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PROFESSOR: So last time, there
was a question that was asked

00:00:23.290 --> 00:00:26.610
about the order of
magnitude rates.

00:00:26.610 --> 00:00:28.390
You asked the question
about the

00:00:28.390 --> 00:00:31.630
concentration of the catalysts.

00:00:31.630 --> 00:00:34.452
So the rates that I gave you,
which were the rates of

00:00:34.452 --> 00:00:36.050
reactions are basically
order of magnitude.

00:00:36.050 --> 00:00:39.380
They're estimates and odd
numbers that, they obviously

00:00:39.380 --> 00:00:41.800
depend, as you pointed out,
depend on the concentration of

00:00:41.800 --> 00:00:42.400
the catalyst.

00:00:42.400 --> 00:00:46.370
Depends on the concentration
of the reactants.

00:00:46.370 --> 00:00:50.500
Depends on where you are on
those concentrations.

00:00:50.500 --> 00:00:54.660
As we saw when we did enzyme
catalysis, is that depending

00:00:54.660 --> 00:00:59.000
on KM, the Michaelis constant,
and the substrate

00:00:59.000 --> 00:01:02.350
concentration, if your
substrate's concentration is

00:01:02.350 --> 00:01:08.400
much higher than KM, you are in
the maximum velocity limit,

00:01:08.400 --> 00:01:12.730
where the rate depends on
k2 times the initial

00:01:12.730 --> 00:01:14.370
concentration of enzyme.

00:01:14.370 --> 00:01:17.640
In that case the rate of the
reaction depends just on the

00:01:17.640 --> 00:01:19.800
concentration of enzyme,
and not on the

00:01:19.800 --> 00:01:21.870
concentration of substrate.

00:01:21.870 --> 00:01:27.410
But if you are at the small
concentration of substrate

00:01:27.410 --> 00:01:31.630
relative to KM, then your rate
depends linearly with the

00:01:31.630 --> 00:01:33.190
concentration of substrate.

00:01:33.190 --> 00:01:38.780
And the same thing happens
with the non-enzymatic

00:01:38.780 --> 00:01:43.470
catalyst, that there will be
some sort of mechanism that

00:01:43.470 --> 00:01:44.820
goes along with it.

00:01:44.820 --> 00:01:46.660
And most likely it's
going to be second

00:01:46.660 --> 00:01:47.650
order in the catalyst.

00:01:47.650 --> 00:01:54.630
And the reactant if it's just
the two body process.

00:01:54.630 --> 00:01:56.580
But it could be more
complicated.

00:01:56.580 --> 00:01:57.660
So definitely those

00:01:57.660 --> 00:01:59.430
concentrations will go in there.

00:01:59.430 --> 00:02:04.430
But a priori, it's not clear
how they will go in there.

00:02:04.430 --> 00:02:06.870
But the orders of magnitude
are roughly right.

00:02:06.870 --> 00:02:10.450
And the basic idea is that you
change your rate of reactions

00:02:10.450 --> 00:02:14.950
by 15, 18 orders of magnitude
with the right catalyst,

00:02:14.950 --> 00:02:18.810
especially if it's a biological
catalyst.

00:02:18.810 --> 00:02:22.280
One concept that I didn't go
through last time, which is

00:02:22.280 --> 00:02:34.360
also quite useful, it's called
the turnover number.

00:02:34.360 --> 00:02:36.630
Turnover number.

00:02:36.630 --> 00:02:40.920
And this is useful for
any kind of catalyst.

00:02:40.920 --> 00:02:46.910
But for enzymatic, or enzyme
catalysts, the

00:02:46.910 --> 00:02:51.380
turnover number is k cat.

00:02:51.380 --> 00:02:53.900
The number per second.

00:02:53.900 --> 00:02:55.300
It's a rate per second.

00:02:55.300 --> 00:03:01.760
How many times does a reactant
get, the number of times that

00:03:01.760 --> 00:03:04.730
a reactant turns into a product

00:03:04.730 --> 00:03:08.300
per second, per catalyst.

00:03:08.300 --> 00:03:09.760
So let me write it out.

00:03:09.760 --> 00:03:34.330
So it's the number of product
formed per second, per second,

00:03:34.330 --> 00:03:45.030
per enzyme, per molecule
of enzyme.

00:03:45.030 --> 00:03:48.470
OK, so if one enzyme is going to
be there, there's going to

00:03:48.470 --> 00:03:50.180
be some substrate going in.

00:03:50.180 --> 00:03:51.800
Coming in, the pocket
coming out.

00:03:51.800 --> 00:03:54.990
Coming in, the pocket coming
out, so the number of cycles

00:03:54.990 --> 00:04:00.250
of this process per second.
is the turnover number.

00:04:00.250 --> 00:04:03.450
And so if you are where the
concentration of substrate is

00:04:03.450 --> 00:04:06.190
very much larger than the
Michaelis constant, where are

00:04:06.190 --> 00:04:10.000
you are at the saturation limit,
then this turnover

00:04:10.000 --> 00:04:13.710
number doesn't depend on the
substrate concentration.

00:04:13.710 --> 00:04:16.900
It's just k cat.

00:04:16.900 --> 00:04:20.860
So that's why k cat
is important.

00:04:20.860 --> 00:04:33.850
As this turnover number in the
maximum velocity limit.

00:04:33.850 --> 00:04:38.080
Because in the maximum velocity
limit, then the

00:04:38.080 --> 00:04:43.080
process that's dominant is the
second process, ES goes to

00:04:43.080 --> 00:04:44.190
enzyme plus product.

00:04:44.190 --> 00:04:47.170
The first process here is not
the rate limiting step.

00:04:47.170 --> 00:04:48.170
This is the rate
limiting step.

00:04:48.170 --> 00:04:50.840
Because the substrate
concentration is very high.

00:04:50.840 --> 00:04:53.920
So this first part goes very
high, goes very fast.

00:04:53.920 --> 00:04:59.550
And the number of product
formed per second, per

00:04:59.550 --> 00:05:02.680
molecule of enzyme, the number
of molecules formed for

00:05:02.680 --> 00:05:11.150
second, that's the rate
of product formation.

00:05:11.150 --> 00:05:14.790
Rate of product formation
per second.

00:05:14.790 --> 00:05:20.330
Divided by the concentration
of the enzyme.

00:05:20.330 --> 00:05:22.700
If you normalize by the
number of enzyme.

00:05:22.700 --> 00:05:25.240
So this becomes rate of product
formation in moles per

00:05:25.240 --> 00:05:27.870
second, divided by
moles of enzyme.

00:05:27.870 --> 00:05:31.410
It's the same thing as what
we're saying here in words.

00:05:31.410 --> 00:05:36.090
And so the rate of product
formation is dP/dt, and you

00:05:36.090 --> 00:05:39.260
divide that by the concentration
of enzyme, which

00:05:39.260 --> 00:05:42.380
you can write as E0. dp/dt,
and the maximum

00:05:42.380 --> 00:05:47.230
rate limit is k max.

00:05:47.230 --> 00:05:49.840
It's a constant divided by E0.

00:05:49.840 --> 00:05:56.130
And k max is k cat times
E0 divided by E0.

00:05:56.130 --> 00:05:57.340
The E0's come out.

00:05:57.340 --> 00:06:00.320
And this is k cat.

00:06:00.320 --> 00:06:08.200
Which is basically k2.

00:06:08.200 --> 00:06:10.960
So that's the turnover number.

00:06:10.960 --> 00:06:12.290
Any questions on this
turnover number?

00:06:12.290 --> 00:06:17.370
You see that a lot in catalyst
literature and

00:06:17.370 --> 00:06:19.850
especially the enzymes.

00:06:19.850 --> 00:06:27.310
I have to get out these
lecture notes.

00:06:27.310 --> 00:06:32.010
Send these back.

00:06:32.010 --> 00:06:35.650
OK, so the last topic
I want to talk about

00:06:35.650 --> 00:06:38.240
is oscillating reactions.

00:06:38.240 --> 00:06:49.040
It's sort of like an
interesting topic.

00:06:49.040 --> 00:06:51.940
And it's applicable not just
to just chemistry, but it's

00:06:51.940 --> 00:06:54.450
basically playing with the
differential equations.

00:06:54.450 --> 00:07:01.700
And you'll see this sort of
stuff later if you keep doing

00:07:01.700 --> 00:07:05.100
science that involves coupled
differential equations.

00:07:05.100 --> 00:07:09.330
It's just all over the place.

00:07:09.330 --> 00:07:14.230
So normally we have an
equilibrium process.

00:07:14.230 --> 00:07:18.760
A goes to B, B goes back to A.
And if you start out with

00:07:18.760 --> 00:07:24.130
something which is out of
equilibrium, then you will

00:07:24.130 --> 00:07:26.700
eventually reach equilibrium at
a rate which is dependent

00:07:26.700 --> 00:07:30.530
on those two rates. k1
plus k minus one.

00:07:30.530 --> 00:07:33.730
So B will to come up to
some B equilibrium.

00:07:33.730 --> 00:07:38.910
And A will, if you start out
with a lot of A, and a little

00:07:38.910 --> 00:07:43.760
bit of B, and concentration of
A, will come down smoothly,

00:07:43.760 --> 00:07:48.280
monotonically, to the
equilibrium state.

00:07:48.280 --> 00:07:53.790
But sometimes, if you're out
of equilibrium, and this

00:07:53.790 --> 00:07:54.720
doesn't have to be chemistry.

00:07:54.720 --> 00:08:02.550
It could be anything that's
out of equilibrium, your

00:08:02.550 --> 00:08:05.300
emotion's out of equilibrium.

00:08:05.300 --> 00:08:08.090
Stock market is out
of equilibrium.

00:08:08.090 --> 00:08:10.840
Population dynamics are
out of equilibrium.

00:08:10.840 --> 00:08:11.750
The weather.

00:08:11.750 --> 00:08:14.630
Whatever.

00:08:14.630 --> 00:08:17.350
So, you're out of equilibrium,
let's say this is equilibrium

00:08:17.350 --> 00:08:24.590
for A, this is equilibrium for
B, let me try to keep them

00:08:24.590 --> 00:08:28.140
somewhat the same as
here, I messed up.

00:08:28.140 --> 00:08:31.980
You're out of equilibrium, you
start down here for A. And

00:08:31.980 --> 00:08:36.240
you, instead of going linearly
down or monotonically down,

00:08:36.240 --> 00:08:37.380
exponentially down to the

00:08:37.380 --> 00:08:40.590
equilibrium, instead you overshoot.

00:08:40.590 --> 00:08:42.190
And you keep going
back and forth.

00:08:42.190 --> 00:08:44.870
Like a spring.

00:08:44.870 --> 00:08:46.990
And then B does the
same thing.

00:08:46.990 --> 00:08:48.380
Overshoots, comes down.

00:08:48.380 --> 00:08:50.760
Overshoots, comes down.

00:08:50.760 --> 00:08:51.450
Like a pendulum.

00:08:51.450 --> 00:08:58.820
A lot of things work
like that.

00:08:58.820 --> 00:09:00.490
The heart is something that
works like that, right?

00:09:00.490 --> 00:09:00.790
It beats.

00:09:00.790 --> 00:09:04.160
It's not in equilibrium.

00:09:04.160 --> 00:09:08.050
Good thing it's not
in equilibrium.

00:09:08.050 --> 00:09:11.640
So the heart is one example of
something that oscillates,

00:09:11.640 --> 00:09:13.010
back and forth, back
and forth.

00:09:13.010 --> 00:09:16.100
And so there are actually many
examples of complicated

00:09:16.100 --> 00:09:19.690
processes that involve
chemistry, that aren't at

00:09:19.690 --> 00:09:26.970
equilibrium, that instead go
back and forth, in and out, of

00:09:26.970 --> 00:09:32.080
on one side or the other
of equilibrium.

00:09:32.080 --> 00:09:35.460
Now, there's an important
feature of, if you want to

00:09:35.460 --> 00:09:42.310
build a chemical process
that looks like this.

00:09:42.310 --> 00:09:47.170
It's often very useful to have
a particular step in the

00:09:47.170 --> 00:09:56.030
mechanism that's called
an autocatalysis step.

00:09:56.030 --> 00:09:59.490
And that provides feedback.

00:09:59.490 --> 00:10:05.950
It's like, if my microphone
here, the speaker which is

00:10:05.950 --> 00:10:11.450
here, I were to stand right in
front of the speaker and my

00:10:11.450 --> 00:10:15.190
microphone would pick up the
volume from the speaker, it

00:10:15.190 --> 00:10:17.450
would get amplified,
and I get feedback.

00:10:17.450 --> 00:10:21.040
And that would be a bad thing.

00:10:21.040 --> 00:10:22.940
So this is a form of
feedback here.

00:10:22.940 --> 00:10:26.320
You need some sort of feedback
that the reaction knows that

00:10:26.320 --> 00:10:29.930
you're building up, that
this is going too fast.

00:10:29.930 --> 00:10:32.220
And then there's a feedback
process that

00:10:32.220 --> 00:10:33.080
brings it back up.

00:10:33.080 --> 00:10:36.070
And a feedback process that
goes back and forth.

00:10:36.070 --> 00:10:40.130
And the way you get this
feedback is by having a step

00:10:40.130 --> 00:10:53.780
in the mechanism that has both
a molecule as part of the

00:10:53.780 --> 00:10:58.620
reactants and as part
of the product.

00:10:58.620 --> 00:11:02.360
It's not the same as a chain
reaction, where we talked

00:11:02.360 --> 00:11:06.490
about an intermediate being
recycled and building up.

00:11:06.490 --> 00:11:10.140
This is an actual reactant and
an actual product that's part

00:11:10.140 --> 00:11:16.550
of this step here.

00:11:16.550 --> 00:11:20.190
So this is an autocatalytic
step.

00:11:20.190 --> 00:11:24.530
And let's just see what
that step looks like.

00:11:24.530 --> 00:11:26.400
Suppose we just have
that step.

00:11:26.400 --> 00:11:29.800
So we want to find out, as a
function of time, if you start

00:11:29.800 --> 00:11:33.180
out with some A and B, what
happens to the concentration

00:11:33.180 --> 00:11:34.660
of B as a function of time.

00:11:34.660 --> 00:11:36.530
The time dependence of that.

00:11:36.530 --> 00:11:37.920
Clearly it's going
to build up.

00:11:37.920 --> 00:11:40.590
But how is it going
to build up?

00:11:40.590 --> 00:11:46.000
So we need to solve for B
as a function of time.

00:11:46.000 --> 00:11:48.270
So let's go ahead and write
down our rates.

00:11:48.270 --> 00:11:53.300
The rate of the reaction is
d[A]/dt, is, let's put a rate

00:11:53.300 --> 00:12:00.400
constant here. k, k times
[A] times [B].

00:12:00.400 --> 00:12:01.050
Now, [B] and [A]

00:12:01.050 --> 00:12:05.190
are related to each other
through stoichiometry.

00:12:05.190 --> 00:12:11.300
The concentration of B is the
concentration of B that he

00:12:11.300 --> 00:12:18.650
started out with, [B]0, then
every time you destroy an A,

00:12:18.650 --> 00:12:21.800
you create 2B.

00:12:21.800 --> 00:12:25.710
So it's plus two times the
amount of A that you've

00:12:25.710 --> 00:12:31.830
destroyed, [A]0 minus
[A], [A]0 is what

00:12:31.830 --> 00:12:32.720
you started out with.

00:12:32.720 --> 00:12:33.670
This is what's left over.

00:12:33.670 --> 00:12:37.790
So the difference is what
you've destroyed.

00:12:37.790 --> 00:12:40.410
But every time you destroy
an A to form 2B, you

00:12:40.410 --> 00:12:43.170
also destroy a b.

00:12:43.170 --> 00:12:46.470
So you have to subtract away the
B that you've destroyed.

00:12:46.470 --> 00:12:52.450
Which is [A]0 minus [A].

00:12:52.450 --> 00:12:56.540
This is the 2B that you created
by destroying A, and

00:12:56.540 --> 00:13:00.640
this is the B that you destroy
by having to destroy A, for

00:13:00.640 --> 00:13:02.410
the reaction.

00:13:02.410 --> 00:13:05.770
You've got to bean-count
correctly.

00:13:05.770 --> 00:13:07.030
Keep track of everything.

00:13:07.030 --> 00:13:11.270
So [B], then, is just, I'm going
to drop those brackets.

00:13:11.270 --> 00:13:17.860
B0 plus A0 minus A. So you
can plug that in here.

00:13:17.860 --> 00:13:28.890
And is equal to k times A, times
B0 plus B0 minus A. And

00:13:28.890 --> 00:13:34.720
now you have a differential
equation that only

00:13:34.720 --> 00:13:37.510
contains A and time.

00:13:37.510 --> 00:13:39.070
B0 is a constant.

00:13:39.070 --> 00:13:43.080
And the way you solve that
is by partial fractions.

00:13:43.080 --> 00:13:49.050
Use partial fractions,
solve for this.

00:13:49.050 --> 00:13:51.245
And I'm not going to
go through the math

00:13:51.245 --> 00:13:52.140
of solving for it.

00:13:52.140 --> 00:13:59.440
It's not that interesting.

00:13:59.440 --> 00:14:03.520
Let me give you the answer.

00:14:03.520 --> 00:14:07.080
You end up with [B] as a
function of time, is this

00:14:07.080 --> 00:14:16.040
function A0 plus B0 over one
plus A0 over B0 times e to the

00:14:16.040 --> 00:14:24.440
minus k times A0 plus
B0 plus time.

00:14:24.440 --> 00:14:26.010
And that's where the time
component comes in.

00:14:26.010 --> 00:14:28.640
Whoops, I'm sorry.

00:14:28.640 --> 00:14:33.950
Usually I don't have this
on, let's turn this off.

00:14:33.950 --> 00:14:41.660
That's where the time
dependence comes in.

00:14:41.660 --> 00:14:45.460
And clearly, at t is equal to
infinity, t equals infinity

00:14:45.460 --> 00:14:47.300
just goes to zero.

00:14:47.300 --> 00:14:50.570
So you have B as A0, plus B0,
everything goes through to the

00:14:50.570 --> 00:14:54.050
product. t equals to zero,
this is equal to one.

00:14:54.050 --> 00:15:00.590
And B is equal to just B0.

00:15:00.590 --> 00:15:05.490
Alright, and so the curve looks
something like this.

00:15:05.490 --> 00:15:09.180
This is what it turns
out to look like.

00:15:09.180 --> 00:15:11.920
We're starting with B0 here.

00:15:11.920 --> 00:15:14.630
There's an induction period
where very little happens.

00:15:14.630 --> 00:15:17.560
It's like the dormant stage.

00:15:17.560 --> 00:15:20.340
So, locusts that are sitting
in the ground for seventeen

00:15:20.340 --> 00:15:27.230
years waiting for something
to happen.

00:15:27.230 --> 00:15:28.700
Dormant stage, the
induction period.

00:15:28.700 --> 00:15:30.710
And suddenly something
starts to happen.

00:15:30.710 --> 00:15:35.710
There's an inflection point.

00:15:35.710 --> 00:15:39.020
And you get to A0 plus B0.

00:15:39.020 --> 00:15:41.070
The locusts wake up.

00:15:41.070 --> 00:15:47.970
And go to maximum
concentration.

00:15:47.970 --> 00:15:55.250
So there's an s-like shape that
has an inflection point

00:15:55.250 --> 00:16:04.290
and an induction period.

00:16:04.290 --> 00:16:10.310
That's pretty typical of an
auto-catalytic reaction.

00:16:10.310 --> 00:16:13.270
OK so let's apply it
to a process now.

00:16:13.270 --> 00:16:15.340
And again, this is going
to be coupled

00:16:15.340 --> 00:16:16.320
differential equations.

00:16:16.320 --> 00:16:18.890
Broadly, broadly applicable,
the example that I'm

00:16:18.890 --> 00:16:20.560
going to give you.

00:16:20.560 --> 00:16:45.030
This mechanism has been used
for all sorts of things.

00:16:45.030 --> 00:16:45.820
So what do we have here?

00:16:45.820 --> 00:16:51.560
We're going to have an island.

00:16:51.560 --> 00:16:56.310
In the ocean.

00:16:56.310 --> 00:16:58.780
Island here.

00:16:58.780 --> 00:17:03.920
And it's going to be sunny,
so let's get some

00:17:03.920 --> 00:17:05.810
yellow chalk here.

00:17:05.810 --> 00:17:08.160
There's the sun.

00:17:08.160 --> 00:17:11.860
And every now and then
it's going to rain.

00:17:11.860 --> 00:17:15.160
We need some white chalk
for the rain.

00:17:15.160 --> 00:17:22.000
There's a cloud that comes in.

00:17:22.000 --> 00:17:26.600
That's going to be one
of the reactants.

00:17:26.600 --> 00:17:31.820
And then some seeds blow in from
the continent, and grass

00:17:31.820 --> 00:17:35.990
starts to grow on the island.

00:17:35.990 --> 00:17:38.410
Grass.

00:17:38.410 --> 00:17:41.590
And then there's a shipwreck and
a couple of rabbits get on

00:17:41.590 --> 00:17:43.190
the island.

00:17:43.190 --> 00:17:55.040
And now we have rabbits.

00:17:55.040 --> 00:17:55.430
Rabbits.

00:17:55.430 --> 00:17:58.490
A little tail on the back.

00:17:58.490 --> 00:18:00.570
I hate rabbits.

00:18:00.570 --> 00:18:04.280
I have a little vegetable
garden, a little tiny

00:18:04.280 --> 00:18:05.180
vegetable garden.

00:18:05.180 --> 00:18:06.990
And there are rabbits.

00:18:06.990 --> 00:18:10.710
And I have to fence in my
vegetable garden and make it

00:18:10.710 --> 00:18:13.405
impermeable impenetrable
to rabbits.

00:18:13.405 --> 00:18:15.990
And rabbits are very clever.

00:18:15.990 --> 00:18:18.740
And my vegetable garden
looks like a fortress.

00:18:18.740 --> 00:18:21.910
It's got a green fence.

00:18:21.910 --> 00:18:23.380
Heavy duty things.

00:18:23.380 --> 00:18:24.010
It's terrible.

00:18:24.010 --> 00:18:25.070
Rabbits are awful.

00:18:25.070 --> 00:18:27.590
They're very cute but
they're awful.

00:18:27.590 --> 00:18:29.400
Alright, so now what happens.

00:18:29.400 --> 00:18:34.060
Well, given the rain and the
grass, it turns out that this

00:18:34.060 --> 00:18:38.140
island can only support
25 rabbits.

00:18:38.140 --> 00:18:39.200
If there are more rabbits
than that, then

00:18:39.200 --> 00:18:40.540
they all eat too much.

00:18:40.540 --> 00:18:43.550
If there are less rabbits
then that's fine.

00:18:43.550 --> 00:18:45.820
But 25 is about the
maximum that can

00:18:45.820 --> 00:18:46.650
support on this island.

00:18:46.650 --> 00:18:51.020
The rabbits, unfortunately,
are not that smart.

00:18:51.020 --> 00:18:52.850
They don't know that.

00:18:52.850 --> 00:18:56.380
They don't know that only 25
rabbits can live there.

00:18:56.380 --> 00:19:04.750
So, what happens is that the
Year one, Year one there are

00:19:04.750 --> 00:19:06.490
ten rabbits.

00:19:06.490 --> 00:19:11.520
And the food condition
is great.

00:19:11.520 --> 00:19:18.100
Lots of food, number
of rabbits.

00:19:18.100 --> 00:19:20.060
Lots of food.

00:19:20.060 --> 00:19:22.810
So the bunnies, the rabbits
make bunnies.

00:19:22.810 --> 00:19:23.950
Lots of rabbits.

00:19:23.950 --> 00:19:26.630
Year two, they kind
of went overboard.

00:19:26.630 --> 00:19:29.240
Now there are 50 rabbits.

00:19:29.240 --> 00:19:30.950
The food is lousy.

00:19:30.950 --> 00:19:34.120
Terrible.

00:19:34.120 --> 00:19:36.700
Well, there's a feedback here.

00:19:36.700 --> 00:19:40.130
Feedback and rabbits
start to die off.

00:19:40.130 --> 00:19:42.440
Don't reproduce as much.

00:19:42.440 --> 00:19:44.520
Year three, we're back to ten.

00:19:44.520 --> 00:19:47.040
Great food.

00:19:47.040 --> 00:19:48.290
Rabbits don't notice.

00:19:48.290 --> 00:19:50.250
They multiply.

00:19:50.250 --> 00:19:53.040
Food is terrible.

00:19:53.040 --> 00:19:55.390
And they just don't learn.

00:19:55.390 --> 00:19:56.750
Humans are a little
bit like that.

00:19:56.750 --> 00:19:59.990
Except our cycle isn't one
year, it's usually on the

00:19:59.990 --> 00:20:00.870
order of 30 years.

00:20:00.870 --> 00:20:03.870
It's like the memory of a
generation disappearing.

00:20:03.870 --> 00:20:05.660
And we have to relearn the
mistakes of the previous

00:20:05.660 --> 00:20:09.520
generation.

00:20:09.520 --> 00:20:10.440
Science is like that too.

00:20:10.440 --> 00:20:12.090
Science goes in cycles.

00:20:12.090 --> 00:20:13.800
20, 30-year cycles.

00:20:13.800 --> 00:20:17.310
Topics that were out of fashion
20 years ago, 25 years

00:20:17.310 --> 00:20:19.270
ago, come back.

00:20:19.270 --> 00:20:21.070
And suddenly everybody's
excited about them.

00:20:21.070 --> 00:20:24.690
And all the ideas, of course
we make progress.

00:20:24.690 --> 00:20:26.710
We go a little bit further
than we did 20 years ago.

00:20:26.710 --> 00:20:28.630
But all the ideas that were
out of fashion 20

00:20:28.630 --> 00:20:29.780
years ago come back.

00:20:29.780 --> 00:20:32.380
People get really excited, and
they relearn all the mistakes

00:20:32.380 --> 00:20:34.250
that people made at
the beginning.

00:20:34.250 --> 00:20:37.770
Technology has improved, we know
more, and therefore we go

00:20:37.770 --> 00:20:39.370
a little bit further this
cycle than we did

00:20:39.370 --> 00:20:40.060
the previous cycle.

00:20:40.060 --> 00:20:42.786
But it's really interesting to
look at history of science and

00:20:42.786 --> 00:20:52.140
see this sort of cycle.

00:20:52.140 --> 00:20:55.240
So let's write a mechanism.

00:20:55.240 --> 00:20:59.540
Let's write a mechanism here.

00:20:59.540 --> 00:21:07.910
So we have rain plus grass
makes more grass.

00:21:07.910 --> 00:21:09.240
At the rate k1.

00:21:09.240 --> 00:21:10.860
That's autocatalytic.

00:21:10.860 --> 00:21:12.430
It provides some feedback.

00:21:12.430 --> 00:21:21.520
Then we have grass plus rabbits
make more rabbits.

00:21:21.520 --> 00:21:23.870
OK, let's do more because
I don't know

00:21:23.870 --> 00:21:28.820
how much, more, more.

00:21:28.820 --> 00:21:30.900
Also autocatalytic.

00:21:30.900 --> 00:21:33.200
Then we have rabbits.

00:21:33.200 --> 00:21:39.700
Eventually, unfortunately,
rabbits become dead rabbits.

00:21:39.700 --> 00:21:45.330
Rate k2 k3, and that's the
termination of the process.

00:21:45.330 --> 00:21:47.580
OK, so now let's put some
chemistry on this.

00:21:47.580 --> 00:21:51.220
Let's assume that instead of
having objects, live objects

00:21:51.220 --> 00:21:54.440
here, we have chemical
molecules.

00:21:54.440 --> 00:21:57.950
So A plus B goes to 2B.

00:21:57.950 --> 00:21:59.820
That's the first step.

00:21:59.820 --> 00:22:05.350
Then we have B plus C,
C is the rabbits.

00:22:05.350 --> 00:22:10.960
And then C goes to some
sort of products.

00:22:10.960 --> 00:22:15.450
Soil.

00:22:15.450 --> 00:22:18.020
This is a very famous
mechanism.

00:22:18.020 --> 00:22:26.820
It's called the Lotka-Volterra
mechanism.

00:22:26.820 --> 00:22:33.970
It's also called the
predator-prey mechanism.

00:22:33.970 --> 00:22:36.920
In my case here, I made it
sort of warm and fuzzy.

00:22:36.920 --> 00:22:38.480
I didn't have a predator
in there.

00:22:38.480 --> 00:22:41.720
But you could change this.

00:22:41.720 --> 00:22:44.510
Instead of having rabbits
and grass, we could

00:22:44.510 --> 00:22:47.170
have rabbits and foxes.

00:22:47.170 --> 00:22:54.580
Where A is the grass, if you
take A to be the grass.

00:22:54.580 --> 00:22:57.880
B to be the rabbits.

00:22:57.880 --> 00:23:01.630
And C you be the foxes.

00:23:01.630 --> 00:23:02.660
Works the same.

00:23:02.660 --> 00:23:05.580
You have grass plus rabbits
makes more rabbits.

00:23:05.580 --> 00:23:07.870
Rabbit plus foxes
make more foxes.

00:23:07.870 --> 00:23:10.450
Eventually the foxes die off.

00:23:10.450 --> 00:23:12.720
Same idea.

00:23:12.720 --> 00:23:32.770
So now, let's solve this.

00:23:32.770 --> 00:23:38.820
Let's assume that A, the rain
here, it's constant.

00:23:38.820 --> 00:23:40.090
There's a steady
supply of rain.

00:23:40.090 --> 00:23:41.860
It doesn't go away.

00:23:41.860 --> 00:23:44.680
That makes sense in
my example here.

00:23:44.680 --> 00:23:50.780
So the concentration of
A is kept constant.

00:23:50.780 --> 00:23:53.590
In order to get oscillations to
keep going, that turns out

00:23:53.590 --> 00:23:54.230
to be important.

00:23:54.230 --> 00:23:58.410
To have one of the reactants
just keeping being

00:23:58.410 --> 00:23:59.500
reintroduced in the system.

00:23:59.500 --> 00:24:01.040
Because it gets used up.

00:24:01.040 --> 00:24:06.280
And when all the reactant gets
used up, then you're done.

00:24:06.280 --> 00:24:10.060
So in the case of the heart,
we have to keep feeding

00:24:10.060 --> 00:24:12.040
ourselves to produce energy.

00:24:12.040 --> 00:24:18.850
Otherwise the heart
would stop.

00:24:18.850 --> 00:24:22.350
OK, now we want to know what is
the concentration of B and

00:24:22.350 --> 00:24:26.350
C as a function of time.

00:24:26.350 --> 00:24:36.110
So we write down our kinetic
equations. dB/dt gets produced

00:24:36.110 --> 00:24:39.800
through the first step, k1 A
times B, gets destroyed in the

00:24:39.800 --> 00:24:47.850
second step. k2 B times C.
dC/dt, C gets produced in the

00:24:47.850 --> 00:24:52.920
second step, k2 B times C. Gets
destroyed in the third

00:24:52.920 --> 00:24:56.700
step. k3 times C, got to do your
bean-counting correctly.

00:24:56.700 --> 00:24:58.870
And the strategy here that we're
going to use to solve

00:24:58.870 --> 00:25:03.700
the problem is to assume that
we're at steady state.

00:25:03.700 --> 00:25:06.200
We're going to find the solution
at steady state.

00:25:06.200 --> 00:25:08.270
Then we're going to perturb
away from steady state a

00:25:08.270 --> 00:25:09.800
little bit.

00:25:09.800 --> 00:25:13.560
And see whether this creates
an oscillation.

00:25:13.560 --> 00:25:17.230
First we need to find out what
the steady state solution is.

00:25:17.230 --> 00:25:21.950
So first, let's find
steady state.

00:25:21.950 --> 00:25:30.740
Then perturb away from
steady state.

00:25:30.740 --> 00:25:32.470
We're looking at a harmonic
response, if

00:25:32.470 --> 00:25:33.840
you've heard that term.

00:25:33.840 --> 00:25:36.640
When it's going to look like
a spring, close to its

00:25:36.640 --> 00:25:41.300
equilibrium state.

00:25:41.300 --> 00:25:44.290
That means that we're going to
set it these two things equal

00:25:44.290 --> 00:25:47.630
to zero, for this
steady state.

00:25:47.630 --> 00:25:48.970
Things are not changing.

00:25:48.970 --> 00:25:50.520
Concentration of B and
C are not changing.

00:25:50.520 --> 00:25:53.970
A is not changing
on purpose here.

00:25:53.970 --> 00:25:56.380
And so all these guys here
are then steady state

00:25:56.380 --> 00:25:57.920
concentrations.

00:25:57.920 --> 00:26:02.450
If we set it equal to zero.

00:26:02.450 --> 00:26:10.110
We solve for A in this.

00:26:10.110 --> 00:26:11.930
For C in this process here.

00:26:11.930 --> 00:26:13.140
The B's cancel out here.

00:26:13.140 --> 00:26:20.160
We divide by B. And you solve
for C as a function of A. So

00:26:20.160 --> 00:26:27.600
this one here gives you C steady
state is equal to k1

00:26:27.600 --> 00:26:32.750
over k2 times the concentration
of A. And this

00:26:32.750 --> 00:26:37.210
one here, now you put in the C
and you solve for B, B steady

00:26:37.210 --> 00:26:58.740
state, is equal to k3 over k2.

00:26:58.740 --> 00:27:02.740
So the next step is to perturb
away from equilibrium.

00:27:02.740 --> 00:27:03.820
Or from the steady
state, rather.

00:27:03.820 --> 00:27:05.910
This is not equilibrium, it's
not equilibrium because we

00:27:05.910 --> 00:27:18.060
keep adding A. We have to put
some input into the system.

00:27:18.060 --> 00:27:19.890
So here we are.

00:27:19.890 --> 00:27:23.980
We've got some concentration of
B here at the steady state.

00:27:23.980 --> 00:27:28.910
We have some concentration
of C at the steady state.

00:27:28.910 --> 00:27:35.480
And now we're going to add some
delta B to the system, or

00:27:35.480 --> 00:27:37.760
delta C. Or both.

00:27:37.760 --> 00:27:45.240
Delta C, we're going to ask the
question, given our system

00:27:45.240 --> 00:27:47.630
here, how is it going
to respond?

00:27:47.630 --> 00:27:51.280
It has a number of choices.

00:27:51.280 --> 00:27:58.300
It could respond by going back
to the steady state in a

00:27:58.300 --> 00:28:03.250
monotonic fashion.

00:28:03.250 --> 00:28:07.420
Without overshooting.

00:28:07.420 --> 00:28:15.910
This is kind of like an
overdamped system.

00:28:15.910 --> 00:28:18.280
It's like having a shock
absorber on your car.

00:28:18.280 --> 00:28:21.100
You go over a bump, the car
doesn't oscillate up and down.

00:28:21.100 --> 00:28:22.170
It's sort of damped.

00:28:22.170 --> 00:28:23.110
It's a damped oscillator.

00:28:23.110 --> 00:28:27.010
It goes back to the
steady state.

00:28:27.010 --> 00:28:40.320
Or, you could, there we go,
there's B, C. Or you could

00:28:40.320 --> 00:28:44.980
perturb, and it could just
stay where it is.

00:28:44.980 --> 00:28:46.490
That could happen too.

00:28:46.490 --> 00:28:54.020
That's an inelastic response.

00:28:54.020 --> 00:28:58.320
Kind of like the supply
and demand with oil.

00:28:58.320 --> 00:29:02.790
The price of oil goes from $10
a barrel to $120 a barrel.

00:29:02.790 --> 00:29:07.100
The use of oil doesn't seem to
be affected very much by the

00:29:07.100 --> 00:29:07.585
increasing price.

00:29:07.585 --> 00:29:08.870
You're still using
just as much oil.

00:29:08.870 --> 00:29:16.280
There's an inelastic response.

00:29:16.280 --> 00:29:20.690
What we're looking for is
something that has an elastic

00:29:20.690 --> 00:29:22.940
response, or a harmonic
response.

00:29:22.940 --> 00:29:34.010
We perturb it, it tries to
get back to steady state.

00:29:34.010 --> 00:29:38.920
But because of feedback, and not
being over-damped, if it's

00:29:38.920 --> 00:29:42.180
an oscillator, it goes
up and down.

00:29:42.180 --> 00:29:51.540
Like a scale that is not well,
well, like a car that has bad

00:29:51.540 --> 00:29:53.750
shock absorbers, all it
has is the springs.

00:29:53.750 --> 00:29:57.720
Up and down.

00:29:57.720 --> 00:30:03.730
So what we want is, we want
to solve for dB/dt as

00:30:03.730 --> 00:30:22.320
a function of time.

00:30:22.320 --> 00:30:24.580
We want, well, I'm going to
do everything in green.

00:30:24.580 --> 00:30:27.040
Since I lost my white chalk.

00:30:27.040 --> 00:30:33.780
We want delta B, delta C
as a function of time.

00:30:33.780 --> 00:30:39.120
So, that means that we start
with B is B steady state plus

00:30:39.120 --> 00:30:45.280
delta B, C is C steady state
plus delta C. And we put that

00:30:45.280 --> 00:30:46.340
in our equations.

00:30:46.340 --> 00:30:48.640
For dB/dt and dC/dt.

00:30:48.640 --> 00:30:57.190
So now dB/dt is the same thing
as dB steady state plus delta

00:30:57.190 --> 00:31:01.730
B dt, which is d delta B / dt.

00:31:01.730 --> 00:31:04.930
Because this is a
constant here.

00:31:04.930 --> 00:31:11.670
And that's equal to k1 times A,
times B steady state plus

00:31:11.670 --> 00:31:19.110
delta B. Minus k2 times B steady
state plus delta B

00:31:19.110 --> 00:31:23.940
times C steady state plus delta
C. And then you have the

00:31:23.940 --> 00:31:30.090
same thing for dC/dt, it's
going to look exactly

00:31:30.090 --> 00:31:34.330
identical. d delta C / dt.

00:31:34.330 --> 00:31:49.440
Thank you much, this is magic.
k2 times B steady state plus

00:31:49.440 --> 00:31:57.340
delta B, times C steady state
plus delta C, minus k3 times C

00:31:57.340 --> 00:32:00.830
steady state plus delta C.

00:32:00.830 --> 00:32:02.770
So these are two differential
equations.

00:32:02.770 --> 00:32:04.260
And there are coupled
differential equations.

00:32:04.260 --> 00:32:07.300
Because delta B is occurring
here, here, and here.

00:32:07.300 --> 00:32:12.990
So the time derivative of delta
C depends on delta B.

00:32:12.990 --> 00:32:17.530
And the time derivative of delta
B depends on delta C.

00:32:17.530 --> 00:32:21.180
It's a coupled system.

00:32:21.180 --> 00:32:26.140
And if you actually expand this
out and take away all the

00:32:26.140 --> 00:32:34.350
terms that cancel out, you end
up with something that looks

00:32:34.350 --> 00:32:40.160
much simpler but is just as
hard. d delta B / dt is equal

00:32:40.160 --> 00:32:52.900
to minus k3 delta C and d delta
C / dt is equal to k1 A

00:32:52.900 --> 00:32:53.320
delta B.

00:32:53.320 --> 00:32:56.820
And then you really see that
it's coupled, because the time

00:32:56.820 --> 00:33:00.190
derivative of delta B depends
on delta C. And the time

00:33:00.190 --> 00:33:03.680
derivative of delta C depends
on delta B. And there's the

00:33:03.680 --> 00:33:08.430
feedback sitting right here.

00:33:08.430 --> 00:33:09.380
And that's what you do.

00:33:09.380 --> 00:33:11.830
You learn how to
solve in 18.03.

00:33:11.830 --> 00:33:13.460
So I'm not going to solve it
here, I'm just going to give

00:33:13.460 --> 00:33:14.190
you the solutions.

00:33:14.190 --> 00:33:16.050
Because that's what we're
interested in.

00:33:16.050 --> 00:33:18.300
After all, you're not expected
to learn how

00:33:18.300 --> 00:33:19.740
to solve this equation.

00:33:19.740 --> 00:33:22.250
Although you should be able to
put the solutions in and make

00:33:22.250 --> 00:33:33.350
sure that they do work out.

00:33:33.350 --> 00:33:37.190
And so the solutions are
harmonic solutions.

00:33:37.190 --> 00:33:41.180
Delta B is a function of time
is equal to delta B0.

00:33:41.180 --> 00:33:45.990
The amount of the perturbation
times the cosine of omega t,

00:33:45.990 --> 00:33:49.450
where omega is the frequency
of the oscillation.

00:33:49.450 --> 00:33:59.370
Minus some pre-factor here, k3
over k1 A to the 1/2 power.

00:33:59.370 --> 00:34:05.350
Times delta C0 sine
of omega t.

00:34:05.350 --> 00:34:07.810
And then delta C
looks the same.

00:34:07.810 --> 00:34:09.630
The same frequency.

00:34:09.630 --> 00:34:15.770
Delta C0, cosine of omega t.

00:34:15.770 --> 00:34:23.960
Plus k1 over A times k3
to the 1/2 power.

00:34:23.960 --> 00:34:29.820
Delta B0 sine of omega t, where
omega, the frequency of

00:34:29.820 --> 00:34:34.630
the oscillation, is these
rates. k1, k3 and A. So

00:34:34.630 --> 00:34:37.360
obviously, keeping the
concentration of a constant is

00:34:37.360 --> 00:34:40.210
important because we don't want
a time dependence here.

00:34:40.210 --> 00:34:42.680
The rain is constant.

00:34:42.680 --> 00:34:45.440
And then if you do that, then
you have this oscillation.

00:34:45.440 --> 00:34:48.570
Up and down and up and down
with this frequency here.

00:34:48.570 --> 00:34:50.770
You get exactly what we're
trying to get.

00:34:50.770 --> 00:34:58.620
Which is this process
right here.

00:34:58.620 --> 00:35:00.490
So let me give you an
example of this.

00:35:00.490 --> 00:35:08.040
And the best way to do is to
actually go and see a movie of

00:35:08.040 --> 00:35:20.730
a reaction that looks
like this.

00:35:20.730 --> 00:35:27.540
This is the reaction that was
optimized and created for

00:35:27.540 --> 00:35:30.220
demonstration purposes.

00:35:30.220 --> 00:35:35.240
So it's a pretty complicated
reaction.

00:35:35.240 --> 00:35:40.910
There are ten steps
to the mechanism.

00:35:40.910 --> 00:35:47.270
And if we gave you that
mechanism to solve for the

00:35:47.270 --> 00:35:49.200
final exam, you would still
be here probably

00:35:49.200 --> 00:35:51.360
two weeks form now.

00:35:51.360 --> 00:35:52.740
It's pretty complicated.

00:35:52.740 --> 00:35:54.780
You'd need a computer
for sure.

00:35:54.780 --> 00:35:57.040
So, ten steps, well, that's
what's known.

00:35:57.040 --> 00:35:58.520
There are probably
more than that.

00:35:58.520 --> 00:36:01.180
It's probably more complicated
than what we really know in

00:36:01.180 --> 00:36:06.210
terms of fishing out
intermediates.

00:36:06.210 --> 00:36:10.460
And the basic reaction is, you
start out with an oxide of

00:36:10.460 --> 00:36:15.290
iodine, which is clear.

00:36:15.290 --> 00:36:22.200
You create I2, which is
going to look gold.

00:36:22.200 --> 00:36:25.140
In solution, and then it creates
Ii minus, which is

00:36:25.140 --> 00:36:29.160
going to look deep blue.

00:36:29.160 --> 00:36:32.830
And then it'll feed back
and start again.

00:36:32.830 --> 00:36:35.400
So we're going to have this
cycle of products.

00:36:35.400 --> 00:36:38.250
And amongst those ten steps, or
more than ten steps, there

00:36:38.250 --> 00:36:44.700
are a few autocatalytic steps,
which provide the feedback.

00:36:44.700 --> 00:36:52.840
So now I have to start it.

00:36:52.840 --> 00:36:56.260
Let's go ahead and read.

00:37:09.960 --> 00:37:10.290
Alright.

00:37:10.290 --> 00:37:11.770
So here we go.

00:37:11.770 --> 00:37:18.660
So, there are two solutions, A
and solution C. Actually, we

00:37:18.660 --> 00:37:21.420
mix three solutions together.

00:37:21.420 --> 00:37:24.040
The potassium iodide is what is
going to give rise to the

00:37:24.040 --> 00:37:25.300
different colors.

00:37:25.300 --> 00:37:29.330
And now we're in the gold
phase of the reaction.

00:37:29.330 --> 00:37:31.580
It's important to stir,
otherwise it wouldn't, and

00:37:31.580 --> 00:37:33.740
then it turns deep blue.

00:37:33.740 --> 00:37:35.530
And then there's an induction
period, where

00:37:35.530 --> 00:37:37.710
you wait for a while.

00:37:37.710 --> 00:37:43.520
And then eventually it
should go clear.

00:37:43.520 --> 00:37:45.010
So it goes clear.

00:37:45.010 --> 00:37:46.340
And then it turns gold again.

00:37:46.340 --> 00:37:47.850
And then deep blue, and there's
the oscillation.

00:37:47.850 --> 00:37:50.060
And then a long induction
period, and then it goes

00:37:50.060 --> 00:37:54.150
through that cycle over
and over again.

00:37:54.150 --> 00:37:57.260
And the recipe can be
found pretty easily.

00:37:57.260 --> 00:38:01.570
And if you ever are in a lab,
have access to doing this,

00:38:01.570 --> 00:38:04.040
it's a pretty easy reaction
to put together.

00:38:04.040 --> 00:38:08.840
The only problem is that unless
you use fresh hydrogen

00:38:08.840 --> 00:38:11.510
peroxide, it doesn't work.

00:38:11.510 --> 00:38:14.560
Because hydrogen peroxide tends
to go bad over time.

00:38:14.560 --> 00:38:18.290
If you're going to do this,
buy your hydrogen peroxide

00:38:18.290 --> 00:38:21.070
immediately before you try
to do the experiment.

00:38:21.070 --> 00:38:22.420
And then it'll work.

00:38:22.420 --> 00:38:26.350
So what happens is that as
long as there is a steady

00:38:26.350 --> 00:38:31.290
supply of hydrogen peroxide in
here, it'll keep oscillating.

00:38:31.290 --> 00:38:34.340
But as you've used up the
hydrogen peroxide, this is

00:38:34.340 --> 00:38:35.240
going to start to die.

00:38:35.240 --> 00:38:39.050
And eventually it'll have
this pale blue solution.

00:38:39.050 --> 00:38:41.770
And it's like the
heart stopping.

00:38:41.770 --> 00:38:43.640
If you want to start it up
again, add more hydrogen

00:38:43.640 --> 00:38:50.360
peroxide and it'll
start up again.

00:38:50.360 --> 00:38:52.450
OK, any questions?

00:38:52.450 --> 00:38:54.760
There's like an infinite number
of these oscillating

00:38:54.760 --> 00:38:59.210
reactions that people
have discovered.

00:38:59.210 --> 00:39:09.600
And optimized, that you can
find in different places.

00:39:09.600 --> 00:39:13.000
Questions?

00:39:13.000 --> 00:39:18.330
Questions about the final.

00:39:18.330 --> 00:39:23.400
I didn't get to do a review,
but you'll get that

00:39:23.400 --> 00:39:26.610
done with the TAs.

00:39:26.610 --> 00:39:26.870
Yes.

00:39:26.870 --> 00:39:33.590
STUDENT: [INAUDIBLE]

00:39:33.590 --> 00:39:35.780
PROFESSOR: Because there's
enough of the equivalent of A,

00:39:35.780 --> 00:39:37.760
of the rain here.

00:39:37.760 --> 00:39:41.550
So it'll oscillate and there'll
be some damping.

00:39:41.550 --> 00:39:45.060
So what will happen in this
reaction is that the induction

00:39:45.060 --> 00:39:48.910
period will get longer, and
longer, and longer.

00:39:48.910 --> 00:39:55.530
Because the w, this term right
here, the frequency which

00:39:55.530 --> 00:39:58.590
depends on the concentration
of A, that well

00:39:58.590 --> 00:39:59.600
get slower and slower.

00:39:59.600 --> 00:40:06.910
And eventually it'll just die.

00:40:06.910 --> 00:40:08.160
OK?