Maxwell's demon

WEBVTT
Kind: captions
Language: en

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The second law of thermodynamics
tells us that

00:00:03.705 --> 00:00:06.580
the entropy of the universe
is always increasing.

00:00:06.580 --> 00:00:10.610
So the change in entropy for
the universe, when it

00:00:10.610 --> 00:00:13.030
undergoes any process,
is always greater

00:00:13.030 --> 00:00:14.260
than or equal to 0.

00:00:14.260 --> 00:00:17.210
And we showed in the previous
video that it has a lot of

00:00:17.210 --> 00:00:18.460
implications.

00:00:20.940 --> 00:00:23.070
Regardless of how you define
your entropy, whether you

00:00:23.070 --> 00:00:27.520
define entropy as equal to
some constant times the

00:00:27.520 --> 00:00:30.660
natural log of the number of
states your system could take

00:00:30.660 --> 00:00:35.080
on, or whether you define change
in entropy to be equal

00:00:35.080 --> 00:00:37.870
to the heat added to the
system divided by the

00:00:37.870 --> 00:00:40.390
temperature at which it is
added, either of these

00:00:40.390 --> 00:00:42.660
descriptions, combined with
our second law of

00:00:42.660 --> 00:00:46.830
thermodynamics tell us things
like, when you have a hot body

00:00:46.830 --> 00:00:52.890
next to a cold body-- so say
this is T1, and then I have T2

00:00:52.890 --> 00:00:55.970
over here-- that heat will
flow from the hot

00:00:55.970 --> 00:00:57.160
body to the cold body.

00:00:57.160 --> 00:00:57.730
And we showed that

00:00:57.730 --> 00:01:00.010
mathematically in the last video.

00:01:00.010 --> 00:01:02.960
That heat will flow
in this direction.

00:01:02.960 --> 00:01:05.170
Now, one of the commenters on
the last video said, hey,

00:01:05.170 --> 00:01:07.260
could you cover Maxwell's
demon?

00:01:07.260 --> 00:01:10.250
And I will.

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Because it's an interesting
thought experiment that seems

00:01:14.900 --> 00:01:16.540
to defy this principle.

00:01:16.540 --> 00:01:19.420
It seems to defy the second
law of thermodynamics.

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And it has a very tantalizing
name, Maxwell's demon.

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Apparently, though, it was not
Maxwell who called it a demon.

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It was Kelvin.

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All these guys, you know, they
all meddle in everything.

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So Maxwell's demon.

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And this is the same Maxwell
famous for Maxwell's equation,

00:01:40.440 --> 00:01:42.760
so he obviously dealt with
a lot of things.

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He's actually also the first
person to ever generate a

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color image.

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So this is in the mid-1800s.

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So all around a fairly
sharp individual.

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But what's Maxwell's demon?

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So when we say something has
a higher temperature than

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something else, what
are we saying?

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We're saying that its average
kinetic energy of its

00:02:01.730 --> 00:02:05.500
molecules bumping around
here-- that the average

00:02:05.500 --> 00:02:08.690
kinetic energy of the molecules
here-- is higher

00:02:08.690 --> 00:02:12.440
than the average kinetic energy
of the molecules here.

00:02:12.440 --> 00:02:15.550
Now notice, I said its average
kinetic energy.

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And we've talked about
this multiple times.

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Temperature is a macro state.

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We know that at the micro level,
all of these molecules

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have different velocities.

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They're bumping into each other,
transferring momentum

00:02:25.670 --> 00:02:26.630
to each other.

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You know, this guy might
be going super

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fast in that direction.

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This guy might actually
be going quite slow.

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This guy might be going
super fast like that.

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That guy might be going
quite slow.

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It's just a hodgepodge
of things.

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You could actually draw
a distribution.

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If you knew the micro states
of everything, you could

00:02:43.260 --> 00:02:44.650
actually draw a little
histogram.

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We could say, for T1--
let's say this is

00:02:49.450 --> 00:02:51.850
on the Kelvin scale.

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So you could say, look, my
average temperature is here,

00:02:55.280 --> 00:02:58.050
but I have a whole distribution
of particles.

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So let's say this is number
of particles.

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And I won't put a scale there.

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You'll get the idea.

00:03:05.890 --> 00:03:08.220
So I have a bunch of particles
that are at T1, but I have

00:03:08.220 --> 00:03:10.340
some particles that could be
really close to absolute 0.

00:03:10.340 --> 00:03:11.860
I mean, it'd be very few, but.

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And then you have a bunch that
are maybe at T1, and then you

00:03:16.710 --> 00:03:19.120
have a bunch of particles that
could have actually kinetic

00:03:19.120 --> 00:03:20.740
energy higher than T1.

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Higher than the average
kinetic energy.

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Maybe that's this one here.

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Maybe the guy down here is
this guy with barely any

00:03:26.450 --> 00:03:27.010
kinetic energy.

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It means there's some guy
who's almost completely

00:03:29.330 --> 00:03:31.290
stationary, who's, you
know, sitting right

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around there someplace.

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So there's a whole distribution
of particles.

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Likewise, this T2 system right
here, on average, these

00:03:41.230 --> 00:03:43.670
molecules have a lower
kinetic energy.

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But you know, there might be one
particle here that has a

00:03:45.780 --> 00:03:47.370
really high kinetic energy.

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But most of them on
average are lower.

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So if I were to draw the
distribution of T2, my average

00:03:54.500 --> 00:03:56.770
kinetic energy is lower, but
my distribution might look

00:03:56.770 --> 00:03:58.640
something like this.

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It can't go backwards
like that.

00:04:00.800 --> 00:04:03.220
It might look something
like this.

00:04:03.220 --> 00:04:06.100
Oh, I don't know, maybe it looks
something like that.

00:04:06.100 --> 00:04:07.950
Let me try it a little
different.

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I'll make it go just as high.

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Maybe it looks something
like that.

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right?

00:04:13.690 --> 00:04:18.760
So notice, there are some
molecules in T1 that are below

00:04:18.760 --> 00:04:20.970
the average kinetic
energy of T2.

00:04:20.970 --> 00:04:21.269
Right?

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There are these molecules
here.

00:04:22.480 --> 00:04:26.680
These are these slow
guys right there.

00:04:26.680 --> 00:04:30.210
And notice, there are some guys
in T2 that have a higher

00:04:30.210 --> 00:04:33.490
kinetic energy than
the average in T1.

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So these are these
guys right here.

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And so the fast guys in T2--
so even though T2 is, quote

00:04:43.480 --> 00:04:46.320
unquote, colder, it has lower
average kinetic energy,

00:04:46.320 --> 00:04:50.980
there's some molecules, if you
look at the micro state, that

00:04:50.980 --> 00:04:53.390
are actually moving around quite
rapidly, and there are

00:04:53.390 --> 00:04:56.180
some molecules here that are
moving around quite slowly.

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So what Maxwell said is, hey,
what if I had my-- and he

00:05:00.090 --> 00:05:02.510
actually didn't use the word
demon, but we'll use the word

00:05:02.510 --> 00:05:05.940
demon, because it makes it
seem very interesting and

00:05:05.940 --> 00:05:08.820
metaphysical on some level, but
it really isn't-- what if

00:05:08.820 --> 00:05:12.210
I had some dude, let's call him
the demon, with a little

00:05:12.210 --> 00:05:13.630
trapdoor here?

00:05:13.630 --> 00:05:14.880
Let me draw a little
bit neater.

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So between those two
systems, let's say

00:05:22.380 --> 00:05:24.120
that they're insulated.

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Let's say that they're separated
from each other.

00:05:27.480 --> 00:05:31.830
So this is T1 where I have a
bunch of particles, you know,

00:05:31.830 --> 00:05:36.210
with their different
kinetic energies.

00:05:36.210 --> 00:05:38.240
And then, here is T2.

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And I'm making them separated,
and maybe they're connected

00:05:40.470 --> 00:05:44.230
only by this little connection
right here.

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T2.

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These guys have a slower
kinetic energy.

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And what Maxwell, his little
thought experiment was, hey,

00:05:52.350 --> 00:05:56.690
let me say that I have some
dude in charge of a door--

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maybe the door is right
here-- and he has

00:06:00.510 --> 00:06:02.050
control over this door.

00:06:02.050 --> 00:06:05.350
And whenever a really fast
particle in T2, one of these

00:06:05.350 --> 00:06:09.080
particles over here, come near
the door-- so let's say this

00:06:09.080 --> 00:06:14.050
guy is flying-- let's say
that guy right there.

00:06:14.050 --> 00:06:16.750
He's going super fast. He has
super high kinetic energy, and

00:06:16.750 --> 00:06:19.360
he's just going perfectly
for the door.

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So the demon says, hey.

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I see that guy.

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He's coming for the door.

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He's going to lift his hatch,
and he's going to allow this

00:06:27.800 --> 00:06:30.100
particle to get into T1.

00:06:33.170 --> 00:06:35.200
So after he lifts the hatch,
that particle will just keep

00:06:35.200 --> 00:06:36.990
going, and it'll be in T1.

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And then he closes the hatch
again, because he just wants

00:06:40.490 --> 00:06:43.730
the fast particles to
go from T2 to T1.

00:06:43.730 --> 00:06:48.330
And then when he sees a little
slow, you know, pokey little

00:06:48.330 --> 00:06:51.870
particle coming here, one of
these guys down here, he opens

00:06:51.870 --> 00:06:54.510
the trapdoor again, and he
allows that one to go.

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So then that guy shows
up in here.

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So if he just kept doing that,
what's it going to look like

00:07:00.460 --> 00:07:01.410
at the end?

00:07:01.410 --> 00:07:03.820
Well, at the end, you're going
to segregate-- and it could

00:07:03.820 --> 00:07:05.490
take a while.

00:07:05.490 --> 00:07:10.660
But you're going to segregate
all the slow particles on--

00:07:10.660 --> 00:07:11.380
let me draw it.

00:07:11.380 --> 00:07:14.140
I'll make the boundary in brown,
because now it's not

00:07:14.140 --> 00:07:15.850
clear which one is-- well.

00:07:15.850 --> 00:07:16.920
We'll talk about it
a little bit.

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So that's the boundary.

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That's his door.

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What's going to happen
at the end?

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All the fast particles-- some
of them are going to be the

00:07:22.010 --> 00:07:25.930
original fast particles
that are in T1, right?

00:07:25.930 --> 00:07:29.620
There are some original fast
particles in T1 are going to

00:07:29.620 --> 00:07:34.390
be still on the side
of the barrier.

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Let me draw-- make sure you
don't get these two confused.

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This is a separate picture.

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Now all of the fast particles
from T2 are also going to be

00:07:41.990 --> 00:07:42.520
stuck there.

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Because eventually they're all
going to get close to that

00:07:45.020 --> 00:07:47.990
door, if you wait long enough.

00:07:47.990 --> 00:07:50.400
So then this guy's also going
to have a bunch of the, what

00:07:50.400 --> 00:07:54.110
would originally, in the T2 side
of the barrier, they're

00:07:54.110 --> 00:07:55.120
also going to be there.

00:07:55.120 --> 00:07:57.460
So you're going to have a
bunch of fast particles.

00:07:57.460 --> 00:08:01.410
Likewise, all the slow T2
particles are going to be

00:08:01.410 --> 00:08:04.150
remaining on the side
of the barrier.

00:08:04.150 --> 00:08:05.850
So these are the slow guys.

00:08:05.850 --> 00:08:09.290
And he would have let all the
slow T1-- I shouldn't even

00:08:09.290 --> 00:08:11.010
call them T1 anymore.

00:08:11.010 --> 00:08:12.630
I'll call them side 1.

00:08:12.630 --> 00:08:14.110
Side 1 particles here.

00:08:17.470 --> 00:08:19.940
Slow side 1 particles.

00:08:19.940 --> 00:08:23.130
So what just happened here?

00:08:23.130 --> 00:08:26.380
This was the hot body, this
was the cold body.

00:08:26.380 --> 00:08:29.430
The second law of thermodynamics
would have told

00:08:29.430 --> 00:08:31.690
us that heat would have gone
from here to here.

00:08:31.690 --> 00:08:34.000
That their temperatures
should have equalized

00:08:34.000 --> 00:08:35.230
to a certain degree.

00:08:35.230 --> 00:08:38.520
So the hot bodies should get
colder, the cold bodies should

00:08:38.520 --> 00:08:39.190
get hotter.

00:08:39.190 --> 00:08:41.270
They should kind of average
out a little bit.

00:08:41.270 --> 00:08:45.550
But using this little demonic
figure, what did he do?

00:08:45.550 --> 00:08:47.780
He made the hot body
hotter, right?

00:08:47.780 --> 00:08:50.080
Now the average kinetic energy
here is even higher.

00:08:50.080 --> 00:08:54.950
He transferred all of these high
kinetic energy particles

00:08:54.950 --> 00:08:57.730
to that distribution, so now
that distribution is going to

00:08:57.730 --> 00:09:01.170
look-- the way you could think
about it, if you transferred

00:09:01.170 --> 00:09:04.450
all of these guys to this guy
over here, the distribution

00:09:04.450 --> 00:09:07.150
will now look something like--
let me see if I can do it.

00:09:07.150 --> 00:09:10.860
It will look something
like that for T1,

00:09:10.860 --> 00:09:12.200
instead of the old one.

00:09:12.200 --> 00:09:16.510
And T2-- and he took all the
hot ones away, all the cold

00:09:16.510 --> 00:09:17.620
ones away, from T1.

00:09:17.620 --> 00:09:19.740
So these guys are going
to disappear.

00:09:19.740 --> 00:09:21.340
They're not going to
be there anymore.

00:09:21.340 --> 00:09:22.590
And he added them to T2.

00:09:25.545 --> 00:09:28.600
So the distribution of T2 is
going to look like that, and

00:09:28.600 --> 00:09:32.270
he erased, of course,
these from T2.

00:09:32.270 --> 00:09:35.152
He took all of these
guys out of T2.

00:09:35.152 --> 00:09:36.880
Let me erase this right here.

00:09:36.880 --> 00:09:38.410
That was the old distribution
of T1.

00:09:38.410 --> 00:09:40.960
So the T2 distribution now looks
something like this.

00:09:43.460 --> 00:09:47.640
So T2, the new average might
be something like here.

00:09:47.640 --> 00:09:49.640
This is my new T2.

00:09:49.640 --> 00:09:52.590
And my new T1 is going to move
to the right a little bit.

00:09:52.590 --> 00:09:54.260
The average is going
to be higher.

00:09:54.260 --> 00:09:57.450
So this demon seems to have
violated the second law of

00:09:57.450 --> 00:09:58.400
thermodynamics.

00:09:58.400 --> 00:10:00.540
Let me box off this
right here.

00:10:00.540 --> 00:10:04.810
My little diagrams
are overlapping.

00:10:04.810 --> 00:10:07.750
This example shows that the hot
got hotter and the cold

00:10:07.750 --> 00:10:08.400
got colder.

00:10:08.400 --> 00:10:10.950
So Maxwell's thought experiment
says, hey, we

00:10:10.950 --> 00:10:14.030
violated the second law
of thermodynamics.

00:10:14.030 --> 00:10:16.530
And this actually
was a conundrum

00:10:16.530 --> 00:10:17.280
for many, many years.

00:10:17.280 --> 00:10:20.450
Even in this century, people
kind of, hey, you know,

00:10:20.450 --> 00:10:22.340
there's something fudgy
about here,

00:10:22.340 --> 00:10:23.420
something not quite right.

00:10:23.420 --> 00:10:25.260
And the thing that's not quite
right, and I'm not going to

00:10:25.260 --> 00:10:27.470
prove it to you mathematically,
is-- and it's

00:10:27.470 --> 00:10:31.270
kind of analogous to the
refrigerator example-- is, to

00:10:31.270 --> 00:10:37.750
have something here-- to have
some dude, perhaps he's a

00:10:37.750 --> 00:10:43.730
demon, here, pulling this
little door when it's

00:10:43.730 --> 00:10:47.000
convenient, when the fast
particles are going from this

00:10:47.000 --> 00:10:49.690
side or the slow particles are
going from that side-- in

00:10:49.690 --> 00:10:52.270
order for him to do it
correctly, he's going to have

00:10:52.270 --> 00:10:53.870
to keep track of where all
the particles are.

00:10:53.870 --> 00:10:55.540
He'll need to keep track
of particles.

00:10:55.540 --> 00:10:58.860
I mean, these aren't balls,
like, you know, macro balls.

00:10:58.860 --> 00:11:03.640
These are micro molecules or
atoms. He's going to have to

00:11:03.640 --> 00:11:06.030
bounce light off of them, or
he's going to have to bounce

00:11:06.030 --> 00:11:08.200
electrons, use an electron
microscope.

00:11:08.200 --> 00:11:10.290
He's going to have to keep
track of these gazillion

00:11:10.290 --> 00:11:11.950
particles that are there.

00:11:11.950 --> 00:11:13.480
And I mean, think about it.

00:11:13.480 --> 00:11:16.330
He might have to have a super
duper-- if it doesn't occur in

00:11:16.330 --> 00:11:18.940
his head, he might have to have
some kind of hardcore

00:11:18.940 --> 00:11:22.550
computer microchip that's
churning away.

00:11:22.550 --> 00:11:25.740
And this would be, actually,
for a computer to do this,

00:11:25.740 --> 00:11:28.160
this would be intensive
computation power.

00:11:28.160 --> 00:11:30.580
And let your computer run for
a little bit and feel the

00:11:30.580 --> 00:11:35.540
microchip-- this is generating
a lot of heat.

00:11:35.540 --> 00:11:41.110
His bouncing off light, or
whatever he's trying to bounce

00:11:41.110 --> 00:11:44.760
off of the different molecules
to be able to measure how fast

00:11:44.760 --> 00:11:47.520
they're going, that's also
going to generate heat.

00:11:47.520 --> 00:11:49.120
He's going to have to
do work to do that.

00:11:49.120 --> 00:11:50.690
He's going to have to
measure everything.

00:11:50.690 --> 00:11:52.880
There's a lot of stuff that's
going on that he's going to

00:11:52.880 --> 00:11:53.800
have to do.

00:11:53.800 --> 00:11:56.520
So the current answer is-- and
it's not easy to prove

00:11:56.520 --> 00:11:58.830
mathematically-- but the current
answer is, if you

00:11:58.830 --> 00:12:01.660
actually wanted to build a demon
like this-- and probably

00:12:01.660 --> 00:12:04.760
in our world today, you'd use
some type of computer with

00:12:04.760 --> 00:12:07.760
some type of sensors-- to
attempt to do this-- and there

00:12:07.760 --> 00:12:12.860
are people who have attempted
to do this, on some level.

00:12:12.860 --> 00:12:16.220
This computer and this whole
system is going to generate

00:12:16.220 --> 00:12:21.340
more entropy-- so the delta S
here is going to generate more

00:12:21.340 --> 00:12:27.790
entropy than the entropy that's
lost by making the cold

00:12:27.790 --> 00:12:30.160
side colder and the
hot side hotter.

00:12:30.160 --> 00:12:31.960
So Maxwell's demon,
and I didn't do

00:12:31.960 --> 00:12:32.900
anything rigorous here.

00:12:32.900 --> 00:12:34.210
I didn't prove it to you.

00:12:34.210 --> 00:12:36.080
But Maxwell's demon, it's
an interesting thought

00:12:36.080 --> 00:12:37.450
experiment, because it gives
you a little bit more

00:12:37.450 --> 00:12:38.710
intuition about the difference
between macro

00:12:38.710 --> 00:12:40.440
states and micro states.

00:12:40.440 --> 00:12:43.630
And what happens at the
molecular level in terms of

00:12:43.630 --> 00:12:45.970
temperature, and how you can
make a cold body colder and a

00:12:45.970 --> 00:12:47.330
hot body hotter.

00:12:47.330 --> 00:12:49.920
But answer is, it really
isn't a paradox or

00:12:49.920 --> 00:12:51.860
anything like that.

00:12:51.860 --> 00:12:54.960
When you think about the entropy
of the entire system,

00:12:54.960 --> 00:12:57.510
you have to include
the demon himself.

00:12:57.510 --> 00:13:00.680
And if you include the demon
himself, he's generating more

00:13:00.680 --> 00:13:04.685
entropy every time he opens that
door-- and maybe there's

00:13:04.685 --> 00:13:06.750
some energy required to
open the door itself.

00:13:06.750 --> 00:13:10.360
But he generates more entropy
when he does all of this than

00:13:10.360 --> 00:13:14.700
the entropy that might be lost,
when say, for example,

00:13:14.700 --> 00:13:17.130
one of these slowpoke particles
kind of just

00:13:17.130 --> 00:13:20.280
traverses onto that side
of the barrier.

00:13:20.280 --> 00:13:22.350
Anyway, I thought I would just
expose you to that, because

00:13:22.350 --> 00:13:25.130
it's a really neat thought
experiment.

00:13:25.130 --> 00:13:27.270
So I'll see in the next video.


Multiplication 2: The multiplication tables

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:02.990
At this point I think you
know a little bit about

00:00:02.990 --> 00:00:05.050
what multiplication is.

00:00:08.740 --> 00:00:11.900
What we're going to do in this
video is to give you just a ton

00:00:11.900 --> 00:00:16.660
of more practice and start you
on your memorization of the

00:00:16.660 --> 00:00:17.940
multiplication tables.

00:00:17.940 --> 00:00:20.450
And if you watch enough Khan
Academy videos, and hopefully

00:00:20.450 --> 00:00:22.520
you will in the future, you'll
realize that I'm normally not

00:00:22.520 --> 00:00:24.550
a big fan of memorization.

00:00:24.550 --> 00:00:26.610
But the one thing about
multiplication is if you

00:00:26.610 --> 00:00:30.240
memorize your multiplication
tables that we'll start to do

00:00:30.240 --> 00:00:33.860
in this video, it'll pay huge
benefits the rest of your life.

00:00:33.860 --> 00:00:37.570
So I promise you, do it now,
you'll never forget it, and the

00:00:37.570 --> 00:00:40.270
rest of your life everything
will be-- well, I don't want to

00:00:40.270 --> 00:00:42.640
make false promises to you, but
they'll be better than if you

00:00:42.640 --> 00:00:45.800
didn't memorize your
multiplication tables.

00:00:45.800 --> 00:00:47.300
So what are the
multiplication tables?

00:00:47.300 --> 00:00:49.740
Well that's all of the
different numbers

00:00:49.740 --> 00:00:50.860
times each other.

00:00:50.860 --> 00:00:53.600
So let's actually do a
little bit of review.

00:00:53.600 --> 00:00:58.910
So if I say what is 2 times 1?

00:00:58.910 --> 00:01:02.280
That is equal to 2
plus itself one time.

00:01:02.280 --> 00:01:05.010
So this is equal to just 2.

00:01:05.010 --> 00:01:07.070
That's 2 plus itself one time.

00:01:07.070 --> 00:01:08.290
I don't have to say plus
anything because there's

00:01:08.290 --> 00:01:09.430
only one 2 there.

00:01:09.430 --> 00:01:13.470
I could also write this as
1 plus itself two times.

00:01:13.470 --> 00:01:15.400
So that's also 1 plus 1.

00:01:15.400 --> 00:01:17.590
Well that also equals 2.

00:01:17.590 --> 00:01:18.220
Fair enough.

00:01:18.220 --> 00:01:19.690
So 2 times 1 is 2.

00:01:19.690 --> 00:01:22.445
And if you watched the last
video, what's 2 times 0?

00:01:22.445 --> 00:01:23.480
Well that's 0.

00:01:23.480 --> 00:01:26.670
So you don't have to memorize
your 0 multiplication tables

00:01:26.670 --> 00:01:30.560
because everything times 0 is
0, or 0 times anything is 0.

00:01:30.560 --> 00:01:31.100
So let's see.

00:01:31.100 --> 00:01:32.430
What's 2 times 2?

00:01:35.750 --> 00:01:37.910
Well, this is equal to--
we're going to add 2

00:01:37.910 --> 00:01:39.470
to itself two times.

00:01:39.470 --> 00:01:41.850
So that's 2 plus 2.

00:01:41.850 --> 00:01:42.780
And there's only a
way to do that.

00:01:42.780 --> 00:01:45.490
I could say take this 2 and
add it to itself two times,

00:01:45.490 --> 00:01:46.690
but it's the same thing.

00:01:46.690 --> 00:01:47.620
And what's 2 plus 2?

00:01:47.620 --> 00:01:49.460
That's equal to 4.

00:01:49.460 --> 00:01:51.410
What's 2 times 3?

00:01:51.410 --> 00:01:57.910
2 times 3 is equal
to 2 plus 2 plus 2.

00:01:57.910 --> 00:02:03.370
It can also be
equal to 3 plus 3.

00:02:03.370 --> 00:02:06.120
We learned in a previous video
this statement can be written

00:02:06.120 --> 00:02:07.165
either of these ways.

00:02:07.165 --> 00:02:09.390
And in either case,
what's it equal to?

00:02:09.390 --> 00:02:12.000
Well, 3 plus 3 is the same
thing as 2 plus 2 plus 2,

00:02:12.000 --> 00:02:14.780
and that's equal to 6.

00:02:14.780 --> 00:02:15.670
All right.

00:02:15.670 --> 00:02:17.440
Now what is 2 times 4?

00:02:20.670 --> 00:02:26.060
Well that's equal to 2
plus 2 plus 2 plus 2.

00:02:26.060 --> 00:02:29.880
And notice, it's exactly
what 2 times 3 was.

00:02:29.880 --> 00:02:32.620
2 times 3 was that.

00:02:32.620 --> 00:02:36.490
I have that here, but now I'm
just adding another 2 to it.

00:02:36.490 --> 00:02:39.740
So if we're too lazy to sit
here and add 2 plus 2 is 4.

00:02:39.740 --> 00:02:40.720
4 plus 2 is 6.

00:02:40.720 --> 00:02:42.740
Instead of doing that, we could
say, hey look, we already know

00:02:42.740 --> 00:02:45.690
that this thing over
here, this was 6.

00:02:45.690 --> 00:02:48.320
We figured it out in the
previous line right there.

00:02:48.320 --> 00:02:51.560
We figured out this is 6, so we
could just say, oh, 2 times 4

00:02:51.560 --> 00:02:55.610
is going to be 2 more than
that, which is equal to 8.

00:02:55.610 --> 00:02:57.490
And you should hopefully
see that pattern.

00:02:57.490 --> 00:03:02.640
As we go from 2 times 1,
to 2 times 2, to 2 times

00:03:02.640 --> 00:03:03.770
3, what's happening?

00:03:03.770 --> 00:03:05.510
How much are we going up by?

00:03:05.510 --> 00:03:08.130
From 2 to 4 we're going plus 2.

00:03:08.130 --> 00:03:11.310
From 4 to 6 we're going plus 2.

00:03:11.310 --> 00:03:13.340
And then from 6 to 8
we're going plus 2.

00:03:13.340 --> 00:03:16.270
So you could figure out what
2 times 5 is, even without

00:03:16.270 --> 00:03:16.960
doing the addition.

00:03:16.960 --> 00:03:23.440
2 times 5 is equal to 2 plus
2 plus 2 plus 2 plus 2.

00:03:23.440 --> 00:03:26.080
It could also be
written as 5 plus 5.

00:03:26.080 --> 00:03:29.120
2 times 4 could've been
written as 4 plus 4 as well.

00:03:29.120 --> 00:03:30.500
And what's that equal to?

00:03:30.500 --> 00:03:33.100
We could add all of these up
or we could add these two up.

00:03:33.100 --> 00:03:35.882
Or we could just say it's going
to be two more than 2 times 4.

00:03:35.882 --> 00:03:39.210
So it's going to be 10.

00:03:39.210 --> 00:03:41.790
I'll finish the 2 times tables.

00:03:41.790 --> 00:03:45.290
And I think you see all of the
patterns that emerge from it.

00:03:45.290 --> 00:03:47.720
So 2 times 6.

00:03:47.720 --> 00:03:51.590
That's going to be 2
plus itself six times.

00:03:51.590 --> 00:03:51.820
Let's see.

00:03:51.820 --> 00:03:56.400
1, 2, 3, 4, 5, 6, which
should also be equal to

00:03:56.400 --> 00:03:58.700
6 plus itself two times.

00:03:58.700 --> 00:04:00.630
This could be
interpreted either way.

00:04:00.630 --> 00:04:02.990
And that's going to
be equal to 12.

00:04:02.990 --> 00:04:08.170
Once again, two more than 2
times 5 because we're adding

00:04:08.170 --> 00:04:09.720
2 to itself one more time.

00:04:09.720 --> 00:04:12.300
So it's going to be two more.

00:04:12.300 --> 00:04:13.890
Let's keep going.

00:04:13.890 --> 00:04:16.550
So 2 times 7.

00:04:16.550 --> 00:04:23.450
2 times 7 is equal to-- well, I
could write 2 plus 2 plus 2

00:04:23.450 --> 00:04:27.240
plus 2-- this is getting
tiring-- plus 2 plus 2.

00:04:27.240 --> 00:04:27.810
Is that 7?

00:04:27.810 --> 00:04:31.210
1, 2, 3, 4, 5, 6, 7.

00:04:31.210 --> 00:04:34.480
And that's the same thing as 7
plus 7, which you may or may

00:04:34.480 --> 00:04:37.380
not know is equal to 14.

00:04:37.380 --> 00:04:39.560
You could just say hey, that's
going to be two more than 12.

00:04:39.560 --> 00:04:43.640
So 12 plus 1 plus 2
is-- 12 plus 1 is 13.

00:04:43.640 --> 00:04:45.540
12 plus 2 is 14.

00:04:45.540 --> 00:04:47.590
All right, let's
just keep going.

00:04:47.590 --> 00:04:50.750
2 times 8.

00:04:50.750 --> 00:04:53.820
I could do all of this business
here where I add the 2's or I

00:04:53.820 --> 00:04:56.670
could say look, it's just going
to be two more than 2 times 7.

00:04:56.670 --> 00:04:58.790
So I could say it's
going to be 14 plus 2.

00:04:58.790 --> 00:05:00.480
I'm just adding
two to that one.

00:05:00.480 --> 00:05:01.930
So I could say it's 16.

00:05:01.930 --> 00:05:05.620
Or I could also say
that's 8 plus 8.

00:05:05.620 --> 00:05:06.810
That's also 16.

00:05:06.810 --> 00:05:09.420
I could have done all the 2's
out, but if you like you

00:05:09.420 --> 00:05:15.040
could do that for your
own benefit and learning.

00:05:15.040 --> 00:05:18.150
We're almost-- well, we could
go forever because there

00:05:18.150 --> 00:05:19.370
is no largest number.

00:05:19.370 --> 00:05:21.520
I can keep going.

00:05:21.520 --> 00:05:24.980
2 times 9 times 10 times 100
times 1,000 times 1,000,000.

00:05:24.980 --> 00:05:27.590
But I'm going to stop at 12
because that tends to be what

00:05:27.590 --> 00:05:28.780
people need to memorize.

00:05:28.780 --> 00:05:32.340
But if you really want
to be a mathlete you

00:05:32.340 --> 00:05:33.670
want to go up to 20.

00:05:33.670 --> 00:05:36.790
But let's go to 2 times 9.

00:05:36.790 --> 00:05:38.815
That's going to be two
more than 2 times 8.

00:05:38.815 --> 00:05:40.995
It's going to be 18.

00:05:40.995 --> 00:05:42.700
Or that's 9 plus 9.

00:05:42.700 --> 00:05:44.070
Also 18.

00:05:44.070 --> 00:05:45.950
What's 2 times 10?

00:05:45.950 --> 00:05:47.950
And 10 times tables
are interesting.

00:05:47.950 --> 00:05:49.560
And we're going to see a
pattern there in a second

00:05:49.560 --> 00:05:53.040
when we try to complete
an entire times tables.

00:05:53.040 --> 00:05:54.650
So 2 times 10?

00:05:54.650 --> 00:05:56.810
Two more than 2 times 9.

00:05:56.810 --> 00:05:59.050
It's 20.

00:05:59.050 --> 00:06:01.060
Or we could also say
that's 10 plus 10.

00:06:01.060 --> 00:06:03.360
10 plus itself two times.

00:06:03.360 --> 00:06:05.070
Now what's interesting
about this?

00:06:05.070 --> 00:06:09.140
This looks just like
a 2 with a 0 added.

00:06:09.140 --> 00:06:10.700
And you're going to see
that anything times 10.

00:06:10.700 --> 00:06:12.310
You just put a 0 on the right.

00:06:12.310 --> 00:06:14.140
And you can think
about why that is.

00:06:14.140 --> 00:06:16.420
You can view this
as two 10's is 20.

00:06:16.420 --> 00:06:18.010
That's what 20 is.

00:06:18.010 --> 00:06:19.550
We're almost done.

00:06:19.550 --> 00:06:21.830
Let's do 2 times 11.

00:06:21.830 --> 00:06:25.930
2 times 11 is going to be 2
more than this right here.

00:06:25.930 --> 00:06:27.840
It's going to be 22.

00:06:27.840 --> 00:06:29.540
Another interesting pattern.

00:06:29.540 --> 00:06:32.220
I have the number repeated
twice-- a 2 and a 2.

00:06:32.220 --> 00:06:33.170
Interesting.

00:06:33.170 --> 00:06:37.680
Something to watch out
for as we look at other

00:06:37.680 --> 00:06:38.700
multiplication tables.

00:06:38.700 --> 00:06:42.000
And then finally-- it's not
finally, we could keep going.

00:06:42.000 --> 00:06:44.750
2 times-- that's too
dark of a color.

00:06:44.750 --> 00:06:47.140
2 times 12.

00:06:47.140 --> 00:06:50.550
2 times 12 is going to be
two more than 2 times 11.

00:06:50.550 --> 00:06:51.390
That's 24.

00:06:51.390 --> 00:06:54.170
We could have also written
that as 12 plus 12.

00:06:54.170 --> 00:06:56.320
Or we could've said 2 plus
2 plus 2 plus 2 plus

00:06:56.320 --> 00:06:57.850
2 twelve times.

00:06:57.850 --> 00:06:59.670
It all gets you to 24.

00:06:59.670 --> 00:07:01.340
So that's the 2 times
tables and I think

00:07:01.340 --> 00:07:01.980
you see the pattern.

00:07:01.980 --> 00:07:04.950
Every time you multiply it by
one higher number you just

00:07:04.950 --> 00:07:06.680
add 2 to that number.

00:07:06.680 --> 00:07:08.810
So now that we see that
pattern, let's see

00:07:08.810 --> 00:07:11.960
if we can complete a
multiplication table.

00:07:11.960 --> 00:07:16.280
So what I want to do, I'm going
to write all the numbers.

00:07:16.280 --> 00:07:17.685
Let's see.

00:07:17.685 --> 00:07:19.340
I hope I have space for this.

00:07:19.340 --> 00:07:29.440
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:07:29.440 --> 00:07:30.920
Actually, I'll just
do it till 9.

00:07:30.920 --> 00:07:31.840
I'll just keep going.

00:07:31.840 --> 00:07:32.580
9.

00:07:32.580 --> 00:07:34.465
Actually I won't have space to
do that because I want you

00:07:34.465 --> 00:07:35.690
to see the entire table.

00:07:35.690 --> 00:07:37.660
So I'm just going up till
9 here, but I encourage

00:07:37.660 --> 00:07:39.600
you after this video to
complete it on your own.

00:07:39.600 --> 00:07:42.890
Maybe if we have time I'll
complete it here as well.

00:07:42.890 --> 00:07:45.225
So these are the first numbers
that I'm going to multiply.

00:07:45.225 --> 00:07:52.070
And I'm going to multiply
it times 1, 2, 3, 4,

00:07:52.070 --> 00:07:57.270
5, 6, 7, 8, and 9.

00:08:00.950 --> 00:08:03.280
Actually I should have
written this 1 under-

00:08:03.280 --> 00:08:04.720
well, what's 1 times 1?

00:08:04.720 --> 00:08:06.120
So this is the way I'm
going to view it.

00:08:06.120 --> 00:08:08.730
Whatever is 1 times 1 I'm
going to write here.

00:08:08.730 --> 00:08:09.970
Well that's 1.

00:08:09.970 --> 00:08:11.520
What's 1 times 2?

00:08:11.520 --> 00:08:12.410
That's 2.

00:08:12.410 --> 00:08:13.740
What's 1 times 3?

00:08:13.740 --> 00:08:14.360
That's 3.

00:08:14.360 --> 00:08:16.220
1 times anything is that
number, so I can just

00:08:16.220 --> 00:08:21.370
write 4, 5, 6, 7, 8, 9.

00:08:21.370 --> 00:08:23.540
1 times 9 is 9.

00:08:23.540 --> 00:08:24.540
Fair enough.

00:08:24.540 --> 00:08:26.310
Now let's do the
2 times tables.

00:08:26.310 --> 00:08:27.580
I'll do that in a blue.

00:08:27.580 --> 00:08:32.070
Actually, let me do 1 in that
color and now in maybe a

00:08:32.070 --> 00:08:33.990
darker blue I'll do
the 2 times tables.

00:08:33.990 --> 00:08:35.260
What's 2 times 1?

00:08:35.260 --> 00:08:36.440
That's 2.

00:08:36.440 --> 00:08:37.700
It's the same thing
as 1 times 2.

00:08:37.700 --> 00:08:40.010
Notice, these two numbers
are the same thing.

00:08:40.010 --> 00:08:41.710
What's 2 times 2?

00:08:41.710 --> 00:08:43.010
That's 4.

00:08:43.010 --> 00:08:44.620
2 times 3 is 6.

00:08:44.620 --> 00:08:45.840
We just did this.

00:08:45.840 --> 00:08:49.600
Every time you increment or
you multiply by a higher

00:08:49.600 --> 00:08:51.190
number, you just add by 2.

00:08:51.190 --> 00:08:52.850
2 times 4 is 8.

00:08:52.850 --> 00:08:55.120
Same thing as 4 times 2.

00:08:55.120 --> 00:08:57.270
2 times 5 is 10.

00:08:57.270 --> 00:08:59.020
2 times 6 is 12.

00:08:59.020 --> 00:09:00.680
I'm just adding 2 every time.

00:09:00.680 --> 00:09:03.980
Up here I added 1 from every
step, here I'm adding 2.

00:09:03.980 --> 00:09:06.830
2 times 7, 14.

00:09:06.830 --> 00:09:09.496
2 times 8, 16.

00:09:09.496 --> 00:09:12.980
2 two times 9, 18.

00:09:12.980 --> 00:09:17.620
All right, let's do
our 3 times tables.

00:09:17.620 --> 00:09:18.410
I'll do it in yellow.

00:09:21.700 --> 00:09:23.820
3 times 1 is 3.

00:09:23.820 --> 00:09:25.360
Notice, 3 times 1 is 3.

00:09:25.360 --> 00:09:26.920
1 times 3 is 3.

00:09:26.920 --> 00:09:28.750
These are the same values.

00:09:28.750 --> 00:09:33.430
3 times 2 is the same
thing as 2 times 3.

00:09:33.430 --> 00:09:37.650
3 times 2 should be the
same thing as 2 times 3.

00:09:37.650 --> 00:09:39.690
So it's 6.

00:09:39.690 --> 00:09:40.480
And that makes sense.

00:09:40.480 --> 00:09:45.690
3 plus 3 is 6 or 2
plus 2 plus 2 is 6.

00:09:45.690 --> 00:09:47.790
So every time here we're
going to increase by 3.

00:09:47.790 --> 00:09:48.890
You see the pattern.

00:09:48.890 --> 00:09:51.020
3 times 3 is 9.

00:09:51.020 --> 00:09:52.870
3 plus 3 plus 3.

00:09:52.870 --> 00:09:54.710
So we went from 3 to 6 to 9.

00:09:54.710 --> 00:09:57.000
So 3 times 4 is going to be 12.

00:09:57.000 --> 00:09:59.000
I'm just adding 3 every time.

00:09:59.000 --> 00:10:00.690
12 plus 3 is 15.

00:10:00.690 --> 00:10:03.230
15 plus 3 is 18.

00:10:03.230 --> 00:10:05.580
18 plus 3 is 21.

00:10:05.580 --> 00:10:08.090
21 plus 3 is 24.

00:10:08.090 --> 00:10:10.510
24 plus 3 is 27.

00:10:10.510 --> 00:10:13.120
So 3 times 9 is 27.

00:10:13.120 --> 00:10:14.840
3 times 8 is 24.

00:10:14.840 --> 00:10:19.550
So if you were to say 8 plus
8 plus 8, it would be 24.

00:10:19.550 --> 00:10:21.800
So now I'm going to speed it
up a little bit now that

00:10:21.800 --> 00:10:22.840
we see the pattern.

00:10:22.840 --> 00:10:25.350
And you should do this on your
own and you really should

00:10:25.350 --> 00:10:26.970
memorize everything
we're doing.

00:10:26.970 --> 00:10:29.000
You should actually go
all the way up to 12

00:10:29.000 --> 00:10:30.510
in both directions.

00:10:30.510 --> 00:10:30.870
So let's see.

00:10:30.870 --> 00:10:35.330
4 times 1 is 4.

00:10:35.330 --> 00:10:37.650
I'm just going to go up
by increments of 4.

00:10:37.650 --> 00:10:40.090
So 4 plus 4 is 8.

00:10:40.090 --> 00:10:41.970
8 plus 4 is 12.

00:10:41.970 --> 00:10:43.930
12 plus 4 is 16.

00:10:43.930 --> 00:10:46.460
16 plus 4 is 20.

00:10:46.460 --> 00:10:48.460
20 plus 4 is 24.

00:10:48.460 --> 00:10:51.470
4 times 6 is 24.

00:10:51.470 --> 00:10:52.980
4 times 7, 28.

00:10:52.980 --> 00:10:54.260
I'm just going up by 4.

00:10:54.260 --> 00:10:58.750
32 and 365.

00:10:58.750 --> 00:11:01.490
All right, 5 times 1.

00:11:01.490 --> 00:11:06.680
5 times 1 is going to be 5.

00:11:06.680 --> 00:11:08.790
Actually, we know that
anything-- well, I want to keep

00:11:08.790 --> 00:11:11.420
changing colors, so I'll just
do it in rows like this.

00:11:11.420 --> 00:11:12.580
5 times 1 is 5.

00:11:12.580 --> 00:11:15.790
5 times 2 is 10.

00:11:15.790 --> 00:11:17.270
5 times 3 is 15.

00:11:17.270 --> 00:11:18.440
I'm just going to
increase by 5.

00:11:18.440 --> 00:11:21.210
5 times tables are very fun as
well because every number

00:11:21.210 --> 00:11:24.130
you're going to add-- if we
multiply 5 times-- well, we'll

00:11:24.130 --> 00:11:25.890
learn about even and
odd in the future.

00:11:25.890 --> 00:11:29.620
But every other number in its
times tables is going to end

00:11:29.620 --> 00:11:32.250
with a 5, and then every other
one's going to end with a 0.

00:11:32.250 --> 00:11:34.800
Because if you add 5
to 15 you get 20.

00:11:34.800 --> 00:11:41.450
You get 25, 30, 35, 40, 45.

00:11:41.450 --> 00:11:43.390
Fair enough.

00:11:43.390 --> 00:11:47.130
6 times tables, let
me do it in green.

00:11:47.130 --> 00:11:48.220
6 times 1 is 6.

00:11:48.220 --> 00:11:49.080
That's easy.

00:11:49.080 --> 00:11:50.810
You add 6 to that, you get 12.

00:11:50.810 --> 00:11:52.500
You add 6 to that, you get 18.

00:11:52.500 --> 00:11:54.250
You add 6 to that, you get 24.

00:11:54.250 --> 00:11:56.220
You add 6 to that, you get 30.

00:11:56.220 --> 00:12:01.220
Then you go 6 more, 36, 42, 48.

00:12:01.220 --> 00:12:04.630
48 plus 6 is 54.

00:12:04.630 --> 00:12:07.630
So 6 times 9 is 54.

00:12:07.630 --> 00:12:09.010
All right, we're almost there.

00:12:09.010 --> 00:12:11.620
7 times 1, that's 7.

00:12:11.620 --> 00:12:13.530
7 times 1 is 7.

00:12:13.530 --> 00:12:15.620
7 times 2 is 14.

00:12:15.620 --> 00:12:17.740
7 times 3, 21.

00:12:17.740 --> 00:12:20.050
7 times 4, 28.

00:12:20.050 --> 00:12:23.670
7 times 5, what's 28 plus 7?

00:12:23.670 --> 00:12:23.820
Let's see.

00:12:23.820 --> 00:12:25.140
If you add 2 you get to 30.

00:12:25.140 --> 00:12:27.710
Then you have 5, 35.

00:12:27.710 --> 00:12:29.490
7 times 6, 42.

00:12:29.490 --> 00:12:32.650
7 times 7, 49.

00:12:32.650 --> 00:12:34.990
7 times 8.

00:12:34.990 --> 00:12:37.830
Seven times is going to be
7 plus this, so it's 56.

00:12:37.830 --> 00:12:41.590
I always used to get confused
between 7 times 8 being 56

00:12:41.590 --> 00:12:43.620
and 6 times 9 being 54.

00:12:43.620 --> 00:12:45.690
So now that I pointed out to
you that I always got confused

00:12:45.690 --> 00:12:48.230
between those two, it's your
job not to be confused

00:12:48.230 --> 00:12:49.400
by those two.

00:12:49.400 --> 00:12:52.620
7 times 8 you could
say has the 6 in it.

00:12:52.620 --> 00:12:54.850
6 times 9 doesn't
have the 6 in it.

00:12:54.850 --> 00:12:56.080
That's the way I think of it.

00:12:56.080 --> 00:12:57.580
Anyway, 7 times 9.

00:12:57.580 --> 00:12:59.320
We're going to add
another 7 here.

00:12:59.320 --> 00:13:00.665
It's going to be 63.

00:13:00.665 --> 00:13:04.840
I'll do it in the same color.

00:13:04.840 --> 00:13:08.360
All right, we're at
our 8 times tables.

00:13:08.360 --> 00:13:10.670
8 times 1 is 8.

00:13:10.670 --> 00:13:12.530
8 times 2 is 16.

00:13:12.530 --> 00:13:14.100
24.

00:13:14.100 --> 00:13:15.690
8 times 3 is 24.

00:13:15.690 --> 00:13:18.310
And if we go to 3 times 8
we should also see the 24.

00:13:18.310 --> 00:13:19.510
Yep, it's there.

00:13:19.510 --> 00:13:21.120
These values are the same.

00:13:21.120 --> 00:13:22.810
So we're actually
doing things twice.

00:13:22.810 --> 00:13:25.150
We're doing it when you do
8 times 3 and we're doing

00:13:25.150 --> 00:13:27.480
it when we did 3 times 8.

00:13:27.480 --> 00:13:27.790
Let's see.

00:13:27.790 --> 00:13:31.220
8 times 4, you're going
to add 8 to it-- 32.

00:13:31.220 --> 00:13:32.490
40.

00:13:32.490 --> 00:13:34.690
Add another 8, 48.

00:13:34.690 --> 00:13:37.280
Notice, 8 times 6, 48.

00:13:37.280 --> 00:13:40.240
6 times 8, 48.

00:13:40.240 --> 00:13:42.450
All right, 8 times 7.

00:13:42.450 --> 00:13:45.870
Well, we already pointed
that one out, that was 56.

00:13:45.870 --> 00:13:48.200
8 times 8, 64.

00:13:48.200 --> 00:13:52.020
8 times 9, add 8
to this, is 72.

00:13:52.020 --> 00:13:54.660
Now we're at the
9 times tables.

00:13:54.660 --> 00:13:57.270
I'm running out of colors.

00:13:57.270 --> 00:13:59.540
Maybe I'll reuse
a color or two.

00:13:59.540 --> 00:14:01.110
I'll use the blue again.

00:14:01.110 --> 00:14:03.290
9 times 1 is 9.

00:14:03.290 --> 00:14:06.710
9 times 2, 18 9 times 3-- we
actually know all of these.

00:14:06.710 --> 00:14:09.425
We could look it up in the rest
of the table because 9 times 3

00:14:09.425 --> 00:14:11.300
is the same thing as 3 times 9.

00:14:11.300 --> 00:14:13.290
It's 27.

00:14:13.290 --> 00:14:14.510
Add 9 to that.

00:14:14.510 --> 00:14:18.400
27 plus 9 is 36.

00:14:18.400 --> 00:14:21.540
36 plus 9 is 45.

00:14:21.540 --> 00:14:24.910
Notice, every time you add
9, you go almost up by

00:14:24.910 --> 00:14:26.250
10, but 1 less than that.

00:14:26.250 --> 00:14:29.970
So up by 10 would be 46, and
then one less than that is 45.

00:14:29.970 --> 00:14:33.380
But anyway, notice, the
1's-- well, I'll talk more

00:14:33.380 --> 00:14:33.900
about it in the future.

00:14:33.900 --> 00:14:37.530
But we go from a 9, 8,
7, 6, 5 on this digit,

00:14:37.530 --> 00:14:39.250
on the second digit.

00:14:39.250 --> 00:14:42.520
And on this digit here
you go 1, 2, 3, 4.

00:14:42.520 --> 00:14:44.030
So it's an interesting pattern.

00:14:44.030 --> 00:14:47.250
Another interesting pattern is
the digits will add up to 9.

00:14:47.250 --> 00:14:49.280
3 plus 6 is 9, 2 plus 7 is 9.

00:14:49.280 --> 00:14:51.010
We'll talk more about that
in the future and maybe

00:14:51.010 --> 00:14:52.640
prove that to you.

00:14:52.640 --> 00:14:56.350
9 times 6, 54.

00:14:56.350 --> 00:14:58.160
That was this one as well.

00:14:58.160 --> 00:15:01.730
9 times 7, 63.

00:15:01.730 --> 00:15:03.860
9 times 8, 72.

00:15:03.860 --> 00:15:05.920
9 times 9 is 81.

00:15:05.920 --> 00:15:07.050
I don't know if
you can see that.

00:15:07.050 --> 00:15:08.370
81.

00:15:08.370 --> 00:15:09.480
There you go.

00:15:09.480 --> 00:15:11.080
Now, I could keep going.

00:15:11.080 --> 00:15:13.790
Actually, I should keep going.

00:15:13.790 --> 00:15:17.550
Well, I realize this video
is already pretty long.

00:15:17.550 --> 00:15:19.680
I want you to memorize this
right now because this is

00:15:19.680 --> 00:15:21.210
going to get you pretty far.

00:15:21.210 --> 00:15:25.052
In the next video I'm going to
do the times tables past 9.

00:15:25.052 --> 00:15:26.970
See you soon.


Entropy intuition

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.660
I've now supplied you with two
definitions of the state

00:00:03.660 --> 00:00:04.600
variable entropy.

00:00:04.600 --> 00:00:06.390
And it's S for entropy.

00:00:06.390 --> 00:00:09.110
The thermodynamic definition
said that the change in

00:00:09.110 --> 00:00:12.440
entropy is equal to the heat
added to the system divided by

00:00:12.440 --> 00:00:14.560
the temperature at which
the heat is added.

00:00:14.560 --> 00:00:16.590
So obviously, if the temperature
is changing while

00:00:16.590 --> 00:00:18.600
we add the heat, which is
normally the case, we're going

00:00:18.600 --> 00:00:20.200
to have to do a little
bit of calculus.

00:00:20.200 --> 00:00:22.790
And then you can view this as
the mathematical, or the

00:00:22.790 --> 00:00:24.700
statistical, or the
combinatorical

00:00:24.700 --> 00:00:26.710
definition of entropy.

00:00:26.710 --> 00:00:29.630
And this essentially says that
entropy is equal to some

00:00:29.630 --> 00:00:32.320
constant times the natural log
of the number of states the

00:00:32.320 --> 00:00:33.840
system can take on.

00:00:33.840 --> 00:00:35.990
And this is the case when all
the states are equally

00:00:35.990 --> 00:00:37.720
probable, which is a pretty
good assumption.

00:00:37.720 --> 00:00:41.180
If you have just a gazillion
molecules that could have a

00:00:41.180 --> 00:00:43.270
gazillion gazillion states,
you can assume they're all

00:00:43.270 --> 00:00:44.760
roughly equally likely.

00:00:44.760 --> 00:00:46.970
There's a slightly more involved
definition if they

00:00:46.970 --> 00:00:48.480
had different probabilities,
but we won't

00:00:48.480 --> 00:00:49.750
worry about that now.

00:00:49.750 --> 00:00:52.570
So given that we've seen these
two definitions, it's a good

00:00:52.570 --> 00:00:56.560
time to introduce you to the
second law of thermodynamics.

00:01:00.000 --> 00:01:01.220
And that's this.

00:01:01.220 --> 00:01:04.959
And it's a pretty simple law,
but it explains a whole range

00:01:04.959 --> 00:01:06.110
of phenomena.

00:01:06.110 --> 00:01:16.410
It tells us that the change in
entropy for the universe when

00:01:16.410 --> 00:01:19.760
any process is undergone
is always greater

00:01:19.760 --> 00:01:22.940
than or equal to 0.

00:01:22.940 --> 00:01:26.530
So that tells us that when
anything ever happens in the

00:01:26.530 --> 00:01:30.560
universe, the net effect is that
there's more entropy in

00:01:30.560 --> 00:01:31.830
the universe itself.

00:01:31.830 --> 00:01:34.830
And this seems very deep,
and it actually is.

00:01:34.830 --> 00:01:38.440
So let's see if we can apply it
to see why it explains, or

00:01:38.440 --> 00:01:41.400
why it makes sense, relative
to some examples.

00:01:41.400 --> 00:01:44.970
So let's say I have two
reservoirs that are in contact

00:01:44.970 --> 00:01:45.610
with each other.

00:01:45.610 --> 00:01:49.110
So I have T1.

00:01:49.110 --> 00:01:52.140
And let's call this
our hot reservoir.

00:01:52.140 --> 00:01:53.390
And then I have T2.

00:01:56.100 --> 00:01:58.290
I'll call this our
cold reservoir.

00:01:58.290 --> 00:01:59.720
Well, we know from experience.

00:01:59.720 --> 00:02:03.170
What happens if I put a hot cup
of water, and it's sharing

00:02:03.170 --> 00:02:07.120
a wall with a cold glass of
water, or cold cube of water,

00:02:07.120 --> 00:02:08.090
what happens?

00:02:08.090 --> 00:02:09.729
Well, their temperatures
equalize.

00:02:09.729 --> 00:02:11.660
If these are the same substance,
we'll end up

00:02:11.660 --> 00:02:13.960
roughly in between, if they're
in the same phase.

00:02:13.960 --> 00:02:15.960
So essentially, we have a
transfer of heat from the

00:02:15.960 --> 00:02:18.720
hotter substance to the
colder substance.

00:02:18.720 --> 00:02:23.290
So we have some heat, Q, that
goes from the hotter substance

00:02:23.290 --> 00:02:24.550
to the colder substance.

00:02:24.550 --> 00:02:27.960
You don't see, in everyday
reality, heat going from a

00:02:27.960 --> 00:02:29.350
colder substance to a
hotter substance.

00:02:29.350 --> 00:02:33.850
If I put an ice cube in, let's
say, some hot tea, you don't

00:02:33.850 --> 00:02:35.760
see the ice cube getting
colder and the

00:02:35.760 --> 00:02:36.780
hot tea getting hotter.

00:02:36.780 --> 00:02:39.880
You see them both getting to
some equal temperature, which

00:02:39.880 --> 00:02:42.550
essentially the tea is giving
heat to the ice cube.

00:02:42.550 --> 00:02:44.490
Now in this situation there
are reservoirs, so I'm

00:02:44.490 --> 00:02:46.810
assuming that their temperatures
stay constant.

00:02:46.810 --> 00:02:49.780
Which would only be the case
if they were both infinite,

00:02:49.780 --> 00:02:51.300
which we know doesn't exist
in the real world.

00:02:51.300 --> 00:02:53.950
In the real world, T1's
temperature as it gave heat

00:02:53.950 --> 00:02:56.390
would go down, and T2's
temperature would go up.

00:02:56.390 --> 00:02:59.720
But let's just see whether the
second law of thermodynamics

00:02:59.720 --> 00:03:01.770
says that this should happen.

00:03:01.770 --> 00:03:04.010
So what's happening here?

00:03:04.010 --> 00:03:07.490
What's the net change
in entropy for T1?

00:03:07.490 --> 00:03:09.790
So the second law of
thermodynamics says that the

00:03:09.790 --> 00:03:13.370
change in entropy for the
universe is greater than 0.

00:03:13.370 --> 00:03:16.670
But in this case, that's equal
to the change in entropy for

00:03:16.670 --> 00:03:23.310
T1 plus the change in entropy
for-- oh, I shouldn't--

00:03:23.310 --> 00:03:28.150
instead of T1, let me
call it just 1.

00:03:28.150 --> 00:03:31.490
For system 1, that's this hot
system up here, plus the

00:03:31.490 --> 00:03:33.730
change in entropy
for system 2.

00:03:33.730 --> 00:03:35.990
So what's the change in
entropy for system 1?

00:03:35.990 --> 00:03:43.330
It loses Q1 at a high
temperature.

00:03:43.330 --> 00:03:50.850
So this equals minus the heat
given to the system is Q over

00:03:50.850 --> 00:03:53.690
some hot temperature T1.

00:03:53.690 --> 00:03:57.780
And then we have the heat being
added to the system T2.

00:03:57.780 --> 00:04:02.460
So plus Q over T2.

00:04:02.460 --> 00:04:05.230
This is the change in entropy
for the system 2, right?

00:04:05.230 --> 00:04:08.260
This guy loses the heat, and is
at temperature 1, which is

00:04:08.260 --> 00:04:09.430
a higher temperature.

00:04:09.430 --> 00:04:12.340
This guy gains the heat, and
he is at a temperature 2,

00:04:12.340 --> 00:04:13.430
which is a colder temperature.

00:04:13.430 --> 00:04:18.399
Now, is this going to
be greater than 0?

00:04:18.399 --> 00:04:19.950
Let's think about
it a little bit.

00:04:19.950 --> 00:04:22.070
If I divide-- let
me rewrite this.

00:04:22.070 --> 00:04:25.900
So I can rearrange them, so that
we can write this as Q

00:04:25.900 --> 00:04:28.980
over T2 minus this one.

00:04:28.980 --> 00:04:30.740
I'm just rearranging it.

00:04:30.740 --> 00:04:33.830
Minus Q over T1.

00:04:33.830 --> 00:04:35.520
Now, which number is bigger?

00:04:35.520 --> 00:04:36.720
T2 to T1?

00:04:36.720 --> 00:04:39.130
Well, T1 is bigger, right?

00:04:39.130 --> 00:04:40.380
This is bigger.

00:04:43.310 --> 00:04:46.060
Now, if I have a bigger number,
bigger than this--

00:04:46.060 --> 00:04:47.480
when we use the word
bigger, you have to

00:04:47.480 --> 00:04:48.870
compare it to something.

00:04:48.870 --> 00:04:50.590
Now, T1 is bigger than this.

00:04:50.590 --> 00:04:52.340
We have the same number
in the numerator

00:04:52.340 --> 00:04:54.050
in both cases, right?

00:04:54.050 --> 00:04:58.370
So if I take, let's say, 1
over some, let's say, 1/2

00:04:58.370 --> 00:05:01.350
minus 1/3, we're going
to be bigger than 0.

00:05:01.350 --> 00:05:03.830
This is a larger number than
this number, because this has

00:05:03.830 --> 00:05:04.635
a bigger denominator.

00:05:04.635 --> 00:05:06.840
You're dividing by
a larger number.

00:05:06.840 --> 00:05:08.190
That's a good way to
think about it.

00:05:08.190 --> 00:05:11.610
You're dividing this Q by some
number here to get something,

00:05:11.610 --> 00:05:13.000
and then you're subtracting
this Q

00:05:13.000 --> 00:05:14.470
divided by a larger number.

00:05:14.470 --> 00:05:16.890
So this fraction is going to be
a smaller absolute number.

00:05:16.890 --> 00:05:20.770
So this is going to
be greater than 0.

00:05:20.770 --> 00:05:23.490
So that tells us the second
law of thermodynamics, it

00:05:23.490 --> 00:05:28.630
verifies this observation we
see in the real world, that

00:05:28.630 --> 00:05:32.360
heat will flow from the hot
body to the cold body.

00:05:32.360 --> 00:05:33.990
Now, you might say, hey, Sal.

00:05:33.990 --> 00:05:38.100
I have a case that will show
you that you are wrong.

00:05:38.100 --> 00:05:39.270
You could say, look.

00:05:39.270 --> 00:05:45.660
If I put an air conditioner in a
room-- Let's say this is the

00:05:45.660 --> 00:05:50.780
room, and this is outside.

00:05:50.780 --> 00:05:54.230
You'll say, look what the
air conditioner does.

00:05:54.230 --> 00:05:59.410
The room is already cold, and
outside is already hot.

00:05:59.410 --> 00:06:01.620
But what the air conditioner
does, is it makes the cold

00:06:01.620 --> 00:06:04.490
even colder, and it makes
the hot even hotter.

00:06:04.490 --> 00:06:08.680
It takes some Q and it goes
in that direction.

00:06:08.680 --> 00:06:09.010
Right?

00:06:09.010 --> 00:06:11.670
It takes heat from the cold
room, and puts it out

00:06:11.670 --> 00:06:12.625
into the hot air.

00:06:12.625 --> 00:06:14.560
And you're saying, this defies
the second law of

00:06:14.560 --> 00:06:15.690
thermodynamics.

00:06:15.690 --> 00:06:17.040
You have just disproved it.

00:06:17.040 --> 00:06:18.790
You deserve a Nobel Prize.

00:06:18.790 --> 00:06:24.460
And I would say to you, you're
forgetting one small fact.

00:06:24.460 --> 00:06:27.340
This air conditioner inside
here, it has some type of a

00:06:27.340 --> 00:06:29.470
compressor, some type
of an engine, that's

00:06:29.470 --> 00:06:31.210
actively doing this.

00:06:31.210 --> 00:06:33.600
It's putting in work to
make this happen.

00:06:33.600 --> 00:06:38.040
And this engine right here--
I'll do it in magenta-- it's

00:06:38.040 --> 00:06:40.630
also expelling some more heat.

00:06:40.630 --> 00:06:47.490
So let's call that
Q of the engine.

00:06:47.490 --> 00:06:52.970
So if you wanted to figure out
the total entropy created for

00:06:52.970 --> 00:07:02.870
the universe, it would be the
entropy of the cold room plus

00:07:02.870 --> 00:07:06.870
the change in entropy for
outside-- I'll call it

00:07:06.870 --> 00:07:10.050
outside, maybe I'll call
this, for the room.

00:07:10.050 --> 00:07:10.870
Right?

00:07:10.870 --> 00:07:11.590
So you might say, OK.

00:07:11.590 --> 00:07:14.520
This change in entropy for the
room, it's giving away heat--

00:07:14.520 --> 00:07:16.700
let's see the room is roughly at
a constant temperature for

00:07:16.700 --> 00:07:19.290
that one millisecond we're
looking at it.

00:07:19.290 --> 00:07:23.860
It's giving away some Q at
some temperature T1.

00:07:23.860 --> 00:07:26.460
And then-- so that's a minus.

00:07:26.460 --> 00:07:30.940
And then this the outside is
gaining some heat at some

00:07:30.940 --> 00:07:32.480
temperature T2.

00:07:32.480 --> 00:07:33.440
And so you'll immediately
say, hey.

00:07:33.440 --> 00:07:39.110
This number right here is a
smaller number than this one.

00:07:39.110 --> 00:07:39.530
Right?

00:07:39.530 --> 00:07:41.610
Because the denominator
is higher.

00:07:41.610 --> 00:07:43.570
So if you just look at this,
this would be negative

00:07:43.570 --> 00:07:46.300
entropy, and you'd say hey, this
defies the second law of

00:07:46.300 --> 00:07:47.150
thermodynamics.

00:07:47.150 --> 00:07:47.650
No!

00:07:47.650 --> 00:07:50.470
But what you have to throw in
here is another notion.

00:07:50.470 --> 00:07:52.660
You have to throw in here the
notion that the outside is

00:07:52.660 --> 00:07:58.900
also getting this heat from the
engine over the outside

00:07:58.900 --> 00:07:59.990
temperature.

00:07:59.990 --> 00:08:03.980
And this term, I can guarantee
you-- I'm not giving you

00:08:03.980 --> 00:08:05.460
numbers right now--
will make this

00:08:05.460 --> 00:08:07.330
whole expression positive.

00:08:07.330 --> 00:08:12.430
This term will turn the total
net entropy to the universe to

00:08:12.430 --> 00:08:13.770
be positive.

00:08:13.770 --> 00:08:17.050
Now let's think a little bit how
about what entropy is and

00:08:17.050 --> 00:08:18.830
what entropy isn't in
terms of words.

00:08:18.830 --> 00:08:22.840
So when you take an intro
chemistry class, the teacher

00:08:22.840 --> 00:08:28.000
often says, entropy
equals disorder.

00:08:28.000 --> 00:08:30.770
Which is not incorrect.

00:08:30.770 --> 00:08:34.210
It is disorder, but you have
to be very careful what we

00:08:34.210 --> 00:08:35.830
mean by disorder.

00:08:35.830 --> 00:08:39.429
Because the very next example
that's often given is that

00:08:39.429 --> 00:08:40.090
they'll say, look.

00:08:40.090 --> 00:08:45.090
A clean room-- let's say your
bedroom is clean, and then it

00:08:45.090 --> 00:08:45.920
becomes dirty.

00:08:45.920 --> 00:08:46.950
And they'll say, look.

00:08:46.950 --> 00:08:48.850
The universe became
more disordered.

00:08:48.850 --> 00:08:53.370
The dirty room has more disorder
than the clean room.

00:08:53.370 --> 00:08:56.980
And this is not a case
of entropy increase.

00:08:56.980 --> 00:08:58.673
So this is not a good example.

00:09:01.230 --> 00:09:04.230
Why is that?

00:09:04.230 --> 00:09:07.670
Because clean and dirty are
just states of the room.

00:09:07.670 --> 00:09:11.720
Remember, entropy is a
macro state variable.

00:09:14.600 --> 00:09:16.720
It's something you use to
describe a system where you're

00:09:16.720 --> 00:09:19.430
not in the mood to sit there and
tell me what exactly every

00:09:19.430 --> 00:09:20.800
particle is doing.

00:09:20.800 --> 00:09:23.200
And this is a macro variable
that actually tells me how

00:09:23.200 --> 00:09:27.430
much time would it take for
me to tell you what every

00:09:27.430 --> 00:09:28.290
particle is doing.

00:09:28.290 --> 00:09:30.610
It actually tells you how many
states there are, or how much

00:09:30.610 --> 00:09:32.820
information I would have
to give you to tell

00:09:32.820 --> 00:09:34.470
you the exact state.

00:09:34.470 --> 00:09:36.480
Now, when you have a clean room
and a dirty room, these

00:09:36.480 --> 00:09:39.590
are two different states
of the same room.

00:09:39.590 --> 00:09:42.800
If the room has the same
temperature, and it has the

00:09:42.800 --> 00:09:46.580
same number of molecules in it
and everything, then they have

00:09:46.580 --> 00:09:47.450
the same entropy.

00:09:47.450 --> 00:09:49.930
So clean to dirty, it's
not more entropy.

00:09:49.930 --> 00:09:55.530
Now, for example, I could
have a dirty, cold room.

00:09:58.530 --> 00:10:01.630
And let's say I were to go into
that room and, you know,

00:10:01.630 --> 00:10:03.950
I work really hard
to clean it up.

00:10:03.950 --> 00:10:06.980
And by doing so, I add a lot of
heat to the system, and my

00:10:06.980 --> 00:10:09.810
sweat molecules drop all over
the place, and so there's just

00:10:09.810 --> 00:10:13.240
more stuff in that room, and
it's all warmed up to me-- so

00:10:13.240 --> 00:10:22.810
to a hot, clean room with sweat
in it-- so it's got more

00:10:22.810 --> 00:10:25.350
stuff in here that can be
configured in more ways, and

00:10:25.350 --> 00:10:28.620
because it's hot, every molecule
in the room can take

00:10:28.620 --> 00:10:29.850
on more states, right?

00:10:29.850 --> 00:10:32.420
Because the average kinetic
energy is up, so they can kind

00:10:32.420 --> 00:10:36.070
of explore the spaces of how
many kinetic energies it can

00:10:36.070 --> 00:10:39.700
have. There's more potential
energies that each molecule

00:10:39.700 --> 00:10:40.240
can take on.

00:10:40.240 --> 00:10:42.610
This is actually an increase
in entropy.

00:10:42.610 --> 00:10:45.410
From a dirty, cold room
to a hot, clean room.

00:10:45.410 --> 00:10:47.410
And this actually goes well
with what we know.

00:10:47.410 --> 00:10:50.720
I mean, when I go into room and
I start cleaning it, I am

00:10:50.720 --> 00:10:52.080
in putting heat into the room.

00:10:52.080 --> 00:10:55.140
And the universe is becoming
more-- I guess we could say

00:10:55.140 --> 00:10:57.720
it's the entropy
is increasing.

00:10:57.720 --> 00:11:00.030
So where does the term
disorder apply?

00:11:03.110 --> 00:11:07.330
Well, let's take a situation
where I take a ball.

00:11:07.330 --> 00:11:10.520
I take a ball, and it
falls to the ground.

00:11:10.520 --> 00:11:12.570
And then it hits the ground.

00:11:12.570 --> 00:11:14.050
And there should have been a
question that you've been

00:11:14.050 --> 00:11:16.160
asking all the time, since the
first law of thermodynamics.

00:11:20.170 --> 00:11:21.360
So the ball hits the
ground, right?

00:11:21.360 --> 00:11:24.520
It got thrown up, it had some
potential energy at the top,

00:11:24.520 --> 00:11:26.380
then that all gets turned into
kinetic energy and it hits the

00:11:26.380 --> 00:11:28.120
ground, and then it stops.

00:11:28.120 --> 00:11:30.410
And so your obvious question is,
what happened to all that

00:11:30.410 --> 00:11:31.880
energy, right?

00:11:31.880 --> 00:11:33.430
Law of conservation of energy.

00:11:33.430 --> 00:11:34.380
Where did all of it go?

00:11:34.380 --> 00:11:35.890
It had all that kinetic energy
right before it hit the

00:11:35.890 --> 00:11:37.350
ground, then it stopped.

00:11:37.350 --> 00:11:37.760
Right?

00:11:37.760 --> 00:11:39.290
It seems like it disappeared.

00:11:39.290 --> 00:11:40.950
But it didn't disappear.

00:11:40.950 --> 00:11:44.260
So when the ball was falling, it
had a bunch of-- you know,

00:11:44.260 --> 00:11:45.470
everything had a little
bit of heat.

00:11:45.470 --> 00:11:47.960
But let's say the ground
was reasonably ordered.

00:11:51.380 --> 00:11:56.260
The ground molecules were
vibrating with some kinetic

00:11:56.260 --> 00:11:58.100
energy and potential energies.

00:11:58.100 --> 00:12:00.650
And then our ball molecules
were also

00:12:00.650 --> 00:12:02.480
vibrating a little bit.

00:12:02.480 --> 00:12:06.340
But most of their motion
was downwards, right?

00:12:06.340 --> 00:12:09.160
Most of the ball molecules'
motion was downwards.

00:12:09.160 --> 00:12:12.220
Now, when it hits the ground,
what happens-- let me show you

00:12:12.220 --> 00:12:14.230
the interface of the ball.

00:12:14.230 --> 00:12:17.790
So the ball molecules at the
front of the ball are going to

00:12:17.790 --> 00:12:19.400
look like that.

00:12:19.400 --> 00:12:20.730
And there's a bunch of them.

00:12:20.730 --> 00:12:21.460
It's a solid.

00:12:21.460 --> 00:12:24.100
It will maybe be some
type of lattice.

00:12:24.100 --> 00:12:25.740
And then it hits the ground.

00:12:25.740 --> 00:12:30.150
And when it hits the ground-- so
the ground is another solid

00:12:30.150 --> 00:12:35.960
like that-- All right, we're
looking at the microstate.

00:12:35.960 --> 00:12:37.370
What's going to happen?

00:12:37.370 --> 00:12:39.390
These guys are going to rub
up against these guys, and

00:12:39.390 --> 00:12:41.880
they're going to transfer
their-- what was downward

00:12:41.880 --> 00:12:45.090
kinetic energy, and a very
ordered downward kinetic

00:12:45.090 --> 00:12:47.510
energy-- they're going
to transfer it to

00:12:47.510 --> 00:12:49.110
these ground particles.

00:12:49.110 --> 00:12:50.950
And they're going to bump into
the ground particles.

00:12:50.950 --> 00:12:54.040
And so when this guy bumps into
that guy, he might start

00:12:54.040 --> 00:12:55.910
moving in that direction.

00:12:55.910 --> 00:12:57.910
This guy will start oscillating
in that direction,

00:12:57.910 --> 00:12:59.770
and go back and forth
like that.

00:12:59.770 --> 00:13:02.300
That guy might bounce off of
this guy, and go in that

00:13:02.300 --> 00:13:04.190
direction, and bump
into that guy, and

00:13:04.190 --> 00:13:05.380
go into that direction.

00:13:05.380 --> 00:13:09.410
And then, because that guy
bumped here, this guy bumps

00:13:09.410 --> 00:13:11.860
here, and because this guy bumps
here, this guy bumps

00:13:11.860 --> 00:13:12.760
over there.

00:13:12.760 --> 00:13:16.820
And so what you have is, what
was relatively ordered motion,

00:13:16.820 --> 00:13:19.210
especially from the ball's point
of view, when it starts

00:13:19.210 --> 00:13:22.280
rubbing up against these
molecules of the ground, it

00:13:22.280 --> 00:13:24.855
starts making the kinetic
energy, or their movement, go

00:13:24.855 --> 00:13:26.760
in all sorts of random
directions.

00:13:26.760 --> 00:13:27.090
Right?

00:13:27.090 --> 00:13:29.130
This guy's going to make this
guy go like that, and that guy

00:13:29.130 --> 00:13:30.170
go like that.

00:13:30.170 --> 00:13:33.570
And so when the movement is no
longer ordered, if I have a

00:13:33.570 --> 00:13:36.550
lot of molecules-- let me do it
in a different color-- if I

00:13:36.550 --> 00:13:38.520
have a lot of molecules, and
they're all moving in the

00:13:38.520 --> 00:13:41.910
exact same direction, then my
micro state looks like my

00:13:41.910 --> 00:13:42.590
macro state.

00:13:42.590 --> 00:13:44.850
The whole thing moves
in that direction.

00:13:44.850 --> 00:13:47.860
Now, if I have a bunch of
molecules, and they're all

00:13:47.860 --> 00:13:51.900
moving in random directions,
my ball as a whole will be

00:13:51.900 --> 00:13:52.320
stationary.

00:13:52.320 --> 00:13:54.330
I could have the exact same
amount of kinetic energy at

00:13:54.330 --> 00:13:56.740
the molecular level, but
they're all going to be

00:13:56.740 --> 00:13:58.670
bouncing into each other.

00:13:58.670 --> 00:14:02.520
And in this case, we described
the kinetic energy as internal

00:14:02.520 --> 00:14:06.140
energy, or we describe it as
temperature, where temperature

00:14:06.140 --> 00:14:07.980
is the average kinetic energy.

00:14:07.980 --> 00:14:10.500
So in this case, when we talk
about, the world is becoming

00:14:10.500 --> 00:14:13.650
more disordered, you think about
the order of maybe the

00:14:13.650 --> 00:14:16.110
velocities or the energies
of the molecules.

00:14:16.110 --> 00:14:18.810
Before they were reasonably
ordered, the molecules-- they

00:14:18.810 --> 00:14:20.210
might have been vibrating a
little bit, but they were

00:14:20.210 --> 00:14:22.050
mainly going down in the ball.

00:14:22.050 --> 00:14:24.490
But when they bump into the
ground, all of a sudden they

00:14:24.490 --> 00:14:26.740
start vibrating in random
directions a little bit more.

00:14:26.740 --> 00:14:28.760
And they make the ground
vibrate in more random

00:14:28.760 --> 00:14:29.600
directions.

00:14:29.600 --> 00:14:32.800
So it makes-- at the
microstate-- everything became

00:14:32.800 --> 00:14:36.010
just that much more
disordered.

00:14:36.010 --> 00:14:37.530
Now there's an interesting
question here.

00:14:37.530 --> 00:14:41.830
There is some probability you
might think-- Look, this ball

00:14:41.830 --> 00:14:43.250
came down and hit the ground.

00:14:43.250 --> 00:14:46.280
Why doesn't the ball just--
isn't there some probability

00:14:46.280 --> 00:14:50.950
that if I have a ground, that
these molecules just rearrange

00:14:50.950 --> 00:14:56.880
themselves in just the right
way to just hit these ball

00:14:56.880 --> 00:14:58.190
molecules in just
the right way?

00:14:58.190 --> 00:15:00.810
There's some probability, just
from the random movement, that

00:15:00.810 --> 00:15:04.520
at get some second, all the
ground molecules just hit the

00:15:04.520 --> 00:15:08.710
ball molecules just right to
send the ball back up.

00:15:08.710 --> 00:15:10.200
And the answer is yes.

00:15:10.200 --> 00:15:13.400
There's actually some
infinitesimally small chance

00:15:13.400 --> 00:15:14.340
that that happens.

00:15:14.340 --> 00:15:16.790
That you could have a ball
that's sitting on the ground--

00:15:16.790 --> 00:15:19.390
and this is interesting--
could have a ball that's

00:15:19.390 --> 00:15:21.990
sitting on the ground, and while
you're looking, you'll

00:15:21.990 --> 00:15:24.740
probably have to wait a few
gazillion years for it to

00:15:24.740 --> 00:15:28.440
happen, if it happens at all--
it could just randomly pop up.

00:15:28.440 --> 00:15:33.160
And there's some random, very
small chance that these

00:15:33.160 --> 00:15:36.310
molecules just randomly vibrate
in just the right way

00:15:36.310 --> 00:15:39.960
to be ordered for a second, and
then the ball will pop up.

00:15:39.960 --> 00:15:43.270
But the probability of this
happening, relative to

00:15:43.270 --> 00:15:46.740
everything else, is
essentially 0.

00:15:46.740 --> 00:15:49.520
So when people talk about
order and disorder, the

00:15:49.520 --> 00:15:52.040
disorder is increasing, because
now these molecules

00:15:52.040 --> 00:15:54.640
are going in more random
directions, and they can take

00:15:54.640 --> 00:15:57.770
on more potential states.

00:15:57.770 --> 00:15:59.460
And we saw that here.

00:15:59.460 --> 00:16:01.970
And you know, on some level,
entropy seems something kind

00:16:01.970 --> 00:16:04.630
of magical, but on some
level, it seems

00:16:04.630 --> 00:16:06.900
relatively common sense.

00:16:06.900 --> 00:16:10.210
In that video-- I think was the
last video-- I had a case

00:16:10.210 --> 00:16:13.580
where I had a bunch of
molecules, and then I had this

00:16:13.580 --> 00:16:18.000
extra space here, and then
I removed the wall.

00:16:18.000 --> 00:16:23.760
And we saw that these molecules
will-- we know,

00:16:23.760 --> 00:16:26.220
there's always some modules
that are bouncing off this

00:16:26.220 --> 00:16:28.430
wall before, because we probably
had some pressure

00:16:28.430 --> 00:16:29.640
associated with it.

00:16:29.640 --> 00:16:31.790
And then as soon as we remove
that wall, the molecule that

00:16:31.790 --> 00:16:34.910
would have bounced there
just keeps going.

00:16:34.910 --> 00:16:36.170
There's nothing to stop
it from there.

00:16:36.170 --> 00:16:37.410
In that direction, there's
a lot of stuff.

00:16:37.410 --> 00:16:39.180
It could bump into other
molecules, and it could

00:16:39.180 --> 00:16:40.410
bumping into these walls.

00:16:40.410 --> 00:16:42.190
But in this direction, the
odds of it bumping into

00:16:42.190 --> 00:16:45.020
everything is, especially for
these leading molecules, is

00:16:45.020 --> 00:16:45.710
essentially 0.

00:16:45.710 --> 00:16:48.690
So it's going to expand
to fill the container.

00:16:48.690 --> 00:16:50.640
So that's kind of
common sense.

00:16:50.640 --> 00:16:53.500
But the neat thing is that the
second law of thermodynamics,

00:16:53.500 --> 00:16:57.170
as we saw in that video, also
says that this will happen.

00:16:57.170 --> 00:17:00.520
That the molecules will all
expand to fill the container.

00:17:00.520 --> 00:17:05.270
And that the odds of this
happening are very low.

00:17:05.270 --> 00:17:11.030
That they all come back and
go into a ordered state.

00:17:11.030 --> 00:17:13.470
Now there is some chance, just
from the random movements once

00:17:13.470 --> 00:17:15.710
they fill, that they all just
happen to come back here.

00:17:15.710 --> 00:17:18.099
But it's a very, very
small probability.

00:17:18.099 --> 00:17:22.160
And even more-- and I want to
make this very clear-- S is a

00:17:22.160 --> 00:17:23.410
macro state.

00:17:25.970 --> 00:17:28.410
We never talk about
the entropy for

00:17:28.410 --> 00:17:30.010
an individual molecule.

00:17:30.010 --> 00:17:32.310
If we know what an individual
molecule is doing, we

00:17:32.310 --> 00:17:33.500
shouldn't be worrying
about entropy.

00:17:33.500 --> 00:17:35.610
We should be worrying about
the system as a whole.

00:17:38.270 --> 00:17:43.400
So even if we're looking at
the system, if we're not

00:17:43.400 --> 00:17:45.470
looking directly at the
molecules, we won't even know

00:17:45.470 --> 00:17:48.940
that this actually happened.

00:17:48.940 --> 00:17:50.920
All we can do is look
at the statistical

00:17:50.920 --> 00:17:52.110
properties of the molecules.

00:17:52.110 --> 00:17:54.460
How many molecules they are,
what their temperature is, all

00:17:54.460 --> 00:17:56.750
their macro dynamics,
the pressure, and

00:17:56.750 --> 00:17:57.560
say, you know what?

00:17:57.560 --> 00:18:01.700
A box that has these molecules
has more state than a smaller

00:18:01.700 --> 00:18:03.740
box, than the box when we
had the wall there.

00:18:03.740 --> 00:18:06.520
Even if, by chance, all of the
molecules happened to be

00:18:06.520 --> 00:18:08.580
collecting over there, we
wouldn't know that that

00:18:08.580 --> 00:18:10.920
happened, because we're not
looking at the micro states.

00:18:10.920 --> 00:18:13.150
And that's a really important
thing to consider.

00:18:13.150 --> 00:18:17.130
When someone says that a dirty
room has a higher entropy than

00:18:17.130 --> 00:18:20.510
a clean room, they're looking
at the micro states.

00:18:20.510 --> 00:18:23.070
And entropy essentially is
a macro state variable.

00:18:23.070 --> 00:18:24.890
You could just say
that a room has a

00:18:24.890 --> 00:18:26.650
certain amount of entropy.

00:18:26.650 --> 00:18:32.160
So entropy is associated with
the room, and it's only useful

00:18:32.160 --> 00:18:33.640
when you really don't
know exactly what's

00:18:33.640 --> 00:18:34.390
going on in the room.

00:18:34.390 --> 00:18:37.040
You just have a general sense of
how much stuff there is in

00:18:37.040 --> 00:18:39.020
the room, what's the temperature
of the room,

00:18:39.020 --> 00:18:40.550
what's the pressure
in the room.

00:18:40.550 --> 00:18:42.750
Just the general macro
properties.

00:18:42.750 --> 00:18:45.910
And then entropy will
essentially tell us how many

00:18:45.910 --> 00:18:50.120
possible micro states that
macro system can actually

00:18:50.120 --> 00:18:53.380
have. Or how much information--
and there's a

00:18:53.380 --> 00:18:56.150
notion of information entropy--
how much information

00:18:56.150 --> 00:19:00.970
would I have to give you to tell
you what the exact micro

00:19:00.970 --> 00:19:03.250
state is of a system at
that point in time.

00:19:03.250 --> 00:19:03.780
Well anyway.

00:19:03.780 --> 00:19:05.330
Hopefully you found this
discussion a little bit

00:19:05.330 --> 00:19:08.710
useful, and it clears up some
misconceptions about entropy,

00:19:08.710 --> 00:19:11.720
and gives you a little bit more
intuition about what it

00:19:11.720 --> 00:19:12.780
actually is.

00:19:12.780 --> 00:19:14.590
See you in the next video.