ChemWiki - User contributions [en]

JEL3117 MolecularReactionDynamics

2019-05-17T16:16:57Z

Jel3117: /* Reaction Dynamics */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillaton. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential energy of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the potential energy of the molecule increases until it reaches the plateau. The difference between the energy of the plateau and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition where time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

For the transition state of a symmetric molecule, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning maximum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning minimum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because at the local minimum the second derivative of the potential energy with respect to one of the distances would not give positive value (not a maximum point).

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the stationary maximum point on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 Å and r2 = 0.908 Å and p1 = p2 = 0 kg m/s.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 Å and r2 = 0.909 Å, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then reaches the plateau which means that the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the above data supports that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot / KJ mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows<ref name="transition state"/> :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

Experimentally, the above mechanism can be confirmed by measuring the heat released during the reaction with a calorimeter and the vibrational energy of HF molecule can be measured by using IR.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.<ref name="polany's rule"/>

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. ''The Morse Potential graph''. Available from : https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/05._The_Harmonic_Oscillator_and_the_Rigid_Rotator%3A_Two_Spectroscopic_Models/5.3%3A_The_Harmonic_Oscillator_Approximates_Vibrations </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. ''Atkin's Physical Chemistry''. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. ''Chemical Kinetics and Dynamics''. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T16:13:28Z

Jel3117: /* Reactive and unreactive trajectories */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillaton. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential energy of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the potential energy of the molecule increases until it reaches the plateau. The difference between the energy of the plateau and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition where time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

For the transition state of a symmetric molecule, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning maximum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning minimum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because at the local minimum the second derivative of the potential energy with respect to one of the distances would not give positive value (not a maximum point).

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the stationary maximum point on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 Å and r2 = 0.908 Å and p1 = p2 = 0 kg m/s.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 Å and r2 = 0.909 Å, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then reaches the plateau which means that the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the above data supports that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot / KJ mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows<ref name="transition state"/> :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.<ref name="polany's rule"/>

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. ''The Morse Potential graph''. Available from : https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/05._The_Harmonic_Oscillator_and_the_Rigid_Rotator%3A_Two_Spectroscopic_Models/5.3%3A_The_Harmonic_Oscillator_Approximates_Vibrations </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. ''Atkin's Physical Chemistry''. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. ''Chemical Kinetics and Dynamics''. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T16:12:34Z

Jel3117: /* Exercise 1 : H + H2 System */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillaton. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential energy of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the potential energy of the molecule increases until it reaches the plateau. The difference between the energy of the plateau and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition where time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

For the transition state of a symmetric molecule, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning maximum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning minimum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because at the local minimum the second derivative of the potential energy with respect to one of the distances would not give positive value (not a maximum point).

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the stationary maximum point on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 Å and r2 = 0.908 Å and p1 = p2 = 0 kg m/s.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 Å and r2 = 0.909 Å, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then reaches the plateau which means that the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the above data supports that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows<ref name="transition state"/> :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.<ref name="polany's rule"/>

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. ''The Morse Potential graph''. Available from : https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/05._The_Harmonic_Oscillator_and_the_Rigid_Rotator%3A_Two_Spectroscopic_Models/5.3%3A_The_Harmonic_Oscillator_Approximates_Vibrations </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. ''Atkin's Physical Chemistry''. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. ''Chemical Kinetics and Dynamics''. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T16:02:04Z

Jel3117: /* Introduction to Molecular Reaction Dynamics */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillaton. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential energy of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the potential energy of the molecule increases until it reaches the plateau. The difference between the energy of the plateau and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows<ref name="transition state"/> :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.<ref name="polany's rule"/>

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. ''The Morse Potential graph''. Available from : https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/05._The_Harmonic_Oscillator_and_the_Rigid_Rotator%3A_Two_Spectroscopic_Models/5.3%3A_The_Harmonic_Oscillator_Approximates_Vibrations </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. ''Atkin's Physical Chemistry''. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. ''Chemical Kinetics and Dynamics''. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T15:58:35Z

Jel3117:

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows<ref name="transition state"/> :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.<ref name="polany's rule"/>

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. ''The Morse Potential graph''. Available from : https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/05._The_Harmonic_Oscillator_and_the_Rigid_Rotator%3A_Two_Spectroscopic_Models/5.3%3A_The_Harmonic_Oscillator_Approximates_Vibrations </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. ''Atkin's Physical Chemistry''. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. ''Chemical Kinetics and Dynamics''. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T15:55:11Z

Jel3117: /* References */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.

== References ==

<references>
<ref name="MorsePotential"> Chemistry LibreTexts. The Morse Potential (blue) </ref>
<ref name="transition state">
Atkins, P.W., and Paula J. De. Atkin's Physical Chemistry. Oxford: Oxford University Press. 2006</ref>
<ref name="polany's rule"> Jeffrey I.Steinfeld, Joseph S. Francisco, William L.Hase. Chemical Kinetics and Dynamics. United States: A Paramount Communications Company, 1989 </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T15:46:54Z

Jel3117: /* Reaction Dynamics */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

=== Polanyi's empirical rule ===

The position of the transition state is important in determining the distribution of energy between different modes. For the early transition (exothermic reaction), having translational energy in majority is sufficient to form the product because having vibrational energy would cause the molecule to slide the potential energy surface side to side such that it does not have enough energy to overcome the activation energy. However, for the late transition (endothermic reaction, having vibrational energy helps to climb the potential energy surface up to the transition state. Having too much vibrational energy bounces back the product to the reactant.

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

File:JEL 3117 energy time graph endothermic.png

2019-05-17T15:28:46Z

Jel3117:

File:JEL 3117 energy time graph exothermic.png

2019-05-17T15:28:38Z

Jel3117:

File:JEL3117 F H2 approximate TS potential surface2.png

2019-05-17T15:28:23Z

Jel3117:

File:JEL3117 F H2 approximate TS intermoleculardistance2.png

2019-05-17T15:27:30Z

Jel3117:

File:JEL3117 H HF potential curve1.png

2019-05-17T15:27:00Z

Jel3117:

File:JEL3117 F HH potential curve1.png

2019-05-17T15:26:46Z

Jel3117:

File:JEL 3117 momentum time graph exothermic.png

2019-05-17T15:26:27Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-17T15:17:48Z

Jel3117: /* Reaction Dynamics */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

By the conservation of energy, the total energy is always conserved. At the start, H2 and and F atom are stationary thus they only have potential energies. As they are brought together, the potential energies between them decrease and the kinetic energies of H2 and F atom subsequently increase. As they collide, they start to oscillate and H-H bond is broken and new H-F bond is formed. Since the reaction is exothermic, the excess energy is released from the reaction as heat and the remaining energy is converted to the vibrational energy of HF molecule and kinetic energy of H atom.

[[File:JEL_3117_momentum_time_graph_exothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T14:39:03Z

Jel3117: /* Exercise 2 : F - H - H system */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

=== Approximate Position of the transition state ===

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

With trial and errors, the position of the transition state for the exothermic F + H2 reaction was found at r1(AB Distance) = 1.8106929 Å and r2(BC Distance) = 0.745 Å.

The position of the transition state for the endothermic H + HF reaction was found at r1(AB Distance) = 0.745 Å and r2(BC Distance) = 1.8106929 Å.

The activation energies of both of the reactions were found by displacing the activated complex slightly from the transition state.

[[File:JEL_3117_energy_time_graph_exothermic.png|frame|none|Figure 18. Energy vs time graph of the exothermic reaction which illustrates the activation energy of the reaction.]]

[[File:JEL_3117_energy_time_graph_endothermic.png|frame|none|Figure 19. Energy vs time graph of the endothermic reaction which illustrates the activation energy of the reaction.]]

The activation energy of the exothermic reaction F + H2 was calculated by displacing r1 to 1.8107 Å. However, the activation energy of the exothermic reaction was negligible such that was hard to be observed from the energy vs time graph.

The activation energy of the endothermic reaction was calculated by displacing r2 to 1.8106 Å. The activation energy was found as 29.5 KJ/mol.

=== Reaction Dynamics ===

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T13:48:03Z

Jel3117:

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

''' Approximate Position of the transition state '''

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time). Likewise, the transition state of the endothermic H + HF reaction lies near the product.

By manipulating the bond distances at zero momenta, the position of the transition state for the exothermic F + H2 reaction is at r1(AB Distance) = 1.8085786 Å and r2(BC Distance) = 0.75 Å.

The position of the transition state for the endothermic H + HF reaction is at r1(AB Distance) = 0.75 Å and r2(BC Distance) = 1.8085786 Å.

The activation energy of the exothermic reaction is

B

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T13:17:50Z

Jel3117: /* Exercise 2 : F - H - H system */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve1.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve1.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

''' Approximate Position of the transition state '''

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance2.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface2.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time.

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T13:16:50Z

Jel3117: /* Exercise 2 : F - H - H system */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is exothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is endothermic because the energy of the product is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + HF reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic F + H2 reaction .

''' Approximate Position of the transition state '''

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the exothermic reaction F + H2 lies near the reactant. Looking at F + H2 graph, the transition state therefore lies near the reactant and it is supported by the intermolecular vs time graph as shown above (No change in the intermolecular distances for a subsequent period of time.

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-17T12:55:30Z

Jel3117: /* Exercise 2 : F - H - H system */

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is endothermic because the energy of the product is higher than the energy of the reactant and the H + HF reaction is exothermic because the energy of the reactant is higher than the energy of the reactant.

This means that the bond strength of H2 is weaker than the bond strength of HF because the energy released from breaking H-H bond is lower than the energy required to form the H-F bond for the endothermic F + H2 reaction and the energy released from breaking H-F bond is higher than the energy required to form the H-H bond for the exothermic H + HF reaction .

''' Approximate Position of the transition state '''

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the endothermic reaction lies near the product. Looking at F + H2 graph, the transition state therefore lies near the product and it is supported by the intermolecular vs time graph as shown above.

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

File:JEL3117 F H2 approximate TS potential surface.png

2019-05-16T15:55:42Z

Jel3117:

File:JEL3117 F H2 approximate TS intermoleculardistance.png

2019-05-16T15:55:29Z

Jel3117:

File:JEL3117 H HF potential curve.png

2019-05-16T15:55:12Z

Jel3117:

File:JEL3117 F HH potential curve.png

2019-05-16T15:54:51Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T15:54:31Z

Jel3117:

== Introduction to Molecular Reaction Dynamics ==

'''Potential Energy Curve'''

The relative motions of the atoms in a molecule can be described by using an anharmonic oscillator which is derived from a simple harmonic oscillator. The potential energy curve shows how the potential of the molecule (ex. diatomic molecule) changes as the nuclei are displaced from the equilibrium position which is located at the minimum point of the potential curve. In the diatomic case, the potential of the molecule increases exponentially as the separation between the nuclei decreases. As the distance between the nuclei increases, the energy of the molecule increases until it reaches a flat limiting potential. The difference between the energy of the flat line and the energy of the equilibrium position is the dissociation energy. When the separation is really small, the oscillator follows the simple harmonic oscillator but as the nuclei are further apart, they deviate from the ideality more and follows the anharmonic oscillator.

[[File:JEL3117_Morse_potential.png|frame|none|Figure 1. A potential energy curve of a diatomic molecule.<ref name="MorsePotential"/>]]

== Exercise 1 : H + H2 System ==

In order for H2 to react with H, the H atom should approach to the H2 with a sufficient amount of energy to overcome the activation energy. At the initial condition when time = 0. the bond distance between B and C is represented as r2 and the distance between A and B is represented as r1 as shown in the diagrams below

[[File:JEL_3117_animation_reactant.png|frame|none|Figure 2. positions of the H and H2 in the reactant state.]]

[[File:JEL_3117_animation_product.png|frame|none|Figure 3. positions of the H and H2 in the product state.]]

In the transition state, r1 and r2 should be equal. If there is a sufficient energy to overcome the activation energy, A and B will form a new bond and C will move away as shown above in figure 3.

[[File:JEL_3117_animation_transition_statte.png|frame|none|Figure 4. positions of the H and H2 in the transition state.]]

=== Defining transition state in a potential energy surface diagram ===

On a potential energy surface diagram, the transition state is mathematically defined by:

[[File:JEL3117_transition_state.png]] - (1)

[[File:JEL3117_transition_state1.png]] - (2)

[[File:JEL3117_transition_state2.png]] - (3)

All three equations above should be satisfied to identify the transition state. The first condition states that the product of the first derivative of the potential energy with respect to r1 and the first derivative of the potential energy with respect to r2 should be zero. The second condition states that the second derivative of the potential energy with respect to r1 should be less than zero (meaning minimum point). The third condition states that the second derivative of the potential energy with respect to r2 should me more than zero (meaning maximum point). Thus, the transition state is defined as the maximum point on the minimum energy path linking reactants and the products.

The transition states are graphically represented below.

[[File:JEL3117_transition_graph1.png|framed|none|300px|Figure 5. the potential energy curve where r1 is constant and r2 is a variable. It shows the minimum energy path along the black line.]]

[[File:JEL3117_transition_graph2.png|frame|none|Figure 6. the potential energy curve where r2 is constant and r1 is a variable. It shows the maximum energy path.]]

The transition states can be distinguished from a local minimum of the potential energy surface because the local minimum will not satisfy the first condition of the equation.

=== Trajectories from r1 = r2: locating the transition state ===

The estimated distance transition state position rts is 0.908 Å and this can be shown by "internuclear Distance vs Time" plot as shown below

[[File:JEL3117_transition_0.98.png|frame|none|Figure 7. internuclear Distance vs Time plot at the transition state position rts.]]

At the transition state, I expect the the atoms to vibrate minimum near to zero because the transition state is the maximum on the minimum energy path. As the they vibrate more and more, they will slide down the potential surface curve and separate to form either product or reactant.

=== Trajectories from r1 = rts + δ, r2 = rts===

[[File:JEL3117_transition1_MEP.png|frame|none|Figure 8. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition_MEP_momentavstime_graph.png|frame|none|Figure 9. momentum vs time graph of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in MEP.]]

[[File:JEL3117_transition1_DYNAMIC.png|frame|none|Figure 10. contour plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

[[File:JEL3117_transition_DYNAMIC_momentavstime_graph.png|frame|none|Figure 11. surface plot of the reaction H + H2 when r1 = 0.909, r2 = 0.908 in Dynamic.]]

The above graphs were plotted by setting r1 = 0.909 and r2 = 0.908 and p1 = p2 = 0.

the first two graphs show the reaction path of the reaction with the MEP mode. MEP sets the momenta and velocities of the molecule to zero (shown in figure 9) in each time every step and process the reaction in a infinitely slow motion. The last two graphs show the reaction path of the reaction with the dynamics mode. Dynamics shows the vibrational motion of the molecule which is derived from the internal energy of the molecule. Thus, oscillation of the energy of the reaction path is shown in the potential energy surface. By setting the momentum to zero at every point, the reaction path will stop when the exchange of the atoms is completed. However, when the momentum is not zero, the products will move away from each other continuously due to conservation of momentum. This is shown in figure 8 and 10 where the reaction path ends earlier for MEP than Dynamics.

If r1 = 0.908 and r2 = 0.909, A and B would form the H2 molecule and C would move away from the molecule. Thus, the reaction path would head towards the other end of the contour plot.

[[File:JEL3117_transition_DYNAMIC_intermolecular.png|frame|none|Figure 12. Intermolecular distance vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

[[File:JEL3117_transition_DYNAMIC_Momenta.png|frame|none|Figure 13. momenta vs time graph of the reaction H + H2 when r1 = 0.908, r2 = 0.909.]]

Figure 12 shows that the distance between B and C increases but the distance between A and B decreases with a little oscillation. Moreover, Figure 13 shows that the momentum of BC increases then plateaus meaning the B and C are separated and the momentum of AB increases then start to oscillate as it plateaus. It means that AB bonds are formed and they are oscillating. Thus, the two above data support that AB bond is formed instead of BC bond when r1 and r2 are reversed

=== Reactive and unreactive trajectories ===

{| class="wikitable" border="1"
|+ Reactive and unreactive trajectories
! p1 !! p2 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -1.25 || -2.5 || -99.018 || Reactive || Initially, C approaches AB molecule. As C approaches, the distance between A and B remains almost constant with no oscillations and the potential energy between B and C increases. As it reaches the transition state, the system has an enough kinetic energy to overcome the activation energy. Therefore, a new BC bond is formed and the AB bond is broken. Then, as the reaction path passes the transition state, the BC molecule oscillates and A moves away from the new product BC. || [[File:JEL3117_reactive_nonreactive_contour_first.png]]
|-
| -1.5 || -2.0 || -100.456 || Unreactive || Initially, the AB molecule oscillates and C approaches AB molecule. As the reaction path reaches the transition state, the system does not have an enough amount of energy to overcome the activation energy and the new BC bond is not formed. Thus, C moves away from the AB molecule and AB molecule remains oscillating and keep their bonds. ||[[File:JEL3117_reactive_nonreactive_contour_second.png]]
|-
| -1.5 || -2.5 || -98.956 || Reactive || Initially, the AB molecule oscillates and C approaches the AB molecule. As the reaction path reaches the transition state, the system has the enough energy to overcome the activation energy and a new BC bond is formed and A moves away from the product as the reaction proceeds. || [[File:JEL3117_reactive_nonreactive_contour_third.png]]
|-
| -2.5 || -5.0 || -84.956 || Unreactive || Initially, the AB molecule does not oscillate and C approaches the molecule AB. As the reaction approaches the transition state, the system has an enough energy to overcome the transition state and the new BC bond is formed and A moves away from the BC molecule. However, the oscillation of the BC bond is too big such that the BC bond is broken and AB bond is formed again. As a result, the AB molecule remains as it is and C moves away from the AB molecule. || [[File:JEL3117_reactive_nonreactive_contour_fourth.png]]
|-
| -2.5 || -5.2 || -83.416 || Reactive || Initially, the AB molecules does not oscillate and C approaches the molecule. As the reaction approaches the transition state, the AB molecule starts to oscillate dramatically such that AB bond is broken. As a result,the BC bond is formed and A moves away from the the molecule BC || [[File:JEL3117_reactive_nonreactive_contour_fifth.png]]
|}

From the table, it can be concluded that activation energy is not the only factor that affects the success of the reaction but other factors such as collision in a correct geometry also contributes to the success of the reaction.

=== Transition state Theory ===

The main assumptions of transition state theory are as follows :

1, For each elementary step in a multi-step reaction, the intermediates are long lived enough to reach a Boltzmann distribution of energies before proceeding to the next step.

2, The atomic nuclei behave according to the classic mechanics.

3, The reaction is successful if the reaction path passes over the lowest energy saddle point on the potential energy surface (transition state).

However, in reality, the assumptions start to break down. The transition states are very short lived and the Boltzmann distribution of energies at each step near activated complexes is hard to be obtained. Moreover, in reality, atoms behave in a quantum mechanical manner and the reaction might be successful to form the product even though the energy of the reactants is not enough to overcome the activation energy via quantum tunnelling. Moreover, in reality, the transition state is not necessarily at the lowest saddle point at high temperatures. Thus, there are alternative pathway to the reaction.

As a result, the predicted reaction rate values by using Transition State Theory would be lower than the experimental values due to limitations of the assumptions.

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

=== PES inspection ===

[[File:JEL3117_F_HH_potential_curve.png|frame|none|Figure 14. the potential energy surface of F + H2 reaction]]

[[File:JEL3117_H_HF_potential_curve.png|frame|none|Figure 15. the potential energy surface of H + HF reaction]]

Based on the potential energy surfaces, the F + H2 reaction is endothermic because the energy of the reactant is higher than the energy of the product and the H + HF reaction is exothermic because the energy of the reactant is lower than the energy of the product.

This means that the bond strength of H2 is weaker than the bond strength of HF because looking at the F + H2 reaction, the energy released from breaking H-H bond is lower than the energy required to form the H-F bond.

'' Approximate Position of the transition state ''

[[File:JEL3117_F_H2_approximate_TS_intermoleculardistance.png|frame|none|Figure 16. The intermolecular distance vs time graph of F + H2 reaction]]

[[File:JEL3117_F_H2_approximate_TS_potential_surface.png|frame|none|Figure 17. The potential energy surface of F + H2 reaction indicating the approximate position of the transition state.]]

Based on the Hammond's postulate, the transition state of the endothermic reaction near the product. Looking at F + H2 graph, the transition state therefore lies near the product and it is supported by the intermolecular vs time graph as shown above.

== References ==

<references>
<ref name="MorsePotential"> This is the image taken from chem libre text </ref>
</references>

JEL3117 MolecularReactionDynamics

2019-05-16T15:10:06Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T15:00:23Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T14:22:33Z

Jel3117:

File:JEL3117 transition DYNAMIC Momenta.png

2019-05-16T13:50:04Z

Jel3117:

File:JEL3117 transition DYNAMIC intermolecular.png

2019-05-16T13:49:47Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T13:48:25Z

Jel3117:

File:JEL3117 transition1 MEP.png

2019-05-16T13:32:12Z

Jel3117:

File:JEL3117 transition1 DYNAMIC.png

2019-05-16T13:30:04Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T13:29:42Z

Jel3117:

File:JEL3117 transition MEP.png

2019-05-16T13:28:35Z

Jel3117: Jel3117 uploaded a new version of File:JEL3117 transition MEP.png

File:JEL3117 transition MEP.png

2019-05-16T13:28:22Z

Jel3117: Jel3117 uploaded a new version of File:JEL3117 transition MEP.png

File:JEL3117 transition MEP.png

2019-05-16T13:28:06Z

Jel3117: Jel3117 uploaded a new version of File:JEL3117 transition MEP.png

JEL3117 MolecularReactionDynamics

2019-05-16T13:27:54Z

Jel3117: /* Trajectories from r1 = rts + δ, r2 = rts */

File:JEL3117 transition MEP momentavstime graph.png

2019-05-16T13:20:33Z

Jel3117:

File:JEL3117 reactive nonreactive contour fifth.png

2019-05-16T13:17:51Z

Jel3117:

File:JEL3117 reactive nonreactive contour fourth.png

2019-05-16T13:17:40Z

Jel3117:

File:JEL3117 reactive nonreactive contour third.png

2019-05-16T13:16:21Z

Jel3117:

File:JEL3117 reactive nonreactive contour second.png

2019-05-16T13:16:07Z

Jel3117:

File:JEL3117 reactive nonreactive contour first.png

2019-05-16T13:15:53Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T13:15:00Z

Jel3117: /* Reactive and unreactive trajectories */

JEL3117 MolecularReactionDynamics

2019-05-16T13:10:13Z

Jel3117: /* Reactive and unreactive trajectories */

JEL3117 MolecularReactionDynamics

2019-05-16T13:08:34Z

Jel3117: /* Reactive and unreactive trajectories */

JEL3117 MolecularReactionDynamics

2019-05-16T13:07:43Z

Jel3117:

File:JEL3117 transition DYNAMIC momentavstime graph.png

2019-05-16T12:43:08Z

Jel3117:

JEL3117 MolecularReactionDynamics

2019-05-16T12:42:32Z

Jel3117: /* Trajectories from r1 = rts + δ, r2 = rts */