Quasistatic and reversible processes

WEBVTT
Kind: captions
Language: en

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SAL: In the last video, where
we talked about macrostates,

00:00:03.760 --> 00:00:07.460
we set up this situation where
I had this canister, or the

00:00:07.460 --> 00:00:09.980
cylinder, and had this
movable ceiling.

00:00:09.980 --> 00:00:11.000
I call that a piston.

00:00:11.000 --> 00:00:15.000
And the piston is being kept
up by the pressure from the

00:00:15.000 --> 00:00:15.905
gas in the canister.

00:00:15.905 --> 00:00:19.960
And it's being kept down by, in
the last example I had, a

00:00:19.960 --> 00:00:21.520
rock or a weight on top.

00:00:21.520 --> 00:00:22.820
And above that I had a vacuum.

00:00:22.820 --> 00:00:26.090
So essentially there's some
force per area, or pressure,

00:00:26.090 --> 00:00:27.790
being applied by the
bumps of the

00:00:27.790 --> 00:00:30.260
particles into this piston.

00:00:30.260 --> 00:00:32.409
And if this weight wasn't here--
let's assume that the

00:00:32.409 --> 00:00:35.920
piston itself or this movable
ceiling itself, it has no

00:00:35.920 --> 00:00:38.490
mass-- if that weight wasn't
there it would just be pushed

00:00:38.490 --> 00:00:41.000
indefinitely far, because
there'd be no pressure from

00:00:41.000 --> 00:00:41.460
the vacuum.

00:00:41.460 --> 00:00:44.360
But this weight is applying
some force on

00:00:44.360 --> 00:00:45.960
that same area downwards.

00:00:45.960 --> 00:00:49.160
So we're at some equilibrium
point, some stability.

00:00:49.160 --> 00:00:53.430
And we plotted that on this
PV diagram right here.

00:00:53.430 --> 00:00:55.910
I'll do it in magenta.

00:00:55.910 --> 00:01:00.390
So that's our state 1 that
we were in right there.

00:01:00.390 --> 00:01:03.410
And then what I did in the last
video, I just blew away

00:01:03.410 --> 00:01:04.750
half of this block.

00:01:04.750 --> 00:01:07.210
And as soon as I blew away half
of this block, obviously

00:01:07.210 --> 00:01:10.080
the force that's being applied
by the block will immediately

00:01:10.080 --> 00:01:13.660
go down by half, and so the
gas will push up on it.

00:01:13.660 --> 00:01:16.900
And it happened so fast that,
al of a sudden the gas is

00:01:16.900 --> 00:01:17.940
pushing up.

00:01:17.940 --> 00:01:20.700
Right when it happens, the gas
near the top of the canister

00:01:20.700 --> 00:01:23.260
is going to have lower pressure,
because it has less

00:01:23.260 --> 00:01:24.250
pushing up against it.

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The molecules that are down here
don't even know that I

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blew away this block yet.

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It's going to take some time.

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And essentially the gas is going
to push it up, and then

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maybe it'll oscillate down,
and then push it up, and

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oscillate down a little bit.

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It'll take some time eventually
until we get to

00:01:39.910 --> 00:01:43.890
another equilibrium state, where
we have a new, probably,

00:01:43.890 --> 00:01:45.510
or definitely lower pressure.

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We definitely have
a higher volume.

00:01:49.150 --> 00:01:51.300
I won't talk too much about it
yet, but we probably have a

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lower temperature as well.

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And this is our new state.

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And our macrostate's pressure
and volume are defined once

00:02:00.110 --> 00:02:03.770
we're at the new equilibrium,
so we're right here.

00:02:03.770 --> 00:02:05.690
So my question in the
last video was,

00:02:05.690 --> 00:02:06.780
how did we get here?

00:02:06.780 --> 00:02:09.900
Is there any way to have defined
a path to get from our

00:02:09.900 --> 00:02:13.970
first state-- where pressure and
volume were well defined,

00:02:13.970 --> 00:02:17.520
because the system wasn't
thermodynamic equilibrium-- to

00:02:17.520 --> 00:02:19.330
get to our second state?

00:02:19.330 --> 00:02:21.400
And the answer was no.

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Because between this state
and this state

00:02:24.460 --> 00:02:25.790
all hell broke loose.

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I had different temperatures
at different

00:02:27.960 --> 00:02:31.110
points in the system.

00:02:31.110 --> 00:02:33.580
I could have had a different
pressure here

00:02:33.580 --> 00:02:35.030
than I had up here.

00:02:35.030 --> 00:02:36.770
The volume might have
been fluctuating

00:02:36.770 --> 00:02:38.080
from moment to moment.

00:02:38.080 --> 00:02:40.220
So when you're outside of
equilibrium-- and I had

00:02:40.220 --> 00:02:44.310
written it down over there--
you cannot define, or you

00:02:44.310 --> 00:02:48.160
can't say that those macro
variables are well defined.

00:02:48.160 --> 00:02:52.710
So there was no path that you
could say how we got from--

00:02:52.710 --> 00:02:57.510
erase this-- how we got from
state 1 to state 2.

00:02:57.510 --> 00:02:59.470
You could just say, OK, we
were in some type of

00:02:59.470 --> 00:03:00.380
equilibrium.

00:03:00.380 --> 00:03:01.840
So we were in state 1.

00:03:01.840 --> 00:03:03.860
Then I blew away
half the rock.

00:03:03.860 --> 00:03:06.430
The pressure went down,
the volume went up.

00:03:06.430 --> 00:03:08.800
The temperature also
probably went down.

00:03:08.800 --> 00:03:11.100
And so I ended up in this other
state once I reached

00:03:11.100 --> 00:03:12.440
equilibrium.

00:03:12.440 --> 00:03:15.290
And that's all fair and good,
but wouldn't it have been nice

00:03:15.290 --> 00:03:16.780
if there was some way?

00:03:16.780 --> 00:03:18.400
If we could have said, look,
you know, there's some way

00:03:18.400 --> 00:03:21.750
that we got from this
point to this point?

00:03:21.750 --> 00:03:25.080
If we could perform my little
rock experiment in a slightly

00:03:25.080 --> 00:03:27.850
different manner, so that all
this hell didn't break loose,

00:03:27.850 --> 00:03:32.600
so that maybe at every point in
between my macro variables

00:03:32.600 --> 00:03:33.940
are actually defined?

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So how could I do that?

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Remember, I said that the
macro variables, the

00:03:37.190 --> 00:03:40.030
macrostates, whether it's
pressure, temperature, volume,

00:03:40.030 --> 00:03:42.740
and there are others, but I said
these are only defined

00:03:42.740 --> 00:03:46.090
when we are in a thermodynamic
equilibrium.

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And that just means that
things have reached a

00:03:48.940 --> 00:03:49.830
stability point.

00:03:49.830 --> 00:03:52.510
That, for example, the
temperature is consistent

00:03:52.510 --> 00:03:53.440
throughout the system.

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If it's not consistent
throughout the system, I

00:03:55.690 --> 00:03:56.720
shouldn't be talking about it.

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If the temperature is different
here than it is up

00:03:59.840 --> 00:04:01.750
here, I shouldn't say that
the temperature of

00:04:01.750 --> 00:04:02.630
the system is x.

00:04:02.630 --> 00:04:03.850
It's different at different
points.

00:04:03.850 --> 00:04:06.010
I really can't make a
well-defined statement about

00:04:06.010 --> 00:04:10.890
temperature, similar for
pressure or for volume,

00:04:10.890 --> 00:04:12.650
because the volume is
also fluctuating.

00:04:12.650 --> 00:04:15.830
But what if I perform that
same experiment?

00:04:15.830 --> 00:04:18.279
That same process,
I should call it.

00:04:18.279 --> 00:04:20.740
Let me draw it again.

00:04:20.740 --> 00:04:23.660
So I have my canister.

00:04:23.660 --> 00:04:27.250
And instead of starting with a
rock, just one big rock-- let

00:04:27.250 --> 00:04:31.830
me draw, this is my piston right
here, at the top of the

00:04:31.830 --> 00:04:33.850
movable ceiling of
the cylinder.

00:04:33.850 --> 00:04:37.960
And I have some gas
inside of it.

00:04:37.960 --> 00:04:42.290
Instead of having just one big
rock like I had over here, how

00:04:42.290 --> 00:04:44.990
about I start with an equal
weight of rock?

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But let's say I have a bunch of
small pebbles that add up

00:04:48.660 --> 00:04:49.600
to that same rock.

00:04:49.600 --> 00:04:54.540
So just a bunch of, well, you
know, just a pile of pebbles.

00:04:54.540 --> 00:04:55.435
You know, maybe they're sand.

00:04:55.435 --> 00:05:01.520
They're super duper small.

00:05:01.520 --> 00:05:05.410
Instead of just blowing away
half of the sand all at once,

00:05:05.410 --> 00:05:08.480
like I did with that rock over
there, and immediately jumping

00:05:08.480 --> 00:05:11.230
to that state and throwing the
whole system into this

00:05:11.230 --> 00:05:14.340
undefined state of
non-equilibrium.

00:05:14.340 --> 00:05:16.950
Instead of doing that, let me
just do things very slowly and

00:05:16.950 --> 00:05:17.670
very gently.

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Let me just take out one grain
of sand at a time.

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So if I just take out
one grain of sand.

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And so I took out an

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infinitesimal amount of weight.

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So what's going to happen?

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Well this piston's going to
move up a little bit.

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And let me draw that.

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So let me copy and paste it.

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So I just took out one
little piece of sand.

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The force pushing down will
be a little bit less.

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The pressure pushing down
will be a little less.

00:05:49.770 --> 00:05:54.740
And so my piston-- let me see
if I can draw this-- it will

00:05:54.740 --> 00:05:59.610
have moved up-- let me erase
it-- it will have moved up a

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very infinitesimal--
infinitesimal means an

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infinitely small amount-- it
would have moved an infinitely

00:06:08.720 --> 00:06:10.120
small amount of time.

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And so you wouldn't have thrown
that system into this,

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you know, havoc that I
did this last time.

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Of course, we haven't moved
all the way here yet.

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But what we have done is, we
would have moved from that

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point maybe to this other point
right here that's just a

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little bit closer to there.

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I've just removed a little
bit of the weight.

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So my pressure went down
just a little bit.

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And my volume went up
a just a little bit.

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Temperature probably
went down.

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And the key here is I'm trying
to do it in such small

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increments that as I do it, my
system is pretty much super

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close to equilibrium.

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I'm just doing it just slow
enough that at every step it

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achieves equilibrium
almost immediately.

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Or it's almost in equilibrium
the whole time I'm doing it.

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And then I do it again,
and do it again.

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And I'll just draw my drawings
a little less neat, just for

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the sake of time.

00:07:02.440 --> 00:07:06.620
Let's say I remove another
little dot of sand that's

00:07:06.620 --> 00:07:09.380
infinitely small mass.

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And now my little piston will
move just a little bit higher.

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And I have, remember I have one
less sand up here than I

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had over here.

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And then my volume in my gas
increases a little bit.

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My pressure goes down
a little bit.

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And I've moved to
this point here.

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What I'm doing here is I'm
setting up what's called a

00:07:30.260 --> 00:07:32.120
quasi-static process.

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And the reason why it's
called that is

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because it's almost static.

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It's almost in equilibrium
the whole time.

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Every time I move a grain of
sand I'm just moving a little

00:07:48.280 --> 00:07:49.920
bit closer.

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And obviously even a grain of
sand, the reality is if I were

00:07:52.820 --> 00:07:55.220
to do this in real life, even
a grain of sand on a small

00:07:55.220 --> 00:07:56.530
scale is going to reek
a little bit

00:07:56.530 --> 00:07:58.040
of havoc on my system.

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This piston is going to
go up a little bit.

00:08:00.820 --> 00:08:03.525
So say, let me just do even a
smaller grain of sand, and do

00:08:03.525 --> 00:08:05.400
it even a little bit slower
so that I'm always in

00:08:05.400 --> 00:08:06.270
equilibrium.

00:08:06.270 --> 00:08:08.440
So you can imagine this is kind
of a theoretical thing.

00:08:08.440 --> 00:08:11.840
If I did an infinitely small
grains of sand, and did it

00:08:11.840 --> 00:08:15.000
just slow enough so that it's
just gently moved from this

00:08:15.000 --> 00:08:16.200
point to this point.

00:08:16.200 --> 00:08:19.210
But we like to think of it
theoretically, because it

00:08:19.210 --> 00:08:21.390
allows us to describe a path.

00:08:21.390 --> 00:08:23.960
Because remember, why am I
being so careful here?

00:08:23.960 --> 00:08:26.890
Why am I so careful to make
sure that the state, the

00:08:26.890 --> 00:08:29.790
system is in equilibrium the
whole time when I get from

00:08:29.790 --> 00:08:31.150
there to there?

00:08:31.150 --> 00:08:35.700
Because our macrostates, our
macro variables like pressure,

00:08:35.700 --> 00:08:38.049
volume, and temperature, our
only defined when we're in

00:08:38.049 --> 00:08:39.000
equilibrium.

00:08:39.000 --> 00:08:42.659
So if I do this process super
slowly, in super small

00:08:42.659 --> 00:08:46.200
increments, it allows me to keep
my pressure and volume

00:08:46.200 --> 00:08:49.270
and actually my temperature
of macrostates at

00:08:49.270 --> 00:08:50.070
any point in time.

00:08:50.070 --> 00:08:51.930
So I could actually
plot a path.

00:08:51.930 --> 00:08:53.860
So if I keep doing it small,
small, small, I could actually

00:08:53.860 --> 00:08:59.280
plot a path to say, how did I
get from state 1 to state 2 on

00:08:59.280 --> 00:09:01.960
this on this PV diagram.

00:09:01.960 --> 00:09:05.270
And you might say, hey, you
know, Sal, this is all-- And

00:09:05.270 --> 00:09:06.580
I'll take a little
step back here.

00:09:06.580 --> 00:09:08.480
I always found this
really confusing.

00:09:08.480 --> 00:09:13.210
You know, you'll see a lot of
talk in thermodynamic circles,

00:09:13.210 --> 00:09:15.920
or even in your book about--
it has to be a quasi-static

00:09:15.920 --> 00:09:18.540
process, and I always used to
wonder, why are people going

00:09:18.540 --> 00:09:20.870
through these pains to describe
this process where

00:09:20.870 --> 00:09:22.340
you're removing sand
after sand?

00:09:22.340 --> 00:09:25.070
And the whole point is because
you want to get as close to

00:09:25.070 --> 00:09:27.960
equilibrium the whole time
you're doing it as possible so

00:09:27.960 --> 00:09:31.780
that your pressure and volume
are defined the whole time.

00:09:31.780 --> 00:09:33.970
The reality is, in the real
world you can never get

00:09:33.970 --> 00:09:36.870
something that's continuously
defined, but you can just do

00:09:36.870 --> 00:09:39.095
really, really, really
small increments.

00:09:39.095 --> 00:09:41.950
So that at each small increment
you're at some

00:09:41.950 --> 00:09:42.490
equilibrium.

00:09:42.490 --> 00:09:44.460
And if you're not happy with
that, you can do even smaller

00:09:44.460 --> 00:09:44.870
increments.

00:09:44.870 --> 00:09:47.120
So at some point, at some
limiting point, you do have

00:09:47.120 --> 00:09:51.970
some type of continuous state
change, while you're always in

00:09:51.970 --> 00:09:52.600
equilibrium.

00:09:52.600 --> 00:09:55.040
It's almost an oxymoron, because
you're saying you're

00:09:55.040 --> 00:09:59.690
static, you're saying that
you're in equilibrium the

00:09:59.690 --> 00:10:02.640
whole time, but clearly you're
also changing the whole time.

00:10:02.640 --> 00:10:05.990
You keep removing little
pieces of sand.

00:10:05.990 --> 00:10:07.770
But you're moving them just
slowly enough that all that

00:10:07.770 --> 00:10:11.130
crazy up and down motion, and
all of the flux, and all of

00:10:11.130 --> 00:10:13.730
the weird temperature changes
don't happen.

00:10:13.730 --> 00:10:16.450
And it just, you know, just
that it slowly, slowly,

00:10:16.450 --> 00:10:18.950
slowly creeps up.

00:10:18.950 --> 00:10:21.180
The reason why I'm even going
through this exercise is

00:10:21.180 --> 00:10:24.250
because it's key when we start
talking about thermodynamics

00:10:24.250 --> 00:10:27.410
and these PV diagrams, and
we'll start talking about

00:10:27.410 --> 00:10:30.980
carnot engines and all of that,
that we be able to at

00:10:30.980 --> 00:10:34.250
least theoretically describe the
path that we take on this

00:10:34.250 --> 00:10:35.280
PV diagram.

00:10:35.280 --> 00:10:37.670
And we wouldn't have been able
to do that if we can't assume

00:10:37.670 --> 00:10:41.210
that we're dealing with a
quasi-static process.

00:10:41.210 --> 00:10:45.230
Now there's another term that
you'll hear in thermodynamic

00:10:45.230 --> 00:10:48.240
circles that really, I mean, to
me it really, I don't know,

00:10:48.240 --> 00:10:50.830
I had trouble comprehending it
the first time I heard it,

00:10:50.830 --> 00:10:52.080
called reversible.

00:10:54.430 --> 00:10:57.820
And sometimes these terms
quasi-static and reversible

00:10:57.820 --> 00:11:00.275
are used interchangeably, but
there is a difference.

00:11:03.190 --> 00:11:06.550
Reversible processes are
quasi-static, and most

00:11:06.550 --> 00:11:08.620
quasi-static processes are
reversible, but there are a

00:11:08.620 --> 00:11:10.150
few special cases that aren't.

00:11:10.150 --> 00:11:12.660
But the idea of a reversible
process is something that

00:11:12.660 --> 00:11:14.620
happens so slowly.

00:11:14.620 --> 00:11:17.420
So in this example I took off
a grain of sand and I got to

00:11:17.420 --> 00:11:20.950
the state, but if I assume that
no friction when, you

00:11:20.950 --> 00:11:23.470
know, when this piston moved up
a little bit, in the real

00:11:23.470 --> 00:11:27.130
world, let's say if this piston
was metal, when this

00:11:27.130 --> 00:11:29.660
rubs against the canister,
there'd be a little bit of

00:11:29.660 --> 00:11:32.170
friction generated and a little
bit of energy would be

00:11:32.170 --> 00:11:33.930
dissipated as friction
or heat.

00:11:33.930 --> 00:11:36.270
But in a reversible process,
we're assuming that, look,

00:11:36.270 --> 00:11:37.520
this is frictionless.

00:11:37.520 --> 00:11:40.500
When anything happens in the
system, when we go from this

00:11:40.500 --> 00:11:42.300
state right here-- let's
say this the state a,

00:11:42.300 --> 00:11:43.080
this is state b.

00:11:43.080 --> 00:11:46.000
So this is state a,
this is state b.

00:11:46.000 --> 00:11:48.710
When we go from this state
to this state, one, we're

00:11:48.710 --> 00:11:51.760
infinitesimally close to
equilibrium the whole time, so

00:11:51.760 --> 00:11:54.910
all of our macrostates
are well defined.

00:11:54.910 --> 00:11:57.670
And even more, when we move from
one state to the other,

00:11:57.670 --> 00:11:59.840
there's no loss or dissipation
of energy.

00:11:59.840 --> 00:12:03.570
So those are two important
characteristics.

00:12:03.570 --> 00:12:06.850
One, infinitely close to
equilibrium at all times, and

00:12:06.850 --> 00:12:07.870
no loss of energy.

00:12:07.870 --> 00:12:10.540
And the reason why that matters
for a reversible

00:12:10.540 --> 00:12:13.120
process is because if we wanted,
if we were sitting in

00:12:13.120 --> 00:12:18.790
state b, we could just add
another grain of sand back in,

00:12:18.790 --> 00:12:22.660
push down this piston infinitely
slowly, at an

00:12:22.660 --> 00:12:26.180
infinitely small increment,
and get back to state a.

00:12:26.180 --> 00:12:27.340
So that's why it's called
reversible.

00:12:27.340 --> 00:12:32.380
You could be at this point right
here, and take out a

00:12:32.380 --> 00:12:34.740
little bit of sand, and get
to this point right here.

00:12:34.740 --> 00:12:37.440
But if you want, since no energy
was lost, you could add

00:12:37.440 --> 00:12:40.970
a little bit of sand, and get
back to this point right here.

00:12:40.970 --> 00:12:44.290
Now the reality in the real
world is, there is no such

00:12:44.290 --> 00:12:46.540
thing as a perfectly
reversible process.

00:12:46.540 --> 00:12:48.790
There will always be, whenever
you do anything, there will

00:12:48.790 --> 00:12:54.090
always be some energy or heat
lost to the process.

00:12:54.090 --> 00:12:58.400
In the real world, if I moved
down here, if I tried to put

00:12:58.400 --> 00:13:00.750
the sand back I would lose some
energy and probably get

00:13:00.750 --> 00:13:02.360
to a little slightly
different point.

00:13:02.360 --> 00:13:03.645
But you don't have to
worry about that.

00:13:03.645 --> 00:13:07.900
The important takeaway from
this video is that, in the

00:13:07.900 --> 00:13:11.400
situation I described there,
there was no intermediate

00:13:11.400 --> 00:13:14.480
macrostate variables, because
our system was in flux, it

00:13:14.480 --> 00:13:16.000
wasn't in equilibrium.

00:13:16.000 --> 00:13:18.200
So if we wanted to get
intermediate states, we just

00:13:18.200 --> 00:13:21.600
have to essentially do
this process slower.

00:13:21.600 --> 00:13:24.060
And so slow, I mean, it
theoretically would take you

00:13:24.060 --> 00:13:26.090
forever, so we can only
approximate it.

00:13:26.090 --> 00:13:29.100
But the sand gives you an idea
of what we're talking about.

00:13:29.100 --> 00:13:32.000
And if we did it slowly with
these infinitesimally small

00:13:32.000 --> 00:13:35.700
particles of sand, then we can
define the state at every

00:13:35.700 --> 00:13:39.480
point along the process.

00:13:39.480 --> 00:13:41.490
And that's why we call it
quasi-static, because at any

00:13:41.490 --> 00:13:43.000
point it's almost static.

00:13:43.000 --> 00:13:44.480
It's almost in equilibrium.

00:13:44.480 --> 00:13:48.730
So our pressures, volumes, and
temperatures can be defined.

00:13:48.730 --> 00:13:51.890
And if we add to that the notion
that we haven't lost

00:13:51.890 --> 00:13:55.510
any heat when we're going in one
direction or another, we

00:13:55.510 --> 00:13:57.420
could say it's reversible,
because if we took a piece of

00:13:57.420 --> 00:14:02.050
sand away, we can always add
a little bit of sand next.

00:14:02.050 --> 00:14:04.280
Now, actually, with that said,
let me give you the one

00:14:04.280 --> 00:14:07.440
example of maybe a
quasi-static-- no, actually

00:14:07.440 --> 00:14:08.910
I'll save that for
future video.

00:14:08.910 --> 00:14:11.370
Anyway, hopefully you understand
that these are two

00:14:11.370 --> 00:14:14.560
concepts that used to really
confuse me, and hopefully this

00:14:14.560 --> 00:14:15.670
clears it up a little bit.

00:14:15.670 --> 00:14:18.180
And I think more than what it
is, I think the first time I

00:14:18.180 --> 00:14:20.210
read about them I'm like, OK,
well what's the big deal?

00:14:20.210 --> 00:14:25.130
The big deal is, it allows you
to define your macrostates for

00:14:25.130 --> 00:14:27.770
every state in between
these two states

00:14:27.770 --> 00:14:28.540
that you care about.

00:14:28.540 --> 00:14:30.880
When you just did it as
a regular kind of

00:14:30.880 --> 00:14:34.040
non-quasi-static process,
in between you

00:14:34.040 --> 00:14:35.980
don't know what happened.


Macrostates and microstates

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.860
SAL: I've done a bunch of videos
where I use words like

00:00:03.860 --> 00:00:09.150
pressure and-- let me write
these down-- pressure and

00:00:09.150 --> 00:00:15.950
temperature and volume.

00:00:15.950 --> 00:00:17.840
And I've done them in the
chemistry and physics

00:00:17.840 --> 00:00:20.780
playlist. Especially the physics
playlist, but even in

00:00:20.780 --> 00:00:27.220
the chemistry playlist, I also
use words like kinetic energy.

00:00:27.220 --> 00:00:29.120
I'll just write e for energy.

00:00:29.120 --> 00:00:35.280
Or I use force and velocity.

00:00:35.280 --> 00:00:38.940
And you know, a whole bunch of
other types of, I guess,

00:00:38.940 --> 00:00:42.300
properties of things, for
better or for worse.

00:00:42.300 --> 00:00:43.910
And in this video what I want
to do is I want to make a

00:00:43.910 --> 00:00:44.510
distinction.

00:00:44.510 --> 00:00:47.800
Because it becomes important
when we start getting a little

00:00:47.800 --> 00:00:50.660
bit more precise, especially
when we get more precise in

00:00:50.660 --> 00:00:54.110
thermodynamics, or, I guess,
you know, the study of how

00:00:54.110 --> 00:00:55.740
heat moves around.

00:00:55.740 --> 00:00:57.320
So these properties right
here, these are

00:00:57.320 --> 00:00:58.655
properties of a system.

00:00:58.655 --> 00:01:02.085
Or we could call them
macrostates of a system.

00:01:09.870 --> 00:01:12.340
And these could be
macrostates.

00:01:12.340 --> 00:01:15.170
So for example, let me make it
clear, when I call a system,

00:01:15.170 --> 00:01:23.750
if I have some balloon like
this, and it has a little tie

00:01:23.750 --> 00:01:26.410
there and, you know, maybe
it has a string.

00:01:26.410 --> 00:01:29.350
This has these macrostates
associated with it.

00:01:29.350 --> 00:01:32.060
There is some pressure
in that balloon.

00:01:32.060 --> 00:01:34.020
Remember that's force
per area.

00:01:34.020 --> 00:01:39.030
There is some temperature
for that balloon.

00:01:39.030 --> 00:01:41.850
And there's some volume to
the balloon, obviously.

00:01:41.850 --> 00:01:45.170
But all of these, these help
us relate what's going on

00:01:45.170 --> 00:01:48.200
inside that balloon, or what
that balloon is doing in kind

00:01:48.200 --> 00:01:50.560
of an every day reality.

00:01:50.560 --> 00:01:54.020
Before people even knew about
what an atom was, or maybe

00:01:54.020 --> 00:01:55.830
they thought that there might
be such an atom but they had

00:01:55.830 --> 00:01:57.910
never proved it, they were
dealing with these

00:01:57.910 --> 00:01:58.820
macrostates.

00:01:58.820 --> 00:02:01.070
They could measure pressure,
they could measure

00:02:01.070 --> 00:02:03.210
temperature, they could
measure volume.

00:02:03.210 --> 00:02:09.030
Now we know that that pressure
is due to things like, you

00:02:09.030 --> 00:02:12.020
have a bunch of atoms
bumping around.

00:02:12.020 --> 00:02:14.870
And let's say that this is a
gas-- it's a balloon- it's

00:02:14.870 --> 00:02:16.170
going to be a gas.

00:02:16.170 --> 00:02:18.910
And we know that the pressure is
actually caused-- and I've

00:02:18.910 --> 00:02:21.320
done several, I think I did
the same video in both the

00:02:21.320 --> 00:02:23.750
chemistry and the physics
playlist. I did them a year

00:02:23.750 --> 00:02:27.670
apart, so you can see if my
thinking has evolved at all.

00:02:27.670 --> 00:02:30.790
But we know that the pressure's
really due by the

00:02:30.790 --> 00:02:34.280
bumps of these particles as they
bump into the walls and

00:02:34.280 --> 00:02:35.740
the side of the balloon.

00:02:35.740 --> 00:02:38.570
And we have so many particles
at any given point of time,

00:02:38.570 --> 00:02:41.530
some of them are bumping into
the wall the balloon, and

00:02:41.530 --> 00:02:44.280
that's what's essentially
keeping the balloon pushed

00:02:44.280 --> 00:02:47.260
outward, giving it its pressure
and its volume.

00:02:47.260 --> 00:02:49.790
We've talked about temperature,
as essentially

00:02:49.790 --> 00:02:54.290
the average kinetic energy of
these-- which is a function of

00:02:54.290 --> 00:02:56.430
these particles, which could
be either the molecules of

00:02:56.430 --> 00:02:59.650
gas, or if it's an ideal gas,
it could be just the

00:02:59.650 --> 00:03:00.650
atoms of the gas.

00:03:00.650 --> 00:03:06.260
Maybe it's atoms of helium or
neon, or something like that.

00:03:06.260 --> 00:03:10.030
And all of these things, these
describe the microstates.

00:03:10.030 --> 00:03:12.440
So for example, I could describe
what's going on with

00:03:12.440 --> 00:03:12.890
the balloon.

00:03:12.890 --> 00:03:15.510
I could say, hey, you know,
there are-- I could just make

00:03:15.510 --> 00:03:18.500
up some numbers.

00:03:18.500 --> 00:03:29.040
The pressure is five newtons
per meters squared, or some

00:03:29.040 --> 00:03:29.760
number of pascals.

00:03:29.760 --> 00:03:32.320
The units aren't what's
important.

00:03:32.320 --> 00:03:33.490
In this video I really
just want to make the

00:03:33.490 --> 00:03:36.430
differentiation between these
two ways of describing

00:03:36.430 --> 00:03:37.150
what's going on.

00:03:37.150 --> 00:03:42.640
I could say the temperature
is 300 kelvin.

00:03:42.640 --> 00:03:47.230
I could say that the volume
is, I don't know,

00:03:47.230 --> 00:03:48.660
maybe it's one liter.

00:03:48.660 --> 00:03:51.020
And I've described a system,
but I've described in on a

00:03:51.020 --> 00:03:52.520
macro level.

00:03:52.520 --> 00:03:55.400
Now I could get a lot more
precise, especially now that

00:03:55.400 --> 00:03:58.600
we know that things like atoms
and molecules exist. What I

00:03:58.600 --> 00:04:01.690
could do, is I could essentially
label every one of

00:04:01.690 --> 00:04:06.930
these molecules, or let's say
atoms, in the gas that's

00:04:06.930 --> 00:04:07.810
contained in the balloon.

00:04:07.810 --> 00:04:11.570
And I could say, at exactly this
moment in time, I could

00:04:11.570 --> 00:04:21.399
say at time equals 0, atom 1
has-- its momentum is equal to

00:04:21.399 --> 00:04:27.760
x, and its position, in
three-dimensional coordinates,

00:04:27.760 --> 00:04:31.480
is x, y, and z.

00:04:31.480 --> 00:04:35.910
And then I could say, atom
number 2-- its momentum-- I'm

00:04:35.910 --> 00:04:39.700
just using rho for momentum--
it's equal to y.

00:04:39.700 --> 00:04:44.050
And its position is a, b, c.

00:04:44.050 --> 00:04:49.270
And I could list every atom
in this molecule.

00:04:49.270 --> 00:04:52.300
Obviously we're dealing with a
huge number of atoms, on the

00:04:52.300 --> 00:04:54.320
order of 10 to the
20 something.

00:04:54.320 --> 00:04:57.580
So it's a massive list I would
have to give you, but I could

00:04:57.580 --> 00:05:02.300
literally give you the state of
every atom in this balloon.

00:05:02.300 --> 00:05:04.280
And then if I did that, I
would be giving you the

00:05:04.280 --> 00:05:05.790
microstates.

00:05:05.790 --> 00:05:08.550
Or I would give you a specific
microstate of the

00:05:08.550 --> 00:05:10.280
balloon at this time.

00:05:10.280 --> 00:05:14.300
Now when a system-- and I'm
going to introduce a word

00:05:14.300 --> 00:05:16.020
here, because this word is
important, especially as we

00:05:16.020 --> 00:05:19.400
go-- is in thermodynamic
equilibrium.

00:05:19.400 --> 00:05:20.680
So let me write that down.

00:05:23.300 --> 00:05:24.550
Equilibrium.

00:05:27.190 --> 00:05:29.330
We learned about equilibrium
from the

00:05:29.330 --> 00:05:30.390
chemistry point of view.

00:05:30.390 --> 00:05:33.680
And that tells you, that the
amount of something going into

00:05:33.680 --> 00:05:35.620
forward reaction is equivalent
to the amount going in the

00:05:35.620 --> 00:05:36.890
reverse reaction.

00:05:36.890 --> 00:05:39.300
And when we talk about
macrostates, thermodynamic

00:05:39.300 --> 00:05:40.820
equilibrium essentially
says that the

00:05:40.820 --> 00:05:42.300
macrostate is defined.

00:05:42.300 --> 00:05:43.580
That they're not changing.

00:05:43.580 --> 00:05:47.260
If this balloon is in
equilibrium, at time 1 its

00:05:47.260 --> 00:05:50.240
pressure, temperature, and
volume will be these things.

00:05:50.240 --> 00:05:52.690
And if we look at it a second
later, its pressure,

00:05:52.690 --> 00:05:55.340
temperature, and volume will
also be these things.

00:05:55.340 --> 00:05:56.870
It's in equilibrium.

00:05:56.870 --> 00:05:59.550
None of the macrostates
have changed.

00:05:59.550 --> 00:06:01.620
And actually, I'll talk about
in a second, in order for

00:06:01.620 --> 00:06:05.180
these macrostates to even be
defined, to be well defined,

00:06:05.180 --> 00:06:06.270
you have to be in equilibrium.

00:06:06.270 --> 00:06:07.970
I'll talk about that
in a second.

00:06:07.970 --> 00:06:12.110
Now, at second number, at time
equals 0, you might have this

00:06:12.110 --> 00:06:14.750
whole set of-- I went and
I listed 10 to the

00:06:14.750 --> 00:06:17.680
20th-something microstates of
all the different atoms in

00:06:17.680 --> 00:06:18.900
this molecule.

00:06:18.900 --> 00:06:22.120
But then if I look at these
gases a second later, I'm

00:06:22.120 --> 00:06:25.280
going to have a completely
different microstate right?

00:06:25.280 --> 00:06:27.210
Because all of these guys are
going to have bumped into each

00:06:27.210 --> 00:06:30.640
other, and given each other
their momentum.

00:06:30.640 --> 00:06:32.770
And all sorts of crazy things
could have happened in a

00:06:32.770 --> 00:06:34.530
second here, so I would have
a completely different

00:06:34.530 --> 00:06:35.510
microstate.

00:06:35.510 --> 00:06:38.520
So even though we're at
thermodynamic equilibrium, and

00:06:38.520 --> 00:06:41.320
our macrostate stayed the same,
our microstates are

00:06:41.320 --> 00:06:43.770
changing every gazillionth
of a second.

00:06:43.770 --> 00:06:45.440
They're constantly changing.

00:06:45.440 --> 00:06:48.590
And that's why, for the most
part, in thermodynamic, we

00:06:48.590 --> 00:06:52.370
tend to deal with these
macrostates.

00:06:52.370 --> 00:06:54.430
And actually most of
thermodynamics, or at least

00:06:54.430 --> 00:06:57.510
most of what you'll learn in
a first-year chemistry or

00:06:57.510 --> 00:07:02.210
physics course, it was devised
or it was thought about well

00:07:02.210 --> 00:07:05.280
before people even had a sense
of what was going on at the

00:07:05.280 --> 00:07:06.410
macro level.

00:07:06.410 --> 00:07:09.350
That's often a very important
thing to think about.

00:07:09.350 --> 00:07:12.430
And we'll go into concepts
like entropy and internal

00:07:12.430 --> 00:07:13.930
energy, and things like that.

00:07:13.930 --> 00:07:17.040
And you can rack your brain, how
does it relate to atoms?

00:07:17.040 --> 00:07:20.320
And we will relate them to
atoms and molecules.

00:07:20.320 --> 00:07:22.980
But it's useful to think that
the people who first came up

00:07:22.980 --> 00:07:25.680
with these concepts came up
with them not really being

00:07:25.680 --> 00:07:27.890
sure of what was going on
at the micro level.

00:07:27.890 --> 00:07:30.860
They were just measuring
everything at the macro level.

00:07:30.860 --> 00:07:35.490
Now I want to go back to this
idea here, of equilibrium.

00:07:39.000 --> 00:07:42.370
Because in order for these
macrostates to be defined, the

00:07:42.370 --> 00:07:44.140
system has to be
in equilibrium.

00:07:44.140 --> 00:07:47.800
And let me explain
what that means.

00:07:47.800 --> 00:07:52.220
If I were to take a cylinder.

00:07:52.220 --> 00:07:55.000
And we will be using this
cylinder a lot, so it's good

00:07:55.000 --> 00:07:56.685
to get used to this cylinder.

00:07:59.810 --> 00:08:02.210
And it's got a piston in it.

00:08:02.210 --> 00:08:04.840
And that's just, it's kind of
the roof of the cylinder can

00:08:04.840 --> 00:08:07.830
move up and down.

00:08:07.830 --> 00:08:09.510
This is the roof of
the cylinder.

00:08:09.510 --> 00:08:11.780
The cylinder's bigger, but let's
say this is a, kind of a

00:08:11.780 --> 00:08:13.850
roof of the cylinder.

00:08:13.850 --> 00:08:15.550
And we can move this
up and down.

00:08:15.550 --> 00:08:17.420
And essentially we'll just be
changing the volume of the

00:08:17.420 --> 00:08:18.260
cylinder, right?

00:08:18.260 --> 00:08:19.150
I could have drawn
it this way.

00:08:19.150 --> 00:08:20.400
I could have drawn it
like a cylinder.

00:08:23.740 --> 00:08:27.950
I could have drawn it like this,
and then I could have

00:08:27.950 --> 00:08:29.490
drawn the piston like this.

00:08:29.490 --> 00:08:31.210
So there's some depth here
that I'm not showing.

00:08:31.210 --> 00:08:34.220
We're just looking at the
cylinder front on.

00:08:34.220 --> 00:08:37.130
And so, at any point in time,
let's say the gas is between

00:08:37.130 --> 00:08:39.099
the cylinder and the floor
of our container.

00:08:42.120 --> 00:08:45.140
You know, we have a bunch of
molecules of gas here, a huge

00:08:45.140 --> 00:08:47.250
number of molecules.

00:08:47.250 --> 00:08:52.200
And let's say that we have
a rock on the cylinder.

00:08:52.200 --> 00:08:54.170
We're doing this in space
so everything above

00:08:54.170 --> 00:08:56.480
the piston is a vacuum.

00:08:56.480 --> 00:08:59.070
Actually just let me erase
everything above.

00:08:59.070 --> 00:09:02.110
Let me just erase this stuff,
just so you see.

00:09:02.110 --> 00:09:05.410
We're doing this in space and
we're doing it in a vacuum.

00:09:05.410 --> 00:09:06.990
Just let me write that down.

00:09:06.990 --> 00:09:13.500
So all of this stuff up here is
a vacuum, which essentially

00:09:13.500 --> 00:09:14.940
says there's nothing there.

00:09:14.940 --> 00:09:17.380
There's no pressure from here,
there's no particles here,

00:09:17.380 --> 00:09:18.410
just empty space.

00:09:18.410 --> 00:09:21.410
And in order to keep this-- we
know already, we've studied it

00:09:21.410 --> 00:09:24.710
multiple times, that this gas is
generating, you know things

00:09:24.710 --> 00:09:28.800
are bumping into the wall,
the floor of this

00:09:28.800 --> 00:09:30.240
piston all the time.

00:09:30.240 --> 00:09:32.390
They're bumping into
everything, right?

00:09:32.390 --> 00:09:33.850
We know that's continuously
happening.

00:09:33.850 --> 00:09:36.970
So we would apply some pressure
to offset the

00:09:36.970 --> 00:09:38.350
pressure being generated
by the gas.

00:09:38.350 --> 00:09:40.110
Otherwise the piston
would just expand.

00:09:40.110 --> 00:09:42.380
It would just move up and the
whole gas would expand.

00:09:42.380 --> 00:09:47.920
So let's just say we stick a
big rock or a big weight on

00:09:47.920 --> 00:09:51.440
top of-- let me do it in a
different color-- We put a big

00:09:51.440 --> 00:09:58.580
weight on top of this piston,
where the force-- completely

00:09:58.580 --> 00:10:02.100
offsets the force being
applied by the gas.

00:10:02.100 --> 00:10:05.370
And obviously this is some force
over some area-- right,

00:10:05.370 --> 00:10:08.240
the area of the piston-- over
some areas so that we could

00:10:08.240 --> 00:10:09.070
figure out its pressure.

00:10:09.070 --> 00:10:11.020
And that pressure will
completely offset the pressure

00:10:11.020 --> 00:10:11.940
of the gas.

00:10:11.940 --> 00:10:13.720
But the pressure of the gas,
just as a reminder, is going

00:10:13.720 --> 00:10:14.370
in every direction.

00:10:14.370 --> 00:10:17.230
The pressure on this plate is
the same as the pressure on

00:10:17.230 --> 00:10:22.250
that side, or on that side, or
on the bottom of the container

00:10:22.250 --> 00:10:23.510
that we're dealing with.

00:10:23.510 --> 00:10:27.450
Now let's say that we were to
just evaporate this-- well

00:10:27.450 --> 00:10:29.330
let's not say that we
evaporate the rock.

00:10:29.330 --> 00:10:32.220
Let's say that we just evaporate
half of the rock

00:10:32.220 --> 00:10:33.560
immediately.

00:10:33.560 --> 00:10:36.630
So all of a sudden our weight
that's being pushed down, or

00:10:36.630 --> 00:10:38.700
the force that's being pushed
down just goes to half

00:10:38.700 --> 00:10:41.270
immediately.

00:10:41.270 --> 00:10:43.630
Let me draw that.

00:10:43.630 --> 00:10:46.850
So I have-- maybe I would be
better off just cut and

00:10:46.850 --> 00:10:48.980
pasting this right here.

00:10:48.980 --> 00:10:52.610
So if I copy and paste it.

00:10:52.610 --> 00:10:56.392
So now I'm going to evaporate
half of that rock magically.

00:10:56.392 --> 00:10:59.980
So let me take my eraser tool.

00:10:59.980 --> 00:11:03.190
And I just evaporate
half of it.

00:11:03.190 --> 00:11:05.180
And now what's going
to happen?

00:11:05.180 --> 00:11:11.110
Well, this piston is now
applying half the force.

00:11:11.110 --> 00:11:14.150
It can't offset the pressure
due to this gas.

00:11:14.150 --> 00:11:16.430
So this whole thing is going
to be pushed upwards.

00:11:16.430 --> 00:11:19.240
But I did it so fast. I did it
so fast. And you could try it.

00:11:19.240 --> 00:11:21.220
I mean, this would be truth
of a lot of things.

00:11:21.220 --> 00:11:23.300
If you had a weight hanging from
a spring, and you would

00:11:23.300 --> 00:11:26.610
just remove half the weight, it
wouldn't just go very, you

00:11:26.610 --> 00:11:28.940
know, nice and smoothly
to another state.

00:11:28.940 --> 00:11:30.920
What's going to happen is--
and let me see if I can do

00:11:30.920 --> 00:11:34.070
this using their cut and paste
tool-- it'll essentially,

00:11:34.070 --> 00:11:37.660
right when I evaporate half of
it, the gas is going to expand

00:11:37.660 --> 00:11:39.945
a bunch, and then this weight
is going to come back down,

00:11:39.945 --> 00:11:41.590
it's going to spring
and go down.

00:11:41.590 --> 00:11:42.570
So let me do it again.

00:11:42.570 --> 00:11:45.260
It's going to expand, because
that gas is going to push up,

00:11:45.260 --> 00:11:46.750
and then it's going
to come back down.

00:11:46.750 --> 00:11:49.020
And then, it's just going to
oscillate a little bit.

00:11:49.020 --> 00:11:52.330
And then eventually it'll come
back to some stable and maybe

00:11:52.330 --> 00:11:53.490
it'll go back.

00:11:53.490 --> 00:11:56.680
It'll look, like right
about there.

00:11:56.680 --> 00:11:58.380
And let me fill this in.

00:11:58.380 --> 00:12:01.605
This shouldn't be white,
it should be black.

00:12:01.605 --> 00:12:04.605
Let me put some walls on
it, on the container.

00:12:09.100 --> 00:12:11.240
So if we wait long enough,
eventually we'll get to

00:12:11.240 --> 00:12:14.720
another equilibrium state, where
this thing, the piston

00:12:14.720 --> 00:12:17.570
on top isn't, or the ceiling
isn't moving anymore.

00:12:17.570 --> 00:12:21.810
And now the gas has filled
this container.

00:12:21.810 --> 00:12:24.850
Now, at this point in time
we were in equilibrium.

00:12:24.850 --> 00:12:27.520
The pressure throughout
the gas was the same.

00:12:27.520 --> 00:12:29.960
The temperature throughout
the gas was the same.

00:12:29.960 --> 00:12:32.720
The volume was in a
stable situation.

00:12:32.720 --> 00:12:34.860
It wasn't changing from
second to second.

00:12:34.860 --> 00:12:38.050
So because of that, our
macrostates were well defined.

00:12:47.530 --> 00:12:51.750
Now, when we wait long enough,
this thing will get to some

00:12:51.750 --> 00:12:53.600
stability where this
thing stops moving.

00:12:53.600 --> 00:12:56.440
When this thing stops moving
our volume stops changing.

00:12:56.440 --> 00:12:59.760
And hopefully our pressure will
start to become uniform

00:12:59.760 --> 00:13:00.800
throughout the container.

00:13:00.800 --> 00:13:02.290
And our temperature will
become uniform.

00:13:02.290 --> 00:13:05.240
And we'll now be a higher volume
or lower pressure,

00:13:05.240 --> 00:13:07.160
probably a lower temperature if
we assume that there's no

00:13:07.160 --> 00:13:08.870
other heat being added
to the system.

00:13:08.870 --> 00:13:12.580
And then we'll be well
defined again.

00:13:12.580 --> 00:13:14.140
So we could say what the
pressure, and the volume, and

00:13:14.140 --> 00:13:16.070
the temperature's going to be.

00:13:16.070 --> 00:13:18.880
But what about right when
I removed this rock?

00:13:18.880 --> 00:13:22.030
And this thing flew up and it
oscillated, and for a while

00:13:22.030 --> 00:13:24.080
the pressure at the top
was lower than the

00:13:24.080 --> 00:13:24.790
pressure down here.

00:13:24.790 --> 00:13:26.750
Maybe the temperature at the
top was lower than the

00:13:26.750 --> 00:13:27.930
temperature down here.

00:13:27.930 --> 00:13:29.740
The whole thing was in
a state of flux.

00:13:29.740 --> 00:13:31.440
It was not an equilibrium.

00:13:31.440 --> 00:13:34.390
And at that point, when we're--
let me let me draw

00:13:34.390 --> 00:13:37.110
that really-- so you know, when
we were in that state,

00:13:37.110 --> 00:13:39.140
where everything was just
crazy, right when we

00:13:39.140 --> 00:13:40.840
evaporated the rock.

00:13:40.840 --> 00:13:42.560
You know, we have a little
rock up here.

00:13:42.560 --> 00:13:44.490
Everything is going
up and down.

00:13:44.490 --> 00:13:48.320
Maybe the pressure up here
was lower than the

00:13:48.320 --> 00:13:53.820
pressure down here.

00:13:53.820 --> 00:13:55.610
Everything did not have
a chance to reach an

00:13:55.610 --> 00:13:56.560
equilibrium.

00:13:56.560 --> 00:13:58.950
At this state-- and this is
important, especially as we go

00:13:58.950 --> 00:14:03.570
into talking about things like
reversible reactions, and

00:14:03.570 --> 00:14:07.580
reversible processes, and
quasi-static processes.

00:14:07.580 --> 00:14:11.100
At this point in the reaction,
when we just did this, none of

00:14:11.100 --> 00:14:14.030
these macrostates were
well defined.

00:14:14.030 --> 00:14:16.350
You couldn't tell me what the
volume of this system is,

00:14:16.350 --> 00:14:18.720
because it's changing for every
second to second, or

00:14:18.720 --> 00:14:21.000
microsecond to microsecond,
it's fluctuating.

00:14:21.000 --> 00:14:23.110
You couldn't tell me what the
pressure of the system is,

00:14:23.110 --> 00:14:24.950
because it's changing
every second.

00:14:24.950 --> 00:14:27.300
You couldn't tell me what
the temperature is.

00:14:27.300 --> 00:14:31.380
Maybe the temperature could
be something there.

00:14:31.380 --> 00:14:32.880
It could be something there.

00:14:32.880 --> 00:14:34.900
All sorts of crazy things
are happening.

00:14:34.900 --> 00:14:37.500
So when the system is in
a state of flux, your

00:14:37.500 --> 00:14:40.670
macrostates are not
well defined.

00:14:40.670 --> 00:14:42.690
And I really want to hit
that point home.

00:14:42.690 --> 00:14:45.190
So me just draw that
in a diagram.

00:14:45.190 --> 00:14:47.450
Let me draw that in
a PV diagram.

00:14:47.450 --> 00:14:50.340
And we're going to use
these fairly heavily.

00:14:50.340 --> 00:14:53.300
So on my y-axis I'm going
to put pressure.

00:14:53.300 --> 00:14:57.500
In my x-axis I'm going
to put volume.

00:14:57.500 --> 00:15:00.720
So our initial state here, when
we had the rock sitting

00:15:00.720 --> 00:15:03.420
on top of the ceiling, this
movable ceiling or this

00:15:03.420 --> 00:15:07.100
piston, maybe we had
some well-defined

00:15:07.100 --> 00:15:08.090
pressure and volume.

00:15:08.090 --> 00:15:11.420
So my y, this is pressure
and this is volume.

00:15:11.420 --> 00:15:14.260
So this is where
we started off.

00:15:14.260 --> 00:15:15.800
So it was well defined.

00:15:15.800 --> 00:15:17.050
This is state 1.

00:15:17.050 --> 00:15:20.500
Let me label it right there.

00:15:20.500 --> 00:15:25.270
Now when we evaporated half the
rock, we eventually waited

00:15:25.270 --> 00:15:26.580
long enough, and this got
to an equilibrium.

00:15:26.580 --> 00:15:28.560
We got to state 2, and our
pressure volume and out

00:15:28.560 --> 00:15:30.310
temperature was well defined.

00:15:30.310 --> 00:15:32.670
And I'll just put it on
this pressure volume.

00:15:32.670 --> 00:15:35.630
So maybe this is state 2.

00:15:35.630 --> 00:15:37.500
We got down here.

00:15:37.500 --> 00:15:40.920
And just as an aside, I could
maybe put temperature as an

00:15:40.920 --> 00:15:42.690
extra dimension, but temperature
is completely

00:15:42.690 --> 00:15:45.420
determined by pressure and
volume, especially if we're

00:15:45.420 --> 00:15:47.030
dealing with an ideal gas.

00:15:47.030 --> 00:15:50.470
Remember, and we did this in
multiple videos, you have PV

00:15:50.470 --> 00:15:55.300
is equal to nRT.

00:15:55.300 --> 00:15:56.750
These are constants.

00:15:56.750 --> 00:15:58.260
The number of moles
isn't changing.

00:15:58.260 --> 00:16:01.710
This is the universal gas
constant, not changing.

00:16:01.710 --> 00:16:03.180
So if you know P and
V you know T.

00:16:03.180 --> 00:16:05.110
So that's the only two things
we have to plot.

00:16:05.110 --> 00:16:08.460
But I'll talk a lot more about
that in future videos.

00:16:08.460 --> 00:16:11.020
But the important thing to
realize is, I started off at

00:16:11.020 --> 00:16:14.750
this state, where pressure and
volume were well defined.

00:16:14.750 --> 00:16:17.980
I finished in this state, where
pressure and volume were

00:16:17.980 --> 00:16:19.210
well defined.

00:16:19.210 --> 00:16:20.810
But how did I get there?

00:16:20.810 --> 00:16:23.220
And because this reaction I
did, all of a sudden it

00:16:23.220 --> 00:16:25.960
happened super fast, and it was
essentially thrown out of

00:16:25.960 --> 00:16:27.210
equilibrium.

00:16:33.640 --> 00:16:35.650
I don't know how I got here.

00:16:35.650 --> 00:16:38.960
The pressure and volume were
not well defined from going

00:16:38.960 --> 00:16:41.090
from that state to this state.

00:16:41.090 --> 00:16:43.500
Pressure, volume, and
temperature are only well

00:16:43.500 --> 00:16:50.580
defined if every intermediate
step is still almost in

00:16:50.580 --> 00:16:51.190
equilibrium.

00:16:51.190 --> 00:16:52.950
And we'll talk a lot more about
that in the next video.

00:16:52.950 --> 00:16:54.500
But I want to really make
this point home.

00:16:54.500 --> 00:16:57.390
It would be nice if we
could draw some path.

00:16:57.390 --> 00:17:02.240
We could say, we moved from some
pressure and volume to

00:17:02.240 --> 00:17:04.339
some other pressure and volume,
and we moved along a

00:17:04.339 --> 00:17:05.720
well-defined path.

00:17:05.720 --> 00:17:07.410
But we cannot say that.

00:17:07.410 --> 00:17:10.240
Because when we went from there
there, our definitions

00:17:10.240 --> 00:17:11.910
just disappeared for pressure
and volume.

00:17:11.910 --> 00:17:16.839
We cannot define those
macrostates in these

00:17:16.839 --> 00:17:19.000
intermediate non-equilibrium
states.

00:17:19.000 --> 00:17:21.560
Now, just as a little aside,
we could have defined the

00:17:21.560 --> 00:17:22.800
microstates.

00:17:22.800 --> 00:17:24.230
The microstates never change.

00:17:24.230 --> 00:17:27.950
At any given snapshot in time,
I could have listed every

00:17:27.950 --> 00:17:29.470
particle that's in this thing.

00:17:29.470 --> 00:17:31.310
And I could have given you
its kinetic energy.

00:17:31.310 --> 00:17:32.890
I could have given
you its position.

00:17:32.890 --> 00:17:34.780
I could have given
you its momentum.

00:17:34.780 --> 00:17:36.900
And there's no reason why I
couldn't have done that.

00:17:36.900 --> 00:17:39.920
So I could have actually
made a plot of

00:17:39.920 --> 00:17:41.270
one particular particle.

00:17:41.270 --> 00:17:43.980
And I could have said what its
kinetic energy, and over a

00:17:43.980 --> 00:17:46.170
course of time, is at any
given moment in time.

00:17:46.170 --> 00:17:47.110
And this is really important.

00:17:47.110 --> 00:17:50.690
So microstates are always
well defined.

00:17:50.690 --> 00:17:53.330
The microstate is what's exactly
happening to the atom

00:17:53.330 --> 00:17:56.970
in terms of its force and its
velocity and its momentum.

00:17:56.970 --> 00:18:01.800
While macrostates are only
defined, I should say well

00:18:01.800 --> 00:18:05.260
defined, when the system-- in
this case it's the balloon, in

00:18:05.260 --> 00:18:08.060
this case it's this piston on
top of this cylinder, this

00:18:08.060 --> 00:18:10.730
movable ceiling-- the
macrostates are only well

00:18:10.730 --> 00:18:13.700
defined when the system is in
equilibrium, or when you can

00:18:13.700 --> 00:18:16.390
essentially say, the pressure is
x, the pressure is the same

00:18:16.390 --> 00:18:16.920
throughout.

00:18:16.920 --> 00:18:19.950
Or the volume isn't changing
from moment to moment.

00:18:19.950 --> 00:18:23.000
Or the temperature is the
same thing throughout.

00:18:23.000 --> 00:18:25.870
Anyway, I'll leave you there and
we'll talk more about why

00:18:25.870 --> 00:18:29.320
I went through all this pain
in the next video.


More exponential decay examples

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.340
SAL: Let's do a couple more
of these exponential decay

00:00:03.340 --> 00:00:05.500
problems, because a lot of this
really is just practice

00:00:05.500 --> 00:00:08.220
and being very comfortable with
the general formula, and

00:00:08.220 --> 00:00:09.270
I'll write it again.

00:00:09.270 --> 00:00:13.000
Where the amount of the element
that's decaying, that

00:00:13.000 --> 00:00:15.760
we have at any period in time,
is equal to the amount that we

00:00:15.760 --> 00:00:21.150
started with, times
e to the minus kt.

00:00:21.150 --> 00:00:24.660
Where the k value is specific to
any certain element with a

00:00:24.660 --> 00:00:26.560
certain half-life, and sometimes
they don't even give

00:00:26.560 --> 00:00:27.600
you the half-life.

00:00:27.600 --> 00:00:30.020
So let's try this interesting
situation.

00:00:30.020 --> 00:00:32.940
Let's say that I have
an element.

00:00:32.940 --> 00:00:34.890
Let me just give
you a formula.

00:00:34.890 --> 00:00:39.180
Let's say that I have some magic
element here, where its

00:00:39.180 --> 00:00:45.560
formula is, its k value I give
to you, k is equal to minus,

00:00:45.560 --> 00:00:47.440
let me think of a-- [coughs]

00:00:47.440 --> 00:00:50.720
Excuse me, I just had a lot of
walnuts and my throat is dry.

00:00:50.720 --> 00:00:53.530
Let's say that k is equal to,
well k, we're putting a minus

00:00:53.530 --> 00:00:54.940
in front of it, so I'll
say the k value

00:00:54.940 --> 00:00:58.440
is a positive 0.05.

00:00:58.440 --> 00:01:01.950
So its exponential decay formula
would be the amount

00:01:01.950 --> 00:01:07.760
that you start off with, times
e to the minus 0.05t.

00:01:07.760 --> 00:01:13.520
My question to you is, given
this, what is the half-life of

00:01:13.520 --> 00:01:14.910
the compound that we're
talking about?

00:01:14.910 --> 00:01:16.770
What is the half-life?

00:01:19.710 --> 00:01:22.630
So to figure that out, we need
to figure out what t value can

00:01:22.630 --> 00:01:26.330
we put here, so that if we start
off with whatever value

00:01:26.330 --> 00:01:28.370
here, we end up with 1/2
of that value there.

00:01:28.370 --> 00:01:29.960
So let's do that.

00:01:29.960 --> 00:01:34.370
So we're starting off with N sub
0 This is just some value,

00:01:34.370 --> 00:01:35.410
our initial starting point.

00:01:35.410 --> 00:01:36.790
We could put 100 there.

00:01:36.790 --> 00:01:41.050
Actually, let's do that, just to
keep things less abstract.

00:01:41.050 --> 00:01:42.840
So let's say we start
with 100.

00:01:42.840 --> 00:01:44.200
I'm just picking
100 out of air.

00:01:44.200 --> 00:01:46.340
I could have left it
abstract with N.

00:01:46.340 --> 00:01:48.220
Let's say I'm starting
with 100.

00:01:48.220 --> 00:01:54.720
And I take the 100 times e to
the minus 0.05, times t.

00:01:54.720 --> 00:01:55.850
t is whatever our half-life.

00:01:55.850 --> 00:01:57.900
So after our half-life we're
going to have 1/2 of this

00:01:57.900 --> 00:01:58.940
stuff left.

00:01:58.940 --> 00:02:02.510
So this should be equal to 50.

00:02:02.510 --> 00:02:03.500
We just solved for t.

00:02:03.500 --> 00:02:05.170
Divide both sides by 100.

00:02:05.170 --> 00:02:10.259
You get e to the minus 0.05t,
is equal to 1/2.

00:02:10.259 --> 00:02:13.265
You take the natural log
of both sides of this.

00:02:13.265 --> 00:02:17.960
The natural log of this, the
natural log of that.

00:02:17.960 --> 00:02:19.770
And then you get-- the natural
log of e to anything, I've

00:02:19.770 --> 00:02:21.790
said it before, is just
the anything.

00:02:21.790 --> 00:02:29.450
So it is minus 0.05t is equal
to the natural log of 1/2.

00:02:29.450 --> 00:02:33.710
And then you get t is equal
to the natural log of 1/2,

00:02:33.710 --> 00:02:37.900
divided by minus 0.05.

00:02:37.900 --> 00:02:39.440
So let's figure out
what that is.

00:02:43.190 --> 00:02:44.490
Actually, someone just
made a comment, and I

00:02:44.490 --> 00:02:45.820
might as well do that.

00:02:45.820 --> 00:02:47.570
I could just put this
minus up here.

00:02:47.570 --> 00:02:50.170
I could make this a plus, and
this a minus, if I just

00:02:50.170 --> 00:02:53.110
multiply the numerator and the
denominator by negative 1.

00:02:53.110 --> 00:02:55.436
And if I want to, just to make
the calculator math a little

00:02:55.436 --> 00:02:58.390
easier, if you put a minus in
front of a natural log, or any

00:02:58.390 --> 00:03:01.730
logarithm, that's the same
thing as the log of the

00:03:01.730 --> 00:03:05.330
inverse of 2 over 0.05.

00:03:05.330 --> 00:03:07.635
It makes the calculator math
a little bit easier.

00:03:07.635 --> 00:03:08.860
The same thing.

00:03:08.860 --> 00:03:16.550
So if I do 2 natural log,
divided by 0.05,

00:03:16.550 --> 00:03:23.230
it is equal to 13.86.

00:03:23.230 --> 00:03:29.310
So when t is equal to 13.86.

00:03:29.310 --> 00:03:32.270
And I'm assuming that we're
dealing with time in years.

00:03:32.270 --> 00:03:34.070
That tends to be the convention,
although sometimes

00:03:34.070 --> 00:03:35.580
it could be something else
and you'd always have

00:03:35.580 --> 00:03:36.890
to convert to years.

00:03:36.890 --> 00:03:39.600
But assuming that this original
formula, where they

00:03:39.600 --> 00:03:44.790
gave this k value 0.05, that was
with the assumption that t

00:03:44.790 --> 00:03:47.410
is in years, and I've just
solved its half-life.

00:03:47.410 --> 00:03:52.430
I just solved that after 13.86
years, you can expect to have

00:03:52.430 --> 00:03:53.720
1/2 of the substance left.

00:03:53.720 --> 00:03:55.840
We started with 100, we
ended up with 50.

00:03:55.840 --> 00:03:59.370
I could have started with x and
ended up with x over 2.

00:03:59.370 --> 00:04:01.140
Let's do one more of these
problems, just so that we're

00:04:01.140 --> 00:04:03.520
really comfortable
with the formula.

00:04:03.520 --> 00:04:13.060
Let's say that I have something
with a half-life of,

00:04:13.060 --> 00:04:17.399
I don't know, let's say I
have it as one month.

00:04:17.399 --> 00:04:19.010
Half-life of one month.

00:04:19.010 --> 00:04:27.170
And after, well let's say that
I-- well let me just for the

00:04:27.170 --> 00:04:28.910
sake of time, let me make
it a little bit simpler.

00:04:28.910 --> 00:04:33.670
Let's say I just have my k value
is equal to-- I mean you

00:04:33.670 --> 00:04:36.010
can go from half-life to a k
value, we did that in the

00:04:36.010 --> 00:04:36.480
previous video.

00:04:36.480 --> 00:04:41.140
Let's say my k value
is equal to 0.001.

00:04:41.140 --> 00:04:45.130
So my general formula is the
amount of product I have, is

00:04:45.130 --> 00:04:48.180
equal to the amount that I
started with times e to the

00:04:48.180 --> 00:04:51.410
minus 0.001 times t.

00:04:51.410 --> 00:04:53.590
And I gave you this, if you
have to figure it out from

00:04:53.590 --> 00:04:57.190
half-life, I did that in the
previous video with carbon-14.

00:04:57.190 --> 00:04:58.840
But let's say this
is the formula.

00:04:58.840 --> 00:05:06.160
And let's say that after, I
don't know, let's say after

00:05:06.160 --> 00:05:14.520
1000 years I have 500 grams of
whatever element is described.

00:05:14.520 --> 00:05:16.320
The decay formula for
whatever element is

00:05:16.320 --> 00:05:17.710
described by this formula.

00:05:17.710 --> 00:05:19.110
How much did I start off with?

00:05:22.090 --> 00:05:25.340
So essentially I need to figure
out N sub 0, right?

00:05:25.340 --> 00:05:32.320
I'm saying that after 1000
years, so N of 1000, which is

00:05:32.320 --> 00:05:35.330
equal to N sub naught
times e to the minus

00:05:35.330 --> 00:05:40.720
0.001, times 1000.

00:05:40.720 --> 00:05:41.020
Right?

00:05:41.020 --> 00:05:43.730
That's the N of 1000.

00:05:43.730 --> 00:05:47.350
And I'm saying that that's
equal to 500 grams. That

00:05:47.350 --> 00:05:50.560
equals 500 grams. So
I just have to

00:05:50.560 --> 00:05:52.590
solve for N sub naught.

00:05:52.590 --> 00:05:53.830
So what's the e value?

00:05:53.830 --> 00:06:02.560
So if I have 0.0001 times 1000,
so this is N sub naught.

00:06:02.560 --> 00:06:06.160
This is 1/1000 of a 1000-- so
times e to the minus 1 is

00:06:06.160 --> 00:06:11.075
equal to 500 grams. Or I could
multiply both sides by e, and

00:06:11.075 --> 00:06:17.410
I have N sub naught is equal to
500e, which is about 2.71.

00:06:17.410 --> 00:06:23.280
So it's 500 times 2.71.

00:06:23.280 --> 00:06:25.460
I don't actually have e on this
calculator or at least I

00:06:25.460 --> 00:06:26.555
don't see it.

00:06:26.555 --> 00:06:33.710
So we'll have 1,355 grams. So
it's equal to 1,355 grams, or

00:06:33.710 --> 00:06:37.940
1.355 kilograms. That's
what I started with.

00:06:41.310 --> 00:06:42.230
So hopefully you see now.

00:06:42.230 --> 00:06:44.310
I mean, I think we've approached
this pretty much at

00:06:44.310 --> 00:06:47.590
almost any direction that a
chemistry test or teacher

00:06:47.590 --> 00:06:49.610
could throw the problem
at you.

00:06:49.610 --> 00:06:52.870
But you really just need to
remember this formula.

00:06:52.870 --> 00:06:54.370
And this applies to
a lot of things.

00:06:54.370 --> 00:06:56.800
Later you'll learn, you know,
when you do compound interest

00:06:56.800 --> 00:06:59.690
in finance, the k will just be
a positive value, but it's

00:06:59.690 --> 00:07:01.230
essentially the same formula.

00:07:01.230 --> 00:07:04.350
And there's a lot of things
that this formula actually

00:07:04.350 --> 00:07:07.210
describes well beyond just
radioactive decay.

00:07:07.210 --> 00:07:10.330
But the simple idea is, use
information they give you to

00:07:10.330 --> 00:07:12.690
solve for as many of these
constants as you can.

00:07:12.690 --> 00:07:15.160
And then whatever they're
asking for, solve for

00:07:15.160 --> 00:07:16.160
whatever's left over.

00:07:16.160 --> 00:07:18.550
And hopefully I've given you
enough examples of that.

00:07:18.550 --> 00:07:21.060
But let me know, I'm
happy to do more.


Introduction to exponential decay

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.200
SAL: Two videos ago we learned
about half-lives.

00:00:03.200 --> 00:00:05.750
And we saw that they're good if
we are trying to figure out

00:00:05.750 --> 00:00:09.020
how much of a compound we have
left after one half-life, or

00:00:09.020 --> 00:00:10.830
two half-lives, or
three half-lives.

00:00:10.830 --> 00:00:13.250
We can just take 1/2 of the
compound at every period.

00:00:13.250 --> 00:00:15.940
But it's not as useful if we're
trying to figure out how

00:00:15.940 --> 00:00:20.960
much of a compound we have after
1/2 of a half-life, or

00:00:20.960 --> 00:00:24.610
after one day, or 10 seconds,
or 10 billion years.

00:00:24.610 --> 00:00:28.030
And to address that issue in the
last video, I proved that

00:00:28.030 --> 00:00:30.720
it involved a little bit
of sophisticated math.

00:00:30.720 --> 00:00:32.490
And if you haven't taken
calculus, you can really just

00:00:32.490 --> 00:00:33.220
skip that video.

00:00:33.220 --> 00:00:35.990
You don't have to watch it
for an intro math class.

00:00:35.990 --> 00:00:38.180
But if you're curious, that's
where we proved

00:00:38.180 --> 00:00:38.950
the following formula.

00:00:38.950 --> 00:00:41.050
That at any given point of
time, if you have some

00:00:41.050 --> 00:00:46.220
decaying atom, some element,
it can be described as the

00:00:46.220 --> 00:00:49.220
amount of element you have at
any period of time is equal to

00:00:49.220 --> 00:00:53.080
the amount you started off
with, times e to some

00:00:53.080 --> 00:00:55.280
constant-- in the last
video I use lambda.

00:00:55.280 --> 00:00:59.450
I could use k this time--
minus k times t.

00:00:59.450 --> 00:01:01.670
And then for a particular
element with a particular

00:01:01.670 --> 00:01:04.980
half-life you can just solve for
the k, and then apply it

00:01:04.980 --> 00:01:05.660
to your problem.

00:01:05.660 --> 00:01:08.030
So let's do that in this video,
just so that all of

00:01:08.030 --> 00:01:12.110
these variables can become a
little bit more concrete.

00:01:12.110 --> 00:01:14.790
So let's figure out the general
formula for carbon.

00:01:14.790 --> 00:01:17.660
Carbon-14, that's the one that
we addressed in the half-life.

00:01:17.660 --> 00:01:24.530
We saw that carbon-14 has a
half-life of 5,730 years.

00:01:24.530 --> 00:01:28.200
So let's see if we can somehow
take this information and

00:01:28.200 --> 00:01:31.390
apply it to this equation.

00:01:31.390 --> 00:01:35.500
So this tells us that after
one half-life-- so

00:01:35.500 --> 00:01:39.360
t is equal to 5,730.

00:01:39.360 --> 00:01:45.930
N of 5,730 is equal to the
amount we start off with.

00:01:45.930 --> 00:01:48.320
So we're starting off with,
well, we're starting off with

00:01:48.320 --> 00:01:52.590
N sub 0 times e to the minus--
wherever you see the t you put

00:01:52.590 --> 00:02:01.180
the minus 5,730-- so minus
k, times 5,730.

00:02:01.180 --> 00:02:02.860
That's how many years
have gone by.

00:02:02.860 --> 00:02:06.660
And half-life tells us that
after 5,730 years we'll have

00:02:06.660 --> 00:02:08.690
1/2 of our initial
sample left.

00:02:08.690 --> 00:02:12.050
So we'll have 1/2 of our
initial sample left.

00:02:12.050 --> 00:02:14.580
So if we try to solve
this equation for

00:02:14.580 --> 00:02:16.000
k, what do we get?

00:02:16.000 --> 00:02:18.180
Divide both sides by N naught.

00:02:18.180 --> 00:02:22.110
Get rid of that variable, and
then we're left with e to the

00:02:22.110 --> 00:02:27.620
minus 5,730k-- I'm just
switching these two around--

00:02:27.620 --> 00:02:29.075
is equal to 1/2.

00:02:29.075 --> 00:02:32.060
If we take the natural log of
both sides, what do we get?

00:02:34.702 --> 00:02:38.220
The natural log of e to
anything, the natural log of e

00:02:38.220 --> 00:02:39.540
to the a is just a.

00:02:39.540 --> 00:02:47.170
So the natural log of this is
minus 5,730k is equal to the

00:02:47.170 --> 00:02:49.030
natural log of 1/2.

00:02:49.030 --> 00:02:50.950
I just took the natural
log of both sides.

00:02:50.950 --> 00:02:55.940
The natural log and natural
log of both sides of that.

00:02:55.940 --> 00:03:01.400
And so to solve for k, we could
just say, k is equal to

00:03:01.400 --> 00:03:08.170
the natural log of 1/2 over
minus 5,730, which we did in

00:03:08.170 --> 00:03:09.030
the previous video.

00:03:09.030 --> 00:03:12.860
But let's see if we can do that
again here, to avoid--

00:03:12.860 --> 00:03:15.560
for those who might
have skipped it.

00:03:15.560 --> 00:03:21.590
So if you have 1/2, 0.5, take
the natural log, and then you

00:03:21.590 --> 00:03:29.580
divide it by 5,730, it's a
negative 5,730, you get 1.2

00:03:29.580 --> 00:03:33.380
times 10 to the negative 4.

00:03:33.380 --> 00:03:38.830
So it equals 1.2 times
10 to the minus 4.

00:03:38.830 --> 00:03:42.250
So now we have the general
formula for carbon-14, given

00:03:42.250 --> 00:03:43.480
its half-life.

00:03:43.480 --> 00:03:47.990
At any given point in time,
after our starting point-- so

00:03:47.990 --> 00:03:52.940
this is for, let's call this for
carbon-14, for c-14-- the

00:03:52.940 --> 00:03:55.560
amount of carbon-14 we're going
to have left is going to

00:03:55.560 --> 00:04:00.690
be the amount that we started
with times e to the minus k. k

00:04:00.690 --> 00:04:02.070
we just solved for.

00:04:02.070 --> 00:04:06.930
1.2 times 10 to the minus 4,
times the amount of time that

00:04:06.930 --> 00:04:08.360
has passed by.

00:04:08.360 --> 00:04:11.700
This is our formula for
carbon, for carbon-14.

00:04:11.700 --> 00:04:13.650
If we were doing this for some
other element, we would use

00:04:13.650 --> 00:04:16.610
that element's half-life to
figure out how much we're

00:04:16.610 --> 00:04:18.660
going to have at any given
period of time to figure out

00:04:18.660 --> 00:04:20.170
the k value.

00:04:20.170 --> 00:04:21.880
So let's use this to
solve a problem.

00:04:21.880 --> 00:04:25.680
Let's say that I start off
with, I don't know, say I

00:04:25.680 --> 00:04:34.530
start off with 300 grams
of carbon, carbon-14.

00:04:34.530 --> 00:04:39.150
And I want to know, how much do
I have after, I don't know,

00:04:39.150 --> 00:04:43.250
after 2000 years?

00:04:43.250 --> 00:04:44.510
How much do I have?

00:04:44.510 --> 00:04:46.940
Well I just plug into
the formula.

00:04:46.940 --> 00:04:53.900
N of 2000 is equal to the amount
that I started off

00:04:53.900 --> 00:05:02.670
with, 300 grams, times e to the
minus 1.2 times 10 to the

00:05:02.670 --> 00:05:07.990
minus 4, times t, is times
2000, times 2000.

00:05:07.990 --> 00:05:10.330
So what is that?

00:05:10.330 --> 00:05:13.230
So I already have that 1.2 times
10 the minus 4 there.

00:05:13.230 --> 00:05:18.920
So let me say, times 2000
equals-- and of course, this

00:05:18.920 --> 00:05:20.790
throws a negative out there,
so let me put the negative

00:05:20.790 --> 00:05:21.730
number out there.

00:05:21.730 --> 00:05:23.060
So there's a negative.

00:05:23.060 --> 00:05:25.130
And I have to raise
e to this power.

00:05:25.130 --> 00:05:27.430
So it's 0.241.

00:05:27.430 --> 00:05:30.790
So this is equal to N of 2000.

00:05:30.790 --> 00:05:34.110
The amount of the substance I
can expect after 2000 years is

00:05:34.110 --> 00:05:42.630
equal to 300 times e to
the minus 0.2419.

00:05:42.630 --> 00:05:48.370
And let's see, my calculator
doesn't have an e to the

00:05:48.370 --> 00:05:50.660
power, so Let me just take e.

00:05:50.660 --> 00:05:51.900
I need to get a better
calculator.

00:05:51.900 --> 00:05:53.840
I should get my scientific
calculator back.

00:05:53.840 --> 00:05:57.830
But e is, let's say 2.71-- I
can keep adding digits but

00:05:57.830 --> 00:06:05.130
I'll just do 2.71-- to the 0.24
negative, which is equal

00:06:05.130 --> 00:06:13.830
to 0.78 times the amount that I
started off with, times 300,

00:06:13.830 --> 00:06:21.550
which is equal to 236 grams. So
this is equal to 236 grams.

00:06:21.550 --> 00:06:24.480
So just like that, using this
exponential decay formula, I

00:06:24.480 --> 00:06:27.140
was able to figure out how
much of the carbon I have

00:06:27.140 --> 00:06:30.890
after kind of an unusual
period of time, a

00:06:30.890 --> 00:06:32.540
non-half-life period of time.

00:06:32.540 --> 00:06:35.080
Let's do another
one like this.

00:06:35.080 --> 00:06:36.170
Let's go the other way around.

00:06:36.170 --> 00:06:39.430
Let's say, I'm trying
to figure out.

00:06:39.430 --> 00:06:48.980
Let's say I start off with
400 grams of c-14.

00:06:48.980 --> 00:06:53.440
And I want to know how long--
so I want to know a certain

00:06:53.440 --> 00:06:55.790
amount of time-- does it
take for me to get to

00:06:55.790 --> 00:06:59.380
350 grams of c-14?

00:06:59.380 --> 00:07:03.040
So, you just say that 350
grams is how much

00:07:03.040 --> 00:07:03.870
I'm ending up with.

00:07:03.870 --> 00:07:07.570
It's equal to the amount that I
started off with, 400 grams,

00:07:07.570 --> 00:07:10.430
times e to the minus k.

00:07:10.430 --> 00:07:15.160
That's minus 1.2 times 10 to
the minus 4, times time.

00:07:15.160 --> 00:07:18.190
And now we solve for time.

00:07:18.190 --> 00:07:19.620
How do we do that?

00:07:19.620 --> 00:07:22.840
Well we could divide
both sides by 400.

00:07:22.840 --> 00:07:24.960
What's 350 divided by 400?

00:07:24.960 --> 00:07:27.180
350 by 400.

00:07:27.180 --> 00:07:29.340
It's 7/8.

00:07:29.340 --> 00:07:31.230
So 0.875.

00:07:31.230 --> 00:07:37.780
So you get 0.875 is equal to e
to the minus 1.2 times 10 to

00:07:37.780 --> 00:07:40.100
the minus 4t.

00:07:40.100 --> 00:07:41.890
You take the natural
log of both sides.

00:07:41.890 --> 00:07:46.460
You get the natural log of
0.875 is equal to-- the

00:07:46.460 --> 00:07:49.100
natural log of e to anything is
just the anything-- so it's

00:07:49.100 --> 00:07:54.990
equal to minus 1.2 times
10 to the minus 4t.

00:07:54.990 --> 00:08:00.280
And so t is equal to this
divided by 1.2 times 10

00:08:00.280 --> 00:08:01.130
to the minus 4.

00:08:01.130 --> 00:08:07.560
So the natural log, 0.875
divided by minus 1.2 times 10

00:08:07.560 --> 00:08:10.650
to the minus 4, is equal to the
amount of time it would

00:08:10.650 --> 00:08:13.620
take us to get from
400 grams to 350.

00:08:13.620 --> 00:08:14.535
[PHONE RINGS]

00:08:14.535 --> 00:08:17.930
My cell phone is ringing,
let me turn that off.

00:08:17.930 --> 00:08:19.350
To 350.

00:08:19.350 --> 00:08:22.120
So let me do the math.

00:08:22.120 --> 00:08:26.030
So if you have 0.875, and we
want to take the natural log

00:08:26.030 --> 00:08:37.120
of it, and divide it by minus
1.-- So divided by 1.2e 4

00:08:37.120 --> 00:08:38.480
negative, 10 to the
negative 4.

00:08:38.480 --> 00:08:41.169
This is all a negative number.

00:08:41.169 --> 00:08:43.250
Oh, I'll just divide it by this,
and then just take the

00:08:43.250 --> 00:08:44.290
negative of that.

00:08:44.290 --> 00:08:47.170
Equals that and then I have
to take a negative.

00:08:47.170 --> 00:08:55.410
So this is equal to 1,112 years
to get from 400 to 350

00:08:55.410 --> 00:08:57.440
grams of my substance.

00:08:57.440 --> 00:09:00.970
This might seem a little
complicated, but if there's

00:09:00.970 --> 00:09:02.890
one thing you just have to do,
is you just have to remember

00:09:02.890 --> 00:09:04.570
this formula.

00:09:04.570 --> 00:09:06.160
And if you want to know where
it came from, watch the

00:09:06.160 --> 00:09:07.230
previous video.

00:09:07.230 --> 00:09:12.370
For any particular element you
solve for this k value.

00:09:12.370 --> 00:09:15.000
And then you just substitute
what you know, and then solve

00:09:15.000 --> 00:09:16.230
for what you don't know.

00:09:16.230 --> 00:09:19.030
I'll do a couple more of these
in the next video.


Exponential decay formula proof (can skip, involves calculus)

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:05.150
SAL: The notion of a half-life
is useful, if we're dealing

00:00:05.150 --> 00:00:07.950
with increments of time that are
multiples of a half-life.

00:00:07.950 --> 00:00:11.980
For example, where time
equals zero, we

00:00:11.980 --> 00:00:14.190
have 100% of our substance.

00:00:14.190 --> 00:00:18.580
Then after time equals one
half-life, we'd have 50% of

00:00:18.580 --> 00:00:19.550
our substance.

00:00:19.550 --> 00:00:23.350
At time is equal to two
half-lives, we'd have 25% of

00:00:23.350 --> 00:00:25.440
our substance, and so
on and so forth.

00:00:25.440 --> 00:00:28.300
So if I say that three
half-lives have gone by-- in

00:00:28.300 --> 00:00:30.740
the case of carbon that would
be, what, roughly 15,000

00:00:30.740 --> 00:00:34.840
years-- I can tell you roughly,
or almost exactly,

00:00:34.840 --> 00:00:39.030
what percentage of my original
element I still have. In the

00:00:39.030 --> 00:00:41.390
case of carbon-14, I'll tell
you what percentage of my

00:00:41.390 --> 00:00:44.570
original carbon-14 has not
decayed into nitrogen, as yet,

00:00:44.570 --> 00:00:45.880
nitrogen-14.

00:00:45.880 --> 00:00:49.110
And that's useful, but what if I
care about how much carbon I

00:00:49.110 --> 00:00:52.460
have after 1/2 a year, or after
1/2 a half life, or

00:00:52.460 --> 00:00:54.510
after three billion years,
or after 10 minutes?

00:00:54.510 --> 00:00:56.980
What if I want a general
function.

00:00:56.980 --> 00:01:00.190
A general function, as a
function of time, that tells

00:01:00.190 --> 00:01:05.430
me the number, or the amount,
of my decaying substance I

00:01:05.430 --> 00:01:07.670
have. So that's what we're going
to do in this video.

00:01:07.670 --> 00:01:09.460
And it's going to be a little
mathy, but I think the math is

00:01:09.460 --> 00:01:12.180
pretty straightforward,
especially if you've taken a

00:01:12.180 --> 00:01:14.030
first-year course in calculus.

00:01:14.030 --> 00:01:16.650
And this is actually a pretty
neat application of it.

00:01:16.650 --> 00:01:20.050
So let's just think a little bit
about the rate of change,

00:01:20.050 --> 00:01:23.090
or the probability, or the
number particles that are

00:01:23.090 --> 00:01:24.390
changing at any given time.

00:01:24.390 --> 00:01:28.910
So if we say, the difference
or change in our number of

00:01:28.910 --> 00:01:32.750
particles, or the amount of
particles, in any very small

00:01:32.750 --> 00:01:38.140
period of time, what's this
going to be dependent on?

00:01:38.140 --> 00:01:39.850
This is the number particles
we have in a

00:01:39.850 --> 00:01:40.860
given period time.

00:01:40.860 --> 00:01:43.160
This is our rate of change.

00:01:43.160 --> 00:01:46.420
So one thing, we know that our
rate of change is going down.

00:01:46.420 --> 00:01:48.960
We know it's a negative
number.

00:01:48.960 --> 00:01:51.390
We know that, in the case of
radioactive decay, I could do

00:01:51.390 --> 00:01:55.150
the same exercise with
compounding growth, where I

00:01:55.150 --> 00:01:57.470
would say, oh no, it's not a
negative number, that our

00:01:57.470 --> 00:02:00.680
growth is dependent on how much
we have. In this case the

00:02:00.680 --> 00:02:03.800
amount we're decaying is
proportional, but it's going

00:02:03.800 --> 00:02:07.400
to be the negative of how much
of the actual compound we

00:02:07.400 --> 00:02:08.229
already have.

00:02:08.229 --> 00:02:09.800
Let me explain that.

00:02:09.800 --> 00:02:12.620
So what I'm saying is, look,
our amount of decay is

00:02:12.620 --> 00:02:15.830
proportional to the amount of
the substance that we already

00:02:15.830 --> 00:02:17.200
are dealing with.

00:02:17.200 --> 00:02:20.700
And just to maybe make that a
little bit more intuitive,

00:02:20.700 --> 00:02:23.270
imagine a situation here
where you have 1

00:02:23.270 --> 00:02:24.320
times 10 to the 9th.

00:02:24.320 --> 00:02:29.750
You have a billion carbon atoms.
And let's say over here

00:02:29.750 --> 00:02:35.020
you have 1 times 10 to the 6th
carbon atoms. And if you look

00:02:35.020 --> 00:02:38.260
at it at over some small period
of time, let's say, if

00:02:38.260 --> 00:02:41.250
you look at it over one second,
let's say our dt.

00:02:41.250 --> 00:02:43.800
dt as an infinitesimally small
time, but let's say it's a

00:02:43.800 --> 00:02:44.670
change in time.

00:02:44.670 --> 00:02:45.610
It's a delta t.

00:02:45.610 --> 00:02:50.080
And let's say over one second,
you observe that this sample

00:02:50.080 --> 00:02:54.720
had, I don't know, let's say you
saw 1000 carbon particles.

00:02:54.720 --> 00:02:56.970
You really wouldn't see that
with carbon-14, but this is

00:02:56.970 --> 00:02:59.840
just for the sake of
our intuition.

00:02:59.840 --> 00:03:02.910
Let's say over one second you
saw 1000 carbon particles per

00:03:02.910 --> 00:03:04.020
second here.

00:03:04.020 --> 00:03:06.990
Well here you have 1000th of the
number particles in this

00:03:06.990 --> 00:03:08.600
sample as this one.

00:03:08.600 --> 00:03:11.340
So, for every thousand particles
you saw decaying

00:03:11.340 --> 00:03:14.810
here, you'd really expect to
see one carbon particle per

00:03:14.810 --> 00:03:15.380
second here.

00:03:15.380 --> 00:03:17.260
Just because you have
a smaller amount.

00:03:17.260 --> 00:03:19.570
Now I don't know what the
actual constant is.

00:03:19.570 --> 00:03:22.040
But we know that no matter what
substance we're talking

00:03:22.040 --> 00:03:25.060
about, this constant is
dependent on the substance.

00:03:25.060 --> 00:03:27.510
Carbon's going to be different
from uranium, is going to be

00:03:27.510 --> 00:03:30.810
different from, you know,
we looked at radon.

00:03:30.810 --> 00:03:32.400
They're all going to
have different

00:03:32.400 --> 00:03:33.710
quantities right here.

00:03:33.710 --> 00:03:34.390
And we can see that.

00:03:34.390 --> 00:03:36.670
We'll actually do it in the next
video, you can actually

00:03:36.670 --> 00:03:39.170
calculate this from
the half-life.

00:03:39.170 --> 00:03:41.930
But the rate of change is always
going to be dependent

00:03:41.930 --> 00:03:45.380
on the number of particles
you have, right?

00:03:45.380 --> 00:03:46.950
I mean, we saw that here
with half-life.

00:03:46.950 --> 00:03:50.340
When you have 1/2 the
number of particles,

00:03:50.340 --> 00:03:53.270
you lose 1/2 as much.

00:03:53.270 --> 00:03:56.180
Here, if we start with 100
particles here, we went to 50

00:03:56.180 --> 00:04:00.070
particles, then we went to 25.

00:04:00.070 --> 00:04:03.115
When you start with 50, in a
period of time you lose 25.

00:04:03.115 --> 00:04:06.230
When you start with
100, you lose 50.

00:04:06.230 --> 00:04:08.700
So clearly the amount you lose
is dependent on the amount you

00:04:08.700 --> 00:04:10.400
started with, right?

00:04:10.400 --> 00:04:12.270
Over any fraction of
time, and here it's

00:04:12.270 --> 00:04:13.840
a very small fraction.

00:04:13.840 --> 00:04:19.279
So what I set up here is really
fairly simple, but it

00:04:19.279 --> 00:04:20.940
doesn't sound so simple to a lot
of people if you say it's

00:04:20.940 --> 00:04:22.400
a differential equation.

00:04:22.400 --> 00:04:25.600
We can actually solve this using
pretty straightforward

00:04:25.600 --> 00:04:26.860
techniques.

00:04:26.860 --> 00:04:30.750
This is actually a separation
of variables problem.

00:04:30.750 --> 00:04:32.490
And so, what can we do?

00:04:32.490 --> 00:04:37.162
Let's divide both sides by N.

00:04:37.162 --> 00:04:39.920
We want to get all the N's on
this side and all the t stuff

00:04:39.920 --> 00:04:41.200
on the other side.

00:04:41.200 --> 00:04:48.460
So if we have 1 over
N, dN over dt is

00:04:48.460 --> 00:04:49.910
equal to minus lambda.

00:04:49.910 --> 00:04:53.560
I just divided both sides
of this by N.

00:04:53.560 --> 00:04:57.230
And then I can multiply both
sides of this by dt, and I get

00:04:57.230 --> 00:05:05.520
1 over N dN is equal
to minus lambda dt.

00:05:05.520 --> 00:05:08.530
Now I can take the integral of
both sides of this equation.

00:05:11.110 --> 00:05:12.740
And what do I get?

00:05:12.740 --> 00:05:14.200
What's the antiderivative?

00:05:14.200 --> 00:05:16.370
I'm taking the indefinite
integral or the

00:05:16.370 --> 00:05:16.540
antiderivative.

00:05:16.540 --> 00:05:19.930
What's the antiderivative
of 1 over N?

00:05:19.930 --> 00:05:29.460
Well that's the natural log of
N plus some constant-- I'll

00:05:29.460 --> 00:05:32.400
just do that in blue--
plus some constant.

00:05:32.400 --> 00:05:37.050
And then that equals-- What's
the antiderivative of just

00:05:37.050 --> 00:05:37.570
some constant?

00:05:37.570 --> 00:05:40.880
Well it's just that
constant times the

00:05:40.880 --> 00:05:41.980
derivative, the variable.

00:05:41.980 --> 00:05:43.360
We're taking the antiderivative

00:05:43.360 --> 00:05:43.725
with respect to.

00:05:43.725 --> 00:05:48.840
So minus lambda, times t,
plus some constant.

00:05:48.840 --> 00:05:51.630
These are different constants,
but they're arbitrary.

00:05:51.630 --> 00:05:53.930
So if we want, we can just
subtract that constant from

00:05:53.930 --> 00:05:56.130
that constant, and put them all
on one side and then we

00:05:56.130 --> 00:05:57.120
just get another constant.

00:05:57.120 --> 00:06:00.260
So this boils down to our
solution to our differential

00:06:00.260 --> 00:06:06.490
equation is the natural log of
N is equal to minus lambda-t,

00:06:06.490 --> 00:06:10.410
plus some other constant, I call
it c3, it doesn't matter.

00:06:10.410 --> 00:06:14.110
And now if we want to just make
this a function of N in

00:06:14.110 --> 00:06:19.460
terms of t, let's take both of
these, or both take e to the

00:06:19.460 --> 00:06:20.920
power of both sides of this.

00:06:20.920 --> 00:06:24.350
You can view that as kind of
the inverse natural log.

00:06:24.350 --> 00:06:29.150
So e to the power of ln of N,
ln of N is just saying what

00:06:29.150 --> 00:06:31.420
power do you raise
e to to get to N?

00:06:31.420 --> 00:06:34.720
So if you raise e to that
power, you get N.

00:06:34.720 --> 00:06:39.120
So I'm just raising both
sides of this equation.

00:06:39.120 --> 00:06:42.220
I'm raising e to both sides
of this equation.

00:06:42.220 --> 00:06:44.650
e to the ln of N is just N.

00:06:44.650 --> 00:06:50.570
And that is equal to e to the
minus lambda-t, plus c3.

00:06:50.570 --> 00:06:53.770
And now this can be rewritten
as, N is equal to e to the

00:06:53.770 --> 00:06:58.350
minus lambda-t, times
e to the c3.

00:06:58.350 --> 00:07:00.630
And now once again this is an
arbitrary constant, so we can

00:07:00.630 --> 00:07:03.450
just really rename that as,
I don't know, let me

00:07:03.450 --> 00:07:04.700
rename it as c4.

00:07:07.200 --> 00:07:11.660
So, our solution to our
differential equation, N, as a

00:07:11.660 --> 00:07:17.870
function of t, is equal to
our c4 constant, c4e

00:07:17.870 --> 00:07:20.030
to the minus lambda-t.

00:07:20.030 --> 00:07:23.330
Now let's say, even better,
let's say is N equals 0.

00:07:23.330 --> 00:07:26.400
Let's say that N equals 0.

00:07:26.400 --> 00:07:31.130
We have N sub 0 of our sample.

00:07:31.130 --> 00:07:32.680
That's how much we're
starting off with.

00:07:32.680 --> 00:07:34.830
So let's see if we can
substitute that into our

00:07:34.830 --> 00:07:37.990
equation to solve for c4.

00:07:37.990 --> 00:07:45.510
So we said N sub-0 is equal to,
let's put 0 in here, so

00:07:45.510 --> 00:07:48.930
let's see, that's equal
to N sub naught.

00:07:48.930 --> 00:07:56.750
And that's equal to c4 times e
to the minus lambda, times 0.

00:07:56.750 --> 00:07:58.770
Well, minus anything
times 0 is 0.

00:07:58.770 --> 00:08:00.300
So it's e to the 0.

00:08:00.300 --> 00:08:01.710
So that's just 1.

00:08:01.710 --> 00:08:04.760
So c4 is equal to N naught,
our starting

00:08:04.760 --> 00:08:05.770
amount for the sample.

00:08:05.770 --> 00:08:09.020
So we've actually got
an expression.

00:08:09.020 --> 00:08:12.560
We have the number of particles,
or the amount as a

00:08:12.560 --> 00:08:16.080
function of t, is equal to the
amount that we start off with,

00:08:16.080 --> 00:08:20.220
at time is equal to 0,
times e to the minus

00:08:20.220 --> 00:08:23.190
lambda, times time.

00:08:23.190 --> 00:08:25.010
And we just have to be careful
that we're always using the

00:08:25.010 --> 00:08:27.610
time constant when we solve
for the different

00:08:27.610 --> 00:08:28.590
coefficients.

00:08:28.590 --> 00:08:30.830
So this seems all abstract.

00:08:30.830 --> 00:08:32.049
How does this relate
to half-life?

00:08:32.049 --> 00:08:35.390
Well let's try to figure out
this equation for carbon.

00:08:35.390 --> 00:08:37.190
This'll be true for
anything where we

00:08:37.190 --> 00:08:38.200
have radioactive decay.

00:08:38.200 --> 00:08:41.150
If we actually had a plus sign
here it'd be exponential

00:08:41.150 --> 00:08:43.039
growth as well.

00:08:43.039 --> 00:08:57.260
We know that carbon, c-14, has
a 5,700-year half-life.

00:08:57.260 --> 00:09:00.060
So the way you could think about
it, is if at time equals

00:09:00.060 --> 00:09:08.240
0 you start off with t-- So time
equals 0. t equals-- let

00:09:08.240 --> 00:09:09.020
me write that down.

00:09:09.020 --> 00:09:14.000
If at N of 0 is equal to--
and we could write

00:09:14.000 --> 00:09:15.240
100 there if we want.

00:09:15.240 --> 00:09:16.090
Actually why don't we do that?

00:09:16.090 --> 00:09:18.530
If N of 0 we start
off with 100.

00:09:18.530 --> 00:09:25.800
And then at N of 5,700 years--
so we're going to take t to be

00:09:25.800 --> 00:09:28.620
in years, you just have to be
consistent with your units--

00:09:28.620 --> 00:09:29.740
how much will we have left?

00:09:29.740 --> 00:09:31.440
We'll have 50 left.

00:09:31.440 --> 00:09:34.096
We could have written x and x
over two here, and it would

00:09:34.096 --> 00:09:35.690
have all have worked
out in the end.

00:09:35.690 --> 00:09:37.980
So let's see, let's apply that
to this equation and try to

00:09:37.980 --> 00:09:40.460
solve this for lambda.

00:09:40.460 --> 00:09:42.770
So we know N of 0
is equal to 100.

00:09:42.770 --> 00:09:51.060
So we immediately know that we
can write this equation as N

00:09:51.060 --> 00:09:57.520
of t is equal to 100e, to the
minus lambda-t, at least in

00:09:57.520 --> 00:09:59.080
this exact circumstance.

00:10:02.800 --> 00:10:10.450
And we also know that N of
5,700-- so that means, N of

00:10:10.450 --> 00:10:15.600
5,700-- that is equal to,
we just said, that's one

00:10:15.600 --> 00:10:16.450
half-life away.

00:10:16.450 --> 00:10:19.020
So we have 1/2 as much
of our compound left.

00:10:19.020 --> 00:10:25.560
That's equal to 50, which is
equal to the 5,700th power

00:10:25.560 --> 00:10:26.830
times lambda.

00:10:26.830 --> 00:10:30.500
So it's equal to 100 times
e, to the minus

00:10:30.500 --> 00:10:34.770
lambda, times 5,700.

00:10:34.770 --> 00:10:35.940
And now we just solve
for lambda.

00:10:35.940 --> 00:10:38.240
Then we'll have a general
equation for how much carbon

00:10:38.240 --> 00:10:40.470
we have at any given
moment in time.

00:10:40.470 --> 00:10:44.850
So if you divide both sides
of this by 100.

00:10:44.850 --> 00:10:46.160
What do we get?

00:10:46.160 --> 00:10:52.230
We get 0.5, we have 1/2, is
equal to e to the-- let me

00:10:52.230 --> 00:10:56.200
just write minus 5,700 lambda,
and then we could take the

00:10:56.200 --> 00:10:58.010
natural log of both sides.

00:10:58.010 --> 00:11:02.350
So then we get-- scroll down a
bit-- the natural log of 1/2

00:11:02.350 --> 00:11:05.140
is equal to the-- the natural
log of this is just minus

00:11:05.140 --> 00:11:07.890
5,700 lambda.

00:11:07.890 --> 00:11:10.510
To solve for lambda, you get
lambda is equal to the natural

00:11:10.510 --> 00:11:16.210
log of 1/2, over minus 5,700.

00:11:16.210 --> 00:11:19.990
So let me see what that is.

00:11:19.990 --> 00:11:20.980
Let's see what that is.

00:11:20.980 --> 00:11:30.980
So 0.5 natural log is that,
divided by minus 5,700.

00:11:30.980 --> 00:11:38.456
5,700 negative is equal to 1.2
times 10 to the negative 4.

00:11:38.456 --> 00:11:43.410
Is equal to 1.21 times
10 to the minus 4.

00:11:43.410 --> 00:11:45.370
So there you have it, we
figured out our lambda.

00:11:45.370 --> 00:11:49.440
So the general equation for
how much carbon-14 we can

00:11:49.440 --> 00:11:55.340
expect at any moment in time,
t, where t is in years, is N

00:11:55.340 --> 00:11:58.340
of t is equal to the amount of
carbon we start off with,

00:11:58.340 --> 00:12:01.280
times e to the minus lambda.

00:12:01.280 --> 00:12:04.780
The minus lambda is 1.21
times 10 to the minus

00:12:04.780 --> 00:12:08.350
4, times t in years.

00:12:08.350 --> 00:12:11.500
So now if you say after 1/2 a
year, you just plug it in and,

00:12:11.500 --> 00:12:13.630
you have to tell me how much you
started off with, and then

00:12:13.630 --> 00:12:15.870
I can tell you how much you
have after 1/2 a year, or

00:12:15.870 --> 00:12:18.060
after a billion years, or
after a gazillion years.

00:12:18.060 --> 00:12:19.360
And we'll do a lot
more of these

00:12:19.360 --> 00:12:21.350
problems in the next video.