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

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00:00:23 --> 00:00:26
about the order of
magnitude rates.

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00:00:26 --> 00:00:29
You asked the question
about the concentration

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00:00:29 --> 00:00:31
of the catalysts.
 

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00:00:31 --> 00:00:34
So the rates that I gave you,
which were the rates of

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00:00:34 --> 00:00:36
reactions are basically
order of magnitude.

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They're estimates and odd
numbers that, they obviously

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depend, as you pointed out,
depend on the concentration

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00:00:41 --> 00:00:42
of the catalyst.
 

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Depends on the concentration
of the reactants.

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00:00:46 --> 00:00:50
Depends on where you are
on those concentrations.

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00:00:50 --> 00:00:55
As we saw when we did enzyme
catalysis, is that depending on

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00:00:55 --> 00:01:00
KM, the Michaelis constant, and
the substrate concentration, if

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00:01:00 --> 00:01:05
your substrate's concentration
is much higher than KM, you are

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00:01:05 --> 00:01:10
in the maximum velocity limit,
where the rate depends

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00:01:10 --> 00:01:14
on k2 times the initial
concentration of enzyme.

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00:01:14 --> 00:01:17
In that case the rate of the
reaction depends just on the

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00:01:17 --> 00:01:20
concentration of enzyme, and
not on the concentration

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00:01:20 --> 00:01:21
of substrate.
 

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00:01:21 --> 00:01:27
But if you are at the small
concentration of substrate

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00:01:27 --> 00:01:31
relative to KM, then your rate
depends linearly with the

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00:01:31 --> 00:01:33
concentration of substrate.
 

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00:01:33 --> 00:01:39
And the same thing happens with
the non-enzymatic catalyst,

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00:01:39 --> 00:01:43
that there will be some sort of
mechanism that goes

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along with it.
 

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00:01:44 --> 00:01:46
And most likely it's
going to be second

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00:01:46 --> 00:01:47
order in the catalyst.
 

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00:01:47 --> 00:01:54
And the reactant if it's
just the two body process.

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00:01:54 --> 00:01:56
But it could be
more complicated.

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00:01:56 --> 00:01:58
So definitely those
concentrations

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00:01:58 --> 00:01:59
will go in there.
 

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00:01:59 --> 00:02:04
But a priori, it's not clear
how they will go in there.

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00:02:04 --> 00:02:06
But the orders of magnitude
are roughly right.

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00:02:06 --> 00:02:10
And the basic idea is that you
change your rate of reactions

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by 15, 18 orders of magnitude
with the right catalyst,

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especially if it's a
biological catalyst.

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00:02:18 --> 00:02:22
One concept that I didn't go
through last time, which

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is also quite useful, it's
called the turnover number.

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Turnover number.
 

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And this is useful for
any kind of catalyst.

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But for enzymatic, or enzyme
catalysts, the turnover

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number is k cat.
 

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The number per second.
 

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00:02:53 --> 00:02:55
It's a rate per second.
 

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How many times does a reactant
get, the number of times that a

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reactant turns into a product
per second, per catalyst.

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So let me write it out.
 

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So it's the number of product
formed per second, per

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second, per enzyme, per
molecule of enzyme.

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00:03:45 --> 00:03:48
OK, so if one enzyme is going
to be there, there's going to

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be some substrate going in.
 

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Coming in, the
pocket coming out.

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Coming in, the pocket coming
out, so the number of cycles

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of this process per second.
is the turnover number.

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And so if you are where the
concentration of substrate is

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very much larger than the
Michaelis constant, where are

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you are at the saturation
limit, then this turnover

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number doesn't depend on the
substrate concentration.

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It's just k cat.
 

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So that's why k
cat is important.

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As this turnover number in
the maximum velocity limit.

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Because in the maximum velocity
limit, then the process that's

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dominant is the second process,
ES goes to enzyme plus product.

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The first process here is
not the rate limiting step.

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This is the rate limiting step.
 

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Because the substrate
concentration is very high.

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So this first part goes
very high, goes very fast.

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And the number of product
formed per second, per molecule

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of enzyme, the number of
molecules formed for second,

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that's the rate of
product formation.

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Rate of product
formation per second.

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Divided by the concentration
of the enzyme.

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If you normalize by
the number of enzyme.

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00:05:22 --> 00:05:25
So this becomes rate of product
formation in moles per second,

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divided by moles of enzyme.
 

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It's the same thing as what
we're saying here in words.

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And so the rate of product
formation is dP/dt, and you

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divide that by the
concentration of enzyme, which

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you can write as E0. dp/dt, and
the maximum rate

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limit is k max.
 

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It's a constant divided by E0.
 

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And k max is k cat times
E0 divided by E0.

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The E0's come out.
 

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00:05:57 --> 00:06:00
And this is k cat.
 

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Which is basically k2.
 

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00:06:08 --> 00:06:10
So that's the turnover number.
 

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00:06:10 --> 00:06:12
Any questions on this
turnover number?

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You see that a lot in
catalyst literature and

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especially the enzymes.
 

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00:06:19 --> 00:06:27
I have to get out
these lecture notes.

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Send these back.
 

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OK, so the last topic I
want to talk about is

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oscillating reactions.
 

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It's sort of like an
interesting topic.

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And it's applicable not just
to just chemistry, but it's

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basically playing with the
differential equations.

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And you'll see this sort of
stuff later if you keep doing

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science that involves coupled
differential equations.

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It's just all over the place.
 

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So normally we have an
equilibrium process.

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A goes to B, B goes back to A.
 

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00:07:16 --> 00:07:19
And if you start out with
something which is out of

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equilibrium, then you will
eventually reach equilibrium at

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00:07:25 --> 00:07:29
a rate which is dependent on
those two rates. k1

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plus k minus one.
 

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So B will to come up to
some B equilibrium.

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00:07:33 --> 00:07:38
And A will, if you start out
with a lot of A, and a little

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00:07:38 --> 00:07:43
bit of B, and concentration of
A, will come down smoothly,

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monotonically, to the
equilibrium state.

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00:07:48 --> 00:07:54
But sometimes, if you're out of
equilibrium, and this doesn't

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00:07:54 --> 00:07:54
have to be chemistry.
 

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00:07:54 --> 00:08:02
It could be anything that's
out of equilibrium, your

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00:08:02 --> 00:08:05
emotion's out of equilibrium.
 

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Stock market is out
of equilibrium.

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Population dynamics are
out of equilibrium.

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The weather.
 

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00:08:11 --> 00:08:14
Whatever.
 

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So, you're out of equilibrium,
let's say this is equilibrium

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for A, this is equilibrium for
B, let me try to keep them

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00:08:24 --> 00:08:28
somewhat the same as
here, I messed up.

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You're out of equilibrium,
you start down here for A.

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And you, instead of going
linearly down or monotonically

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down, exponentially down to the
equilibrium, instead

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you overshoot.
 

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And you keep going
back and forth.

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00:08:42 --> 00:08:44
Like a spring.
 

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00:08:44 --> 00:08:46
And then B does the same thing.
 

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Overshoots, comes down.
 

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00:08:48 --> 00:08:50
Overshoots, comes down.
 

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00:08:50 --> 00:08:51
Like a pendulum.
 

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00:08:51 --> 00:08:58
A lot of things work like that.
 

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The heart is something that
works like that, right?

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It beats.
 

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It's not in equilibrium.
 

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00:09:04 --> 00:09:08
Good thing it's not
in equilibrium.

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So the heart is one example of
something that oscillates, back

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00:09:11 --> 00:09:13
and forth, back and forth.
 

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00:09:13 --> 00:09:16
And so there are actually many
examples of complicated

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00:09:16 --> 00:09:19
processes that involve
chemistry, that aren't at

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equilibrium, that instead go
back and forth, in and out,

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of on one side or the
other of equilibrium.

150
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Now, there's an important
feature of, if you want to

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00:09:35 --> 00:09:42
build a chemical process
that looks like this.

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It's often very useful to have
a particular step in the

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mechanism that's called
an autocatalysis step.

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00:09:56 --> 00:09:59
And that provides feedback.
 

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00:09:59 --> 00:10:05
It's like, if my microphone
here, the speaker which is

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00:10:05 --> 00:10:11
here, I were to stand right in
front of the speaker and my

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00:10:11 --> 00:10:15
microphone would pick up the
volume from the speaker, it

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00:10:15 --> 00:10:17
would get amplified,
and I get feedback.

159
00:10:17 --> 00:10:21
And that would be a bad thing.
 

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00:10:21 --> 00:10:22
So this is a form
of feedback here.

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You need some sort of feedback
that the reaction knows that

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00:10:26 --> 00:10:29
you're building up, that
this is going too fast.

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00:10:29 --> 00:10:33
And then there's a feedback
process that brings it back up.

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00:10:33 --> 00:10:36
And a feedback process
that goes back and forth.

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00:10:36 --> 00:10:40
And the way you get this
feedback is by having a step in

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00:10:40 --> 00:10:53
the mechanism that has both a
molecule as part of the

167
00:10:53 --> 00:10:58
reactants and as part
of the product.

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00:10:58 --> 00:11:02
It's not the same as a chain
reaction, where we talked

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00:11:02 --> 00:11:06
about an intermediate being
recycled and building up.

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00:11:06 --> 00:11:09
This is an actual reactant
and an actual product that's

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00:11:09 --> 00:11:16
part of this step here.
 

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00:11:16 --> 00:11:20
So this is an
autocatalytic step.

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00:11:20 --> 00:11:24
And let's just see what
that step looks like.

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00:11:24 --> 00:11:26
Suppose we just have that step.
 

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00:11:26 --> 00:11:29
So we want to find out, as a
function of time, if you start

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out with some A and B, what
happens to the concentration

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00:11:33 --> 00:11:34
of B as a function of time.
 

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00:11:34 --> 00:11:36
The time dependence of that.
 

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00:11:36 --> 00:11:37
Clearly it's going to build up.
 

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00:11:37 --> 00:11:40
But how is it going
to build up?

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00:11:40 --> 00:11:46
So we need to solve for B
as a function of time.

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00:11:46 --> 00:11:48
So let's go ahead and
write down our rates.

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00:11:48 --> 00:11:53
The rate of the reaction is
d[A]/dt, is, let's put a rate

184
00:11:53 --> 00:11:58
constant here. k, k times [A]
 

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00:11:58 --> 00:12:00
times [B].
 

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00:12:00 --> 00:12:00
Now, [B]
 

187
00:12:00 --> 00:12:01
and [A]
 

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00:12:01 --> 00:12:05
are related to each other
through stoichiometry.

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00:12:05 --> 00:12:11
The concentration of B is the
concentration of B that he

190
00:12:11 --> 00:12:17
started out with, [B]0, then
every time you destroy

191
00:12:17 --> 00:12:21
an A, you create 2B.
 

192
00:12:21 --> 00:12:25
So it's plus two times the
amount of A that you've

193
00:12:25 --> 00:12:32
destroyed, [A]0 minus [A], [A]0
is what you started out with.

194
00:12:32 --> 00:12:33
This is what's left over.
 

195
00:12:33 --> 00:12:37
So the difference is
what you've destroyed.

196
00:12:37 --> 00:12:40
But every time you destroy
an A to form 2B, you

197
00:12:40 --> 00:12:43
also destroy a b.
 

198
00:12:43 --> 00:12:46
So you have to subtract away
the B that you've destroyed.

199
00:12:46 --> 00:12:52
Which is [A]0 minus [A].
 

200
00:12:52 --> 00:12:56
This is the 2B that you created
by destroying A, and this is

201
00:12:56 --> 00:12:59
the B that you destroy by
having to destroy A,

202
00:12:59 --> 00:13:02
for the reaction.
 

203
00:13:02 --> 00:13:05
You've got to
bean-count correctly.

204
00:13:05 --> 00:13:07
Keep track of everything.
 

205
00:13:07 --> 00:13:11
So [B], then, is just, I'm
going to drop those brackets.

206
00:13:11 --> 00:13:15
B0 plus A0 minus A.
 

207
00:13:15 --> 00:13:17
So you can plug that in here.
 

208
00:13:17 --> 00:13:28
And is equal to k times A,
times B0 plus B0 minus A.

209
00:13:28 --> 00:13:34
And now you have a differential
equation that only

210
00:13:34 --> 00:13:37
contains A and time.
 

211
00:13:37 --> 00:13:39
B0 is a constant.
 

212
00:13:39 --> 00:13:43
And the way you solve that
is by partial fractions.

213
00:13:43 --> 00:13:49
Use partial fractions,
solve for this.

214
00:13:49 --> 00:13:52
And I'm not going to go through
the math of solving for it.

215
00:13:52 --> 00:13:59
It's not that interesting.
 

216
00:13:59 --> 00:14:03
Let me give you the answer.
 

217
00:14:03 --> 00:14:04
You end up with [B]
 

218
00:14:04 --> 00:14:11
as a function of time, is this
function A0 plus B0 over one

219
00:14:11 --> 00:14:19
plus A0 over B0 times e to
the minus k times A0

220
00:14:19 --> 00:14:24
plus B0 plus time.
 

221
00:14:24 --> 00:14:26
And that's where the time
component comes in.

222
00:14:26 --> 00:14:28
Whoops, I'm sorry.
 

223
00:14:28 --> 00:14:33
Usually I don't have this
on, let's turn this off.

224
00:14:33 --> 00:14:41
That's where the time
dependence comes in.

225
00:14:41 --> 00:14:45
And clearly, at t is equal to
infinity, t equals infinity

226
00:14:45 --> 00:14:47
just goes to zero.
 

227
00:14:47 --> 00:14:50
So you have B as A0, plus B0,
everything goes through to the

228
00:14:50 --> 00:14:54
product. t equals to zero,
this is equal to one.

229
00:14:54 --> 00:15:00
And B is equal to just B0.
 

230
00:15:00 --> 00:15:05
Alright, and so the curve
looks something like this.

231
00:15:05 --> 00:15:09
This is what it turns
out to look like.

232
00:15:09 --> 00:15:11
We're starting with B0 here.
 

233
00:15:11 --> 00:15:14
There's an induction period
where very little happens.

234
00:15:14 --> 00:15:17
It's like the dormant stage.
 

235
00:15:17 --> 00:15:21
So, locusts that are sitting in
the ground for seventeen years

236
00:15:21 --> 00:15:27
waiting for something
to happen.

237
00:15:27 --> 00:15:28
Dormant stage, the
induction period.

238
00:15:28 --> 00:15:30
And suddenly something
starts to happen.

239
00:15:30 --> 00:15:35
There's an inflection point.
 

240
00:15:35 --> 00:15:39
And you get to A0 plus B0.
 

241
00:15:39 --> 00:15:41
The locusts wake up.
 

242
00:15:41 --> 00:15:47
And go to maximum
concentration.

243
00:15:47 --> 00:15:55
So there's an s-like shape
that has an inflection point

244
00:15:55 --> 00:16:04
and an induction period.
 

245
00:16:04 --> 00:16:10
That's pretty typical of an
auto-catalytic reaction.

246
00:16:10 --> 00:16:13
OK so let's apply it
to a process now.

247
00:16:13 --> 00:16:16
And again, this is going to be
coupled differential equations.

248
00:16:16 --> 00:16:18
Broadly, broadly applicable,
the example that I'm

249
00:16:18 --> 00:16:20
going to give you.
 

250
00:16:20 --> 00:16:45
This mechanism has been used
for all sorts of things.

251
00:16:45 --> 00:16:45
So what do we have here?
 

252
00:16:45 --> 00:16:51
We're going to have an island.
 

253
00:16:51 --> 00:16:56
In the ocean.
 

254
00:16:56 --> 00:16:58
Island here.
 

255
00:16:58 --> 00:17:03
And it's going to be
sunny, so let's get

256
00:17:03 --> 00:17:05
some yellow chalk here.
 

257
00:17:05 --> 00:17:08
There's the sun.
 

258
00:17:08 --> 00:17:11
And every now and then
it's going to rain.

259
00:17:11 --> 00:17:15
We need some white
chalk for the rain.

260
00:17:15 --> 00:17:22
There's a cloud that comes in.
 

261
00:17:22 --> 00:17:26
That's going to be one
of the reactants.

262
00:17:26 --> 00:17:31
And then some seeds blow in
from the continent, and grass

263
00:17:31 --> 00:17:35
starts to grow on the island.
 

264
00:17:35 --> 00:17:38
Grass.
 

265
00:17:38 --> 00:17:41
And then there's a shipwreck
and a couple of rabbits

266
00:17:41 --> 00:17:43
get on the island.
 

267
00:17:43 --> 00:17:55
And now we have rabbits.
 

268
00:17:55 --> 00:17:55
Rabbits.
 

269
00:17:55 --> 00:17:58
A little tail on the back.
 

270
00:17:58 --> 00:18:00
I hate rabbits.
 

271
00:18:00 --> 00:18:04
I have a little vegetable
garden, a little tiny

272
00:18:04 --> 00:18:05
vegetable garden.
 

273
00:18:05 --> 00:18:06
And there are rabbits.
 

274
00:18:06 --> 00:18:10
And I have to fence in my
vegetable garden and make it

275
00:18:10 --> 00:18:13
impermeable impenetrable
to rabbits.

276
00:18:13 --> 00:18:15
And rabbits are very clever.
 

277
00:18:15 --> 00:18:18
And my vegetable garden
looks like a fortress.

278
00:18:18 --> 00:18:21
It's got a green fence.
 

279
00:18:21 --> 00:18:23
Heavy duty things.
 

280
00:18:23 --> 00:18:24
It's terrible.
 

281
00:18:24 --> 00:18:25
Rabbits are awful.
 

282
00:18:25 --> 00:18:27
They're very cute
but they're awful.

283
00:18:27 --> 00:18:29
Alright, so now what happens.
 

284
00:18:29 --> 00:18:33
Well, given the rain and the
grass, it turns out that

285
00:18:33 --> 00:18:38
this island can only
support 25 rabbits.

286
00:18:38 --> 00:18:39
If there are more rabbits
than that, then they

287
00:18:39 --> 00:18:40
all eat too much.
 

288
00:18:40 --> 00:18:43
If there are less rabbits
then that's fine.

289
00:18:43 --> 00:18:45
But 25 is about the
maximum that can

290
00:18:45 --> 00:18:46
support on this island.
 

291
00:18:46 --> 00:18:51
The rabbits, unfortunately,
are not that smart.

292
00:18:51 --> 00:18:52
They don't know that.
 

293
00:18:52 --> 00:18:56
They don't know that only
25 rabbits can live there.

294
00:18:56 --> 00:19:04
So, what happens is that
the Year one, Year one

295
00:19:04 --> 00:19:06
there are ten rabbits.
 

296
00:19:06 --> 00:19:11
And the food
condition is great.

297
00:19:11 --> 00:19:18
Lots of food,
number of rabbits.

298
00:19:18 --> 00:19:20
Lots of food.
 

299
00:19:20 --> 00:19:22
So the bunnies, the
rabbits make bunnies.

300
00:19:22 --> 00:19:23
Lots of rabbits.
 

301
00:19:23 --> 00:19:26
Year two, they kind
of went overboard.

302
00:19:26 --> 00:19:29
Now there are 50 rabbits.
 

303
00:19:29 --> 00:19:30
The food is lousy.
 

304
00:19:30 --> 00:19:34
Terrible.
 

305
00:19:34 --> 00:19:36
Well, there's a feedback here.
 

306
00:19:36 --> 00:19:40
Feedback and rabbits
start to die off.

307
00:19:40 --> 00:19:42
Don't reproduce as much.
 

308
00:19:42 --> 00:19:44
Year three, we're back to ten.
 

309
00:19:44 --> 00:19:47
Great food.
 

310
00:19:47 --> 00:19:48
Rabbits don't notice.
 

311
00:19:48 --> 00:19:50
They multiply.
 

312
00:19:50 --> 00:19:53
Food is terrible.
 

313
00:19:53 --> 00:19:55
And they just don't learn.
 

314
00:19:55 --> 00:19:56
Humans are a little
bit like that.

315
00:19:56 --> 00:19:59
Except our cycle isn't one
year, it's usually on

316
00:19:59 --> 00:20:00
the order of 30 years.
 

317
00:20:00 --> 00:20:03
It's like the memory of a
generation disappearing.

318
00:20:03 --> 00:20:05
And we have to relearn
the mistakes of the

319
00:20:05 --> 00:20:09
previous generation.
 

320
00:20:09 --> 00:20:10
Science is like that too.
 

321
00:20:10 --> 00:20:12
Science goes in cycles.
 

322
00:20:12 --> 00:20:13
20, 30-year cycles.
 

323
00:20:13 --> 00:20:17
Topics that were out of
fashion 20 years ago, 25

324
00:20:17 --> 00:20:19
years ago, come back.
 

325
00:20:19 --> 00:20:21
And suddenly everybody's
excited about them.

326
00:20:21 --> 00:20:24
And all the ideas, of
course we make progress.

327
00:20:24 --> 00:20:26
We go a little bit further
than we did 20 years ago.

328
00:20:26 --> 00:20:28
But all the ideas that
were out of fashion 20

329
00:20:28 --> 00:20:29
years ago come back.
 

330
00:20:29 --> 00:20:31
People get really excited,
and they relearn all the

331
00:20:31 --> 00:20:34
mistakes that people
made at the beginning.

332
00:20:34 --> 00:20:37
Technology has improved, we
know more, and therefore we go

333
00:20:37 --> 00:20:40
a little bit further this cycle
than we did the previous cycle.

334
00:20:40 --> 00:20:42
But it's really interesting to
look at history of science

335
00:20:42 --> 00:20:52
and see this sort of cycle.
 

336
00:20:52 --> 00:20:55
So let's write a mechanism.
 

337
00:20:55 --> 00:20:59
Let's write a mechanism here.
 

338
00:20:59 --> 00:21:07
So we have rain plus
grass makes more grass.

339
00:21:07 --> 00:21:09
At the rate k1.
 

340
00:21:09 --> 00:21:10
That's autocatalytic.
 

341
00:21:10 --> 00:21:12
It provides some feedback.
 

342
00:21:12 --> 00:21:21
Then we have grass plus
rabbits make more rabbits.

343
00:21:21 --> 00:21:24
OK, let's do more because
I don't know how

344
00:21:24 --> 00:21:28
much, more, more.
 

345
00:21:28 --> 00:21:30
Also autocatalytic.
 

346
00:21:30 --> 00:21:33
Then we have rabbits.
 

347
00:21:33 --> 00:21:39
Eventually, unfortunately,
rabbits become dead rabbits.

348
00:21:39 --> 00:21:45
Rate k2 k3, and that's the
termination of the process.

349
00:21:45 --> 00:21:47
OK, so now let's put
some chemistry on this.

350
00:21:47 --> 00:21:50
Let's assume that instead
of having objects, live

351
00:21:50 --> 00:21:54
objects here, we have
chemical molecules.

352
00:21:54 --> 00:21:57
So A plus B goes to 2B.
 

353
00:21:57 --> 00:21:59
That's the first step.
 

354
00:21:59 --> 00:22:05
Then we have B plus
C, C is the rabbits.

355
00:22:05 --> 00:22:10
And then C goes to some
sort of products.

356
00:22:10 --> 00:22:15
Soil.
 

357
00:22:15 --> 00:22:18
This is a very
famous mechanism.

358
00:22:18 --> 00:22:26
It's called the
Lotka-Volterra mechanism.

359
00:22:26 --> 00:22:33
It's also called the
predator-prey mechanism.

360
00:22:33 --> 00:22:36
In my case here, I made it
sort of warm and fuzzy.

361
00:22:36 --> 00:22:38
I didn't have a
predator in there.

362
00:22:38 --> 00:22:41
But you could change this.
 

363
00:22:41 --> 00:22:44
Instead of having rabbits
and grass, we could

364
00:22:44 --> 00:22:47
have rabbits and foxes.
 

365
00:22:47 --> 00:22:54
Where A is the grass, if you
take A to be the grass.

366
00:22:54 --> 00:22:57
B to be the rabbits.
 

367
00:22:57 --> 00:23:01
And C you be the foxes.
 

368
00:23:01 --> 00:23:02
Works the same.
 

369
00:23:02 --> 00:23:05
You have grass plus rabbits
makes more rabbits.

370
00:23:05 --> 00:23:07
Rabbit plus foxes
make more foxes.

371
00:23:07 --> 00:23:10
Eventually the foxes die off.
 

372
00:23:10 --> 00:23:12
Same idea.
 

373
00:23:12 --> 00:23:32
So now, let's solve this.
 

374
00:23:32 --> 00:23:38
Let's assume that A, the
rain here, it's constant.

375
00:23:38 --> 00:23:40
There's a steady
supply of rain.

376
00:23:40 --> 00:23:41
It doesn't go away.
 

377
00:23:41 --> 00:23:44
That makes sense in
my example here.

378
00:23:44 --> 00:23:50
So the concentration of
A is kept constant.

379
00:23:50 --> 00:23:53
In order to get oscillations
to keep going, that turns

380
00:23:53 --> 00:23:54
out to be important.
 

381
00:23:54 --> 00:23:58
To have one of the reactants
just keeping being

382
00:23:58 --> 00:23:59
reintroduced in the system.
 

383
00:23:59 --> 00:24:01
Because it gets used up.
 

384
00:24:01 --> 00:24:06
And when all the reactant gets
used up, then you're done.

385
00:24:06 --> 00:24:10
So in the case of the heart,
we have to keep feeding

386
00:24:10 --> 00:24:12
ourselves to produce energy.
 

387
00:24:12 --> 00:24:18
Otherwise the heart would stop.
 

388
00:24:18 --> 00:24:22
OK, now we want to know what
is the concentration of B

389
00:24:22 --> 00:24:26
and C as a function of time.
 

390
00:24:26 --> 00:24:36
So we write down our kinetic
equations. dB/dt gets produced

391
00:24:36 --> 00:24:39
through the first step, k1 A
times B, gets destroyed in the

392
00:24:39 --> 00:24:47
second step. k2 B times C.
dC/dt, C gets produced in the

393
00:24:47 --> 00:24:51
second step, k2 B times C.
 

394
00:24:51 --> 00:24:54
Gets destroyed in the third
step. k3 times C, got to do

395
00:24:54 --> 00:24:56
your bean-counting correctly.
 

396
00:24:56 --> 00:24:58
And the strategy here that
we're going to use to solve

397
00:24:58 --> 00:25:03
the problem is to assume
that we're at steady state.

398
00:25:03 --> 00:25:06
We're going to find the
solution at steady state.

399
00:25:06 --> 00:25:07
Then we're going to
perturb away from steady

400
00:25:07 --> 00:25:09
state a little bit.
 

401
00:25:09 --> 00:25:13
And see whether this
creates an oscillation.

402
00:25:13 --> 00:25:17
First we need to find out what
the steady state solution is.

403
00:25:17 --> 00:25:21
So first, let's
find steady state.

404
00:25:21 --> 00:25:30
Then perturb away
from steady state.

405
00:25:30 --> 00:25:32
We're looking at a harmonic
response, if you've

406
00:25:32 --> 00:25:33
heard that term.
 

407
00:25:33 --> 00:25:36
When it's going to look like
a spring, close to its

408
00:25:36 --> 00:25:41
equilibrium state.
 

409
00:25:41 --> 00:25:44
That means that we're going to
set it these two things equal

410
00:25:44 --> 00:25:47
to zero, for this steady state.
 

411
00:25:47 --> 00:25:48
Things are not changing.
 

412
00:25:48 --> 00:25:50
Concentration of B and
C are not changing.

413
00:25:50 --> 00:25:53
A is not changing
on purpose here.

414
00:25:53 --> 00:25:56
And so all these guys here
are then steady state

415
00:25:56 --> 00:25:57
concentrations.
 

416
00:25:57 --> 00:26:02
If we set it equal to zero.
 

417
00:26:02 --> 00:26:10
We solve for A in this.
 

418
00:26:10 --> 00:26:11
For C in this process here.
 

419
00:26:11 --> 00:26:13
The B's cancel out here.
 

420
00:26:13 --> 00:26:14
We divide by B.
 

421
00:26:14 --> 00:26:19
And you solve for C
as a function of A.

422
00:26:19 --> 00:26:25
So this one here gives you
C steady state is equal

423
00:26:25 --> 00:26:31
to k1 over k2 times the
concentration of A.

424
00:26:31 --> 00:26:35
And this one here, now you put
in the C and you solve for

425
00:26:35 --> 00:26:58
B, B steady state, is
equal to k3 over k2.

426
00:26:58 --> 00:27:02
So the next step is to perturb
away from equilibrium.

427
00:27:02 --> 00:27:03
Or from the steady
state, rather.

428
00:27:03 --> 00:27:05
This is not equilibrium, it's
not equilibrium because

429
00:27:05 --> 00:27:11
we keep adding A.
 

430
00:27:11 --> 00:27:18
We have to put some
input into the system.

431
00:27:18 --> 00:27:19
So here we are.
 

432
00:27:19 --> 00:27:23
We've got some concentration of
B here at the steady state.

433
00:27:23 --> 00:27:28
We have some concentration
of C at the steady state.

434
00:27:28 --> 00:27:34
And now we're going to
add some delta B to the

435
00:27:34 --> 00:27:36
system, or delta C.
 

436
00:27:36 --> 00:27:37
Or both.
 

437
00:27:37 --> 00:27:44
Delta C, we're going to ask
the question, given our

438
00:27:44 --> 00:27:47
system here, how is
it going to respond?

439
00:27:47 --> 00:27:51
It has a number of choices.
 

440
00:27:51 --> 00:27:57
It could respond by going
back to the steady state

441
00:27:57 --> 00:28:03
in a monotonic fashion.
 

442
00:28:03 --> 00:28:07
Without overshooting.
 

443
00:28:07 --> 00:28:15
This is kind of like
an overdamped system.

444
00:28:15 --> 00:28:18
It's like having a shock
absorber on your car.

445
00:28:18 --> 00:28:21
You go over a bump, the car
doesn't oscillate up and down.

446
00:28:21 --> 00:28:22
It's sort of damped.
 

447
00:28:22 --> 00:28:23
It's a damped oscillator.
 

448
00:28:23 --> 00:28:27
It goes back to
the steady state.

449
00:28:27 --> 00:28:38
Or, you could, there
we go, there's B, C.

450
00:28:38 --> 00:28:44
Or you could perturb, and it
could just stay where it is.

451
00:28:44 --> 00:28:46
That could happen too.
 

452
00:28:46 --> 00:28:54
That's an inelastic response.
 

453
00:28:54 --> 00:28:58
Kind of like the supply
and demand with oil.

454
00:28:58 --> 00:29:02
The price of oil goes from $10
a barrel to $120 a barrel.

455
00:29:02 --> 00:29:06
The use of oil doesn't seem
to be affected very much

456
00:29:06 --> 00:29:07
by the increasing price.
 

457
00:29:07 --> 00:29:08
You're still using
just as much oil.

458
00:29:08 --> 00:29:16
There's an inelastic response.
 

459
00:29:16 --> 00:29:20
What we're looking for is
something that has an

460
00:29:20 --> 00:29:22
elastic response, or
a harmonic response.

461
00:29:22 --> 00:29:34
We perturb it, it tries to
get back to steady state.

462
00:29:34 --> 00:29:38
But because of feedback,
and not being over-damped,

463
00:29:38 --> 00:29:42
if it's an oscillator,
it goes up and down.

464
00:29:42 --> 00:29:51
Like a scale that is not well,
well, like a car that has bad

465
00:29:51 --> 00:29:53
shock absorbers, all it
has is the springs.

466
00:29:53 --> 00:29:57
Up and down.
 

467
00:29:57 --> 00:30:03
So what we want is, we want
to solve for dB/dt as

468
00:30:03 --> 00:30:22
a function of time.
 

469
00:30:22 --> 00:30:24
We want, well, I'm going to
do everything in green.

470
00:30:24 --> 00:30:27
Since I lost my white chalk.
 

471
00:30:27 --> 00:30:33
We want delta B, delta C
as a function of time.

472
00:30:33 --> 00:30:39
So, that means that we start
with B is B steady state plus

473
00:30:39 --> 00:30:44
delta B, C is C steady
state plus delta C.

474
00:30:44 --> 00:30:46
And we put that in
our equations.

475
00:30:46 --> 00:30:48
For dB/dt and dC/dt.
 

476
00:30:48 --> 00:30:57
So now dB/dt is the same thing
as dB steady state plus delta B

477
00:30:57 --> 00:31:01
dt, which is d delta B / dt.
 

478
00:31:01 --> 00:31:04
Because this is a
constant here.

479
00:31:04 --> 00:31:10
And that's equal to k1
times A, times B steady

480
00:31:10 --> 00:31:13
state plus delta B.
 

481
00:31:13 --> 00:31:21
Minus k2 times B steady
state plus delta B times C

482
00:31:21 --> 00:31:23
steady state plus delta C.
 

483
00:31:23 --> 00:31:29
And then you have the same
thing for dC/dt, it's going

484
00:31:29 --> 00:31:34
to look exactly identical.
d delta C / dt.

485
00:31:34 --> 00:31:49
Thank you much, this is magic.
k2 times B steady state plus

486
00:31:49 --> 00:31:57
delta B, times C steady state
plus delta C, minus k3 times

487
00:31:57 --> 00:32:00
C steady state plus delta C.
 

488
00:32:00 --> 00:32:02
So these are two
differential equations.

489
00:32:02 --> 00:32:04
And there are coupled
differential equations.

490
00:32:04 --> 00:32:07
Because delta B is occurring
here, here, and here.

491
00:32:07 --> 00:32:12
So the time derivative of
delta C depends on delta B.

492
00:32:12 --> 00:32:17
And the time derivative of
delta B depends on delta C.

493
00:32:17 --> 00:32:21
It's a coupled system.
 

494
00:32:21 --> 00:32:31
And if you actually expand this
out and take away all the terms

495
00:32:31 --> 00:32:34
that cancel out, you end up
with something that looks much

496
00:32:34 --> 00:32:40
simpler but is just as hard. d
delta B / dt is equal to minus

497
00:32:40 --> 00:32:53
k3 delta C and d delta C / dt
is equal to k1 A delta B.

498
00:32:53 --> 00:32:56
And then you really see that
it's coupled, because the

499
00:32:56 --> 00:32:59
time derivative of delta
B depends on delta C.

500
00:32:59 --> 00:33:01
And the time derivative of
delta C depends on delta B.

501
00:33:01 --> 00:33:08
And there's the feedback
sitting right here.

502
00:33:08 --> 00:33:09
And that's what you do.
 

503
00:33:09 --> 00:33:11
You learn how to
solve in 18.03.

504
00:33:11 --> 00:33:13
So I'm not going to solve
it here, I'm just going to

505
00:33:13 --> 00:33:14
give you the solutions.
 

506
00:33:14 --> 00:33:16
Because that's what
we're interested in.

507
00:33:16 --> 00:33:18
After all, you're not
expected to learn how

508
00:33:18 --> 00:33:19
to solve this equation.
 

509
00:33:19 --> 00:33:22
Although you should be able to
put the solutions in and make

510
00:33:22 --> 00:33:33
sure that they do work out.
 

511
00:33:33 --> 00:33:37
And so the solutions are
harmonic solutions.

512
00:33:37 --> 00:33:41
Delta B is a function of
time is equal to delta B0.

513
00:33:41 --> 00:33:45
The amount of the perturbation
times the cosine of omega t,

514
00:33:45 --> 00:33:49
where omega is the frequency
of the oscillation.

515
00:33:49 --> 00:33:59
Minus some pre-factor here, k3
over k1 A to the 1/2 power.

516
00:33:59 --> 00:34:05
Times delta C0 sine of omega t.
 

517
00:34:05 --> 00:34:07
And then delta C
looks the same.

518
00:34:07 --> 00:34:09
The same frequency.
 

519
00:34:09 --> 00:34:15
Delta C0, cosine of omega t.
 

520
00:34:15 --> 00:34:23
Plus k1 over A times
k3 to the 1/2 power.

521
00:34:23 --> 00:34:29
Delta B0 sine of omega t, where
omega, the frequency of the

522
00:34:29 --> 00:34:34
oscillation, is these
rates. k1, k3 and A.

523
00:34:34 --> 00:34:37
So obviously, keeping the
concentration of a constant is

524
00:34:37 --> 00:34:40
important because we don't
want a time dependence here.

525
00:34:40 --> 00:34:42
The rain is constant.
 

526
00:34:42 --> 00:34:45
And then if you do that, then
you have this oscillation.

527
00:34:45 --> 00:34:48
Up and down and up and down
with this frequency here.

528
00:34:48 --> 00:34:50
You get exactly what
we're trying to get.

529
00:34:50 --> 00:34:58
Which is this
process right here.

530
00:34:58 --> 00:35:00
So let me give you
an example of this.

531
00:35:00 --> 00:35:07
And the best way to do is to
actually go and see a movie

532
00:35:07 --> 00:35:20
of a reaction that
looks like this.

533
00:35:20 --> 00:35:27
This is the reaction that was
optimized and created for

534
00:35:27 --> 00:35:30
demonstration purposes.
 

535
00:35:30 --> 00:35:35
So it's a pretty
complicated reaction.

536
00:35:35 --> 00:35:40
There are ten steps
to the mechanism.

537
00:35:40 --> 00:35:47
And if we gave you that
mechanism to solve for the

538
00:35:47 --> 00:35:49
final exam, you would
still be here probably

539
00:35:49 --> 00:35:51
two weeks form now.
 

540
00:35:51 --> 00:35:52
It's pretty complicated.
 

541
00:35:52 --> 00:35:54
You'd need a computer for sure.
 

542
00:35:54 --> 00:35:57
So, ten steps, well,
that's what's known.

543
00:35:57 --> 00:35:58
There are probably
more than that.

544
00:35:58 --> 00:36:01
It's probably more complicated
than what we really know

545
00:36:01 --> 00:36:06
in terms of fishing
out intermediates.

546
00:36:06 --> 00:36:10
And the basic reaction is, you
start out with an oxide of

547
00:36:10 --> 00:36:15
iodine, which is clear.
 

548
00:36:15 --> 00:36:22
You create I2, which is
going to look gold.

549
00:36:22 --> 00:36:25
In solution, and then it
creates Ii minus, which is

550
00:36:25 --> 00:36:29
going to look deep blue.
 

551
00:36:29 --> 00:36:32
And then it'll feed
back and start again.

552
00:36:32 --> 00:36:35
So we're going to have
this cycle of products.

553
00:36:35 --> 00:36:38
And amongst those ten steps, or
more than ten steps, there are

554
00:36:38 --> 00:36:44
a few autocatalytic steps,
which provide the feedback.

555
00:36:44 --> 00:36:52
So now I have to start it.
 

556
00:36:52 --> 00:36:56
Let's go ahead and read.
 

557
00:36:56 --> 00:37:09
 


558
00:37:09 --> 00:37:10
Alright.
 

559
00:37:10 --> 00:37:11
So here we go.
 

560
00:37:11 --> 00:37:18
So, there are two solutions,
A and solution C.

561
00:37:18 --> 00:37:21
Actually, we mix three
solutions together.

562
00:37:21 --> 00:37:23
The potassium iodide is what
is going to give rise to

563
00:37:23 --> 00:37:25
the different colors.
 

564
00:37:25 --> 00:37:29
And now we're in the gold
phase of the reaction.

565
00:37:29 --> 00:37:31
It's important to stir,
otherwise it wouldn't, and

566
00:37:31 --> 00:37:33
then it turns deep blue.
 

567
00:37:33 --> 00:37:35
And then there's an
induction period, where

568
00:37:35 --> 00:37:37
you wait for a while.
 

569
00:37:37 --> 00:37:43
And then eventually
it should go clear.

570
00:37:43 --> 00:37:45
So it goes clear.
 

571
00:37:45 --> 00:37:46
And then it turns gold again.
 

572
00:37:46 --> 00:37:47
And then deep blue, and
there's the oscillation.

573
00:37:47 --> 00:37:49
And then a long induction
period, and then it

574
00:37:49 --> 00:37:54
goes through that cycle
over and over again.

575
00:37:54 --> 00:37:57
And the recipe can be
found pretty easily.

576
00:37:57 --> 00:38:01
And if you ever are in a
lab, have access to doing

577
00:38:01 --> 00:38:04
this, it's a pretty easy
reaction to put together.

578
00:38:04 --> 00:38:08
The only problem is that unless
you use fresh hydrogen

579
00:38:08 --> 00:38:11
peroxide, it doesn't work.
 

580
00:38:11 --> 00:38:14
Because hydrogen peroxide
tends to go bad over time.

581
00:38:14 --> 00:38:18
If you're going to do this,
buy your hydrogen peroxide

582
00:38:18 --> 00:38:21
immediately before you
try to do the experiment.

583
00:38:21 --> 00:38:22
And then it'll work.
 

584
00:38:22 --> 00:38:26
So what happens is that as long
as there is a steady supply of

585
00:38:26 --> 00:38:31
hydrogen peroxide in here,
it'll keep oscillating.

586
00:38:31 --> 00:38:34
But as you've used up the
hydrogen peroxide, this is

587
00:38:34 --> 00:38:35
going to start to die.
 

588
00:38:35 --> 00:38:39
And eventually it'll have
this pale blue solution.

589
00:38:39 --> 00:38:41
And it's like the
heart stopping.

590
00:38:41 --> 00:38:43
If you want to start it up
again, add more hydrogen

591
00:38:43 --> 00:38:50
peroxide and it'll
start up again.

592
00:38:50 --> 00:38:52
OK, any questions?
 

593
00:38:52 --> 00:38:55
There's like an infinite number
of these oscillating reactions

594
00:38:55 --> 00:38:59
that people have discovered.
 

595
00:38:59 --> 00:39:09
And optimized, that you can
find in different places.

596
00:39:09 --> 00:39:13
Questions?
 

597
00:39:13 --> 00:39:18
Questions about the final.
 

598
00:39:18 --> 00:39:23
I didn't get to do a review,
but you'll get that

599
00:39:23 --> 00:39:26
done with the TAs.
 

600
00:39:26 --> 00:39:26
Yes.
 

601
00:39:26 --> 00:39:33
STUDENT: [INAUDIBLE]
 

602
00:39:33 --> 00:39:35
PROFESSOR: Because there's
enough of the equivalent

603
00:39:35 --> 00:39:37
of A, of the rain here.
 

604
00:39:37 --> 00:39:41
So it'll oscillate and
there'll be some damping.

605
00:39:41 --> 00:39:45
So what will happen in this
reaction is that the induction

606
00:39:45 --> 00:39:48
period will get longer,
and longer, and longer.

607
00:39:48 --> 00:39:55
Because the w, this term right
here, the frequency which

608
00:39:55 --> 00:39:58
depends on the concentration
of A, that well get

609
00:39:58 --> 00:39:59
slower and slower.
 

610
00:39:59 --> 00:40:06
And eventually it'll just die.
 

611
00:40:06 --> 00:40:08
OK?
 

612
00:40:08 --> 00:40:09