MRD:phw417

2019-05-22T13:46:26Z

Phw417: /* Reverse reaction, H + HF: */

== Exercise 1: H + H2 system ==

=== Dynamics from the transition state region ===
On a potential energy surface diagram, the transition state is mathematically defined as the saddle point. The transition state has partial derivatives of zero with respect to the molecular distances, (fx = 0 and fy = 0) as it is the maximum on the minimum energy path linking reactants and the products . To distinguish between a local minimum or a local maximum and a transition state, the second derivatives of the potential energy surface with respect to the molecular distances is examined. Compute ''D = fxxfyy - (fxy)2 ''and substitute'' (x,y) ''where it is the parameters for the stationary point. There is a saddle point if D is negative.

=== Trajectories from r1 = r2: locating the transition state ===
[[File:Phw417 r1=r2 Surface Plot.png|centre|thumb|Fig 1 Internuclear distance plot for rBC = rAB = 0.9077421 Å|294x294px]]
The best estimate of the transition state position was 0.9077421 Å, it was found by minimising the forces to zero on the three H atoms. This estimate is good enough as they appear to be straight lines instead of sinusoidal functions.

=== Trajectories from r1 = rts + δ, r2 = rts ===
The conditions were set to r1 = 0.9087421 Å and r2 = 0.9077421 Å:
[[File:Phw417 rts+δ(mep) Surface Plot.png|centre|thumb|379x379px|Fig 2 MEP reaction path]]
[[File:Phw417 rts+δ(dynamics) Surface Plot.png|centre|thumb|381x381px|Fig 3 Dynamic reaction path]]
The difference between MEP and Dynamic reaction path is the absence of oscillatory motion in the MEP calculation. This is because the trajectory flows downhill along the minimum energy path at the transition state with all the inertia effects of the atoms removed.

If the values of r1 and r2 are reversed, the trajectory will be mirrored along the line y=x.
[[File:Phw417 rts+δ(mep,reversed) Surface Plot.png|centre|thumb|372x372px|Fig 4 MEP reaction path with reversed distances]]
If the signs of the momenta are reversed, then the trajectory will approach very closely to the transition state but falls back and follow back the mirror image of the same trajectory. The time was set to be at 0.4 s.
{| class="wikitable"
|AB distance
|0.982206
|-
|BC distance
|0.849663
|-
|AB momentum
|<nowiki>-0.370287 </nowiki>
|-
|BC momentum
|0.205708
|}
[[File:Phw417 rts+δ(mep,reversed,contour) Surface Plot.png|centre|thumb|365x365px|Fig 5 H + H2 dynamics trajectory with reversed momenta]]
[[File:Phw417 rts+δ(mep,reversed,distance) Surface Plot.png|centre|thumb|364x364px|Fig 6 H + H2 internuclear distances with reversed momenta]]

=== Reactive and unreactive trajectories ===
The initial positions were set to be '''r1''' = 0.74 and '''r2''' = 2.0.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics
!Trajectories
|-
| -1.25 || -2.5 ||-99.018||Yes||HA - HB and HC collide with weak oscillations. HB breaks bond with HA and forms bond with HC. Some fluctuations of HA - HB can be seen.
|[[File:Phw417 Reactive and unreactive trajectories(1).png|centre|thumb]]
|-
| -1.5 || -2.0 ||-100.456||No||HA - HB and HC gently collide with weak oscillation, but no reaction occurs. No bonds made or broken in the collision.
|[[File:Phw417 Reactive and unreactive trajectories(2).png|centre|thumb]]
|-
| -1.5 || -2.5 ||-98.956||Yes||HA - HB and HC collide with weak oscillation. HB breaks bond with HA and form bond with HC. HB - Hc leaves with more oscillation.
|[[File:Phw417 Reactive and unreactive trajectories(3).png|centre|thumb]]
|-
| -2.5 || -5.0 ||-84.956||No||HA - HB and HC collide with strong oscillations. HB breaks bonds with HA and bonds with HC, HB - Hc briefly forms but reverts back to HA - HB as HB breaks bond with HC. HA - HB and HC seperates with its original configuration.
|[[File:Phw417 Reactive and unreactive trajectories(4).png|centre|thumb]]
|-
| -2.5 || -5.2 ||-83.416||Yes||HA-HB approaches HC violently (with strong oscillations). Collision between reactants causes initial successful reaction, but the strong oscillation leads to the break-down of the 'first products'. The 'first products' then reacts again to form HA and HB-HC, hence overall reactive. 

|[[File:Phw417 Reactive and unreactive trajectories(5).png|centre|thumb]]
|}
Conclusion from the table: A molecule with energy greater than its activation energy does not always lead to successful reactions . Specific trajectories, including initial momenta and separations are crucial for a successful reaction to occurs. This is shown in reaction 4 where the products are formed but then reform reactants.

=== Main assumptions of Transition State Theory ===
Transition State Theory (TST) is a method for estimating rate constants of transitions. This method identify the dividing surface (transition state) that separates the reactants (initial state) and the products (final state). The assumptions are:
# Once the system passes the transition state into the products, it does not return to the reactants.
# The rate is slow enough that atoms in the reactant state have energies that are Boltzmann distributed.
# Quantum tunneling effects are assumed negligible
# Born-Oppenheimer approximation is introduced
As a conclusion, the reaction rates obtained using transition state theory would be higher than experimental values.

== Exercise 2: F - H - H system ==

=== PES Inspection ===
[[File:Phw417 HF surface plot.png|centre|thumb|347x347px|Fig 7 Surface plot for F + H2 using Dynamic calculation.]]
F + H2 -> H + HF Exothermic as VAB < VBC

H + HF -> F + H2 Endothermic as VAB > VBC

From the PES plot, F + H2 -> H + HF is an exothermic reaction as F and H2 are located at higher potential energy surface. The H2 molecule gains kinetic energy as it loses potential energy when H-H bond starts to break due to conservation of energy. The bond strength of H-F bond is 569 kJmol-1 , which is higher than that of H-H bond of 432 kJmol-1<ref><nowiki>http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html</nowiki></ref>. The bond formed is of higher energy than the bond broken. Therefore, energy is released and the reaction is exothermic.

Transition state is at rHF = 1.81025 Å and rHH = 0.744891 Å. Forces along H-F is -0.002 whilst along H-H is 0.000. Transition state total energy = -103.752 kcal/mol.
[[File:Phw417 HF(distance).png|centre|thumb|343x343px|Fig 8 Internuclear distance against time for the F-H2 system at the transition state]]
The Hammond's Postulate states that the transition state of a reaction resembles the structure of the species (reactant or product) to which it is closer in energy. Since F + H2 -> H + HF is an exothermic reaction, the transition state should resemble its reactants (early TS).

Activation Energy F + H2: -103.752 + 104.017 = +0.265 kcal/mol.[[File:Phw417 TS of F+H2.png|centre|thumb|Fig 9 Determining the activation energy of H + HF|342x342px]]

Activation Energy H + HF: -103.752 + 134.022 = +30.270 kcal/mol.

=== Reaction Dynamics ===

==== Question 1: ====
The set of Initial conditions that results in a reactive trajectory are : rFH = 2.0, rHH = 0.74, pFH = -0.9 and pHH = -0.7

with Etot = -101.928 kcal/mol
[[File:Phw417 contour plot of F+H2.png|centre|thumb|337x337px|Fig 10 F - H - H trajectory with initial trajectories seen above]]
[[File:Phw417 energy plot of F+H2.png|centre|thumb|332x332px|Fig 11 F - H - H energy vs time]]
[[File:Phw417 momentum plot of F+H2.png|centre|thumb|332x332px|Fig 12 F - H - H momentum vs time]]Initially, there is only little vibrational energy and the system is high in potential energy. As the reaction proceeds, there is an increase in the magnitude of momentum and average kinetic energy. The total energy of the system remains constant. This shows that there is an increase in vibrational energy in the HF molecule, which releases as heat in the system. This can be confirmed by performing the reaction and measuring the increase in temperature of the exothermic reaction using a bomb calorimeter.

==== Question 2: ====
For the reaction of F + H2, with rFH = 2.0, rHH = 0.74, pFH = -0.5 and pHH between -3 and 3, the results of calculation can be shown in the table below (The step size was set to be 0.002):

{| class="wikitable"
!pHH
!Etot
!Reactive?
!Description of the dynamics
|-
|<nowiki>-3.00</nowiki>
|<nowiki>-96.159</nowiki>
|No
|HAHB collides with F then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.95</nowiki>
|<nowiki>-96.431</nowiki>
|No
|HAHB collides with F twice then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.90</nowiki>
|<nowiki>-96.699</nowiki>
|No
|HAHB collides with F 4 times then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.85</nowiki>
|<nowiki>-96.961</nowiki>
|No
|HAHB collides with F 3 times then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.80</nowiki>
|<nowiki>-97.219</nowiki>
|No
|HAHB collides with F twice then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.70</nowiki>
|<nowiki>-97.719</nowiki>
|Yes
|HAHB collides with F twice then rebounds back as HB, forming HF as a product.
|-
|<nowiki>-2.60</nowiki>
|<nowiki>-98.199</nowiki>
|No
|HAHB collides with F then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.50</nowiki>
|<nowiki>-98.659</nowiki>
|No
|HAHB collides with F then rebounds back as H2, leaving as H2.
|-
|<nowiki>-2.00</nowiki>
|<nowiki>-100.659</nowiki>
|Yes
|HAHB collides with F and rebounds back as HB to form HF as a product.
|-
|<nowiki>-1.50</nowiki>
|<nowiki>-102.159</nowiki>
|Yes
|HAHB collides with F 3 times and rebounds back as HB to form HF as a product.
|-
|<nowiki>-1.00</nowiki>
|<nowiki>-103.159</nowiki>
|Yes
|HAHB collides with F twice and rebounds back as HB to form HF as a product.
|-
|0.00
|<nowiki>-103.659</nowiki>
|Yes
|HAHB collides with F 3 times and rebounds back as HB to form HF as a product.
|-
|<nowiki>+1.00</nowiki>
|<nowiki>-102.159</nowiki>
|No
|HAHB collides with F twice and rebounds back as H2.
|-
|<nowiki>+1.50</nowiki>
|<nowiki>-100.659</nowiki>
|Yes
|HAHB collides with F and rebounds back as HB to form HF as a product.
|-
|<nowiki>+2.00</nowiki>
|<nowiki>-98.659</nowiki>
|No
|HAHB collides with F and rebounds back as H2.
|-
|<nowiki>+2.50</nowiki>
|<nowiki>-96.159</nowiki>
|No
|HAHB collides with F and rebounds back as H2.
|-
|<nowiki>+2.60</nowiki>
|<nowiki>-95.599</nowiki>
|No
|HAHB collides with F and rebounds back as H2.
|-
|<nowiki>+2.70</nowiki>
|<nowiki>-95.019</nowiki>
|No
|HAHB collides with F 3 times and rebounds back as H2.
|-
|<nowiki>+2.80</nowiki>
|<nowiki>-94.419</nowiki>
|Yes
|HAHB collides with F 9 times and rebounds back as HB to form HF as a product.
|-
|<nowiki>+2.85</nowiki>
|<nowiki>-94.111</nowiki>
|Yes
|HAHB collides with F 3 times then rebounds back as HB, forming HF as a product.
|-
|<nowiki>+2.90</nowiki>
|<nowiki>-93.799</nowiki>
|Yes
|HAHB collides with F twice and rebounds back as HB to form HF as a product.
|-
|<nowiki>+2.95</nowiki>
|<nowiki>-93.481</nowiki>
|Yes
|HAHB collides with F five times and rebounds back as HB to form HF as a product.
|-
|<nowiki>+3.00</nowiki>
|<nowiki>-93.159</nowiki>
|No
|HAHB collides with F and rebounds back as H2.
|}
From the table, it can be concluded that molecules having enough vibrational energy is not sufficient to lead to a successful reaction. The trajectory sometimes is still not able to go pass the transition state to form products even though it possess sufficient energy.

==== Question 3: ====
For the same initial positions (rFH = 2.0, rHH = 0.74), the momentum was increased slightly to pFH = -0.8, and the overall energy of the system was considerably reduced by reducing the momentum pHH = 0.1.
[[File:Phw417 contour plot of F+H2(reduced energy).png|centre|thumb|367x367px|Fig 13 Contour plot of F + H2 system with low vibrational energy ]]
The total energy of the system is at -103.364 kcalmol-1, which is very low. However, it still results in a successful reaction.

==== Reverse reaction, H + HF: ====
The initial conditions were set to be rFH = 0.92, rHH = 2.0, pFH = -10.0 and pHH = -16.0.

The total energy is +14.929 kcalmol-1. The initial kinetic energy is +148.632 kcalmol-1, which is beyond the activation energy. As a result, the reaction is successful producing H2.
[[File:Phw417 contour plot of HF+H.png|centre|thumb|372x372px|Fig 14 Reaction dynamics trajectory of H + HF ]]
Then, another calculation was performed with increased magnitude of momentum with the same initial positions (rFH = 0.92, rHH = 2.0). The initial momenta were set to be pFH = -16.0 and pHH = -16.0.

The total energy is + 1.034kcalmol-1. The initial kinetic energy is + 134.737 kcalmol-1, which is still beyond the activation energy. However, the reaction is unsuccessful.
[[File:Phw417 contour plot of HF+H(increased momenta).png|centre|thumb|391x391px|Fig 15 Unreacted dynamics trajectory of H +HF with increased momenta]]

MRD:phw417

2019-05-21T15:07:20Z

Phw417: /* F - H - H system */

MRD:phw417

2019-05-21T15:06:36Z

Phw417:

== H + H2 system ==

=== Dynamics from the transition state region ===
On a potential energy surface diagram, the transition state is mathematically defined as the saddle point. The transition state has partial derivatives of zero with respect to the molecular distances, (fx = 0 and fy = 0) as it is the maximum on the minimum energy path linking reactants and the products . To distinguish between a local minimum or a local maximum and a transition state, the second derivatives of the potential energy surface with respect to the molecular distances is examined. Compute ''D = fxxfyy - (fxy)2 ''and substitute'' (x,y) ''where it is the parameters for the stationary point. There is a saddle point if D is negative.

=== Trajectories from r1 = r2: locating the transition state ===
[[File:Phw417 r1=r2 Surface Plot.png|centre|thumb|Fig 1 Internuclear distance plot for rBC = rAB = 0.9077421 Å]]
The best estimate of the transition state position is 0.9077421 Å, it was found by minimising the forces to zero on the three H atoms. This estimate is good enough as they appear to be straight lines instead of sinusoidal functions.

=== Trajectories from r1 = rts + δ, r2 = rts ===
The conditions are set to r1 = 0.9087421 Å and r2 = 0.9077421 Å:
[[File:Phw417 rts+δ(mep) Surface Plot.png|centre|thumb|379x379px|Fig 2 MEP reaction path]]
[[File:Phw417 rts+δ(dynamics) Surface Plot.png|centre|thumb|381x381px|Fig 3 Dynamic reaction path]]
The difference between MEP and Dynamic reaction path is the absence of oscillatory motion in the MEP calculation. This is because the trajectory flows downhill along the minimum energy path at the transition state with all the inertia effects of the atoms removed.

If the values of r1 and r2 are reversed, the trajectory will be mirrored along the line y=x.
[[File:Phw417 rts+δ(mep,reversed) Surface Plot.png|centre|thumb|372x372px|Fig 4 MEP reaction path with reversed distances]]
If the signs of the momenta are reversed, then the trajectory will approach very closely to the transition state but falls back and follow back the mirror image of the same trajectory. The time is set to be at 0.4 s.
{| class="wikitable"
|AB distance
|0.982206
|-
|BC distance
|0.849663
|-
|AB momentum
|<nowiki>-0.370287 </nowiki>
|-
|BC momentum
|0.205708
|}
[[File:Phw417 rts+δ(mep,reversed,contour) Surface Plot.png|centre|thumb|365x365px|Fig 5 H + H2 dynamics trajectory with reversed momenta]]
[[File:Phw417 rts+δ(mep,reversed,distance) Surface Plot.png|centre|thumb|364x364px|Fig 6 H + H2 internuclear distances with reversed momenta]]

=== Reactive and unreactive trajectories ===
The initial positions are set to be '''r1''' = 0.74 and '''r2''' = 2.0.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics
!Trajectories
|-
| -1.25 || -2.5 ||-99.018||Y||HA - HB and HC collide with weak oscillations. HB breaks bond with HA and forms bond with HC. Some fluctuations of HA - HB can be seen.
|[[File:Phw417 Reactive and unreactive trajectories(1).png|centre|thumb]]
|-
| -1.5 || -2.0 ||-100.456||N||HA - HB and HC gently collide with weak oscillation, but no reaction occurs. No bonds made or broken in the collision.
|[[File:Phw417 Reactive and unreactive trajectories(2).png|centre|thumb]]
|-
| -1.5 || -2.5 ||-98.956||Y||HA - HB and HC collide with weak oscillation. HB breaks bond with HA and form bond with HC. HB - Hc leaves with more oscillation.
|[[File:Phw417 Reactive and unreactive trajectories(3).png|centre|thumb]]
|-
| -2.5 || -5.0 ||-84.956||N||HA - HB and HC collide with strong oscillations. HB breaks bonds with HA and bonds with HC, HB - Hc briefly forms but reverts back to HA - HB as HB breaks bond with HC. HA - HB and HC seperates with its original configuration.
|[[File:Phw417 Reactive and unreactive trajectories(4).png|centre|thumb]]
|-
| -2.5 || -5.2 ||-83.416||Y||HA-HB approaches HC violently (with strong oscillations). Collision between reactants causes initial successful reaction, but the strong oscillation leads to the break-down of the 'first products'. The 'first products' then reacts again to form HA and HB-HC, hence overall reactive. 

|[[File:Phw417 Reactive and unreactive trajectories(5).png|centre|thumb]]
|}
Conclusion from the table: A molecule with energy greater than its activation energy does not always lead to successful reactions . Specific trajectories, including initial momenta and separations are crucial for a successful reaction to occurs. This is shown in reaction 4 where the products are formed but then reform reactants.

=== Main assumptions of Transition State Theory ===
Transition State Theory (TST) is a method for estimating rate constants of transitions. This method identify the dividing surface (transition state) that separates the reactants (initial state) and the products (final state). The assumptions are:
# Once the system passes the transition state into the products, it does not return to the reactants.
# The rate is slow enough that atoms in the reactant state have energies that are Boltzmann distributed.
# Quantum tunneling effects are assumed negligible
# Born-Oppenheimer approximation is introduced
As a conclusion, the reaction rates obtained using transition state theory would be higher than experimental values.

== F - H - H system ==

MRD:phw417

2019-05-21T14:26:34Z

Phw417: /* Reactive and unreactive trajectories */

MRD:phw417

2019-05-21T14:14:35Z

Phw417: /* Trajectories from r1 = rts+δ, r2 = rts */

MRD:phw417

2019-05-21T14:06:25Z

Phw417: /* Trajectories from r1 = r2: locating the transition state */