 00:07:47.450
So minus q1 minus w 1 plus x.

00:07:47.450 --> 00:07:48.973
If we take a minus, we're
going to subtract it.

00:07:48.973 --> 00:07:50.800
So it's a minus and a plus.

00:07:50.800 --> 00:07:52.450
These cancel out.

00:07:52.450 --> 00:07:55.610
This minus cancels out with--
so this first term could be

00:07:55.610 --> 00:08:00.010
rewritten as q1 plus q1x.

00:08:00.010 --> 00:08:00.830
Right?

00:08:00.830 --> 00:08:01.910
We could rewrite it that way.

00:08:01.910 --> 00:08:03.400
So this cancels out with that.

00:08:03.400 --> 00:08:06.050
And so the net upward movement
when we combine the two

00:08:06.050 --> 00:08:11.270
engines is q1 times x.

00:08:11.270 --> 00:08:12.440
Fair enough.

00:08:12.440 --> 00:08:13.640
Now what about the
work transfer?

00:08:13.640 --> 00:08:16.380
Well, whatever work this guy
produces is exactly the amount

00:08:16.380 --> 00:08:17.090
of work that I need.

00:08:17.090 --> 00:08:20.540
So no outside work has to
be done on the system.

00:08:20.540 --> 00:08:21.650
It just works.

00:08:21.650 --> 00:08:23.960
This guy produces work, this
guy uses the work.

00:08:23.960 --> 00:08:25.730
Now what's in that
heat transferred

00:08:25.730 --> 00:08:28.175
up to our hot reservoir?

00:08:30.690 --> 00:08:33.140
What's the amount of heat?

00:08:33.140 --> 00:08:35.090
Well, it's the difference
between these two.

00:08:35.090 --> 00:08:38.419
And this is clearly a larger
number than this one, so the

00:08:38.419 --> 00:08:40.230
upward movement dominates.

00:08:40.230 --> 00:08:42.299
So what's this minus that?

00:08:42.299 --> 00:08:48.750
So this can be rewritten
as q1 plus q1x, right?

00:08:48.750 --> 00:08:50.380
I just distributed the q1.

00:08:50.380 --> 00:08:51.580
We're going to subtract
that out.

00:08:51.580 --> 00:08:53.400
Minus q1.

00:08:53.400 --> 00:08:54.330
You're left with q1x.

00:08:54.330 --> 00:08:56.090
So the net movement,
when we combine the

00:08:56.090 --> 00:08:59.700
two engines, is q1x.

00:08:59.700 --> 00:09:01.580
So what's happening here?

00:09:01.580 --> 00:09:05.770
I have no external energy or
work has to be expended into

00:09:05.770 --> 00:09:06.795
this system.

00:09:06.795 --> 00:09:11.890
And it's just taking heat from
a cold body, and it's moving

00:09:11.890 --> 00:09:13.970
it to a warm body.

00:09:13.970 --> 00:09:15.140
And it does this indefinitely.

00:09:15.140 --> 00:09:16.260
It'll do this as much
as I want to.

00:09:16.260 --> 00:09:17.290
I can just build a bigger one.

00:09:17.290 --> 00:09:19.840
It'll do it on even a larger
and larger scale.

00:09:19.840 --> 00:09:24.520
So if you think about it, I
could heat my house with ice

00:09:24.520 --> 00:09:26.090
by just making the ice colder.

00:09:26.090 --> 00:09:32.270
I could create steam from things
that arbitrarily cold.

00:09:32.270 --> 00:09:36.570
This goes against the second
law of thermodynamics.

00:09:36.570 --> 00:09:39.770
The net entropy in this
world is going down.

00:09:39.770 --> 00:09:41.620
Because what's happening here?

00:09:41.620 --> 00:09:45.700
This is just a straight up
transfer of q1x from a cold

00:09:45.700 --> 00:09:47.230
body to a hot body.

00:09:47.230 --> 00:09:50.340
So what's the net change
in entropy here?

00:09:50.340 --> 00:09:53.060
The change in entropy
of the universe.

00:09:53.060 --> 00:10:02.430
Well, the hot body is gaining
some heat, so it's q1x over

00:10:02.430 --> 00:10:05.135
the temperature of the hot body,
and then the cold body

00:10:05.135 --> 00:10:06.430
is losing the same amount.

00:10:06.430 --> 00:10:11.450
So it's minus q1x over
the cold body.

00:10:11.450 --> 00:10:17.110
Now, this is a bigger number
than this is, right?

00:10:17.110 --> 00:10:19.150
Because the denominator
is smaller.

00:10:19.150 --> 00:10:20.130
This is a cold body.

00:10:20.130 --> 00:10:22.490
Its temperature in Kelvin will
be a smaller number.

00:10:22.490 --> 00:10:26.355
So this is going to be less than
0, which the second law

00:10:26.355 --> 00:10:28.240
of thermodynamics tells
us cannot be.

00:10:28.240 --> 00:10:30.610
The entropy cannot shrink
in the universe.

00:10:30.610 --> 00:10:33.140
This whole thing is an
independent system, and the

00:10:33.140 --> 00:10:34.180
entropy is shrinking.

00:10:34.180 --> 00:10:36.900
And we can make the entropy
shrink arbitrarily if we just

00:10:36.900 --> 00:10:38.480
scale up our x's enough.

00:10:38.480 --> 00:10:42.670
So this is why the Carnot engine
is the most efficient

00:10:42.670 --> 00:10:43.500
engine possible.

00:10:43.500 --> 00:10:46.690
Because if anyone claimed to
have a more efficient engine,

00:10:46.690 --> 00:10:49.650
you could couple it with a
reverse Carnot engine, and

00:10:49.650 --> 00:10:53.110
then create this perpetual
reverse-- I guess you could

00:10:53.110 --> 00:10:55.450
call it a perpetual
refrigeration machine that

00:10:55.450 --> 00:10:59.610
just out of the blue creates
anti-entropy from anywhere,

00:10:59.610 --> 00:11:01.970
and it would be this perpetual
energy source that creates

00:11:01.970 --> 00:11:03.460
energy out of nothing.

00:11:03.460 --> 00:11:06.390
And so this is just something
that cannot be done in our

00:11:06.390 --> 00:11:08.830
world, especially if you believe
the second law of

00:11:08.830 --> 00:11:10.060
thermodynamics.

00:11:10.060 --> 00:11:13.620
So the most efficient engine
is the Carnot engine, where

00:11:13.620 --> 00:11:17.810
its efficiency is described as 1
minus the temperature of the

00:11:17.810 --> 00:11:20.380
cold body divided
by the hot body.

00:11:20.380 --> 00:11:22.920
So if I have two temperature
reservoirs, let's say that my

00:11:22.920 --> 00:11:30.590
hot one is at 500 Kelvin, and my
cold one is at 300 Kelvin,

00:11:30.590 --> 00:11:33.910
and I have some engine that
takes heat from there, and

00:11:33.910 --> 00:11:35.690
transfers it there, and
does some work.

00:11:35.690 --> 00:11:38.350
The most efficient engine, if
I were to remove all the

00:11:38.350 --> 00:11:42.030
friction in the engine, the
highest efficiency I could get

00:11:42.030 --> 00:11:47.390
would be 1 minus 300 Kelvin over
500 Kelvin, which is 1

00:11:47.390 --> 00:11:53.360
minus 3/5, which is 2/5, which
is equal to 0.4, which is

00:11:53.360 --> 00:11:56.650
equal to 40%.

00:11:56.650 --> 00:11:59.720
So if someone tells you that
they made an engine that

00:11:59.720 --> 00:12:04.570
operates between a reservoir
that's 500 Kelvin and 300

00:12:04.570 --> 00:12:07.090
Kelvin, and they say, oh, I've
achieved 41% percent

00:12:07.090 --> 00:12:07.520
efficiency.

00:12:07.520 --> 00:12:09.460
I've really polished
the thing well.

00:12:09.460 --> 00:12:11.250
You know that they are lying.

00:12:11.250 --> 00:12:14.150
So anyway, hopefully you found
that reasonably interesting,

00:12:14.150 --> 00:12:16.750
and I'll see you in
the next video.


Efficiency of a Carnot engine

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.970
I'm now going to introduce
you to the notion of the

00:00:02.970 --> 00:00:04.505
efficiency of an engine.

00:00:09.780 --> 00:00:13.900
And it's represented by
the Greek letter eta.

00:00:13.900 --> 00:00:16.360
Even though it sounds like an
e, it looks like a kind of

00:00:16.360 --> 00:00:17.820
funny-looking n.

00:00:17.820 --> 00:00:19.920
So Greek letter eta,
this is efficiency.

00:00:19.920 --> 00:00:23.060
And efficiency as applies to
engines is kind of similar to

00:00:23.060 --> 00:00:24.940
the way we use the word
in our everyday life.

00:00:24.940 --> 00:00:28.160
If I said, how efficient are
you with your time, I care

00:00:28.160 --> 00:00:30.610
about, what are you doing with
that hour that I gave you?

00:00:30.610 --> 00:00:32.520
Or if I said, how efficient are
you with your money, I'd

00:00:32.520 --> 00:00:35.570
say, how much were you able to
buy with that $100 I gave you?

00:00:35.570 --> 00:00:37.770
So efficiency with an engine
is the same thing.

00:00:37.770 --> 00:00:41.360
What were you able to do with
the stuff that I gave you?

00:00:41.360 --> 00:00:46.100
So in the engine world, we
define efficiency to be, the

00:00:46.100 --> 00:00:50.470
work you did with the energy
that I gave you to do the

00:00:50.470 --> 00:00:51.780
work, essentially.

00:00:51.780 --> 00:00:54.860
And in our Carnot world, up
here, this is all the stuff

00:00:54.860 --> 00:00:58.560
I'd done before, what was the
energy that I had given you?

00:00:58.560 --> 00:01:00.710
Well, the energy that I'd given
you was the energy that

00:01:00.710 --> 00:01:03.280
came from this first
reservoir up here.

00:01:03.280 --> 00:01:07.690
Remember, when we were moving
the pebbles from A to B, we

00:01:07.690 --> 00:01:10.730
were moving the pebbles to
keep it as a quasistatic

00:01:10.730 --> 00:01:14.520
process, so that we can go back,
if we had to, and so

00:01:14.520 --> 00:01:16.420
that the system stayed in
equilibrium the whole time.

00:01:16.420 --> 00:01:17.980
If we didn't have this
reservoir here, the

00:01:17.980 --> 00:01:19.970
temperature would have gone
down, because we're expanding

00:01:19.970 --> 00:01:21.550
the volume, and all of that.

00:01:21.550 --> 00:01:23.070
So we had to keep that
reservoir there.

00:01:23.070 --> 00:01:25.580
The added heat to the system,
we figured out this multiple

00:01:25.580 --> 00:01:28.540
times, was Q1.

00:01:28.540 --> 00:01:31.250
It was equivalent to the work we
did over that time period,

00:01:31.250 --> 00:01:33.210
so it would have been the whole
shaded region, not just

00:01:33.210 --> 00:01:34.760
what's inside the circle.

00:01:34.760 --> 00:01:37.460
But so the heat that
we gave you was Q1.

00:01:37.460 --> 00:01:39.540
You later gave some heat
back, when some work

00:01:39.540 --> 00:01:41.310
was done for you.

00:01:41.310 --> 00:01:44.950
But what we care about is the
heat that we were given.

00:01:44.950 --> 00:01:48.030
So in this case,
it would be Q1.

00:01:48.030 --> 00:01:49.720
And what was the work
that you did?

00:01:49.720 --> 00:01:53.320
Well, the net work that you
did was the shaded area.

00:01:53.320 --> 00:01:56.160
It was the area inside
of our Carnot cycle.

00:01:56.160 --> 00:01:58.240
So that's our definition
for efficiency.

00:01:58.240 --> 00:01:59.630
It's always going to
be a fraction.

00:01:59.630 --> 00:02:01.970
Sometimes it's given as a
percentage, where you just,

00:02:01.970 --> 00:02:03.380
you know, this is 0.56.

00:02:03.380 --> 00:02:05.500
You would call that
56% efficient.

00:02:05.500 --> 00:02:08.800
Which is essentially saying, you
were able to transfer 56%

00:02:08.800 --> 00:02:12.690
of the heat energy that you were
given, and turn that into

00:02:12.690 --> 00:02:13.720
useful work.

00:02:13.720 --> 00:02:17.320
And so it makes sense, at least
in my head, that that

00:02:17.320 --> 00:02:20.020
would be the definition
for efficiency.

00:02:20.020 --> 00:02:23.550
Now le'ts see if we can play
around with this, and see how

00:02:23.550 --> 00:02:25.720
efficiency would play out with
some of the variables we're

00:02:25.720 --> 00:02:27.720
dealing with in the
Carnot cycle.

00:02:27.720 --> 00:02:30.310
So what was the work
that we did?

00:02:30.310 --> 00:02:32.960
Well, we know our definition
of internal energy.

00:02:32.960 --> 00:02:36.370
Our definition of internal
energy has been more useful

00:02:36.370 --> 00:02:39.380
than you would have thought,
for such a simple equation.

00:02:39.380 --> 00:02:46.840
Our change in internal energy is
the net heat applied to the

00:02:46.840 --> 00:02:55.640
system minus the work
done by the system.

00:02:55.640 --> 00:02:56.420
Right?

00:02:56.420 --> 00:02:59.910
Now, when you complete one
Carnot cycle, when you go from

00:02:59.910 --> 00:03:03.430
a all the way around back to a,
and actually, I'll make a

00:03:03.430 --> 00:03:04.060
little aside here.

00:03:04.060 --> 00:03:05.020
You could have actually
gone the other

00:03:05.020 --> 00:03:05.910
way around the cycle.

00:03:05.910 --> 00:03:09.090
But when you go the way we went
the first time, when you

00:03:09.090 --> 00:03:10.980
go clockwise, you're
a Carnot engine.

00:03:10.980 --> 00:03:13.330
You're doing work, and
you're transferring

00:03:13.330 --> 00:03:16.940
heat from T1 to T2.

00:03:16.940 --> 00:03:20.450
If you went the other way
around the circle,

00:03:20.450 --> 00:03:23.430
essentially, you'd be a Carnot
refrigerator, where you would

00:03:23.430 --> 00:03:26.080
have work being done to you,
and I'll touch on this in a

00:03:26.080 --> 00:03:27.610
second, and you'd
be transferring

00:03:27.610 --> 00:03:28.680
energy the other way.

00:03:28.680 --> 00:03:31.690
And this will be important to
our proof of why the Carnot

00:03:31.690 --> 00:03:34.980
engine is the best engine, at
least theoretically, from an

00:03:34.980 --> 00:03:36.070
efficiency point of view.

00:03:36.070 --> 00:03:38.510
If efficiency is all
you cared about.

00:03:38.510 --> 00:03:39.080
But anyway.

00:03:39.080 --> 00:03:40.110
That's what I was
talking about.

00:03:40.110 --> 00:03:43.670
So if I go a complete cycle in
this PV diagram and I end up

00:03:43.670 --> 00:03:47.260
back at A, what's my change
in internal energy?

00:03:47.260 --> 00:03:48.280
It's 0.

00:03:48.280 --> 00:03:51.360
My internal energy is a state
variable, so my change in

00:03:51.360 --> 00:03:52.070
internal energy is 0.

00:03:52.070 --> 00:03:53.570
My change in entropy
would also be 0.

00:03:53.570 --> 00:03:57.160
It's another state variable,
when I get from A back to A.

00:03:57.160 --> 00:03:59.950
So over the course of this
cycle, we know that my change

00:03:59.950 --> 00:04:02.220
in internal energy is 0.

00:04:02.220 --> 00:04:06.855
What's the net heat applied
to the system?

00:04:09.470 --> 00:04:15.690
Well, we applied Q1 to the
system, and then we

00:04:15.690 --> 00:04:17.720
took out Q2, right?

00:04:17.720 --> 00:04:19.839
We gave that to the
second reservoir.

00:04:19.839 --> 00:04:24.140
We gave that down there to T2,
to that second reservoir.

00:04:24.140 --> 00:04:28.540
And then minus work-- and all
of this is equal to 0.

00:04:28.540 --> 00:04:31.400
This is the net, i just want to
make it clear, this is the

00:04:31.400 --> 00:04:33.210
net heat applied
to our system.

00:04:33.210 --> 00:04:38.080
So the work done by the system--
we just add W to both

00:04:38.080 --> 00:04:44.210
sides of this equation, you get
W is equal to Q1 minus Q2.

00:04:44.210 --> 00:04:44.980
So there we have it.

00:04:44.980 --> 00:04:47.440
Let's just substitute
that back here.

00:04:47.440 --> 00:04:53.050
And instead of W, we can write
Q1 minus Q2 as the numerator

00:04:53.050 --> 00:04:56.050
in our in our efficiency
definition, and then the

00:04:56.050 --> 00:04:59.700
denominator is still Q1.

00:04:59.700 --> 00:05:01.280
And we can do a little
bit of math.

00:05:01.280 --> 00:05:03.090
This simplifies-- this is Q1.

00:05:03.090 --> 00:05:04.890
This is the heat
we put into it.

00:05:04.890 --> 00:05:08.040
So it's the net heat we applied
to the system divided

00:05:08.040 --> 00:05:10.070
by the heat we put into it.

00:05:10.070 --> 00:05:14.460
So this is equal to-- Q1
divided by Q1 is 1,

00:05:14.460 --> 00:05:18.960
minus Q2 over Q1.

00:05:18.960 --> 00:05:23.040
So once again, this is another
interesting definition of

00:05:23.040 --> 00:05:23.590
efficiency.

00:05:23.590 --> 00:05:25.730
They're all algebraic
manipulations using the

00:05:25.730 --> 00:05:28.810
definition of internal energy,
and whatever else.

00:05:28.810 --> 00:05:30.900
Now let's see if we can somehow
relate efficiency to

00:05:30.900 --> 00:05:32.760
our temperatures.

00:05:32.760 --> 00:05:35.240
Now, let me-- this is
Q1 right there.

00:05:35.240 --> 00:05:37.920
So what were Q2 and Q1?

00:05:37.920 --> 00:05:38.600
What were they?

00:05:38.600 --> 00:05:41.420
What were their absolute values,
not looking at the

00:05:41.420 --> 00:05:42.150
signs of them?

00:05:42.150 --> 00:05:46.670
I mean, we know that Q2 was
transferred out of the system,

00:05:46.670 --> 00:05:50.200
so if we said Q2 in terms of
the energy applied to the

00:05:50.200 --> 00:05:51.360
system, it would be
a negative number.

00:05:51.360 --> 00:05:53.320
But if we just wanted to know
the magnitude of Q2, what

00:05:53.320 --> 00:05:55.240
would it be, and the
magnitude of Q1?

00:05:55.240 --> 00:05:59.070
Well, the magnitude of Q1--
let me draw a new Carnot

00:05:59.070 --> 00:06:02.730
cycle, just for cleanliness.

00:06:02.730 --> 00:06:05.570
I'll draw a little small
one over here.

00:06:05.570 --> 00:06:06.970
That's my volume axis.

00:06:06.970 --> 00:06:10.360
That's my pressure axis.

00:06:10.360 --> 00:06:12.790
PV.

00:06:12.790 --> 00:06:16.170
I start here at some state, and
then I go isothermally.

00:06:16.170 --> 00:06:19.410
What's a good color for an
isothermal expansion?

00:06:19.410 --> 00:06:20.100
Maybe purple.

00:06:20.100 --> 00:06:21.196
It's kind of-- let's see.

00:06:21.196 --> 00:06:22.850
An isothermal expansion.

00:06:22.850 --> 00:06:25.050
I'm on an isotherm here.

00:06:25.050 --> 00:06:29.290
So I go down there, and I
go to state, and I went

00:06:29.290 --> 00:06:30.300
down to state B.

00:06:30.300 --> 00:06:31.910
So this is A to B.

00:06:31.910 --> 00:06:33.750
And we know we are
an isotherm.

00:06:33.750 --> 00:06:37.830
This is when Q1 was added.

00:06:37.830 --> 00:06:38.910
This is an isotherm.

00:06:38.910 --> 00:06:41.330
If your temperature didn't
change, your internal energy

00:06:41.330 --> 00:06:42.200
didn't change.

00:06:42.200 --> 00:06:46.110
And like I said before, if your
internal energy is 0,

00:06:46.110 --> 00:06:48.970
then the heat applied to the
system is the same as

00:06:48.970 --> 00:06:49.700
the work you did.

00:06:49.700 --> 00:06:50.760
They cancel each other out.

00:06:50.760 --> 00:06:51.880
That's why you got to 0.

00:06:51.880 --> 00:06:55.220
So this Q1 that we apply to the
system, it must be equal

00:06:55.220 --> 00:06:56.520
to the work we did.

00:06:56.520 --> 00:07:01.270
And the work we did is just
the area under this curve.

00:07:01.270 --> 00:07:02.810
We did this multiple times.

00:07:02.810 --> 00:07:04.350
And why is it the area
under the curve?

00:07:04.350 --> 00:07:06.120
Because it's a bunch
of rectangles of

00:07:06.120 --> 00:07:09.770
pressure times volume.

00:07:09.770 --> 00:07:12.320
And then you just add up all
the rectangles, an infinite

00:07:12.320 --> 00:07:13.990
number of infinitely
narrow rectangles,

00:07:13.990 --> 00:07:15.300
and you get the area.

00:07:15.300 --> 00:07:16.540
And what is that?

00:07:16.540 --> 00:07:18.970
Just a review-- pressure times
volume was the work, right?

00:07:18.970 --> 00:07:22.100
Because we're expanding
the cylinder.

00:07:22.100 --> 00:07:23.520
We're moving up that piston.

00:07:23.520 --> 00:07:25.040
We're doing force
times distance.

00:07:25.040 --> 00:07:29.050
So the amount Q1 is equal to
that integral-- the amount of

00:07:29.050 --> 00:07:34.260
work we did as well-- is equal
to the integral from V final--

00:07:34.260 --> 00:07:36.160
I shouldn't say V final.

00:07:36.160 --> 00:07:40.980
from VB, from our volume
at B-- oh sorry.

00:07:40.980 --> 00:07:44.400
From our volume at A
to our volume at B.

00:07:44.400 --> 00:07:48.080
We're starting here, and we're
going to our volume at B.

00:07:48.080 --> 00:07:52.010
And we're taking the integral
of pressure-- and I've done

00:07:52.010 --> 00:07:55.490
this multiple times, but
I'll do it again.

00:07:55.490 --> 00:07:57.080
So the pressure, our
height, times our

00:07:57.080 --> 00:07:59.930
change in volume, dv.

00:07:59.930 --> 00:08:06.270
We go back to our ideal gas
formula, PV is equal to nRT,

00:08:06.270 --> 00:08:08.740
divide both sides by
V, and you get P is

00:08:08.740 --> 00:08:13.050
equal to nRT over V.

00:08:13.050 --> 00:08:20.190
And so you have Q1 is equal to
the integral of VA from VA to

00:08:20.190 --> 00:08:23.560
VB of this little
thing over here.

00:08:23.560 --> 00:08:27.750
nRT over V dv.

00:08:27.750 --> 00:08:29.950
All of this stuff up here,
these are all constants.

00:08:29.950 --> 00:08:31.210
Remember, we're an isotherm.

00:08:31.210 --> 00:08:32.200
Our temperature isn't
changing.

00:08:32.200 --> 00:08:34.260
We could write T1 there,
because that's our

00:08:34.260 --> 00:08:34.950
temperature.

00:08:34.950 --> 00:08:35.820
We're at T1.

00:08:35.820 --> 00:08:38.770
We're on a T1 isotherm right
there, because we're touching

00:08:38.770 --> 00:08:39.980
the T1 reservoir.

00:08:39.980 --> 00:08:41.370
But this is all a constant,
so we can take

00:08:41.370 --> 00:08:42.870
it out of the equation.

00:08:42.870 --> 00:08:46.130
And then we've evaluated this
multiple times, so I won't go

00:08:46.130 --> 00:08:47.600
into the mechanics
of the integral.

00:08:47.600 --> 00:08:55.970
But so this is Q1 is equal to
our constant terms, nRT1 times

00:08:55.970 --> 00:08:56.790
this definite integral.

00:08:56.790 --> 00:08:59.220
All we have left in the integral
is 1 over V The

00:08:59.220 --> 00:09:01.720
antiderivative of that is
natural log, and then you

00:09:01.720 --> 00:09:03.390
evaluated the two boundaries.

00:09:03.390 --> 00:09:11.040
So you get the natural log of VB
minus VA, which is the same

00:09:11.040 --> 00:09:15.450
thing as the natural
log of VB over VA.

00:09:15.450 --> 00:09:16.480
Fair enough.

00:09:16.480 --> 00:09:18.510
This right here is Q1.

00:09:18.510 --> 00:09:18.860
All right.

00:09:18.860 --> 00:09:20.500
Now what is Q2?

00:09:20.500 --> 00:09:23.220
Q2 was this part of
the Carnot cycle.

00:09:23.220 --> 00:09:25.720
Q2 is when we went
from C to D.

00:09:28.540 --> 00:09:33.080
So the magnitude of Q2 is
the area under this

00:09:33.080 --> 00:09:34.330
curve, right here.

00:09:36.680 --> 00:09:39.650
Now this is the work done to of
the system, so that's why

00:09:39.650 --> 00:09:41.770
we subtracted it out when we
wanted to know the net work

00:09:41.770 --> 00:09:43.050
done by the system.

00:09:43.050 --> 00:09:45.510
And we ended up with this area
over here, when you subtracted

00:09:45.510 --> 00:09:47.460
this as well, over here
on this side.

00:09:47.460 --> 00:09:50.180
But if we just want to know the
magnitude of Q2, we just

00:09:50.180 --> 00:09:55.150
take the integral under
this curve.

00:09:55.150 --> 00:09:58.350
And what's the integral
under that curve?

00:09:58.350 --> 00:10:01.370
This is the heat out of the
system, the heat that had to

00:10:01.370 --> 00:10:04.520
be pushed out of the system
as work was done to it.

00:10:04.520 --> 00:10:07.300
So that integral-- so we could
just say, the magnitude of

00:10:07.300 --> 00:10:16.980
Q2-- I'll do it over here-- is
equal to-- the same exact

00:10:16.980 --> 00:10:19.470
logic applies, just the
boundaries are different.

00:10:19.470 --> 00:10:22.240
We're now going from-- and
remember, when we cared about

00:10:22.240 --> 00:10:25.090
the direction, I would say,
I'm going from VC to VD.

00:10:25.090 --> 00:10:28.140
But if I just want to know the
absolute value of that area,

00:10:28.140 --> 00:10:31.900
because I just want to know the
magnitude, I could go from

00:10:31.900 --> 00:10:34.900
VC to VD and just take the
absolute value of it, or I

00:10:34.900 --> 00:10:38.610
could just go from VD to VC, and
I'd get a positive area.

00:10:38.610 --> 00:10:41.350
So let me just do that.

00:10:41.350 --> 00:10:45.910
So it's the integral
from VD to VC.

00:10:45.910 --> 00:10:49.600
Remember, the cycle we went from
VC to VD, but I just want

00:10:49.600 --> 00:10:50.270
the absolute value.

00:10:50.270 --> 00:10:51.550
I want this to be positive.

00:10:51.550 --> 00:10:54.810
So I'm turning it
the other way.

00:10:54.810 --> 00:10:57.580
Of P dv.

00:10:57.580 --> 00:11:00.200
And we do the exact
same math there.

00:11:00.200 --> 00:11:06.050
You get Q2 is equal to
nR-- this time the

00:11:06.050 --> 00:11:07.070
temperature is T2.

00:11:07.070 --> 00:11:09.130
We're on a T2 reservoir.

00:11:09.130 --> 00:11:14.260
Times the natural log--
and this time,

00:11:14.260 --> 00:11:15.420
what's it going to be?

00:11:15.420 --> 00:11:21.150
Times the natural log of--
instead of VB over VA, it's

00:11:21.150 --> 00:11:28.800
going to be VC divided by VD.

00:11:31.900 --> 00:11:34.680
Now let's use these two pieces
of information and substitute

00:11:34.680 --> 00:11:38.450
them back into that results for
efficiency we just got.

00:11:38.450 --> 00:11:41.220
We just learned that you could
also write the efficiency of

00:11:41.220 --> 00:11:46.170
an engine to be equal
to 1 minus Q-- let's

00:11:46.170 --> 00:11:46.930
look back at it.

00:11:46.930 --> 00:11:48.350
1 minus Q2 over Q1.

00:11:53.070 --> 00:11:56.390
So let's substitute Q2 there
and Q1 over here.

00:11:56.390 --> 00:11:57.820
And what do you get?

00:11:57.820 --> 00:12:01.590
You get the efficiency of your
engine is equal to 1 minus--

00:12:01.590 --> 00:12:08.090
Q2 is this expression over
here-- nRT2 times the natural

00:12:08.090 --> 00:12:20.100
log of VC over VD, all of that
divided by Q1, which is this

00:12:20.100 --> 00:12:21.186
one over here.

00:12:21.186 --> 00:12:29.720
nRT1 times the natural
log of VB over VA.

00:12:29.720 --> 00:12:31.680
Now we can do a little
bit of canceling.

00:12:31.680 --> 00:12:34.800
Obviously the n and
the r is cancel.

00:12:34.800 --> 00:12:36.930
And we have these natural
logs and all of that.

00:12:36.930 --> 00:12:41.300
But I had a whole video
dedicated to show you that VC

00:12:41.300 --> 00:12:46.370
over dv is equal
to VB over VA.

00:12:46.370 --> 00:12:48.450
Now, if we know that these
are equal, then

00:12:48.450 --> 00:12:49.460
this is equal to this.

00:12:49.460 --> 00:12:50.840
So the natural logs
of them are equal,

00:12:50.840 --> 00:12:52.260
so we can just divide.

00:12:52.260 --> 00:12:53.850
And what are we left with?

00:12:53.850 --> 00:12:57.350
We're left with the fact that
efficiency can also be written

00:12:57.350 --> 00:13:08.550
as 1 minus T2 over T1
for a Carnot engine.

00:13:08.550 --> 00:13:11.980
Remember, this time, what we did
over here, this applied to

00:13:11.980 --> 00:13:12.970
any engine.

00:13:12.970 --> 00:13:16.130
This was just a little bit of
math and the definition of

00:13:16.130 --> 00:13:20.950
what work is, and-- well,
I won't go too much

00:13:20.950 --> 00:13:21.850
into it right now.

00:13:21.850 --> 00:13:24.490
But this is for a Carnot
engine, right?

00:13:24.490 --> 00:13:26.580
Because we did a little bit of
work here that involved the

00:13:26.580 --> 00:13:28.020
Carnot cycle.

00:13:28.020 --> 00:13:30.850
But this is a pretty bit
important outcome, because

00:13:30.850 --> 00:13:33.600
we're going to show that the
Carnot engine, in the next

00:13:33.600 --> 00:13:36.800
video, is actually the most
efficient engine that can ever

00:13:36.800 --> 00:13:37.870
be attained.

00:13:37.870 --> 00:13:40.150
Well, you have to be very
careful about that, that when

00:13:40.150 --> 00:13:42.940
we say efficient, it means that
between two temperature

00:13:42.940 --> 00:13:48.430
sources, you can't get a more
efficient engine than the

00:13:48.430 --> 00:13:49.070
Carnot engine.

00:13:49.070 --> 00:13:50.620
Now, I'm not saying it's the
best engine, or it's a

00:13:50.620 --> 00:13:53.630
practical engine, or you'd want
it to power your lawn

00:13:53.630 --> 00:13:56.050
mower or your jet plane.

00:13:56.050 --> 00:13:59.320
I'm just saying that it's the
most efficient engine between

00:13:59.320 --> 00:14:01.340
these two temperature
reservoirs.

00:14:01.340 --> 00:14:03.770
And I'll show you that
in the next video.


Multiplication 3: 10,11,12 times tables

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.150
In the last video we went over
the multiplication tables for 1

00:00:04.150 --> 00:00:06.370
through 9 and I ran out of
time, and actually, it was a

00:00:06.370 --> 00:00:09.820
good thing because 1 through 9
are kind of the core

00:00:09.820 --> 00:00:11.270
multiplication tables.

00:00:11.270 --> 00:00:13.860
And you'll see that if you know
all your multiplication tables

00:00:13.860 --> 00:00:16.930
from 1 to 9, so you know any
number between 1 and 9 times

00:00:16.930 --> 00:00:19.380
any other number between 1 and
9, you can actually do

00:00:19.380 --> 00:00:21.680
any multiplication
problem out there.

00:00:21.680 --> 00:00:23.550
But what I want to do now
is I want to complete the

00:00:23.550 --> 00:00:29.820
multiplication tables
for 10, 11, and 12.

00:00:29.820 --> 00:00:33.690
So what is 10 times--
let's just start with 0.

00:00:33.690 --> 00:00:35.990
10 times 0.

00:00:35.990 --> 00:00:39.040
Anything times 0 is 0.

00:00:39.040 --> 00:00:40.400
Ten 0's are 0.

00:00:40.400 --> 00:00:43.830
0 plus 0 plus 0 ten
times is still 0.

00:00:43.830 --> 00:00:45.246
What's 10 times 1?

00:00:48.130 --> 00:00:49.990
Well that's just 10 one time.

00:00:49.990 --> 00:00:52.750
Or 1 plus itself ten times.

00:00:52.750 --> 00:00:53.900
That's 10.

00:00:53.900 --> 00:00:56.510
I think this is second nature
to you at this point.

00:00:56.510 --> 00:00:59.980
What's 10 times 2?

00:00:59.980 --> 00:01:02.670
I meant to switch
colors, but I didn't.

00:01:02.670 --> 00:01:04.060
10 times 2?

00:01:04.060 --> 00:01:08.960
That's 10 plus 10, which is 20.

00:01:08.960 --> 00:01:09.560
Fair enough.

00:01:09.560 --> 00:01:12.660
And notice, we went up
by 10 the first time.

00:01:12.660 --> 00:01:15.220
We went up by 10
again to get to 20.

00:01:15.220 --> 00:01:17.710
What's 10 times 3?

00:01:17.710 --> 00:01:21.710
Well, that's 10 plus 10 plus
10, or we could view it as 10

00:01:21.710 --> 00:01:26.460
times 2 plus another 10,
which is equal to 30.

00:01:26.460 --> 00:01:27.330
What's 10 times 4?

00:01:27.330 --> 00:01:28.680
I think you start
to see a pattern.

00:01:28.680 --> 00:01:32.460
10 times 4 is equal to 40.

00:01:32.460 --> 00:01:37.410
Notice, 10 times 4
is equal to 40.

00:01:37.410 --> 00:01:42.040
If I were to tell you what
is 10 times-- let me

00:01:42.040 --> 00:01:44.870
do another color-- 5?

00:01:44.870 --> 00:01:50.270
Well that's equal to 50.

00:01:50.270 --> 00:01:56.420
10 times anything is that
anything with a 0 behind it.

00:01:56.420 --> 00:01:58.020
So the 10 times tables
you almost don't

00:01:58.020 --> 00:01:59.320
have to remember it.

00:01:59.320 --> 00:02:01.390
So let's just keep going.

00:02:01.390 --> 00:02:04.000
What's 10 times 6?

00:02:04.000 --> 00:02:06.850
It's equal to 60.

00:02:06.850 --> 00:02:08.150
6 0.

00:02:08.150 --> 00:02:11.100
What's 10 times 7?

00:02:11.100 --> 00:02:12.310
70.

00:02:12.310 --> 00:02:13.350
10 times 8?

00:02:13.350 --> 00:02:14.930
This is almost ridiculous.

00:02:14.930 --> 00:02:16.820
10 times 8 is 80.

00:02:16.820 --> 00:02:19.040
10 times 9?

00:02:19.040 --> 00:02:20.330
90.

00:02:20.330 --> 00:02:21.600
10 times 10?

00:02:21.600 --> 00:02:22.385
Now this is interesting.

00:02:25.300 --> 00:02:28.996
So it'll be a 10-- let's
see me write this.

00:02:28.996 --> 00:02:32.190
Let me do it in
this orange color.

00:02:32.190 --> 00:02:33.250
10 times 10.

00:02:33.250 --> 00:02:38.550
So it'll be ten 10's or a
10 with a 0 behind it.

00:02:38.550 --> 00:02:39.000
There you go.

00:02:39.000 --> 00:02:41.860
Notice everything-- whatever
times 10 I just add a 0 and

00:02:41.860 --> 00:02:42.600
then I get the next number.

00:02:42.600 --> 00:02:43.960
So it's 100.

00:02:43.960 --> 00:02:45.530
And I think you
understand why that is.

00:02:45.530 --> 00:02:47.910
I added 10 to itself ten times.

00:02:47.910 --> 00:02:51.710
That each 10-- you
go from 10, 20, 30.

00:02:51.710 --> 00:02:54.250
30 is just three
10's or 10 times 3.

00:02:54.250 --> 00:02:58.260
90 is just nine 10's
or 9 times 10.

00:02:58.260 --> 00:03:00.460
Let's keep going.

00:03:00.460 --> 00:03:05.130
So 10 times 11 is equal to
11 with a 0 behind it.

00:03:05.130 --> 00:03:06.970
110.

00:03:06.970 --> 00:03:13.080
Finally, 10 times 12
is equal to 120.

00:03:13.080 --> 00:03:17.380
Now, just for fun, these are
kind of your 10 times tables.

00:03:17.380 --> 00:03:20.480
But now that t know the
pattern you can do anything.

00:03:20.480 --> 00:03:29.630
If I asked you what 5,732 times
10 is, what's it going to be?

00:03:29.630 --> 00:03:32.080
It's going to be this number
with just one more 0.

00:03:32.080 --> 00:03:34.660
So it's going to be-- I
won't read it out yet.

00:03:34.660 --> 00:03:39.100
5 7 3 2 with a 0 behind it.

00:03:39.100 --> 00:03:41.640
And just so you know, this
little comma that I wrote in

00:03:41.640 --> 00:03:44.040
the number there, that's
just to make it easier for

00:03:44.040 --> 00:03:45.990
me to read that number.

00:03:45.990 --> 00:03:48.680
And you put the comma-- you
start over here and every third

00:03:48.680 --> 00:03:49.750
number you put the comma.

00:03:49.750 --> 00:03:53.080
So here I'm going to put
the comma right here.

00:03:53.080 --> 00:03:54.970
I'm going to put the
comma right there.

00:03:54.970 --> 00:03:56.650
So now I can read this.

00:03:56.650 --> 00:03:59.500
The comma doesn't really add
or take anything away from

00:03:59.500 --> 00:04:01.200
the number, it just
helps me read it.

00:04:01.200 --> 00:04:07.900
Now 5,732 times 10 is 57,320.

00:04:07.900 --> 00:04:10.420
i just had to add a 0 there,
but that was a pretty

00:04:10.420 --> 00:04:12.770
straightforward multiplication.

00:04:12.770 --> 00:04:16.670
And notice, we had 5,000 times
10 and we got to 50 something

00:04:16.670 --> 00:04:19.080
thousand when we
multiplied them.

00:04:19.080 --> 00:04:21.960
So that's similar to 5
times 10 is equal to 50.

00:04:21.960 --> 00:04:26.520
But instead of 5 I had a 5,000,
and so I got a 50,000 and

00:04:26.520 --> 00:04:28.060
something and all
this other stuff.

00:04:28.060 --> 00:04:29.750
We're going to learn more
about how to do problems

00:04:29.750 --> 00:04:30.950
like this in the future.

00:04:30.950 --> 00:04:33.420
But I thought I would introduce
you to the idea that just from

00:04:33.420 --> 00:04:35.660
this little pattern of adding a
0 you already know your

00:04:35.660 --> 00:04:37.700
10's times tables.

00:04:37.700 --> 00:04:38.855
Now let's do our 11's.

00:04:42.080 --> 00:04:44.670
Well, they start off easy, and
then they get a little more

00:04:44.670 --> 00:04:47.150
difficult as we get
into high numbers.

00:04:47.150 --> 00:04:49.010
11 times 0.

00:04:49.010 --> 00:04:51.520
This is easy, this is 0.

00:04:51.520 --> 00:04:52.840
11 times 1.

00:04:52.840 --> 00:04:54.010
This is also easy.

00:04:54.010 --> 00:04:56.020
It's 11.

00:04:56.020 --> 00:04:57.360
11 times 2.

00:04:57.360 --> 00:04:59.280
We're going to start
seeing a pattern here.

00:04:59.280 --> 00:05:03.430
It's 11 plus 11 or we could've
added 2 to itself eleven times,

00:05:03.430 --> 00:05:06.460
but that is equal to 22.

00:05:06.460 --> 00:05:11.520
If we do 11 times 3
it is equal to 33.

00:05:11.520 --> 00:05:15.560
11 times 4 is equal to 44.

00:05:15.560 --> 00:05:16.970
I think this is becoming
obvious to you.

00:05:16.970 --> 00:05:18.990
What's 11 times 5?

00:05:18.990 --> 00:05:20.940
11 times 5 is 55.

00:05:20.940 --> 00:05:23.010
Notice I put the 5 twice.

00:05:23.010 --> 00:05:24.940
What's 11 times 6?

00:05:24.940 --> 00:05:27.340
It's 66.

00:05:27.340 --> 00:05:30.837
11 times 7 is 84-- no.

00:05:30.837 --> 00:05:32.270
I'm kidding.

00:05:32.270 --> 00:05:33.720
I didn't want to mess
with you like that.

00:05:33.720 --> 00:05:34.250
But no.

00:05:34.250 --> 00:05:35.320
Of course, it's 77.

00:05:37.900 --> 00:05:41.090
You just repeat
the number twice.

00:05:41.090 --> 00:05:42.640
Let me switch colors.

00:05:42.640 --> 00:05:46.270
11 times 8 is equal to 88.

00:05:46.270 --> 00:05:50.420
11 times 9 is equal to 99.

00:05:50.420 --> 00:05:51.800
Now what's 11 times 12?

00:05:54.430 --> 00:05:56.750
Oh sorry, I skipped 10.

00:05:56.750 --> 00:05:58.150
11 times 10.

00:05:58.150 --> 00:06:00.160
You might want to
say it's 1,010.

00:06:00.160 --> 00:06:00.730
No.

00:06:00.730 --> 00:06:01.410
That's wrong.

00:06:01.410 --> 00:06:05.100
It's not 1,010.

00:06:05.100 --> 00:06:07.990
So that little pattern that we
had where you just repeat the

00:06:07.990 --> 00:06:10.000
number, that only worked
for one-digit numbers.

00:06:10.000 --> 00:06:11.490
So it only worked
for 1 through 9.

00:06:11.490 --> 00:06:12.860
11 times 10?

00:06:12.860 --> 00:06:14.580
Well, we could think about
it a couple of ways.

00:06:14.580 --> 00:06:16.750
We can add 11 to 99.

00:06:16.750 --> 00:06:20.355
So we can say it's 99 plus 11.

00:06:20.355 --> 00:06:21.990
And what's that?

00:06:21.990 --> 00:06:23.400
That's equal to 110.

00:06:23.400 --> 00:06:25.650
And I'm going to show you how
to do-- well, hopefully you've

00:06:25.650 --> 00:06:28.320
already watched the video on
how to add two-digit numbers

00:06:28.320 --> 00:06:30.220
like this, but that's 110.

00:06:30.220 --> 00:06:32.640
Or you could just use the
property from the 10's times

00:06:32.640 --> 00:06:36.000
tables that we learned where if
you just take 11 times 10 you

00:06:36.000 --> 00:06:37.680
add a 0 to the 11, you get 110.

00:06:37.680 --> 00:06:39.570
That's the 11 right there.

00:06:39.570 --> 00:06:41.920
Finally, let's do 11 times 12.

00:06:44.740 --> 00:06:46.480
No easy way to remember
this, you just kind of

00:06:46.480 --> 00:06:47.090
should remember it.

00:06:47.090 --> 00:06:49.180
Or you could say look,
it's going to be 11 more

00:06:49.180 --> 00:06:51.320
than 11 times-- sorry.

00:06:51.320 --> 00:06:53.150
I keep skipping things.

00:06:53.150 --> 00:06:55.420
We should do 11 times 11 first.

00:06:55.420 --> 00:06:56.710
Let me make sure this is clear.

00:06:56.710 --> 00:07:00.600
We're doing 11 times 11
before we go to 11 times 12.

00:07:00.600 --> 00:07:05.490
So 11 times 11 is going to be
11 more than 11 times 10.

00:07:05.490 --> 00:07:07.280
So we add 11 to this.

00:07:07.280 --> 00:07:13.130
11 plus 110 is 121.

00:07:13.130 --> 00:07:15.170
And actually, as you'll see,
there actually is an order as

00:07:15.170 --> 00:07:18.800
we get to higher multiples of
11, but I'll leave that

00:07:18.800 --> 00:07:20.250
to a future video.

00:07:20.250 --> 00:07:22.300
And then finally,
we're at 11 times 12.

00:07:26.290 --> 00:07:28.540
And we could add 11 to
itself twelve times.

00:07:28.540 --> 00:07:30.770
We could add 12 to
itself eleven times.

00:07:30.770 --> 00:07:35.990
Or we could just say, hey,
is going to be 11 more

00:07:35.990 --> 00:07:37.910
than 11 times 11.

00:07:37.910 --> 00:07:38.850
So that is what?

00:07:38.850 --> 00:07:40.810
You add 11 to this.

00:07:40.810 --> 00:07:41.350
What do you get?

00:07:41.350 --> 00:07:46.370
You get 132.

00:07:46.370 --> 00:07:51.120
I just added 121 plus
11 and then got 132.

00:07:51.120 --> 00:07:52.605
Now the other way you could
have said it is, well,

00:07:52.605 --> 00:07:54.900
what's 10 times 12?

00:07:54.900 --> 00:07:56.170
We already knew that.

00:07:56.170 --> 00:07:58.580
That was 120.

00:07:58.580 --> 00:08:02.980
So 11 times 12 because we're
multiplying 12 by one more

00:08:02.980 --> 00:08:04.940
should be 12 more than that.

00:08:04.940 --> 00:08:06.710
So that should be 132.

00:08:06.710 --> 00:08:10.100
So two ways to get the
exact same answer.

00:08:10.100 --> 00:08:12.785
All right, now let's do
our 12 times tables.

00:08:15.330 --> 00:08:18.650
And once you know this you
are ready to tackle any type

00:08:18.650 --> 00:08:19.745
of multiplication problem.

00:08:19.745 --> 00:08:21.995
But we'll do that
in future videos.

00:08:21.995 --> 00:08:24.000
So 12 times 0.

00:08:24.000 --> 00:08:24.800
Super easy.

00:08:24.800 --> 00:08:25.350
0.

00:08:25.350 --> 00:08:26.430
12 times 1.

00:08:26.430 --> 00:08:27.335
Also super easy.

00:08:27.335 --> 00:08:28.490
Is 12.

00:08:28.490 --> 00:08:29.930
Now it gets interesting.

00:08:29.930 --> 00:08:33.510
We're going to increase
by 12 every time.

00:08:33.510 --> 00:08:37.000
12 times 2 is equal to 24.

00:08:37.000 --> 00:08:39.900
12 plus 12 is 24.

00:08:39.900 --> 00:08:43.240
12 times-- not 22.

00:08:43.240 --> 00:08:44.170
Let me rewrite that.

00:08:44.170 --> 00:08:50.260
12 times 3 is going to
be 12 plus 12 plus 12.

00:08:50.260 --> 00:08:53.990
Or we could write
that as 12 times 2.

00:08:53.990 --> 00:08:56.070
I see my brain is doing
the wrong things.

00:08:56.070 --> 00:09:00.680
We could rewrite that
as 12 times 2 plus 12.

00:09:00.680 --> 00:09:03.780
Or we could rewrite
that as 24 plus 12.

00:09:03.780 --> 00:09:07.040
Either way, all of
these get us to 36.

00:09:07.040 --> 00:09:10.410
And notice, that's
just that plus 12.

00:09:10.410 --> 00:09:12.250
12 times 4.

00:09:12.250 --> 00:09:17.190
12 times 4 is equal to 48.

00:09:17.190 --> 00:09:18.275
There's a lot of ways you
could think about it.

00:09:18.275 --> 00:09:19.980
You could say 11 times 4 is 44.

00:09:25.670 --> 00:09:30.030
And you go up by one more 4,
so you get to 12 times 4.

00:09:30.030 --> 00:09:33.630
Or you could say 12 times 3
is 36 and you can add one

00:09:33.630 --> 00:09:36.950
12 to it to get to 48.

00:09:36.950 --> 00:09:39.235
Either way works and that's
because you can multiply

00:09:39.235 --> 00:09:40.310
in either direction.

00:09:40.310 --> 00:09:42.310
Let's keep going.

00:09:42.310 --> 00:09:47.850
12 times 5 is equal to 60.

00:09:47.850 --> 00:09:55.340
10 times 5 is 50, 11 times 5
is 55, so 12 times 5 is 60.

00:09:55.340 --> 00:09:58.665
12 times 6 is equal to what?

00:09:58.665 --> 00:10:00.580
It's going to be 12
more than this.

00:10:00.580 --> 00:10:02.640
It's going to equal 72.

00:10:02.640 --> 00:10:04.520
12 times 7.

00:10:04.520 --> 00:10:06.720
12 more than this again.

00:10:06.720 --> 00:10:09.470
12 more than 72 is 84.

00:10:09.470 --> 00:10:15.890
And I'm serious, I'm probably a
lot older than you are and I

00:10:15.890 --> 00:10:19.200
still, in my head to confirm, I
go to some 12 times tables that

00:10:19.200 --> 00:10:20.700
I remember as definitely right.

00:10:20.700 --> 00:10:23.290
Like oh, 12 times 5-- and
sometimes in my head I say,

00:10:23.290 --> 00:10:24.100
oh, let me add another 12.

00:10:24.100 --> 00:10:26.300
Oh yeah, definitely, my
memory was correct.

00:10:26.300 --> 00:10:28.490
12 times 6 is 72.

00:10:28.490 --> 00:10:29.030
All right.

00:10:29.030 --> 00:10:31.330
Then you go to 12 times 8.

00:10:31.330 --> 00:10:34.100
Add 12 to the 12 times 7.

00:10:34.100 --> 00:10:35.410
96.

00:10:35.410 --> 00:10:38.986
12 times 9 .

00:10:38.986 --> 00:10:44.350
Well you add 12 to
this, so it's 108.

00:10:44.350 --> 00:10:46.700
And then 12 times 10.

00:10:46.700 --> 00:10:48.680
This is an easy one.

00:10:48.680 --> 00:10:51.390
We just add a 0 to
the 12 to get 120.

00:10:51.390 --> 00:10:54.210
Or we could've added 12 to 108.

00:10:54.210 --> 00:10:55.570
Either way.

00:10:55.570 --> 00:10:58.300
12 times 11.

00:10:58.300 --> 00:10:59.420
We just did this.

00:10:59.420 --> 00:11:02.000
You add 12 to this
you get to 132.

00:11:02.000 --> 00:11:09.000
And then 12 times 12
is equal to 144.

00:11:09.000 --> 00:11:10.170
And this actually shows up.

00:11:10.170 --> 00:11:13.530
If I had a dozen of a dozen
eggs-- a dozen is 12.

00:11:13.530 --> 00:11:17.020
Or if I had, I think a gross
is actually 12 dozens.

00:11:17.020 --> 00:11:18.420
So that's 144 eggs.

00:11:18.420 --> 00:11:20.710
You'll actually end up
seeing this number a lot.

00:11:20.710 --> 00:11:22.560
More than you would
expect in life.

00:11:22.560 --> 00:11:25.720
But anyway, we've now
completed all of our

00:11:25.720 --> 00:11:27.420
multiplication tables.

00:11:27.420 --> 00:11:29.790
And I really encourage you
to take the time now to

00:11:29.790 --> 00:11:30.930
go and memorize them.

00:11:30.930 --> 00:11:33.600
Make some flash cards.

00:11:33.600 --> 00:11:37.100
Use the little software thing
that I wrote on my website.

00:11:37.100 --> 00:11:38.860
You could try that out.

00:11:38.860 --> 00:11:41.730
As of September
2009 it's working.

00:11:41.730 --> 00:11:43.900
I haven't touched it in a
while, but I'm actually

00:11:43.900 --> 00:11:45.000
probably going to
rebuild it soon.

00:11:45.000 --> 00:11:48.050
So if you're watching this
video in the year 2200.

00:11:48.050 --> 00:11:50.430
Well, I would probably
not exist anymore.

00:11:50.430 --> 00:11:52.260
But hopefully you'll get
a better version of

00:11:52.260 --> 00:11:53.360
the software app.

00:11:53.360 --> 00:11:55.030
But you should practice it.

00:11:55.030 --> 00:11:56.620
You should get your
parents to quiz you.

00:11:56.620 --> 00:11:58.090
You should get notes cards.

00:11:58.090 --> 00:12:00.540
You should just be mumbling
to yourself as you walk to

00:12:00.540 --> 00:12:02.190
school-- what is 12 times 9?

00:12:02.190 --> 00:12:03.750
What is 11 times 11?

00:12:03.750 --> 00:12:07.120
And you should quiz each other
because it'll pay huge rewards

00:12:07.120 --> 00:12:08.690
to you later on in life.

00:12:08.690 --> 00:12:10.640
See you in the next video.


More on entropy

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.820
I'm going to do one more
video on entropy.

00:00:02.820 --> 00:00:04.180
Maybe I'll do more
in the future.

00:00:04.180 --> 00:00:06.970
But I thought I would at least
right now do one more, because

00:00:06.970 --> 00:00:11.080
I really want to make clear
the idea of entropy as a

00:00:11.080 --> 00:00:13.290
macrostate variable.

00:00:13.290 --> 00:00:14.590
So let me write that down.

00:00:14.590 --> 00:00:20.320
An S, which is entropy, is
a macrostate variable.

00:00:20.320 --> 00:00:24.100
I'm going to write macro
in bold red.

00:00:24.100 --> 00:00:28.010
It's a macrostate
state variable.

00:00:28.010 --> 00:00:29.500
And I want to emphasize this.

00:00:29.500 --> 00:00:31.780
And I talked a little bit about
it in my previous video,

00:00:31.780 --> 00:00:35.590
but I think even then I wasn't
as exact as I needed to be.

00:00:35.590 --> 00:00:37.850
And the reason why I want to
say it's macro is because

00:00:37.850 --> 00:00:40.920
there's a very strong temptation
to point to

00:00:40.920 --> 00:00:44.520
particular micro states and
say, does this have higher

00:00:44.520 --> 00:00:46.570
entropy or lower entropy
than another?

00:00:46.570 --> 00:00:48.830
For example, the classic one.

00:00:48.830 --> 00:00:51.430
Even I did this.

00:00:51.430 --> 00:00:55.750
Actually, let me make a
bigger, thicker line.

00:00:55.750 --> 00:01:02.360
So you have this box that
has a divider in it.

00:01:02.360 --> 00:01:05.090
We've gone through this
multiple times.

00:01:05.090 --> 00:01:08.710
Let me draw the divider
like right there.

00:01:08.710 --> 00:01:09.960
OK.

00:01:09.960 --> 00:01:13.530
So at first if we have a
system where all of the

00:01:13.530 --> 00:01:19.550
molecules are over here-- so
that's our first scenario.

00:01:19.550 --> 00:01:21.510
And then our second scenario,
we've studied this a lot.

00:01:21.510 --> 00:01:25.190
We blow up the wall here.

00:01:25.190 --> 00:01:28.870
And we actually calculated
the entropy.

00:01:28.870 --> 00:01:29.630
Control.

00:01:29.630 --> 00:01:29.870
OK.

00:01:29.870 --> 00:01:33.240
Let me copy it and then
I'm going to paste it.

00:01:33.240 --> 00:01:37.210
Let me put these next
to each other.

00:01:37.210 --> 00:01:38.210
All right.

00:01:38.210 --> 00:01:39.540
So I have these two things.

00:01:39.540 --> 00:01:42.360
And then the second time,
I blew away this wall.

00:01:42.360 --> 00:01:45.200
Let me blow it away.

00:01:45.200 --> 00:01:47.120
Let me erase the wall there.

00:01:47.120 --> 00:01:50.230
And then we said, once the
system reaches equilibrium

00:01:50.230 --> 00:01:54.640
again-- remember, these
macrostate variables, like

00:01:54.640 --> 00:01:57.600
pressure, like volume, like
temperature, like entropy, are

00:01:57.600 --> 00:02:01.240
only defined once the system
is in equilibrium.

00:02:01.240 --> 00:02:04.660
So once the system is in
equilibrium again, these

00:02:04.660 --> 00:02:07.305
particles are now-- I wanted the
same number of particles,

00:02:07.305 --> 00:02:11.840
so let me erase some of these
particles, let me move them.

00:02:11.840 --> 00:02:13.160
Let me see if I can
select some.

00:02:23.920 --> 00:02:25.110
There.

00:02:25.110 --> 00:02:29.380
So we'll have a more even--
Let me just redraw.

00:02:29.380 --> 00:02:32.850
1, 2, 3, 4, 5, 6,
7, 8 particles.

00:02:32.850 --> 00:02:37.980
Let me erase what I have. And
I'll make it with a more even

00:02:37.980 --> 00:02:39.230
distribution.

00:02:41.020 --> 00:02:46.770
So then, once I blow away the
wall, I might have 1, 2, 3, 4,

00:02:46.770 --> 00:02:48.930
5, 6, 7, 8.

00:02:48.930 --> 00:02:51.520
Now, the reason why I'm doing
all of this is because there's

00:02:51.520 --> 00:02:56.660
a temptation to say that this
state, what I just drew you

00:02:56.660 --> 00:02:59.080
when I, you know, made sure to
blow these away and draw 8

00:02:59.080 --> 00:03:01.600
more, I drew a microstate.

00:03:01.600 --> 00:03:02.850
This is a microstate.

00:03:05.900 --> 00:03:08.080
Anytime someone is actually
drawing molecules for you,

00:03:08.080 --> 00:03:09.330
they're drawing a microstate.

00:03:12.580 --> 00:03:13.680
Now, I want to be very clear.

00:03:13.680 --> 00:03:17.260
This microstate does not have
more entropy than this

00:03:17.260 --> 00:03:18.310
microstate.

00:03:18.310 --> 00:03:21.380
In fact, microstates you
don't have entropy.

00:03:21.380 --> 00:03:23.140
Entropy does not make sense.

00:03:23.140 --> 00:03:27.670
What you can say is a system--
and this time, I'm going to

00:03:27.670 --> 00:03:29.050
draw it without the particles.

00:03:29.050 --> 00:03:33.900
That if I have a container
that is this big, that

00:03:33.900 --> 00:03:36.210
contains-- so it has
some volume.

00:03:36.210 --> 00:03:38.700
So volume is equal to v1.

00:03:38.700 --> 00:03:42.840
Its temperature is equal
to a t1, and it has 8

00:03:42.840 --> 00:03:43.860
particles in it.

00:03:43.860 --> 00:03:46.230
This has some entropy
associated with it.

00:03:46.230 --> 00:03:48.840
And what we can say, is if we
were to double the size of

00:03:48.840 --> 00:03:54.880
this container, which we did by
blowing away that wall, now

00:03:54.880 --> 00:03:59.160
all of a sudden our volume is
equal to 2 times v1, if we say

00:03:59.160 --> 00:04:00.100
this is double.

00:04:00.100 --> 00:04:02.330
Our temperature is still
equal to t1.

00:04:02.330 --> 00:04:04.080
We saw that a few videos ago.

00:04:04.080 --> 00:04:05.445
And we still have 8 molecules.

00:04:08.030 --> 00:04:10.940
The entropy of this
system is higher.

00:04:10.940 --> 00:04:14.180
So now entropy is higher.

00:04:14.180 --> 00:04:16.500
And I want to make this very
clear, because you never see

00:04:16.500 --> 00:04:17.279
it drawn this way.

00:04:17.279 --> 00:04:20.029
People always want to draw
the actual molecules.

00:04:20.029 --> 00:04:22.079
But that confuses the issue.

00:04:22.079 --> 00:04:24.380
When you draw actual molecules,
you're showing a

00:04:24.380 --> 00:04:26.010
particular state.

00:04:26.010 --> 00:04:30.200
For example, this system, if
we were to actually measure

00:04:30.200 --> 00:04:33.600
the microstate, it could be--
there's a very, very

00:04:33.600 --> 00:04:37.310
infinitesimally probability--
but all of the molecules, all

00:04:37.310 --> 00:04:40.160
8 molecules might
be right there.

00:04:40.160 --> 00:04:43.380
I mean, it's almost, you know,
you could wait for the whole

00:04:43.380 --> 00:04:45.500
universe to come and go, and
it might not happen.

00:04:45.500 --> 00:04:47.270
But there is some probability
it would happen.

00:04:47.270 --> 00:04:50.160
So you can't assign entropy
to a particular state.

00:04:50.160 --> 00:04:54.110
All you can do is assign it
to a particular system.

00:04:54.110 --> 00:04:55.610
I want to be clear about that.

00:04:55.610 --> 00:04:58.410
So even I talked about
a clean and dirty

00:04:58.410 --> 00:05:00.060
room and all of that.

00:05:00.060 --> 00:05:01.690
Clean versus dirty room.

00:05:04.650 --> 00:05:08.220
And the point I was making is,
the entropy of a room is not

00:05:08.220 --> 00:05:10.840
dependent on its cleanliness
or its dirtiness.

00:05:10.840 --> 00:05:11.960
In fact, you could kind
of view these

00:05:11.960 --> 00:05:13.670
as states of a room.

00:05:13.670 --> 00:05:14.980
But even more, these
really aren't even

00:05:14.980 --> 00:05:15.890
states of the room.

00:05:15.890 --> 00:05:19.390
Because when a room is clean, or
a room is static at a macro

00:05:19.390 --> 00:05:20.640
level, they're static.

00:05:23.400 --> 00:05:29.760
If my books or lying like--
you know, sometimes people

00:05:29.760 --> 00:05:31.390
look at a deck of cards and
say, oh, if I have all my

00:05:31.390 --> 00:05:35.940
cards stacked up like this, or
if I have all my cards that

00:05:35.940 --> 00:05:43.320
are all messy like that, that
this has higher entropy.

00:05:43.320 --> 00:05:44.680
And I want to make
it very clear.

00:05:44.680 --> 00:05:46.740
I mean, maybe you can kind
of make an analogy.

00:05:46.740 --> 00:05:49.090
But that's not the case.

00:05:49.090 --> 00:05:51.960
Both of these systems
are macrostates.

00:05:51.960 --> 00:05:55.420
For example, it's not like
these cards are vibrating

00:05:55.420 --> 00:05:57.730
around any more than
these cards are.

00:05:57.730 --> 00:06:00.660
It's not like these can take
on more configurations than

00:06:00.660 --> 00:06:01.970
these cards can.

00:06:01.970 --> 00:06:04.770
So when you talk about entropy,
you're trying to take

00:06:04.770 --> 00:06:07.800
a macro variable that's
describing at a micro level.

00:06:07.800 --> 00:06:10.710
And the cards themselves are not
the micro level, because

00:06:10.710 --> 00:06:13.820
they're not vibrating around
continuously due to some

00:06:13.820 --> 00:06:15.220
kinetic energy or whatever.

00:06:15.220 --> 00:06:19.410
It's the cards' molecules that
are at the micro level.

00:06:19.410 --> 00:06:23.000
And if these cards, if you have
the same mass of cars as

00:06:23.000 --> 00:06:26.490
you have here, and if they're
at the same temperature, the

00:06:26.490 --> 00:06:30.120
molecules in these cards can
take on just as many states as

00:06:30.120 --> 00:06:32.500
the molecules in these cards.

00:06:32.500 --> 00:06:34.230
So they're going to have
the same entropy.

00:06:34.230 --> 00:06:38.380
Entropy is a macrostate
variable, or a macrostate

00:06:38.380 --> 00:06:40.500
function, that describes
the number of states a

00:06:40.500 --> 00:06:41.930
system can take on.

00:06:41.930 --> 00:06:44.790
So here, I would view the
cards as a system.

00:06:44.790 --> 00:06:47.580
And what we care about is not
the number of configurations

00:06:47.580 --> 00:06:48.900
the cards themselves
can take on.

00:06:48.900 --> 00:06:51.280
The cards aren't constantly
vibrating and changing from

00:06:51.280 --> 00:06:52.330
one thing to another.

00:06:52.330 --> 00:06:55.740
It's at the atomic level, at the
molecular level, that as

00:06:55.740 --> 00:06:59.110
long as we're above absolute
zero, which we pretty much

00:06:59.110 --> 00:07:01.460
always are, things are going
to be vibrating around

00:07:01.460 --> 00:07:04.150
continuously, and continuously
changing its state.

00:07:04.150 --> 00:07:06.460
So it's almost impossible
to measure the state.

00:07:06.460 --> 00:07:08.860
And since it's impossible to
measure the state, we use

00:07:08.860 --> 00:07:10.280
something like entropy to
say, well, how many

00:07:10.280 --> 00:07:11.310
states can we have?

00:07:11.310 --> 00:07:14.990
And I mean, all of these things,
entropy, whether we

00:07:14.990 --> 00:07:17.580
call it internal energy, whether
we look at entropy,

00:07:17.580 --> 00:07:21.730
whether we look at pressure,
volume, temperature.

00:07:21.730 --> 00:07:23.810
These are all, if you can think
about it in some way,

00:07:23.810 --> 00:07:26.740
these are shortcuts around
having to actually measure

00:07:26.740 --> 00:07:28.990
what each molecule is doing.

00:07:28.990 --> 00:07:31.600
And entropy, you can kind of
view it as a meta shortcut.

00:07:31.600 --> 00:07:34.060
I mean, temperature tells you
average kinetic energy, this

00:07:34.060 --> 00:07:36.600
tells you all of the energy
that's in it, this tells you,

00:07:36.600 --> 00:07:40.010
you know, how frequently the
molecules are bumping against

00:07:40.010 --> 00:07:41.220
a certain area.

00:07:41.220 --> 00:07:45.030
This tells you, on average, kind
of where the outermost

00:07:45.030 --> 00:07:46.280
molecules are.

00:07:46.280 --> 00:07:48.130
Entropy is kind of a, you
can almost view it

00:07:48.130 --> 00:07:49.260
as a metastate variable.

00:07:49.260 --> 00:07:52.030
It tells you how many states,
how many micro

00:07:52.030 --> 00:07:53.570
states can we take on?

00:07:53.570 --> 00:07:55.490
And so I just want to make this
very clear, because this

00:07:55.490 --> 00:07:59.250
is often confused, and there's
a very, very, very strong

00:07:59.250 --> 00:08:02.280
temptation to point to a
particular state and to say

00:08:02.280 --> 00:08:05.820
that that has higher entropy
than another, that somehow

00:08:05.820 --> 00:08:08.450
this state is more entropic
than that state.

00:08:08.450 --> 00:08:09.330
That's not the case.

00:08:09.330 --> 00:08:13.310
This system is more entropic
than this system,

00:08:13.310 --> 00:08:14.560
than this half box.

00:08:14.560 --> 00:08:15.960
It has more volume.

00:08:15.960 --> 00:08:21.920
If it has the same temperature
with more volume, then its

00:08:21.920 --> 00:08:24.860
particles can take on more
possible scenarios at any

00:08:24.860 --> 00:08:26.560
given moment in time.

00:08:26.560 --> 00:08:28.670
Anyway, hopefully you found
that a little bit useful.

00:08:28.670 --> 00:08:30.900
I just want to make it very,
very, very clear.

00:08:30.900 --> 00:08:34.059
Because this is often, often,
often confused.

00:08:34.059 --> 00:08:37.650
It is a macrostate variable for
a system, where a system

00:08:37.650 --> 00:08:41.429
is composed of things that are
bumping around randomly.

00:08:41.429 --> 00:08:46.210
Every millionth of a second,
they're changing states, so

00:08:46.210 --> 00:08:48.740
it's very hard to even measure
one of the microstates.

00:08:48.740 --> 00:08:51.750
You can't point to a microstate,
and say, oh, this

00:08:51.750 --> 00:08:53.820
has higher entropy
than another.

00:08:53.820 --> 00:08:54.190
Anyway.

00:08:54.190 --> 00:08:55.440
See you in the next video.