This screencast focuses on the relationship between potential energy diagrams of reaction
coordinates and reaction kinetics.
So in courses as early as high school chemistry, we learned to start looking at reactions in
terms of potential energy diagrams, where we have reactants that have to cross some
kind of energy barrier in order to form products.
And we would expect that the rate with which reactants go to form products is fundamentally
related to this energy landscape, and we want to try to explore what that relationship is
in the screencast through the development of what's known as the Eyring equation.
To keep things simple, we'll look at a very simple gas-phase reaction, and that's the
isomerization between CNH and HCN, and so we can draw what the stable reactant and product
structures of that reaction would look like.
We can also imagine what the transition state might look like.
Since this reaction involves a hydrogen atom basically swapping sides of the molecules,
you might expect that the transition state involves the hydrogen being basically halfway
across the CN molecule on it's way to actually being attached to the carbon instead of the
nitrogen.
So we might propose that this is a transition state structure.
It turns out that we can actually use for example quantum chemical software packages
to compute various forms of the energy of these kinds of molecules, so for example,
we'll be able to compute quantities like the Gibbs free energy, the enthalpy, we can use
statistical mechanics relations to compute the entropy, and so forth.
So by having these approximate structures for the reactant, transition state, and product
forms of CNH, we can compute this energy landscape, or these positions from the energy landscape.
So we kind of want to know, how will we use that information to actually tell us something
about the reaction rate?
So the central idea behind transition state theory is that we can determine the concentration,
the density of molecules in a given vessel volume that are in the transition state by
assuming an equilibrium relationship with the concentration of reactant molecules, which
is much easier to measure, so we expect that the density of transition state molecules,
because they're highly energetic, is very low within the reaction vessel.
It's going to be something that's difficult for us to measure, whereas we know in general
the concentration of reactants that we've loaded, so that's a measurable quantity.
And therefore it's useful to keep track of this concentration of the transition state
in terms of this assumed equilibrium relationship.
This is a key assumption in transition state theory, so it's only technically true that
we expect an equilibrium relationship between stable states, so between say reactants and
products, and the transition state has some of the properties of a stable state, and that
because it's at a maximum in the energy, the derivative of energy with respect to the reaction
coordinate is zero at that maximum, just like it is for the reactant and product state,
but the second derivative has a different sign.
Here we have a maximum instead of a minimum, so we don't expect rigorously that an equilibrium
relationship will necessarily apply, but we're going to use it as a model and it has been
used effectively, this kind of equilibrium relationship, in order to predict reaction
rates.
So the approach in transition state theory is that we're going to determine the rate
of product formation in terms of some kind of rate constant times the concentration of
the transition state, and that rate constant takes this particular form.
It's often called the crossing frequency, where we multiply Boltzmann's constant times
the temperature over h times the concentration of the transition state.
The idea here is that once we've achieved enough energy to produce this transition state,
and we've come up to this maximum on the potential energy surface, now we don't need any more
energy for the molecule to fall into the product state where the hydrogen is next to the carbon,
and really the rate with which we form product from the transition state is just related
to how long it takes the hydrogen to traverse from this position to the reactant.
And that ends up being a very fast reaction rate, this rate constant is approximately
10^13 reciprocal seconds when you plug in temperatures of around 500 K and the values
for Boltzmann's constant and Planck's constant.
And this quantity comes from an analysis that involves the kinetic theory of gases, we'll
just accept it, and we'll use another screencast to deal with what is known as the crossing
frequency comes from.
But in any case, it's a very large number, and again that's because it only relies on,
since we've achieved enough energy for the molecule to reach the transition state, it's
just how long does it take to actually fall down into the product state.
We're going to use this equilibrium constant inside this relationship, and just to make
it clearer that we're talking about a transition state, I'll use the typical double-dagger
formalism to denote that pseudo-equilibrium constant.
We can substitute in the top expression to the bottom one to write down an expression
for the reaction rate, and we see that when we do that, we get this collection of constants
and the temperature, which we know the values of for any given experiment that we would
conduct at a particular temperature.
We then have this equilibrium constant that we'll deal with in just a minute, and now
instead of the concentration of the transition state, we've substituted in for the concentration
of the reactant, which is something that's easy for us to measure, and in fact this has
sort of a classic form, that this entire quantity we can refer to as a reaction rate constant,
and we see therefore that the reaction rate is equal to this reaction rate constant times
the concentration of the reactant, which is the form of the reaction rate that we are
used to using for a first-order reaction.
So we're interested in evaluating this reaction rate constant k.
We can again just express this in terms of the crossing frequency, and this equilibrium
constant we can re-express based on the definition of an equilibrium constant, that it's equal
to the exponential of negative delta G for the reaction of going from reactants to transition
state, over RT.
And so if on our potential energy diagram we're actually plotting the Gibbs free energy,
then the quantity that we need is actually this value, which is the difference in the
Gibbs free energy between the reactants and the transition state.
And we see that if we have this value, we can actually plug it straight into this exponential,
and if we have the temperature, all we have to have to evaluate this expression are the
constants, Boltzmann's constant and Planck's constant.
And therefore, we can evaluate the rate constant completely from this potential energy diagram.
To do this we would just have to use for example a quantum chemical software package to estimate
the Gibbs free energy of the reactants and the Gibbs free energy of the transition states
by inputting this linear structure CNH and this triangular structure transition state,
and computing their Gibbs free energies.
Now, there's another transformation that's often useful, and that's to recognize that
delta G is equal to delta H minus the temperature times delta S, and if we just use that relationship
and plug it in to the expression, then we get this expression at bottom, where we've
introduced now two variables instead of one, so in a way we've added complexity, but the
reason this is done is that it turns out delta G is strongly dependent on temperature, whereas
delta H to the transition state has a very weak temperature dependence.
So we get a constant here that doesn't depend very strongly on temperature, and what we
can see is that there's a fundamental similarity here between this expression for the rate
constant and the empirically derived, or empirically determined Arrhenius expression for our rate
constant, where we have a temperature-independent activation energy divided by RT that we're
taking the exponential to, times a pre-exponential factor that is nominally assumed to be a weak
function of temperature in the Arrhenius expression.
We see that delta S to the transition state and of course the temperature itself, those
are actually temperature dependent quantities, obviously.
In the case of the temperature here, this is not an exact correlation, that A is not
exactly equal to this collection of terms, and the activation energy is not exactly equal
to delta H to the transition state, but as we'll show in a future screencast, they actually
are pretty close in most cases, so you can understand the activation energy as coming
mostly from the energy difference between the transition state and the reactants, whereas
the pre-exponential factor is dependent mostly on the entropy difference between the transition
state and the reactants, and we see that these dependencies have opposite signs inside the
exponential, and that makes sense because we expect that as entropy goes up, that's
a more favorable process, a more probable process, and so the rate will tend to go up,
whereas as the enthalpy of the transition state goes down, it's easier to have molecules
achieve that enthalpy, so we'll expect that that will increase the rate of reaction, and
that's the reason for the opposite signs there.
This is an expression that again has several assumptions associated with it, but has actually
been found to be extraordinarily useful and accurate in many applications that people
care about in predicting rate constants, based on just understanding the structure and energies
of reactants in the transition states.