The objective of this screencast is to show the relationship between the Arrhenius relationship
and the thermodynamic interpretation of transition state theory expression, otherwise known as
the Eyring equation.
So I've put up an example that we've considered in another screencast, which is a very simple
reaction, so the isomerization of CNH to HCN, which involves a triangular transition state
where the hydrogen is basically hopping across the rest of the molecule, and the transition
state occurs about where the hydrogen is halfway across, that's the highest point in energy.
We've shown previously that we could determine the rate of this kind of reaction.
So we wrote the rate constant from transition state theory for this kind of reaction, which
would be a first order rate constant, is equal to this collection of constants like Boltzmann's
constant and Planck's constant, with the temperature, and then an equilibrium constant for the transition
state that can be calculated based on the Gibbs free energy difference between the transition
state for the reaction and the reactant, where that equilibrium constant we assume can be
calculated as the exponential of that negative Gibbs free energy change over RT, just as
if we were dealing with a reaction between a stable reactant and a product.
We want to understand something about how this rate constant that's really defined based
on first principles and assumptions about the nature of the potential energy surface
to a purely empirically derived relationship for the rate constant, which is the Arrhenius
expression, which tells us that the rate constant is equal to some temperature-independent pre-factor
times an exponential of the negative activation energy, which is also temperature independent,
over RT.
To do this comparison, we'll actually start by looking at a little bit different definition
of the activation energy, and first we simply take the log of both sides of this equation,
and then we're going to take the derivative of both sides of the equation with respect
to the temperature.
When we take that derivative, the first term is constant with temperature, so it becomes
zero.
The second term, when we take the derivative, we get an E over RT squared, and the sign
changes, so this is another way of defining the activation energy that is equal to RT
squared times the derivative quantity.
We're going to basically take the same kind of derivative with respect to this rate constant,
so first we take the log of both sides, and now we're going to take the derivative with
respect to temperature.
And this first term, when we take the derivative of a natural log, we get 1 over the argument,
so h over k_B T, and we multiply that by the derivative of the argument with respect to
temperature, which is k_B over h.
So these terms cancel with the h and k_B and we're left with just a 1 over temperature
from that first term in the sum.
And for the second term, we can rely on a derivation from classical thermo that's known
as the Van't Hoff relationship, which relates the behavior of the equilibrium constant to
temperature.
And as you might remember, for an endothermic reaction like the one we're dealing with here,
the sign of delta H with our sign convention is that it will be positive.
If we're going from reactants to transition states we're going up in enthalpy, and we
expect that as the temperature increases for an endothermic process, the equilibrium constant
will also increase, and hence we have this positive dependence on the enthalpy.
The opposite, although it doesn't apply for most cases where you're dealing with a transition
state, the opposite would be true for an exothermic reaction, and again the Van't Hoff relation
is what relates the equilibrium constant to temperature, and it does so through the enthalpy
of that reaction.
In this case, that reaction is from the reactant to the transition state, where now we can
also plot enthalpy on this diagram and track what delta H to the transition state is.
That will define how the equilibrium constant, and therefore the rate constant, will change
with temperature.
So if we clean this up and re-express these two terms in the sum, we just get a 1 over
T plus an enthalpy to the transition state, enthalpy change to the transition state, divided
by RT squared.
To make it convenient to combine these terms, we can just go for a common denominator, so
to do that we can just multiply both the numerator and the denominator of the first term by RT,
and that will give us RT over RT squared, and therefore we can see by comparing the
equation on the left and on the right, on the left hand sign, both are d(ln(k))/dT,
so we're going to expect the right hand sides to also be equal, and we see that therefore,
we expect that the Arrhenius activation energy is related to the enthalpy to the transition
state plus RT.
So it turns out that the activation energy is generally very close in value to the enthalpy
difference between the reactants and the transition state, and that's because RT for most reactions
of interest that are conducted at room temperature to 500 Kelvin or so, RT will have a value
of approximately 1 kcal per mole, whereas typical enthalpy change to the transition
state for a reaction we might care about can have various values, but you're usually looking
at something in the range of 10-35 kcals per mole.
Therefore, RT is a relatively small fraction of the activation energy, and the majority
of it in most cases is determined by the enthalpy change to the transition state.
In that way, on one of these plots, we can compute the enthalpy change to the transition
state, and that essentially tells us right away the activation energy, and if we want
to try to be more accurate about it, we can add this small factor of RT.
One thing I want to clarify with this diagram is that the Gibbs free energy and the enthalpy
are related to each other through the addition of the temperature times the entropy change.
They'll both look qualitatively like this, but they won't follow the same trend, and
I just took a shortcut using the same diagram for both G and H. They will not follow the
same shape or have the same values, since they are offset from each other by T delta
S. Finally, the thing I want to note about this relationship is that it is actually dependent
on the molecularity of the reaction.
And what I mean by that is that this is a first order reaction with a first order rate
constant, so that we only have a unimolecular reaction.
In other words, one molecule goes to make one molecule in the product state.
The other common case that we're typically interested in would be a bimolecular reaction,
so where A and B must collide with each other to form a transition state that then leads
to products.
In that case we actually get a different relationship if you go through and do this same kind of
derivation, and that relationship is just that we get an extra factor of 2 in front
of the RT for a bimolecular reaction, and trimolecular reactions are rare, but if you
do have a trimolecular case, then that 2 just becomes a 3.
In other words, the factor that's in front of RT is directly related to the molecularity
of the reaction.