Here is an animated form of the HCN <=> CNH isomerization:
Here is an animated form of the imaginary frequency of the transition state:
(Using values calculated by the PM3 method)
Heat of Reaction
Because Cerius2 calculates heats of formation for each of our calculations we will be able to use the classical definition of heat of reaction:
= 23.65 kcal/mol, a endothermic reaction.
Change in Entropy of Reaction
The change in entropy can be simply calculated by taking the difference of product and reactant entropies.
= 0.491E-3 kcal/mol K
Reaction Gibbs Free Energy Change
The Gibbs free energy is calculated:= 23.503 kcal/molEquilibrium Constant
To calculate the equilibrium constant:
= 7.528E-18
This value of the equilibrium constant shows, as expected, that at equilibrium there is a very small amount of CNH in respect to HCN.Activation Energy
The activation energy can be calculated by taking the difference of the transition state enthalpy and the reactant enthalpy.
= 75.135 kcal/mol
Preexponential Factor
The preexponential factor can be calculated using some transition state theory calculations. Let's just walk through the derivation of the preexponential factor step by step. The first mathematical relationship we will state must be taken as a given since the derivation of the relation is beyond the scope of this explanation.
The relation between the rate constant (k) and a pseudo transition state equilibrium constant:
(1)
where:
(2)
This is very much like the over all equilibrium constant but deals with the transition state instead of the overall reaction. The relation between the transition state equilibrium constant and the Gibbs free energy change of the transition state is:
(3)
The Gibbs free energy can be written as:
(4)
and substituted into the transition state equilibrium constant relation.
(5)
where:
= 4.5024 cal/mol K
and
Finally substitute equation (5) into (1) and
(6)
Examining equation (6) we see that it resembles the Arrhenius rate equation.
We can see that the first and second terms are equal to the preexponential factor and after defining the activation energy as:
we find that the last term is identical to the last term in the Arrhenius rate equation. Therefore, the preexponential is calculated by:= 6.026E13
With these quantities we could readily go ahead with a non-isothermal reactor design.
Lets move on and review what we have learned.
