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PROFESSOR: So we're going to
start directly today and look

00:00:24.140 --> 00:00:24.910
at chain reactions.

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And explosions.

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So chain reactions, how
many people have

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heard of chain reactions?

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Lot of people have heard
of chain reactions.

00:00:36.440 --> 00:00:40.110
Have seen chain reactions
done, in kinetics.

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No.

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Alright, well, then I'll
do chain reactions.

00:00:46.430 --> 00:00:48.290
Chain reactions.

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Chain reactions are reactions
that feed on themselves.

00:00:55.920 --> 00:00:57.540
And there are three parts
to a chain reaction.

00:00:57.540 --> 00:01:05.120
There's an initiation step,
which is where your reactant

00:01:05.120 --> 00:01:06.580
makes an intermediate.

00:01:06.580 --> 00:01:09.040
And that's usually
a slow step.

00:01:09.040 --> 00:01:17.640
There's a propagation step,
where the intermediate reacts

00:01:17.640 --> 00:01:20.520
with your reactant, becomes
a feedstock.

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And then there's some
intermediate plus maybe a

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first product.

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Then there's that second
intermediate, can further

00:01:32.960 --> 00:01:38.330
react to re-form the initial
intermediate.

00:01:38.330 --> 00:01:39.260
And that's the loop.

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That's the loop of the
chain reaction.

00:01:41.150 --> 00:01:43.970
Plus maybe a second product.

00:01:43.970 --> 00:01:48.760
And then there's a
termination step.

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Which is where when your
intermediate, I1, becomes a

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stable product.

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Or I2 becomes a stable
product.

00:01:55.630 --> 00:01:57.320
Basically, where the
intermediates

00:01:57.320 --> 00:01:58.960
come out of the loop.

00:01:58.960 --> 00:02:08.770
So if you look at this loop
here, you have your reactant

00:02:08.770 --> 00:02:11.940
that comes into the loop.

00:02:11.940 --> 00:02:14.020
And then go around the loop.

00:02:14.020 --> 00:02:20.420
You create I1, and maybe
a product comes out.

00:02:20.420 --> 00:02:23.510
Maybe I1 gets terminated, but
most likely it keeps going

00:02:23.510 --> 00:02:25.120
around the loop.

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Creating I2, which reacts with,
the reactant at some

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point gets included
in the circle.

00:02:31.540 --> 00:02:33.940
And I1 comes again, and I2,
and I1 comes, and I2.

00:02:33.940 --> 00:02:36.372
And every time you form
I1 or I2, you may

00:02:36.372 --> 00:02:38.330
form some of the product.

00:02:38.330 --> 00:02:40.010
So this is the initial feed.

00:02:40.010 --> 00:02:42.090
Then you go in circles.

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Eating up R, spitting
out products.

00:02:45.340 --> 00:02:46.160
Keep going in circles.

00:02:46.160 --> 00:02:47.980
And at some point the
termination's going to win,

00:02:47.980 --> 00:02:52.870
and the circle stops, and
the chain has stopped.

00:02:52.870 --> 00:02:57.160
And there are two kinds
of chain reactions.

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There's a stable chain and an
unstable chain, where you get

00:03:00.840 --> 00:03:08.310
your explosion.

00:03:08.310 --> 00:03:15.650
Stable chain, or also called
stationary, chain reaction, is

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where the amount of these
intermediates, I1 and I2, are

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constant in time.

00:03:23.490 --> 00:03:27.420
So you don't build up
any intermediate.

00:03:27.420 --> 00:03:37.110
So I1 constant in time, which is
why it's called stationary.

00:03:37.110 --> 00:03:48.410
Then you have the unstable,
or non-stationary chain.

00:03:48.410 --> 00:03:54.090
Where in that case
your intermediate

00:03:54.090 --> 00:03:59.540
increases in time.

00:03:59.540 --> 00:04:06.000
So in the example I gave here,
you destroy I1 in the first

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step, you create it in
the second step.

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Comes back, you destroy it, you
create it, you destroy it

00:04:10.045 --> 00:04:11.680
you create it.

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For every I1 that you
start out with, you

00:04:14.660 --> 00:04:16.310
end up with an I1.

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There's no change in the
concentration of I1.

00:04:18.580 --> 00:04:27.210
But if instead I put a two here,
than my reactant creates

00:04:27.210 --> 00:04:28.740
an intermediate I1.

00:04:28.740 --> 00:04:30.160
I destroy it.

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Create a new I2, the I2
decomposes, creates two I1's.

00:04:34.220 --> 00:04:41.640
So after the first cycle,
let's look at the

00:04:41.640 --> 00:04:45.260
concentration of P2 here, after
the first cycle here I

00:04:45.260 --> 00:04:48.810
have one I1, and make one P2.

00:04:48.810 --> 00:04:49.930
But I make two I2's.

00:04:49.930 --> 00:04:52.800
So now I have two I1's
coming in here.

00:04:52.800 --> 00:04:55.960
So I have twice this reaction.

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I make two I2's.

00:04:57.050 --> 00:05:01.710
These two I2's form
two products.

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And now four I1's,
two times two.

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These four I1's go through
the process.

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I get four P2's.

00:05:09.980 --> 00:05:12.610
So third cycle, I
get four P2's.

00:05:12.610 --> 00:05:15.390
And I get two times
four, eight I1's.

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These eight I1's go through,
and I get eight here.

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For cycle, et cetera.

00:05:19.980 --> 00:05:24.870
So I end up with something that
goes like, so this is two

00:05:24.870 --> 00:05:25.840
to the zero power.

00:05:25.840 --> 00:05:28.080
This is two to the first, two
to the second, two to the

00:05:28.080 --> 00:05:29.540
third, et cetera.

00:05:29.540 --> 00:05:32.090
Eventually you get two
to the 25 or whatever

00:05:32.090 --> 00:05:33.650
after so many cycles.

00:05:33.650 --> 00:05:36.240
And if these are gas phase
products, so if there's an

00:05:36.240 --> 00:05:41.370
exothermic process where heat
gets created at any of these

00:05:41.370 --> 00:05:46.940
steps here, then you end
up with a problem.

00:05:46.940 --> 00:05:50.530
The thing goes out of control.

00:05:50.530 --> 00:05:51.640
Sometimes that's a good thing.

00:05:51.640 --> 00:05:52.570
If that's what you want to do.

00:05:52.570 --> 00:05:54.650
And sometimes it's
a bad thing.

00:05:54.650 --> 00:05:56.370
Depends on what you
want to do.

00:05:56.370 --> 00:05:58.220
OK, so we're going to
show two examples

00:05:58.220 --> 00:05:59.550
of these chain reactions.

00:05:59.550 --> 00:06:02.510
One is stable chain and the
other one is unstable chain.

00:06:02.510 --> 00:06:24.050
And go through the process of
how you solve for them.

00:06:24.050 --> 00:06:29.720
So the first example
is taking, so

00:06:29.720 --> 00:06:31.430
let's do a stable chain.

00:06:31.430 --> 00:06:38.060
Take acetaldehyde, CH3CHO,
that decomposers to form

00:06:38.060 --> 00:06:42.270
methane and carbon monoxide.

00:06:42.270 --> 00:06:44.740
That's the reaction.

00:06:44.740 --> 00:06:46.320
And here are the observations.

00:06:46.320 --> 00:06:47.580
There are two observations.

00:06:47.580 --> 00:06:52.240
The first one is that methane
and carbon dioxide are not the

00:06:52.240 --> 00:06:53.400
only products that are formed.

00:06:53.400 --> 00:07:05.260
In fact, there's a trace amount
of ethane and hydrogen,

00:07:05.260 --> 00:07:13.180
dihydrogen are also formed.

00:07:13.180 --> 00:07:16.090
That's a clue to something
funny is going on here.

00:07:16.090 --> 00:07:18.140
That there's a mechanism that's
more complicated than

00:07:18.140 --> 00:07:21.240
just this decomposing in
a first order process.

00:07:21.240 --> 00:07:24.460
And the other thing is that
the rate, if you do an

00:07:24.460 --> 00:07:28.240
experiment and you measure,
somehow you do a half time

00:07:28.240 --> 00:07:32.270
experiment, or an initial rate
experiment, or one of the ways

00:07:32.270 --> 00:07:35.320
that we talked about early on,
you measure the rate of the

00:07:35.320 --> 00:07:41.440
reaction and you find that it's
proportional to something

00:07:41.440 --> 00:07:43.590
to the 3/2 power.

00:07:43.590 --> 00:07:45.310
That's pretty weird.

00:07:45.310 --> 00:07:47.560
It's not a simple mechanism.

00:07:47.560 --> 00:07:49.970
It's pretty weird.

00:07:49.970 --> 00:07:53.100
And so those two observations
are clues, usually, that

00:07:53.100 --> 00:07:55.740
you've got a chain reaction.

00:07:55.740 --> 00:08:00.870
The fact that you have a trace
amount of extra things, that's

00:08:00.870 --> 00:08:02.880
usually due to a termination
reaction.

00:08:02.880 --> 00:08:06.460
It's not a major product, it's
a minor product that's the

00:08:06.460 --> 00:08:07.670
termination of the chain.

00:08:07.670 --> 00:08:11.650
And then this funny power
here, in the kinetics.

00:08:11.650 --> 00:08:13.210
Of the rate of reaction,
that's also

00:08:13.210 --> 00:08:17.040
usually a chain reaction.

00:08:17.040 --> 00:08:20.910
So now we have to figure out
if there's a mechanism that

00:08:20.910 --> 00:08:23.760
supports those observations.

00:08:23.760 --> 00:08:26.730
And I'm going to tell you
possible mechanism that

00:08:26.730 --> 00:08:27.990
supports those observations.

00:08:27.990 --> 00:08:30.550
And you can do experiments to
show that you can fish out

00:08:30.550 --> 00:08:31.820
certain intermediates.

00:08:31.820 --> 00:08:34.790
It's not proved that that's the
ultimate mechanism, but

00:08:34.790 --> 00:08:43.940
it's at least consistent with
these observations.

00:08:43.940 --> 00:08:51.490
More hidden chemistry.

00:08:51.490 --> 00:08:53.620
OK, so this is the mechanism
that's proposed.

00:08:53.620 --> 00:08:55.440
And then we'll figure out
whether this proposed

00:08:55.440 --> 00:08:58.370
mechanism fits the data.

00:08:58.370 --> 00:09:01.710
So, the proposed mechanism is
that the initiation step

00:09:01.710 --> 00:09:07.390
consists of the acetaldehyde,
CH3CHO,

00:09:07.390 --> 00:09:09.670
decomposing into a radical.

00:09:09.670 --> 00:09:13.710
Chain reactions often have
radical processes also.

00:09:13.710 --> 00:09:18.530
Because the radicals love
to make chains.

00:09:18.530 --> 00:09:23.660
To the methyl radical, and
then the CHO radical.

00:09:23.660 --> 00:09:24.720
So this is the first step.

00:09:24.720 --> 00:09:27.220
It's the decomposition
step, it's very slow.

00:09:27.220 --> 00:09:31.310
That molecule doesn't really
want to fall apart.

00:09:31.310 --> 00:09:34.280
Then there's a propagation
step.

00:09:34.280 --> 00:09:39.370
And that's the methyl radical
plus the acetaldehyde, your

00:09:39.370 --> 00:09:46.910
reactant, forming with
the system rate k2.

00:09:46.910 --> 00:09:52.440
Forming CH3CO radical,
plus methane.

00:09:52.440 --> 00:09:54.910
So that is their product right
here, coming out of the first

00:09:54.910 --> 00:09:57.020
step of the propagation.

00:09:57.020 --> 00:10:00.840
And then this new intermediate
here, so there's this one here

00:10:00.840 --> 00:10:01.910
would be I1.

00:10:01.910 --> 00:10:03.160
The methyl radical.

00:10:03.160 --> 00:10:05.950
And the CH3CO radical
here, this is I2.

00:10:05.950 --> 00:10:08.280
And you have to use that
up in the second step.

00:10:08.280 --> 00:10:12.030
CH3CO.

00:10:12.030 --> 00:10:14.080
That decomposes on its own.

00:10:14.080 --> 00:10:19.190
Rate k3, to form the first
intermediate back, the methyl

00:10:19.190 --> 00:10:22.580
radical, plus the second
product, CO.

00:10:22.580 --> 00:10:23.990
And then this intermediate
then feeds

00:10:23.990 --> 00:10:27.810
the first step back.

00:10:27.810 --> 00:10:30.760
And you've got your cycle.

00:10:30.760 --> 00:10:35.460
And every point you spit
out your two products.

00:10:35.460 --> 00:10:37.690
And this could go on forever.

00:10:37.690 --> 00:10:40.760
But it doesn't, because at some
point those two methyl

00:10:40.760 --> 00:10:42.530
radicals, two of those
methyl radicals can

00:10:42.530 --> 00:10:45.310
get along, can collide.

00:10:45.310 --> 00:10:48.040
And in a very easy way.

00:10:48.040 --> 00:10:50.860
So we've got a few termination
steps.

00:10:50.860 --> 00:11:00.560
You can form two times CH3
radical ethane, C2H6.

00:11:00.560 --> 00:11:03.810
Rate k4, let's call this k4.

00:11:03.810 --> 00:11:09.990
And then there are other side
reactions that can happen.

00:11:09.990 --> 00:11:13.890
For instance, this radical
here, which we

00:11:13.890 --> 00:11:15.950
haven't used up.

00:11:15.950 --> 00:11:17.310
That can also react.

00:11:17.310 --> 00:11:20.620
It's just sitting around, slowly
building up as this

00:11:20.620 --> 00:11:23.006
thing, very slowly building
up, because the first step

00:11:23.006 --> 00:11:24.440
here is very slow.

00:11:24.440 --> 00:11:26.670
You don't need very much
of this methyl radical

00:11:26.670 --> 00:11:27.880
to start the chain.

00:11:27.880 --> 00:11:30.940
Then you start chewing
up the reactant.

00:11:30.940 --> 00:11:35.400
But you're still building up a
little bit of this guy here.

00:11:35.400 --> 00:11:36.980
This is a side reaction.

00:11:36.980 --> 00:11:39.700
It's not a termination reaction,
because that

00:11:39.700 --> 00:11:43.350
radical's not involved in the
chain propagation step.

00:11:43.350 --> 00:11:48.360
But that radical can react with
some other species, M,

00:11:48.360 --> 00:11:51.330
and it could be any of
the other species.

00:11:51.330 --> 00:11:52.640
Because it's just a chaperone.

00:11:52.640 --> 00:11:58.400
It's just a, to help this
thing to decompose.

00:11:58.400 --> 00:12:04.680
CO plus H radical plus
the chaperone back.

00:12:04.680 --> 00:12:13.290
And then that hydrogen radical
here can itself react with

00:12:13.290 --> 00:12:15.610
your reactant.

00:12:15.610 --> 00:12:23.520
CH3 acetaldehyde, CH3CHO,
with a rate k6.

00:12:23.520 --> 00:12:30.630
To form hydrogen gas plus your
intermediate, CH3CO.

00:12:30.630 --> 00:12:41.060
And this intermediate can feed
the step right here.

00:12:41.060 --> 00:12:46.490
OK, so in terms of the
observations, then, we got the

00:12:46.490 --> 00:12:48.540
right products.

00:12:48.540 --> 00:12:50.880
Methane, and carbon monoxide.

00:12:50.880 --> 00:12:56.360
We've got the right amount of
trace stuff, the ethane coming

00:12:56.360 --> 00:12:58.150
out from the termination step.

00:12:58.150 --> 00:13:04.410
And hydrogen gas coming out
from the side reaction.

00:13:04.410 --> 00:13:07.470
And I should probably have
written CO also forming from

00:13:07.470 --> 00:13:10.570
the side reaction - well, CO
is a real product, so it's

00:13:10.570 --> 00:13:12.895
happening both from the side
reactions as well as from the

00:13:12.895 --> 00:13:14.270
main reaction.

00:13:14.270 --> 00:13:17.230
So we've got all the species
taken into account.

00:13:17.230 --> 00:13:21.530
And now we've got to show that
this mechanism here, in fact,

00:13:21.530 --> 00:13:27.560
gives rise to this 3/2 power.

00:13:27.560 --> 00:13:31.860
And I know I won't be able
to do it on one board.

00:13:31.860 --> 00:13:38.190
I want to keep all up,
so let me use up.

00:13:38.190 --> 00:13:40.480
So the first thing, as we've
been doing in kinetics this

00:13:40.480 --> 00:13:43.590
whole time is, you've got to
write down all your rates.

00:13:43.590 --> 00:13:46.250
Once you've got that down, and
you haven't made a mistake,

00:13:46.250 --> 00:13:49.580
then you basically have
the problem solved.

00:13:49.580 --> 00:13:53.400
So you write down all
your rate laws.

00:13:53.400 --> 00:13:55.110
Eventually, what we're
interested in showing is the

00:13:55.110 --> 00:13:56.790
rate of the reaction,
is that power.

00:13:56.790 --> 00:13:58.720
So that's write down what the
rate of the reaction is.

00:13:58.720 --> 00:14:03.330
The rate of reaction is the rate
of formation of one of

00:14:03.330 --> 00:14:04.370
the products.

00:14:04.370 --> 00:14:09.440
Let's say the methane.

00:14:09.440 --> 00:14:10.440
So we write down that rate.

00:14:10.440 --> 00:14:11.380
How was it formed?

00:14:11.380 --> 00:14:15.300
It's formed through
the k2 step.

00:14:15.300 --> 00:14:17.840
It's formed through this
step right here.

00:14:17.840 --> 00:14:28.580
Which is a second order in
CH3 radical times CH3CHO.

00:14:28.580 --> 00:14:33.040
acetaldehyde.

00:14:33.040 --> 00:14:34.580
We're going to keep that in
the back of our mind.

00:14:34.580 --> 00:14:36.190
Because we're going to
need this again.

00:14:36.190 --> 00:14:39.470
Because we want to show, at the
end, that in fact this is,

00:14:39.470 --> 00:14:44.150
this rate is proportional
to this funny power.

00:14:44.150 --> 00:14:47.280
There's the intermediate
sitting here.

00:14:47.280 --> 00:14:51.440
So what's the rate for this
intermediate here is CH3, dt.

00:14:51.440 --> 00:14:53.700
There are many places that
it's coming from.

00:14:53.700 --> 00:14:56.080
It's being created in the first
step, and so the hard

00:14:56.080 --> 00:14:58.230
part here is bean-counting.

00:14:58.230 --> 00:15:00.710
Making sure that you've found
all the places where the

00:15:00.710 --> 00:15:08.890
species is being created. k1
times acetaldehyde, CH3CHO,

00:15:08.890 --> 00:15:11.210
and then it's destroyed through
the second step, minus

00:15:11.210 --> 00:15:16.230
k2 CH3, better get your
signs right also.

00:15:16.230 --> 00:15:20.370
So there's the destruction,
there's the formation.

00:15:20.370 --> 00:15:23.620
Times acetaldehyde, CH3CHO.

00:15:23.620 --> 00:15:26.030
Then it gets created again,
through the second step of the

00:15:26.030 --> 00:15:31.190
propagation, rate k3.

00:15:31.190 --> 00:15:34.640
Radical CH3CO.

00:15:34.640 --> 00:15:36.370
And then it finally
gets destroyed in

00:15:36.370 --> 00:15:38.700
the termination step.

00:15:38.700 --> 00:15:41.270
We've got to put that in, even
though it may be a very small

00:15:41.270 --> 00:15:44.740
amount compared to these
other amounts there.

00:15:44.740 --> 00:15:47.470
So the termination step is k4.

00:15:47.470 --> 00:15:53.460
And it's a second order
process, CH3 squared.

00:15:53.460 --> 00:15:54.850
We've got another
intermediate.

00:15:54.850 --> 00:15:57.190
We want to write down all
our intermediates here.

00:15:57.190 --> 00:15:58.940
Because they're all
related, it's the

00:15:58.940 --> 00:16:00.000
intermediates sitting here.

00:16:00.000 --> 00:16:00.770
So there are going
to be a couple of

00:16:00.770 --> 00:16:02.150
differential equations.

00:16:02.150 --> 00:16:04.380
So we've got to write
down the equation

00:16:04.380 --> 00:16:12.990
for that CH3CO radical.

00:16:12.990 --> 00:16:15.940
And again, you look
at your mechanism.

00:16:15.940 --> 00:16:17.550
And you find out where
it's coming from.

00:16:17.550 --> 00:16:19.350
It's being created
in this step.

00:16:19.350 --> 00:16:21.110
Destroyed in this step.

00:16:21.110 --> 00:16:24.320
It's also being created
in this step.

00:16:24.320 --> 00:16:28.730
But we're going to ignore that
side reaction for now.

00:16:28.730 --> 00:16:34.630
Just to make our lives a little
bit easier. k2 CH3.

00:16:34.630 --> 00:16:36.200
You've got to put the
termination step in, but the

00:16:36.200 --> 00:16:38.380
side reactions are
not so important.

00:16:38.380 --> 00:16:40.370
CH3.

00:16:40.370 --> 00:16:42.620
Because if you don't put the
termination step in, then your

00:16:42.620 --> 00:16:45.060
reaction's just going
to go on forever.

00:16:45.060 --> 00:16:48.370
That's just not physical.

00:16:48.370 --> 00:16:54.460
Minus k3 CH3CO.

00:16:54.460 --> 00:16:56.860
Creation and destruction.

00:16:56.860 --> 00:16:58.490
OK, so now what do we do?

00:16:58.490 --> 00:17:02.820
Well, we know it's
a chain reaction.

00:17:02.820 --> 00:17:04.190
A stable chain reaction.

00:17:04.190 --> 00:17:08.120
Because we don't create more
intermediates than

00:17:08.120 --> 00:17:09.490
we started out with.

00:17:09.490 --> 00:17:14.460
The first step is always
very slow.

00:17:14.460 --> 00:17:17.450
Initiation at this
step is slow.

00:17:17.450 --> 00:17:21.340
So we're not making very much
of this methyl radical.

00:17:21.340 --> 00:17:24.470
We're not making very
much of it.

00:17:24.470 --> 00:17:27.170
And we're not making any
more of it here.

00:17:27.170 --> 00:17:29.330
So during this whole time that
the chain reaction is

00:17:29.330 --> 00:17:33.350
happening, this methyl radical
here is in small quantities.

00:17:33.350 --> 00:17:35.060
Small quantities and it's
not going to change

00:17:35.060 --> 00:17:36.440
very much at all.

00:17:36.440 --> 00:17:38.530
It's pretty much stationary.

00:17:38.530 --> 00:17:39.610
Magic word is stationary.

00:17:39.610 --> 00:17:42.480
Stationary means it's
not changing.

00:17:42.480 --> 00:17:44.040
Not changing, so which
approximation

00:17:44.040 --> 00:17:45.480
does that belong to?

00:17:45.480 --> 00:17:47.270
That's a steady state
approximation.

00:17:47.270 --> 00:17:48.300
We've got a steady state here.

00:17:48.300 --> 00:17:51.100
Chain, which is going blah,
blah, blah, blah, stationary.

00:17:51.100 --> 00:17:51.900
Steady state.

00:17:51.900 --> 00:17:53.370
All sounds the same.

00:17:53.370 --> 00:17:55.430
So that's the clue here.

00:17:55.430 --> 00:17:57.540
This is going to be a steady
state approximation problem.

00:17:57.540 --> 00:18:00.420
So we're going to put
a steady state here.

00:18:00.420 --> 00:18:02.240
We're going to put a
steady state here.

00:18:02.240 --> 00:18:04.020
We're going to set this
equal to zero.

00:18:04.020 --> 00:18:07.070
We're going to put a steady
state right here.

00:18:07.070 --> 00:18:09.860
We're going to put a
steady state here.

00:18:09.860 --> 00:18:10.860
We're going to set this
equal to zero.

00:18:10.860 --> 00:18:13.360
And suddenly the problem
becomes a

00:18:13.360 --> 00:18:16.170
tractable algebraic problem.

00:18:16.170 --> 00:18:22.020
Now we have, to solve this,
let's do it in one more place.

00:18:22.020 --> 00:18:24.490
Let's put it right here.

00:18:24.490 --> 00:18:28.760
So now we have two equations,
two algebraic equations.

00:18:28.760 --> 00:18:30.640
Two unknowns.

00:18:30.640 --> 00:18:31.740
These two intermediates.

00:18:31.740 --> 00:18:33.280
The concentration of these two

00:18:33.280 --> 00:18:34.920
intermediates are the two unknowns.

00:18:34.920 --> 00:18:40.830
Two equations, two unknowns,
we can solve it.

00:18:40.830 --> 00:18:43.460
It can be messy.

00:18:43.460 --> 00:18:45.410
But not necessarily tricky.

00:18:45.410 --> 00:18:47.360
Just turn the crank.

00:18:47.360 --> 00:18:50.080
And computers could do this.

00:18:50.080 --> 00:18:55.820
So, let me just write
the answer.

00:18:55.820 --> 00:19:01.530
Because I don't want to go
through the algebra.

00:19:01.530 --> 00:19:04.510
After you turn the crank, what
you want is to get this out.

00:19:04.510 --> 00:19:06.060
You've got your two equations,
two unknowns.

00:19:06.060 --> 00:19:08.810
You turn the crank.

00:19:08.810 --> 00:19:17.190
And you get that form for the
concentration of the methyl

00:19:17.190 --> 00:19:20.590
radical two k4.

00:19:20.590 --> 00:19:27.820
CH3CHO turns out to be
to the 1/2 power.

00:19:27.820 --> 00:19:29.720
When you do that.

00:19:29.720 --> 00:19:32.740
There's the funny
power coming up.

00:19:32.740 --> 00:19:35.120
That's going to be here.

00:19:35.120 --> 00:19:44.070
And then this is what you
want to put into here.

00:19:44.070 --> 00:19:46.360
So, and there's the
acetaldehyde,

00:19:46.360 --> 00:19:47.800
sitting right here.

00:19:47.800 --> 00:19:52.110
There's one acetaldehyde
to the 1/2 power, times

00:19:52.110 --> 00:19:54.150
acetaldehyde to the one power,
that's the whole

00:19:54.150 --> 00:19:58.060
thing to a 3/2 power.

00:19:58.060 --> 00:20:00.320
So in fact, the rate is
proportion to the 3/2,

00:20:00.320 --> 00:20:03.410
according to our mechanism.

00:20:03.410 --> 00:20:07.070
And all these rate constants
go into it as well.

00:20:07.070 --> 00:20:18.330
And what we find is that the
rate of reaction is equal to

00:20:18.330 --> 00:20:28.170
k2 times square root
of k1 over two k4.

00:20:28.170 --> 00:20:32.550
CH2CHO to the 3/2 power.

00:20:32.550 --> 00:20:35.750
Basically, k prime to an
effective rate times that.

00:20:35.750 --> 00:20:37.420
It's a typical problem.

00:20:37.420 --> 00:20:44.570
Any questions on this
one right here?

00:20:44.570 --> 00:20:45.000
Yes.

00:20:45.000 --> 00:20:59.850
STUDENT:: [INAUDIBLE]

00:20:59.850 --> 00:21:02.700
PROFESSOR: Here?

00:21:02.700 --> 00:21:04.620
Right here.

00:21:04.620 --> 00:21:05.660
STUDENT:: [INAUDIBLE]

00:21:05.660 --> 00:21:08.100
PROFESSOR: You know, according
to my notes I don't have it,

00:21:08.100 --> 00:21:09.180
yes, you're absolutely right.

00:21:09.180 --> 00:21:11.440
There's a two here.

00:21:11.440 --> 00:21:13.690
Thank you very much.

00:21:13.690 --> 00:21:16.200
It's arbitrary.

00:21:16.200 --> 00:21:21.690
Where the two comes from, I
didn't want to get into this.

00:21:21.690 --> 00:21:31.500
Sometimes it's a matter
of convention.

00:21:31.500 --> 00:21:46.780
When you have a second order
reaction, A plus B going to C,

00:21:46.780 --> 00:21:51.840
you write your rate of the
reaction, as let's write it

00:21:51.840 --> 00:22:00.716
here, k times A times B.
Sometimes we have 2A goes to

00:22:00.716 --> 00:22:07.190
C. You have a rate k2, and the
rate of the reaction is, you

00:22:07.190 --> 00:22:17.810
can write it as dC/dt, which is
minus k2 times A squared.

00:22:17.810 --> 00:22:24.080
But if you write dA/dt, because
of the stoichiometry.

00:22:24.080 --> 00:22:33.890
dA/dt concentration for, so you
have that C is equal to A0

00:22:33.890 --> 00:22:40.820
minus A, twice.

00:22:40.820 --> 00:22:48.950
So dC/dt is equal to
minus two dA/dt.

00:22:48.950 --> 00:22:52.800
So minus 1/2 dC/dt is
equal to dA/dt.

00:22:52.800 --> 00:22:59.980
So dA/dt is equal to minus
1/2 k2 A squared.

00:22:59.980 --> 00:23:03.110
And instead of having the 1/2
here, what we do is we

00:23:03.110 --> 00:23:05.650
redefine our rate constant.

00:23:05.650 --> 00:23:09.180
So that's k2 prime is
equal to twice k2.

00:23:09.180 --> 00:23:12.620
You put a two here,
a prime here.

00:23:12.620 --> 00:23:14.210
Get rid of this here.

00:23:14.210 --> 00:23:16.850
And there's the two sitting
right here.

00:23:16.850 --> 00:23:18.580
Purely convention.

00:23:18.580 --> 00:23:20.070
Not something to really
worry about.

00:23:20.070 --> 00:23:21.620
You've just got to have it right
at the beginning, and

00:23:21.620 --> 00:23:28.610
keep that two in, if you're
going to put it in there.

00:23:28.610 --> 00:23:31.540
Does that make sense?

00:23:31.540 --> 00:23:34.680
It's the stoichiometry, it's
because of this two sitting

00:23:34.680 --> 00:23:35.920
right here.

00:23:35.920 --> 00:23:37.230
Then you have a choice.

00:23:37.230 --> 00:23:40.020
You can either say the rate of
the reaction is the rate of

00:23:40.020 --> 00:23:42.110
formation of this thing.

00:23:42.110 --> 00:23:44.270
Or you can say the rate of
reaction is the rate of

00:23:44.270 --> 00:23:50.030
destruction of A. And then if
you say that this is the rate,

00:23:50.030 --> 00:23:53.540
and you don't want to have any
factors in here, then the two

00:23:53.540 --> 00:23:54.040
shows up here.

00:23:54.040 --> 00:23:55.870
If you want to say this is the
rate, you don't want any

00:23:55.870 --> 00:23:57.290
factors in here, then you're
going to have a

00:23:57.290 --> 00:23:59.420
different rate constant.

00:23:59.420 --> 00:24:03.090
Purely arbitrary choice.

00:24:03.090 --> 00:24:04.900
Good question, though.

00:24:04.900 --> 00:24:11.520
Any other questions?

00:24:11.520 --> 00:24:12.310
OK.

00:24:12.310 --> 00:24:19.895
So, there's another thing about
the chain reactions, a

00:24:19.895 --> 00:24:23.730
bit of nomenclature that's
interesting to talk about.

00:24:23.730 --> 00:24:30.630
It's the chain length.

00:24:30.630 --> 00:24:33.780
And that nomenclature is due to
the fact that many of these

00:24:33.780 --> 00:24:37.190
chain reactions are used
for polymer reactions.

00:24:37.190 --> 00:24:39.890
Where you use radical
polymerization and then you

00:24:39.890 --> 00:24:45.920
have, basically, in a polymer
reaction you'll have an

00:24:45.920 --> 00:24:49.860
initiator and then you'll
add a monomer

00:24:49.860 --> 00:24:52.330
through a radical process.

00:24:52.330 --> 00:24:54.600
And another monomer will add
through a radical process.

00:24:54.600 --> 00:24:57.310
Another monomer will add,
another monomer will add,

00:24:57.310 --> 00:24:58.670
another monomer will
add, and then

00:24:58.670 --> 00:25:02.990
you're going to terminate.

00:25:02.990 --> 00:25:04.310
And then you end up
with a chain.

00:25:04.310 --> 00:25:06.520
A polymer chain.

00:25:06.520 --> 00:25:10.300
So your product, instead of
being molecular products, the

00:25:10.300 --> 00:25:12.900
product is adding a monomer
to the chain

00:25:12.900 --> 00:25:14.820
of the growing polymer.

00:25:14.820 --> 00:25:19.460
And then it's meaningful to
talk about chain length.

00:25:19.460 --> 00:25:24.380
So that's where the nomenclature
comes from.

00:25:24.380 --> 00:25:27.020
It's a number of cycles of the
reaction before you get a

00:25:27.020 --> 00:25:29.070
termination step.

00:25:29.070 --> 00:25:40.950
So we can define the chain
length as the number of

00:25:40.950 --> 00:25:51.230
propagation steps per
initiation step.

00:25:51.230 --> 00:25:54.590
That gives you the chain.

00:25:54.590 --> 00:25:56.770
Each one of these is
a propagation step.

00:25:56.770 --> 00:26:01.970
And there's the initiation
step here.

00:26:01.970 --> 00:26:04.010
And this could also be
termination step.

00:26:04.010 --> 00:26:16.010
In in a stable chain,
or termination step.

00:26:16.010 --> 00:26:17.650
There's a termination
step here.

00:26:17.650 --> 00:26:20.310
For every initiation step in a
stable chain, you've got a

00:26:20.310 --> 00:26:23.590
termination step.

00:26:23.590 --> 00:26:24.760
Because you're not building
up intermediates.

00:26:24.760 --> 00:26:26.600
So you create it.

00:26:26.600 --> 00:26:28.950
At the end of the chain
you terminate it.

00:26:28.950 --> 00:26:31.160
You have as many initiation
as termination steps.

00:26:31.160 --> 00:26:33.640
So you have a choice here,
in the algebra.

00:26:33.640 --> 00:26:37.390
You can use either one.

00:26:37.390 --> 00:26:40.740
So another way, instead
of doing numbers here,

00:26:40.740 --> 00:26:41.820
you can also rate.

00:26:41.820 --> 00:26:44.920
This is also the rate of
propagation divided by the

00:26:44.920 --> 00:26:46.000
rate of initiation.

00:26:46.000 --> 00:26:47.752
You just take the derivative of
this with respect to time

00:26:47.752 --> 00:26:49.110
of the top and bottom.

00:26:49.110 --> 00:26:50.800
Same thing.

00:26:50.800 --> 00:26:58.940
So we can also define it as rate
of propagation divided by

00:26:58.940 --> 00:27:02.700
rate of, let's say,
termination.

00:27:02.700 --> 00:27:07.100
Or initiation, whichever is your
favorite one to use to

00:27:07.100 --> 00:27:08.370
make it simple.

00:27:08.370 --> 00:27:12.760
So, for instance, in our example
here, the rate of

00:27:12.760 --> 00:27:21.770
propagation is the rate, also
the rate of product formation.

00:27:21.770 --> 00:27:26.490
Since we have a product
formation.

00:27:26.490 --> 00:27:29.390
Rate of propagation is
the rate of forming

00:27:29.390 --> 00:27:31.760
this methane here.

00:27:31.760 --> 00:27:35.720
So in our example here, the rate
of forming the methane is

00:27:35.720 --> 00:27:36.680
the rate reaction here.

00:27:36.680 --> 00:27:38.080
This is this right here.

00:27:38.080 --> 00:27:44.240
So you can plug this
here on top.

00:27:44.240 --> 00:27:46.980
I may not write it down because
I don't want to spend

00:27:46.980 --> 00:27:48.320
time writing too much down.

00:27:48.320 --> 00:27:50.640
So this guy goes
right in there.

00:27:50.640 --> 00:27:57.460
And then the rate of termination
is going to be,

00:27:57.460 --> 00:28:02.760
the rate of termination is,
well, let's not use

00:28:02.760 --> 00:28:04.570
termination because that has
intermediates in it it.

00:28:04.570 --> 00:28:07.070
Let's use initiation.

00:28:07.070 --> 00:28:09.250
Since we have a choice.

00:28:09.250 --> 00:28:11.310
Rate of initiation.

00:28:11.310 --> 00:28:12.720
Either one works fine.

00:28:12.720 --> 00:28:21.900
Initiation is k1 times the
acetaldehyde. k1 times CH3CHO.

00:28:21.900 --> 00:28:28.740
So there's the 3/2 power,
which, where is it?

00:28:28.740 --> 00:28:36.730
There's 1/2 plus, so 1/2 plus
one is the 3/2 power here.

00:28:36.730 --> 00:28:38.760
So there's CH3CHO to
the 3/2 power.

00:28:38.760 --> 00:28:46.540
CH3CHO to the 3/2 power, then
some effective rate k prime.

00:28:46.540 --> 00:28:54.250
And you end up with chain length
going to the 1/2 power.

00:28:54.250 --> 00:28:58.450
So there's some k double prime,
so effective rate times

00:28:58.450 --> 00:29:02.860
CH3CHO to the 1/2 power.

00:29:02.860 --> 00:29:03.710
That's the length of chain.

00:29:03.710 --> 00:29:08.240
It depends on the concentration
of the reactant.

00:29:08.240 --> 00:29:13.190
And the typical chain's on the
order of a couple hundred or

00:29:13.190 --> 00:29:13.900
something like this, usually.

00:29:13.900 --> 00:29:19.030
You always have a termination
step at at some point.

00:29:19.030 --> 00:29:24.800
Maybe a few thousand if you're
doing polymer chemistry and

00:29:24.800 --> 00:29:26.580
sometimes millions, if
you're very good at

00:29:26.580 --> 00:29:29.950
doing polymer chemistry.

00:29:29.950 --> 00:29:35.100
OK, any questions on
the stable chain?

00:29:35.100 --> 00:29:41.820
OK, let's do the
explosion now.

00:29:41.820 --> 00:29:55.020
And so this is the
typical example.

00:29:55.020 --> 00:29:56.140
Of an unstable chain.

00:29:56.140 --> 00:30:00.960
It's the formation of water
from hydrogen and oxygen.

00:30:00.960 --> 00:30:05.250
2 H2O.

00:30:05.250 --> 00:30:07.750
That's the reaction.

00:30:07.750 --> 00:30:09.710
Gas phase reaction.

00:30:09.710 --> 00:30:12.360
And the observation is that
that, well, you know the

00:30:12.360 --> 00:30:12.880
observation.

00:30:12.880 --> 00:30:16.590
You put a spark in a balloon
that contains hydrogen and

00:30:16.590 --> 00:30:20.920
oxygen and get an explosion.

00:30:20.920 --> 00:30:24.670
So we're going to have
to account for that.

00:30:24.670 --> 00:30:27.050
Sometimes you don't
get an explosion.

00:30:27.050 --> 00:30:28.320
Sometimes it kind
of peters out.

00:30:28.320 --> 00:30:32.160
You have a drop of
water coming out.

00:30:32.160 --> 00:30:35.380
Not bad.

00:30:35.380 --> 00:30:37.610
And there's a temperature
dependence to it also.

00:30:37.610 --> 00:30:40.970
So, whatever mechanism we put in
there needs to account for

00:30:40.970 --> 00:30:44.320
all the pressure dependence, the
stoichiometry dependence,

00:30:44.320 --> 00:30:46.810
and the temperature
dependence.

00:30:46.810 --> 00:30:50.190
So this is the mechanism
that's proposed.

00:30:50.190 --> 00:30:51.550
Initiation.

00:30:51.550 --> 00:30:56.830
You have your hydrogen
molecule.

00:30:56.830 --> 00:31:01.725
Spark, or something slow
that decomposes it to

00:31:01.725 --> 00:31:03.790
two hydrogen radicals.

00:31:03.790 --> 00:31:08.800
This could be a spark plug,
and every time you make a

00:31:08.800 --> 00:31:11.200
little bit of the radical.

00:31:11.200 --> 00:31:15.040
Then you have a branching
step.

00:31:15.040 --> 00:31:22.100
Where your radical reacts with
the oxygen molecule to form a

00:31:22.100 --> 00:31:25.210
hydroxyl radical.

00:31:25.210 --> 00:31:29.270
Plus an oxygen.

00:31:29.270 --> 00:31:36.335
Oxygen atom plus your hydrogen
gas, your reactant, reacts to

00:31:36.335 --> 00:31:42.540
form more hydroxyl radical plus
the initial intermediate.

00:31:42.540 --> 00:31:45.580
And then the hydroxyl radical,
I don't know why I have

00:31:45.580 --> 00:31:51.960
parentheses here, hydroxyl
radical reacts with the

00:31:51.960 --> 00:32:01.190
reactant again, the hydrogen
gas, k3, to form more of the

00:32:01.190 --> 00:32:05.500
initial intermediate plus
water, your product.

00:32:05.500 --> 00:32:09.180
So there are three steps
to the branching here.

00:32:09.180 --> 00:32:12.040
You get one intermediate, one
of the initial intermediates

00:32:12.040 --> 00:32:13.420
here coming in.

00:32:13.420 --> 00:32:15.240
Forms a second intermediate.

00:32:15.240 --> 00:32:17.220
OH radical.

00:32:17.220 --> 00:32:19.020
Two OH radicals.

00:32:19.020 --> 00:32:22.870
And the OH radicals
form the hydrogen.

00:32:22.870 --> 00:32:27.140
There's also radical hydrogen
that's formed right here.

00:32:27.140 --> 00:32:31.270
So you have this formation.

00:32:31.270 --> 00:32:35.460
So because you're forming two
hydroxyl radicals here, in

00:32:35.460 --> 00:32:38.000
this step here you're actually
forming the equivalent of two

00:32:38.000 --> 00:32:40.230
hydrogen radicals.

00:32:40.230 --> 00:32:43.940
So, one hydrogen atom here.

00:32:43.940 --> 00:32:47.450
After going through the
branching, you end up with

00:32:47.450 --> 00:32:50.470
three hydrogen radicals.

00:32:50.470 --> 00:32:57.340
So you go from one to
three in one cycle.

00:32:57.340 --> 00:32:59.880
And the second time around,
you go from three to nine.

00:32:59.880 --> 00:33:02.740
The next time, you go
from nine to 27.

00:33:02.740 --> 00:33:04.700
Pretty soon you've
got a couple of

00:33:04.700 --> 00:33:07.450
billion hydrogen radicals.

00:33:07.450 --> 00:33:09.830
And that's the problem
right there.

00:33:09.830 --> 00:33:12.260
Or the opportunity.

00:33:12.260 --> 00:33:13.570
The opportunity to
get something

00:33:13.570 --> 00:33:18.040
that goes out of control.

00:33:18.040 --> 00:33:21.210
And there are termination
steps.

00:33:21.210 --> 00:33:23.290
That we need to take
into account.

00:33:23.290 --> 00:33:33.430
The hydrogen radical
can hit the wall.

00:33:33.430 --> 00:33:36.260
Before it finds something
to react with.

00:33:36.260 --> 00:33:38.130
And gets stuck.

00:33:38.130 --> 00:33:40.640
So if the pressure is
low enough, you know

00:33:40.640 --> 00:33:41.530
that's going to happen.

00:33:41.530 --> 00:33:43.370
You're going to make
a hydrogen radical.

00:33:43.370 --> 00:33:44.840
Hardly any gas molecules
around.

00:33:44.840 --> 00:33:46.470
It's going to find a wall,
it's going to get stuck.

00:33:46.470 --> 00:33:47.690
It's very reactive.

00:33:47.690 --> 00:33:51.740
So that's going to terminate
the process.

00:33:51.740 --> 00:33:56.480
Hydrogen radical could re-find
an oxygen and another

00:33:56.480 --> 00:33:58.480
chaperone, that's a termolecular
process.

00:33:58.480 --> 00:33:59.930
That's an unlikely process.

00:33:59.930 --> 00:34:02.650
But nevertheless,
it could happen.

00:34:02.650 --> 00:34:08.520
With another termination step,
to form the HO2 radical plus

00:34:08.520 --> 00:34:12.300
M. And then that HO2 too radical
can react in a side

00:34:12.300 --> 00:34:14.200
reaction to form some
products, but not

00:34:14.200 --> 00:34:15.540
be part of the chain.

00:34:15.540 --> 00:34:15.920
Yes.

00:34:15.920 --> 00:34:18.510
STUDENT:: [INAUDIBLE]

00:34:18.510 --> 00:34:20.030
PROFESSOR: How do we get
three hydrogens?

00:34:20.030 --> 00:34:22.790
Because there are two hydroxyl
radicals here.

00:34:22.790 --> 00:34:25.000
And so in this step here, you
actually have two hydroxyl

00:34:25.000 --> 00:34:33.120
radicals to form two, so I could
multiply this by two.

00:34:33.120 --> 00:34:40.420
But you usually don't do that.

00:34:40.420 --> 00:34:45.110
So this already tells you what
conditions you're going to

00:34:45.110 --> 00:34:47.840
get, with something where the
termination is fast, compared

00:34:47.840 --> 00:34:49.110
to the propagation.

00:34:49.110 --> 00:34:50.640
Because you have low pressure.

00:34:50.640 --> 00:34:53.310
If you can't get that hydrogen
radical to react with

00:34:53.310 --> 00:34:56.420
something before hitting the
wall, you're going to be OK.

00:34:56.420 --> 00:34:58.240
You're not getting
an explosion.

00:34:58.240 --> 00:35:01.080
This tells you that a very
high pressure, where the

00:35:01.080 --> 00:35:04.840
probability of having a
three-body collision becomes

00:35:04.840 --> 00:35:08.210
high enough, then you're
going to terminate

00:35:08.210 --> 00:35:09.410
the chain as well.

00:35:09.410 --> 00:35:13.390
Those hydrogen radicals are
going to find oxygen molecules

00:35:13.390 --> 00:35:18.580
and some other molecule and form
this radical before you

00:35:18.580 --> 00:35:20.610
propagate the chain.

00:35:20.610 --> 00:35:24.700
At least some reasonably
high pressure.

00:35:24.700 --> 00:35:28.730
So that's the clue here.

00:35:28.730 --> 00:35:29.540
What's the strategy?

00:35:29.540 --> 00:35:33.050
The strategy is to assume that
everything's going to be fine.

00:35:33.050 --> 00:35:34.860
That is, stable.

00:35:34.860 --> 00:35:35.790
That's going to be
the strategy.

00:35:35.790 --> 00:35:37.030
And then we're going
to do a proof by

00:35:37.030 --> 00:35:38.340
contradiction, basically.

00:35:38.340 --> 00:35:41.820
We can assume that the
chain is stable.

00:35:41.820 --> 00:35:44.710
That we can use the steady
state approximation.

00:35:44.710 --> 00:35:47.090
We're going to apply the steady
state approximation.

00:35:47.090 --> 00:35:49.240
We're going to find where
that breaks down.

00:35:49.240 --> 00:35:52.410
And where it breaks down means
that it wasn't right.

00:35:52.410 --> 00:35:54.660
That the intermediate
concentration was too high.

00:35:54.660 --> 00:35:57.190
And that means that we're going
to get an explosion.

00:35:57.190 --> 00:36:08.190
That's the logic.

00:36:08.190 --> 00:36:11.180
So strategy is a proof by
contradiction, basically.

00:36:11.180 --> 00:36:20.550
Proof by contradiction.

00:36:20.550 --> 00:36:24.410
So we're going to assume that
the propagation, or the

00:36:24.410 --> 00:36:26.970
intermediates, are in
small quantities.

00:36:26.970 --> 00:36:28.830
We're going to assume that we
can use the steady state

00:36:28.830 --> 00:36:36.000
approximation, the d[O]/dt
steady state is equal to zero.

00:36:36.000 --> 00:36:44.140
That the hydroxyl radical,
d[OH]/dt, steady state is

00:36:44.140 --> 00:36:45.380
equal to zero.

00:36:45.380 --> 00:36:54.006
And that d[H]/dt steady state
is equal to zero.

00:36:54.006 --> 00:36:54.980
These are going to be
our assumptions.

00:36:54.980 --> 00:36:58.900
That all the intermediates
are in small quantities.

00:36:58.900 --> 00:37:05.510
And they don't change
very much.

00:37:05.510 --> 00:37:08.080
And now we're going to write
down, you write down the rates

00:37:08.080 --> 00:37:09.650
that correspond to these guys.

00:37:09.650 --> 00:37:11.910
And then you solve.

00:37:11.910 --> 00:37:14.300
So I'm only going to do
this first one here.

00:37:14.300 --> 00:37:18.270
So you write dO/dt, you see
where it gets formed.

00:37:18.270 --> 00:37:20.550
It gets formed in the
first step here.

00:37:20.550 --> 00:37:26.730
So it's going to be k1 times H
radical times O2, and it gets

00:37:26.730 --> 00:37:33.980
destroyed in the second
step here, minus k2

00:37:33.980 --> 00:37:37.400
times O times H2.

00:37:37.400 --> 00:37:38.790
You set the steady state
approximation, you get a

00:37:38.790 --> 00:37:39.380
steady state.

00:37:39.380 --> 00:37:43.230
Steady state here, you set
that equal to zero.

00:37:43.230 --> 00:37:50.280
You solve for O, radical
O, steady state.

00:37:50.280 --> 00:38:02.930
And you find that it's k1 times
H times O2 times k2 H2.

00:38:02.930 --> 00:38:05.040
Then you do the same thing
for the other radicals.

00:38:05.040 --> 00:38:13.080
You solve for OH steady state.

00:38:13.080 --> 00:38:15.290
And you get some expression
which is in the notes.

00:38:15.290 --> 00:38:16.183
I'm not going to write
it down, it's

00:38:16.183 --> 00:38:17.820
going to get too messy.

00:38:17.820 --> 00:38:19.940
But the expression that I will
write down is the final

00:38:19.940 --> 00:38:23.460
expression, which we're going
to be taking into account.

00:38:23.460 --> 00:38:31.460
So, this expression here,
for the oxygen atom.

00:38:31.460 --> 00:38:34.050
It has the radical in there.

00:38:34.050 --> 00:38:38.020
So it's not, we'd like something
that's just in terms

00:38:38.020 --> 00:38:43.830
of the rates and either
products or reactants.

00:38:43.830 --> 00:38:49.270
And this hydroxyl radical, the
concentration turns out, it

00:38:49.270 --> 00:38:54.520
will also be a function of
the concentration of

00:38:54.520 --> 00:38:57.300
the hydrogen radical.

00:38:57.300 --> 00:39:02.470
So what we really want, I don't
want to cover this up,

00:39:02.470 --> 00:39:05.440
let me go on this board here.

00:39:05.440 --> 00:39:14.780
So when you solve for this, the
hydrogen atom is where you

00:39:14.780 --> 00:39:19.320
get something that's free
of intermediates.

00:39:19.320 --> 00:39:21.534
And has just rates in it.

00:39:21.534 --> 00:39:24.740
There's a termination rate here
going in the denominator.

00:39:24.740 --> 00:39:28.270
Another termination rate
in the denominator.

00:39:28.270 --> 00:39:33.600
Times O2 times M, where M could
be anything, really.

00:39:33.600 --> 00:39:36.810
Minus, and there's the creation
rate of the first

00:39:36.810 --> 00:39:37.900
propagation step.

00:39:37.900 --> 00:39:42.960
O2 here.

00:39:42.960 --> 00:39:44.630
So now we have to see
whether or not our

00:39:44.630 --> 00:39:46.420
approximation was valid.

00:39:46.420 --> 00:39:47.530
We made an approximation.

00:39:47.530 --> 00:39:49.860
We said the steady state
approximation was valid.

00:39:49.860 --> 00:39:53.770
If it is valid, then this
concentration, this pressure

00:39:53.770 --> 00:39:56.810
here, needs to be small.

00:39:56.810 --> 00:39:59.680
Needs to be small
at all times.

00:39:59.680 --> 00:40:03.760
If it's not small, then steady
state is not valid, we have a

00:40:03.760 --> 00:40:07.670
branching non-stable chain and
we've got an explosion.

00:40:07.670 --> 00:40:11.100
So let's look at the different
possibilities here.

00:40:11.100 --> 00:40:14.050
What are the possibilities?

00:40:14.050 --> 00:40:18.020
Well, the first possibility is
that as we saw intuitively,

00:40:18.020 --> 00:40:21.330
when we wrote down this equation
here, the first

00:40:21.330 --> 00:40:25.120
possibility is that we're
at low pressure.

00:40:25.120 --> 00:40:31.610
Where at low pressure, then, k1
times oxygen is very small.

00:40:31.610 --> 00:40:33.820
Because the concentration of
oxygen is small, the pressure

00:40:33.820 --> 00:40:37.530
is small. k5 times the pressure
of oxygen times the

00:40:37.530 --> 00:40:39.740
pressure of M is small also.

00:40:39.740 --> 00:40:45.660
So k5 T O M, O2 times
M is small.

00:40:45.660 --> 00:40:50.530
And they're both, then, much
smaller than k4 times the

00:40:50.530 --> 00:40:53.710
time. k4 termination.

00:40:53.710 --> 00:40:56.140
The hydrogen radical hits
the wall before

00:40:56.140 --> 00:40:58.020
finding anything else.

00:40:58.020 --> 00:41:00.190
End of chain.

00:41:00.190 --> 00:41:02.110
So in the equation here,
these two guys are

00:41:02.110 --> 00:41:05.290
small compared to here.

00:41:05.290 --> 00:41:08.750
RI, the initiation rate,
is very small.

00:41:08.750 --> 00:41:11.170
You control that
by your spark.

00:41:11.170 --> 00:41:12.760
Once a second, once a minute.

00:41:12.760 --> 00:41:15.580
Could be really, slow.

00:41:15.580 --> 00:41:17.660
Steady state is good.

00:41:17.660 --> 00:41:21.720
We don't have an
unstable chain.

00:41:21.720 --> 00:41:24.920
No explosion here.

00:41:24.920 --> 00:41:25.790
Medium pressure.

00:41:25.790 --> 00:41:30.240
You've got a medium pressure
where two k1 times oxygen

00:41:30.240 --> 00:41:32.330
concentration is
approximately.

00:41:32.330 --> 00:41:33.900
You're building up your
concentration.

00:41:33.900 --> 00:41:35.540
This is getting bigger here.

00:41:35.540 --> 00:41:37.200
This is a constant.

00:41:37.200 --> 00:41:38.720
This is getting bigger.

00:41:38.720 --> 00:41:43.525
This is also getting bigger, but
you've got this rate here

00:41:43.525 --> 00:41:44.300
and this rate here.

00:41:44.300 --> 00:41:49.380
At some point, at some point,
this rate here, two k1 times

00:41:49.380 --> 00:41:56.710
oxygen concentration, is going
to be on the order of k4 T

00:41:56.710 --> 00:42:04.780
plus k5 T times oxygen times M.
At some point it's going to

00:42:04.780 --> 00:42:07.550
be like that.

00:42:07.550 --> 00:42:10.790
And then you're in
big trouble.

00:42:10.790 --> 00:42:13.020
You're in big trouble, because
you've got a denominator here

00:42:13.020 --> 00:42:13.900
that's really tiny.

00:42:13.900 --> 00:42:17.575
Could be even, if you hit it
right on the nail, you get

00:42:17.575 --> 00:42:19.500
zero in the denominator and then
you're in real trouble

00:42:19.500 --> 00:42:20.650
from the math perspective.

00:42:20.650 --> 00:42:23.370
But from the physical science
perspective, it's

00:42:23.370 --> 00:42:24.250
not being in trouble.

00:42:24.250 --> 00:42:26.270
Is just that your approximation
is wrong.

00:42:26.270 --> 00:42:28.770
The approximation that
you were in a

00:42:28.770 --> 00:42:30.460
stationary state was wrong.

00:42:30.460 --> 00:42:33.420
And so that means that in this
case here, if you want to have

00:42:33.420 --> 00:42:37.170
an explosion, you've got it.

00:42:37.170 --> 00:42:40.240
So you adjust everything until
you're in a situation.

00:42:40.240 --> 00:42:47.050
And then finally, you keep
increasing the pressure.

00:42:47.050 --> 00:42:51.280
And then if you increase it
high enough, then this guy

00:42:51.280 --> 00:42:55.320
here, which goes as the square
of the pressures, is going to

00:42:55.320 --> 00:42:56.120
overwhelm this.

00:42:56.120 --> 00:42:58.810
Which goes linearly
with pressure.

00:42:58.810 --> 00:43:03.700
And k5, the termination step,
O2 times M, is going to be

00:43:03.700 --> 00:43:10.530
bigger than two k1 times
the pressure of O2.

00:43:10.530 --> 00:43:13.110
And then you've got something
small again.

00:43:13.110 --> 00:43:18.610
This is where that's hydrogen
radical here is able to

00:43:18.610 --> 00:43:26.360
undergo a termolecular process
before finding a reactant.

00:43:26.360 --> 00:43:34.570
Then steady state is valid.

00:43:34.570 --> 00:43:34.655
OK.

00:43:34.655 --> 00:43:36.980
At very high pressures, you get
in trouble again, because

00:43:36.980 --> 00:43:44.850
this hydroxyl radical, this HO2
radical, so at very high

00:43:44.850 --> 00:43:51.220
pressure. then what happens
is that you have this side

00:43:51.220 --> 00:44:01.440
reaction, HO2 radical plus
hydrogen goes to H2O plus

00:44:01.440 --> 00:44:04.650
hydroxyl radical.

00:44:04.650 --> 00:44:08.320
So there's a probability that
this radical here finds a

00:44:08.320 --> 00:44:09.370
hydrogen molecule.

00:44:09.370 --> 00:44:13.230
And then feeds into the
propagation through this

00:44:13.230 --> 00:44:15.810
hydroxyl radical that's
the result.

00:44:15.810 --> 00:44:19.100
And then that keeps the
chain propagating.

00:44:19.100 --> 00:44:21.470
So at very high pressure,
you're in trouble again.

00:44:21.470 --> 00:44:26.320
And you get an explosion
again.

00:44:26.320 --> 00:44:37.640
So you can plot this as a
function of the pressure.

00:44:37.640 --> 00:44:39.080
And we're going to put
pressure, actually,

00:44:39.080 --> 00:44:40.850
on the y axis here.

00:44:40.850 --> 00:44:42.056
Let's put pressure here.

00:44:42.056 --> 00:44:42.810
Put temperature here.

00:44:42.810 --> 00:44:44.760
And so for a given temperature,
let's call it

00:44:44.760 --> 00:44:49.080
temperature T prime, at low
pressure there's no explosion.

00:44:49.080 --> 00:44:53.390
Then there's a range
of pressures.

00:44:53.390 --> 00:44:54.290
Where there's an explosion.

00:44:54.290 --> 00:44:55.100
Medium pressures.

00:44:55.100 --> 00:44:58.080
And then a high pressure
termination wins again.

00:44:58.080 --> 00:45:01.520
And then if it's really high
you get explosion again.

00:45:01.520 --> 00:45:04.500
You do this for, as a function
of temperature.

00:45:04.500 --> 00:45:07.190
And what you find that is a
diagram, looks like a phase

00:45:07.190 --> 00:45:10.930
diagram, that looks something
like this.

00:45:10.930 --> 00:45:15.140
Where at high temperature, you
get an explosion at low

00:45:15.140 --> 00:45:24.850
temperatures, and this steady
state is valid.

00:45:24.850 --> 00:45:35.060
OK, any questions on this?

00:45:35.060 --> 00:45:38.220
Don't try this in
your kitchen.

00:45:38.220 --> 00:45:42.310
Go to the desert if you
want to do this.

00:45:42.310 --> 00:45:44.470
Or don't do it either.

00:45:44.470 --> 00:45:45.720
OK.