MRD:ohoyoureapproachingme

Exercise 1 H+H₂:

In this system, we are simulating if an atom and a diatomic were to collide and by varying the bond distance and atom momenta to see if they would react. The simplest system would be a hydrogen atom colliding with a hydrogen molecule.

Transition state is mathematically defined as a point at which the function graphing the point has a derivative dV.(dr_i/dr_i) equal to 0. It can be identified by being a maxima on a minimum energy pathway, also a Hessian second derivative of the function can be taken and if less than 0, is shown to be a maxima and the transition state. For a symmetrical system like H + H₂, the transition state occurs when r_AB = r_BC shown in figure 1, in this case the transition state occurs when AB distance is 90.8 pm found in figure 2. Evidence to suggest this can be found in figure 3 and 4, figure 3 is a PES plot of the system with the co-ordinate placed on the peak of the minimum energy pathway and figure 4 shows that the distances have remained constant indicating little to no vibrations are occurring.

Transition state plots
Figure 1: Transition state range	Figure 2: Point of lowest energy H2 + H transition state	Figure 3: Surface plot of H2 reaction pathway	Figure 4: Inter-nuclear distance between H2 and H at transition state

From the transition state being known, the minimal energy pathway can be found by using a MEP calculation at transition state bond distances and momenta set to 0, this calculation would set the momenta of the atoms to be 0 at every time step, resulting in an infinitely slow reaction, figure 5 was found by offsetting the distance AB by 1 pm from the transition state distance and let occur for 500 seconds. Comparing a dynamic calculation with identical conditions, it shows that the molecule H₂ is vibrating

MEP calculation	Dynamic calculation

With distance r₁ set at 74 pm and r₂ set at 200 pm and varying the momenta the following table is found.

Set	p1/g.mol-1.pm.fs-1	p2/g.mol-1.pm.fs-1	Etot kJ.mol-1	Reactive?	Description of the dynamics	Illustration of the trajectory
1	-2.56	-5.1	-414.280	Yes
2	-3.1	-4.1	-420.077	No	The approaching hydrogen atom doesn't have enough energy to cause a favourable collision, doesn't reach transition state.
3	-3.1	-5.1	-413.977	Yes
4	-5.1	-10.1	-357.277	No	While the approaching atom has more than enough energy to react, the products also have enough energy to re-enter the transition state and revert to reactants.
5	-5.1	-10.6	-349.477	Yes	This trajectory passes over the transition state 3 times.

From the table, it can be found that from comparing momenta set 1 and set 2, that the reaction requires a certain amount of energy in order to make it to the transition state and possibly form products, as shown in set 3 where the total energy is around -414 kJ.mol^-1 allows the reaction to occur. Set 4 shows that too much energy would allow the products to react again and reverse the reaction to form the reagents.

Transition State Theory will overestimate the rate of reaction compared to the experimentally calculated rate of reaction due to the theory assuming that all trajectories that make it past the transition state will end up as products, which is then shown in set 4 which did have enough energy to make it to the transition state but the trajectory wasn't reactive.

Exercise 2 H + HF and F + H₂

F-H₁ r_ts = 181.15 pm

H₁-H₂ r_ts = 74.2 pm

E_ts = -433.97 kJ.mol^-1

Initial energy of reactants in F + H₂ = -434.71 kJ.mol^-1

Activation energy of F + H₂ = 0.74 kJ.mol^-1

Initial energy of reactants in H + HF = -560.31 kJ.mol^-1

Activation energy of H + HF = 125.6 kJ.mol^-1