Hello. In this video
we're going to talk about reaction rate constants. We'll go through a simple
theoretical derivation of the Arrhenius expression, give you an idea of
where it comes from. Anymore advanced theories of reaction rate constants is a bit beyond this set of courses. In addition to that we'll look at some simple
reaction mechanisms, and later we'll look at some
real reaction mechanisms. So reaction rates have a lot to do with
collision dynamics. Collisions are required
for reaction to occur. There are different types
of collisions and they're classified as to
whether they change the mass of the
colliding particles, the momentum energy, or charge. In chemical kinetics we're concerned with change
of mass of particles. In other words, with an exchange of atoms between molecules. Collisions are also categorized
as elastic and inelastic, and we've already said a
little bit about that when we talked about
transport properties. So elastic collisions
are ones that conserve a momentum and translational energy and those are the only things that change. Inelastic collisions
are ones that can change the internal
state of a molecule, or cause charge exchange, or proton exchange, or reaction. Collisions that result in chemical reactions are
inelastic collisions. There's a series of questions
that you must ask when you talk about
reacting collisions. So for a collision to
cause a reaction process, one of the questions you
ask is what conditions must be satisfied before a given
process can take place? Another question is how often are these conditions satisfied
per unit volume and time? The first question is
answered by analyzing the dynamics of specific
collision trajectories. This can be done classically
or it can be quantum. Quantum collision dynamics
takes into account that the quantum dynamical behavior of molecule-molecule
interactions. The second question is answered
basically by statistics. So the rate at which
some quantity is produced or removed per
unit time as a result of collisions is equal to the integral of the rate at
which a certain trajectory occurs which offers
the possibility of that quantity being changed, times the fraction of collisions that would produce a result, and then that's integrated
over all trajectories. So we can illustrate these
ideas in a simple fashion. If you think back to the concept of a collision
cross section where we talked about two molecules
colliding with each other. If we calculate the
number of collisions of 1A molecule colliding with all B molecules per unit time that was equal
to the cross section Pi d squared times the relative
collision velocity times the concentration
of the B type molecules. Then the number of collisions of all A type molecules
not 1A type molecules. All A type molecules with all B molecules per unit time is the same collision cross section times the relative velocity times the number density or concentration of both A and B. We can incorporate some
collision dynamics into the cross-section or into the effect of collision diameter, if we define it to be an effective
collision dynamics or an effective
cross-section I guess. In which case, the
rate of change of the concentration of molecule A with respect of time
would be equal to the integration of Pi d squared the relative
velocity n_A, n_B integrated over all relative velocities
and orientations. In which case, since
the concentration of A and B do not depend on orientation or
relative velocities, they could come out in
front and that gives you the reaction rate constant. So which is the integral
of Pi d squared V_r over all the V_rs
and orientations. So suppose we consider
a reaction like this the dissociation
of hydrogen. We're going to dissociate
the hydrogen by whacking it with an M type particle whether it's an
atom or a molecule. Then we'd expect that the rate of reaction would be related
to the collision energy, the amount of energy available. We'd expect that at low
collision energy we couldn't cause the hydrogen
molecule to fall apart. But as the collision energy increased some
small percentage of collision might result
in a dissociation. Finally, if the collision
energy is high enough all such collisions will result in dissociation
of the hydrogen. So the maximum amount of energy that's available in
a given collision is 1.5 times the reduced mass times the relative
velocity squared. Although you've got to think
about this a little bit, that amount of energy
is only available if the molecules collide
directly on the line of centers. So taking into account the fact that there might
be an impact parameter, there might be an offset in the directions that the molecules are going as illustrated
in the little cartoon. Then the effect of line of sight relative velocity is the actual relative
velocity times the cosine of the angle Alpha. The energy is the total
energy available times 1 minus b squared over d squared or b squared is
the impact parameter. So for a reaction to occur we
have to have that the line of sight energy has to be greater than some
threshold energy, which I've called Epsilon
sub T super naught. It's going to be about equal to the dissociation
energy for example. So the equation, the energy along the line
of centers equals that threshold energy is equal to the total energy times
1 minus b squared over d naught defines for
any relative velocity the maximum impact parameter at which a reaction can occur. If the impact parameter is
bigger the molecules just miss each other, no
collision occurs. So if we solve for
that maximum impact parameter we get this
simple expression for it. In addition, we can define an effective reaction
cross section which takes into account the impact factor effect on
the collision frequency. So we can show that
this expression dz equals the product of the
concentrations of A and B times Mu over kT to the three halves times
2 over pi to the 1.5, times 2 Pi bdb relative
velocity cubed e__minus mu v relative squared
over 2kT dv sub r is the number of collisions per unit time and volume with B and between B to b plus db and
the relative velocity between some relative velocity and that relative velocity plus db. So to get the total
rate of collisions first integrate over
the impact parameter. If you do that you get
that that portion of the integral 2 Pi bdb
is pi b max squared. Then you need to integrate
over the velocity, although it turns out it's easier that transform velocity to kinetic energy and that
expression is given here. So you end up with the expression
for the velocity cube, the exponential term dv as shown in the right-hand side of the equation in terms of energy. So if you plug that all together you
get an expression for the rate constant
and if you carry out the integration you get
the second equation. The rate constant is Pi d squared the relative velocity e_minus
threshold energy over kT. The relative velocity
is our old friend 8kT over Pi Mu to
the one-half power. So you will recall
that the form of the Arrhenius rate
relation was some A times T to some power, times e to the minus
activation energy over RT. So clearly this has that
form pi d squared is a number and that would lead to that pre-exponential factor A. The T to the power comes from the relative velocity
term and then the activation energy term the exponential e to the
minus activation energy over RT comes from
this exponential term. Here it is and that's why
we see it the way it is. Now of course we haven't considered the complexities
of the reaction. We've only assumed
that if the collision provided enough energy
then reaction would occur. In fact, we've
neglected orientation, internal state, and quantum
state effects and so on. The activation energy
is not exactly the dissociation energy
because when you have a thermal distribution of
quantum states you're going to have the molecule in different rotational
and vibrational levels, and of course the higher the rotational or
vibrational energy the less the dissociation
energy from that state. So the activation
energy is really an effective dissociation
energy and so on. So these are going to
modify the values of a of the the power
of t and so on. In some cases the Arrhenius
expression is insufficient. If more advanced analysis methods produce a functional form and they produce a rate constant, that is the
temperature dependence of the rate constant
cannot be set with this function then
you have to use some different function
and that happens. To get back to the
question of what is required for reaction to occur, it's only if a variety
of conditions are met. As we said there has to be
enough energy available that molecules have to be
appropriately oriented. One thing we haven't mentioned
is that the reaction must be quantum
mechanically allowed. When we talked about radiative transitions we said some of them are not allowed. They cannot happen. That's true of chemical
reactions too. There are some state changes that just aren't allowed
by quantum mechanics, and you must always conserve
mass momentum and energy. Well, at a more advanced level you explore these
questions by examining what's called the
potential energy surface of the combined system of
reactants and products. So here's a simple
cartoon along with just a reminder that
the forces along an inner atomic coordinate
is equal to the negative of the derivative of the potential energy function. This just shows what that potential energy function looks like for a diatomic molecule. But diatomic molecules are
only one kind of molecule. So how do you determine that
potential energy surface? While you typically make the Born-Oppenheimer Approximation, you can calculate using quantum mechanics for
first principles. Sometimes you can use
empirical expressions or you can determine
it spectroscopically. Nowadays it's almost always determined by quantum mechanics. Here is an example for a
potential energy surface for a series of reactions between taking these different states of these particular
molecules and this is a series of reactions of so-called Furanic
type molecule. Furanic has got four
carbons and an oxygen. All of these are functionalized versions
of purine meaning they have functional groups attached to some of the carbons. In any event, so looking at these kinds of
potential energy services and using the data from
them will allow you to calculate reaction
rate constants using various
higher-order theories than we'll all we've looked at., but are beyond the scope of what we're going
to talk about, just know they exist. So that's it for this video. Thanks for listening
and have a great day.