Hello in this video we're going to
talk about Unimolecular Reactions. If you'll recall these
are reactions involving changes for only one reactant,
say some A goes to a product. The reaction could be isomerization, meaning that the physical configuration
of the molecule is changed. Dissociation meaning it's torn apart,
and there may be an energy barrier in either direction or dissociation with
a barrier only for the dissociation. So an example of a unimolecular reaction
is the dissociation of methanol into a methyl radical and Radicals,
and there are many other examples. So here are just some different
potential energy wells or surfaces one showing what
one might look like for isomerization then dissociation with
some barrier to reverse reaction. That means you can dissociate it, but then
if you want to go back you've got to go over another barrier or then the third
one is dissociation without a barrier. So that would be like a diatomic
molecule you pull it apart. There's no barrier to the two atoms that make up the diatomic
molecule coming back together. The first major theory of unimolecular
reactions was due to Lindemann with modifications by Hinshelwood,
the Lindemann version, actually, is what we're going to talk
about it's the most simple. So the theory proposes that
unimolecular reactions take place in the following faction. You have a third body M, our old friend M,
and it whacks a molecule A, and it causes that molecule to
go to an excited quantum state, usually an excited
vibrational quantum state. And then there's enough energy in
that vibrational quantum state so that the molecule is unstable and
it dissociates to products. This particular theory assumes that the
reverse reaction is strong in the sense that it occurs at the gas kinetic rate and that A star is not formed
from the products. So if you apply an approximation called,
steady-state, to the formation of A star,
then the overall rate of reaction becomes equal to
an effective unimolecular reaction rate constant times
the concentration of A. And then this is actually
driven by the second reaction the dissociation of A2 with
some reaction rate constant K2. And if you do the math you
get that the overall rate is k1 k2 times a concentration of
A times the concentration of M divided by K minus 1
the reverse rate constant for the reaction of A star back to A time's
the concentration of M plus k2. So you should note that
the rate is first order at high pressures as M goes to infinity, because M will drop out of
the ratio on the right there. So the consequence is that
there's a high pressure limit and it's usually called k sub infinity or
just k in some cases and it happens to be equal to
k1 k 2 over k minus 1. If you look at the low pressure limit
where the concentration of M is quite small, the concentration of M drops out of
the denominator in the overall expression. And you get that k not which is equal
to K1 times the concentration of M times the concentration of
A times the concentration of M. The unimolecular reaction rate
constant can be written, therefore, as k infinity over 1 plus k infinity
over k1 times the concentration of M, where k infinity is the rate
constant in the high pressure limit. Sometimes people define a pressure called
P one-half, it's the pressure which the unimolecular reaction rate
is one-half of the k infinity, so it's defined as or
it's proportional to k infinity over k1. So if you were to plot
the unimolecular reaction rate and a log-log plot like this,
it would look like this. We're actually plotting it
against the density and so as a density decreases the concentration
of M decreases, and so you have this sort of three domains you have the high
pressure limit or high density limit. You have the low pressure or
low density limit and you have a transition called
the fall off region between them. So here's an example of reaction
that's in many combustion reaction mechanisms,
it's the dissociation of propane into C2H5 plus
methyl radical CH3 and you must provide two reaction rates and
one is the k infinity. This is just the notation in this
particular computer code that I took this from and you have the low
pressure limit kf not and you're given A the temperature coefficient and
the activation energy for each of those. Then there's a line here
called a efficiencies and that takes into account that not
all M's are alike in other words. The effectiveness or the efficiency
of this dissociation reaction depends on what the collision
part partner is and so you can account for
that by inputting different efficiencies. And if it's symbolic,
you'll you don't have listed, I think the program probably
just assumes efficiency of one. You can see they vary quite a bit for
argon it's 0.7 but for methane it's 2.0 and for water, it's 6.0 and that has
to do with the structure of the molecule, the more internal structure a molecule has
the better it is at causing dissociation. And then the last line in this
specification of this reaction in a computer code is the so-called
falloff formula, so I think I mentioned the Lindemann theory
is not that great in the falloff region. And there was a German scientist,
still alive, he's a great guy, named Jurgen Troe, who several decades ago
developed a better functional form for the transition between the two limits and
the parameters for the Troe of fit are given in this line. There're many other falloff
expressions and so on but this is just a good example of how
you would specify in a computer code the data you need to calculate
a unimolecular reaction rate constant. So that's it for this video and
thanks for living and thanks for living, yes indeed, and thanks for
listening and have a great day.