ChemWiki - User contributions [en]

MRD:POLS

2018-05-25T11:39:32Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 6 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 7 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left |Figure 8 Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre |Figure 9 Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left |Figure 10 Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 11 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 12 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:39:09Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 6 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 7 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left |Figure 8 Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre |Figure 9 Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left |Figure 10 Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:38:46Z

Pg1616: /* Transition state of reactions */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 6 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 7 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:38:28Z

Pg1616: /* Reactive and unreactive trajectories */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:38:17Z

Pg1616: /* Reactive and unreactive trajectories */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:38:07Z

Pg1616: /* Calculating the reaction path */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 4 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 5 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:37:51Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy. This coincides with what it should be for an endothermic reaction.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:36:44Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective. The momenta was set to (p1=1.5, p2 = -0.55). This was then reversed. As can be seen this system does obey Polanyi's rules and a reaction occurs when the vibrational energy is high but not when it is low in respect to the translational energy

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:33:29Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:33:18Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

MRD:POLS

2018-05-25T11:32:58Z

Pg1616: /* Polanyi's empirical rules */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome a late transition state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The H + HF reaction is endothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in vibrational energy over translational is more effective.

[[File:PG_H_F_C.png|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High vibrational energy]]
[[File:PG_H_F_C_2.png|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of low vibrational energy]]

File:PG H F C.png

2018-05-25T11:25:17Z

Pg1616:

File:PG H F C 2.png

2018-05-25T11:25:03Z

Pg1616:

File:PG F H C.png

2018-05-25T11:24:09Z

Pg1616: Pg1616 uploaded a new version of File:PG F H C.png

MRD:POLS

2018-05-25T11:23:40Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises. This could be analyzed through IR spectroscopy to see the high overtone bands produced by molecules with higher vibrational states

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

===Polanyi's empirical rules===

The reactants in a reaction must have enough energy to overcome the activation barrier of a reaction. This energy can be in the form of vibrational or translational. Polanyi's rules state that vibrational energy is better at getting your reactants to overcome an late transistion state and they say that translational energy is better at getting your reactants to overcome an early transition state. Therefore for the exothermic reaction, F + H2 , translational energy in the molecule is more effective and for the H + HF reaction, vibrational energy is more effective. (p1=vibrational energy, p2 = translational energy)

The F + H2 reaction is exothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in translational energy over the vibrational energy would result in a reactive reaction.

Figure 16 shows a surface plot of a system that obeys Polanyi's rules, where p1=1.5, p2 = -5.5, producing a system with high translational energy and low vibrational energy. A system of high vibrational energy and low translational energy , where p1=6, p2 = -1.0, was seen not to produce a reactive reaction (Figure 17). This system obeys Polanyi's rules as only systems of high translational energy should cause a successful reaction when it is exothermic.[[File:Pol1.PNG|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High translational energy]]
[[File:Poly2.PNG|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of Low translational energy]]

MRD:POLS

2018-05-25T10:52:21Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''
This is an exothermic reaction and as discussed previously therefore releases energy into the surroundings, usually as heat energy if we assume no work on the surroundings is being done. This heat energy is released to the surroundings through vibrational motion. We can see this increase in vibrational motion on the contour map clearly as when the HF bond is made it has a high vibrational movement. On the energy Vs time plot we can clearly see that the energy is being conserved as as the potential energy decreases the kinetic energy rises.

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

MRD:POLS

2018-05-25T10:46:27Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 1.81, rH-H = 0.75, pF-H = -0.5 , pH-H = 0.8'''

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

MRD:POLS

2018-05-25T10:45:30Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 2.3, rH-H = 0.81, pF-H = -7.0, pH-H = -0.9'''

[[File:PG_F_H_C.png | thumb | 300px | left | Reaction trajectory]]
[[File:PG_F_H_E.png | thumb | 300px | centre | Energy vs Time]]
[[File:PG_F_H_M.png| thumb | 300px | left | Momentum vs Time]]

File:PG F H M.png

2018-05-25T10:43:28Z

Pg1616:

File:PG F H E.png

2018-05-25T10:43:14Z

Pg1616:

File:PG F H C.png

2018-05-25T10:42:58Z

Pg1616:

MRD:POLS

2018-05-25T10:42:33Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F + H2 reaction were:

'''rF-H = 2.3, rH-H = 0.81, pF-H = -7.0, pH-H = -0.9'''

MRD:POLS

2018-05-25T10:31:37Z

Pg1616: /* Reaction Dynamics */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===
The conditions found to get a reactive trajectory of F plus H2 were:

'''rF-H = 2.3, rH-H = 0.81, pF-H = -7.0, pH-H = -0.9'''

MRD:POLS

2018-05-25T10:29:10Z

Pg1616: /* Activation energy of the reaction */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be -104.028

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104.028) = 0.277 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.176 Kcal/mol

This makes sense as the HF bond is lower in energy than the H-H bond and therefore the reaction to form the HF bond would have a lower activation energy as the reactant bond (H-H) is of higher energy and so has less energy to gain to overcome the activation barrier. This is the reverse for the other reaction.

===Reaction Dynamics===

MRD:POLS

2018-05-24T15:50:14Z

Pg1616: /* Transition state of reactions */

==Transition state==
The transition state of a reaction is a transient structure that exists at a saddle point on a potential energy surface. The gradient of the total energy equals zero and is the maximum of the minimum energy path of the potential energy surface. The minimum point of the surface also has a gradient of 0 and so in order to determine which critical point is a saddle or minimum point is to take the second derivative of the potential energy surface. If the second derivative is less than zero than it is a saddle point/transition state and if it is greater than zero it is a minimum.

∂2V('''ri''')/∂2'''ri'''> 0 'Minimum'

∂2V('''ri''')/∂2'''ri'''< 0 'Saddle point'

==Locating The Transition state==
At the transition state there is no force acting on the hydrogen molecule as -∂V(ri)/∂ri=0. Therefore, if you start a trajectory exactly at the transition state with no momentum then it will stay there forever. As Hydrogen is symmetry we can set r1=r2 and set p1=p2=0 to estimate a value for the transition state.
By adjusting the bond distance in response to oscillation on the Inter-nuclear distance Vs Time graph, the transition bond length estimated to be a value of 0.9076 Å. This is because no change in the inter nuclear distance over time shows that the atoms are being held in that position. Also, the kinetic energy equals 0 and so this also indicates the transition state had been reached.

[[File:PG_KE_T_H.png|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:PG_ID_T_H.png|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:PG_ID_T_H_H.png|313x313px|thumb|centre|Figure 2 - A plot of the internuclear distance for the system, where the points intersect is the transition state]]

==Calculating the reaction path==

The reaction path ( minimum energy path) is a trajectory that requires the minimum energy to form the products. This can be mapped from the transition state and can be seen to follow the valley floor. Both this and the dynamic trajectory were run by changing one of the bond distances to 0.9176. The black line seen is the mapped trajectory. The dynamic surface plot shows oscillations of potential energy surface showing that the molecule is vibrating and this agrees with an inertial trajectory whereas the mep trajectory does not show any oscillation which is not an accurate representation of what is actually occurring. Also, the dynamic path shows the full movement hydrogen atom away in the gas phase whereas the mep shows no further movement to get the lowest energy which isn't a realistic account of the motion and is a very slow rate.

[[File:PG_SP_MEP.png|298x298px|thumb|right|Figure 3 - Surface plot using MEP]]
[[File:PG_SP_D.png|298x298px|thumb|left|Figure 1 - A surface plot using dynamic trajectory]]

==Reactive and unreactive trajectories==

It can be concluded that trajectories within the range '''r1''' = 0.74, '''r2''' = 2.0, with -1.5 < '''p1''' < -0.8 and '''p2''' = -2.5 are reactive. We can also assume that any trajectory with a momenta over the activation barrier will reach completion as they have enough kinetic energy to overcome the transition state and reach product formation. This was tested by using the initial positions '''r1''' = 0.74 and '''r2''' = 2.0 and changing the momenta.

{| class="wikitable" border="1"
|+ Effect of inertia (Dynamic calculation type)
! p1 !! p2 !! ETotal !! React/Non!!Contour plot !! Discussion
|-
| -1.25 || -2.5 || -99.199 || Reactive || [[File:PG_M_1.png|315px]] ||Prior to the transition state, the reaction trajectory is in a straight line and so shows that the BC molecule does not oscillate. The A molecule then attacks the diatomic molecule when it is stationary. The new AB diatomic molecule formed however does oscillate and so does vibrate. ||
|-
| -1.5 || -2.0 || -100.456 || Unreactive || [[File:PG_M_2.png|315px]] || This is an unreactive path as the trajectory does not reach the product channel. This is because the momentum is not high enough to overcome the activation barrier and so the reaction does not reach completion.
|-
| -1.5 || -2.5 || -98.956 || Reactive || [[File:PG_M_3.png|315px]] || The BC diatomic molecule oscillates slightly as A approaches but it seems to look like the BC bond length increases closer to when A approaches at the transition state compared to the other reactive trajectory's.
|-
| -2.5 || -5.0 || -84.956 || Unreactive || [[File:PG_M_4.png|315px]] || Initially the BC diatomic molecule is not vibrating but once A approaches there is heavy vibration and a triatomic complex appears to form and an AB bond forms. However this doesn't have enough kinetic energy to overcome the barrier and the complex dissociates and A disperses away from the now vibrating BC diatomic and reverts back down the reactant channel .||
|-
| -2.5 || -5.2 || -83.416 || Reactive || [[File:PG_M_5.png|315px]] || The reaction path again starts with A approaching a non oscillating BC molecule and then another complex is formed but this time it has enough energy to form the products and and the trajectory moves into the product channel. The AB diatomic molecule then has a large amount of oscillation.
||
|}

From this investigation we can then see that the previous assumption is not always correct as when p gave a value of -2.5 and -5.2 the reaction became unreactive.

===Transition State Theory===

Transition state theory is used to explain the rates of reactions and qualitatively how they take place. The three main assumptions are:

1) The reactants and the activated complex are in equilibrium ( quasi equilibrium ) but not the products
2) The reaction trajectory will pass through the saddle point/transition state on the potential energy surface.
3) The reactants nuclei adhere to classical mechanics

Assumption three, assumes that the nuclei obey newtons law's of motion and do not take into account tunneling or the fact that molecular vibrations are quantized as these are quantum mechanical effects and it also doesn't take into account barrier re crossing that can occur as was seen in the table. The complex formed and was high energy but did not produce products, this shows that not all energetic collisions result in product formation but TST assumes this. So, the rates of reaction assumed by TST would be faster than those determined experimentally.

==F-H-H system==
===Reaction energetics===
The F + H2 reaction is exothermic because the HF bond is lower in energy ( lower enthalpy) than the H-H bond. Therefore, when the new bond is formed energy is released into the surroundings making the reaction exothermic. Therefore H + HF is endothermic as the H-H bond is higher in energy ( higher enthalpy) than the H-F bond and so when the new H-H bond it has to take in energy from the surroundings making the reaction endothermic.

===Transition state of reactions===
Locating the transition state in this scenario is a little different as the system is no longer symmetric and so we cannot assume that r1=r2. But, the momentum (kinetic energy) is still 0. In an exothermic reaction the transition state most closely resembles the reactants and in an endothermic reaction the transition state more closely resembles the product. So for F + H-H the values weren't moved very far from the original values and the momentum was set to zero and it was found that the transition state was found at H-H 0.7463 and F-H 1.810 angstroms. For the H + HF system, the transition state was found by moving the bond lengths closer to the products which is the same as the reactants in the previous reaction. The same transition state was found as the same complex is formed in both cases; this is a good representation of Hammond's postulate. As can be seen the kinetic energy is zero showing that the transition state has been reached.

[[File:PG_F_T.png|300x300px|thumb|Figure 5 - A plot of Energy vs. Time for the F-H-H system]][[File:PG_T_F_H.png|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the F + H2 reaction]]

===Activation energy of the reaction===
The activation energy is the energy barrier that the reactants have to reach in order to form products. This barrier is difference between the energy of the transition state and the minimum of the reactants. The energy of the transition state was found to be -103.751. the F + H2 reaction was studied. MEP trajectory was undertaken and the bond length was changed towards the formation of FH just slightly (1.80) this gave an energy minima for the reaction to be -133.927, and for the HF + H reaction the same was done and the energy minima was found to be

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.751 -(-104) = 0.25 Kcal/mol

For the HF + H reaction : -103.751 -(-133.927) = 30.25 Kcal/mol

===Reaction Dynamics===

MRD:POLS

2018-05-24T15:02:51Z

Pg1616: /* Transition state of reactions */

File:PG T F H.png

2018-05-24T15:01:12Z

Pg1616:

MRD:Na2615

2018-05-24T14:37:36Z

Pg1616: /* The position of the transition state */

==H + H2 system Analysis==

=== Transition states and Minima ===
The transition state is a transient state of a reaction coordinate, which can also be referred to as a saddle point on a potential energy surface. Analysing the potential energy surface of the H + H2 system, the total energy gradient for the transition state and the minimum were both found to be zero, as ∂V(ri)/∂ri=0. The transition state can be distinguished from the minimum point by taking the second derivative of the gradient.

<center>∆ =〖((∂2z)/∂x∂y)〗2 - ((∂2z)/(∂x2))((∂2z)/(∂y2))</center>

where x and y are the critical points in question and z is the derivative of the critical points.

Using this determinant the critical point produced can be identified as a minima or a saddle point/ transition state. For if ∆ > 0 the point will be a saddle point but if ∆ < 0 and ((∂2z)/(∂x2)) > 0 then the point will be a local minima.<ref>C. M. Bender and S. A. Orszag, ''Advanced Mathematical
Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation
Theory'', Springer New York, 2013.</ref>

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 13:51, 9 June 2017 (BST) This is correct but it needs to be in the basis of the normal modes. one of these will be the reaction coordinate.

=== Locating the Transition state ===
The force acting on the hydrogen molecule is equal to -∂V(ri)/∂ri. As the gradient of the potential energy surface at the transition state is equal to zero, the force will also be equal to zero at the Transition state. Therefore there would be no energy in the form of kinetic energy and all would be converted into potential energy. As the the H + H2 surface is symmetric, the distances between the three atoms is equal at the transition state and as there is not force acting on the atoms, the momentum for both bond distances is equal to zero. The bond distance where the transition state occurs was found to be 0.90775 Å. Looking at the inter-nuclear distance vs time plot at this value (Figure 1), the inter-nuclear distance is initially constant, void of any oscillations, indicating that there is no/little kinetic energy (Figure 2) as the complex is in its transition state. Beyond 9 seconds, the complex is no longer in its transition state as the the inter-nuclear distances are no longer constant, and oscillations can be seen for the A-B bond distance.

The estimation for the transition state bond length was achieved by optimising the value obtained when the the inter-nuclear distances of AB and BC intersect (Figure 3).

[[File:Tstry1.PNG|281x281px|thumb|right|Figure 3 - A plot inter-nuclear distance vs. Time for the H + H2 system]]

[[File:H2_Trans_distance.PNG|323x323px|thumb|left|Figure 1 - A plot of the inter-nuclear distance vs. Time]]

[[File:H2_Trans_KE.PNG|313x313px|thumb|centre|Figure 2 - A plot of the kinetic energy vs. Time]]

=== MEP and Dynamic Trajectories ===
The minimum energy path (mep) is the lowest energy reaction path taken to permit the formation of products from reactants. The mep has a dependency on the transition state and can be mapped out by following the downhill gradient in both the forward and backward direction from the transition state. The reaction paths were run for a dynamic and a minimum energy path; the system used was the same as that of the transition state except that the bond length r1 was changed slightly to 0.91775 Å. The data obtained from these two reaction paths differed.[[File:mepspna.PNG|thumb|left|Figure 4 - MEP surface plot|309x309px]][[File:Dynamspna.PNG|thumb|Figure 5 - Dynamic Surface Plot|259x259px]]
==== The Surface plot ====
The black line on the surface plot illustrates the reaction path. For the MEP plot, the reaction is seen to start at the transition state and progresses slightly towards product formation. As the black line is shown not to be wavy, it indicates that the diatomic molecule is not vibrating.
For the dynamic plot, the reaction path also starts at the transition state, however, unlike the MEP plot, the reaction goes into completion and the product is formed (this can be inferred as the A-B bond length is small and the B-C bond length continues to increase). The reaction path for this plot is wavy, indicating vibrations of the diatomic molecule, this vibrational motion coincides with an inertial trajectory.[[File:interdynamna.PNG|thumb|left|Figure 6 - Dynamic Inter-nuclear Distance vs. Time plot|271x271px]][[File:intermepna.PNG|thumb|Figure 7 - MEP Inter-nuclear Distance vs. Time plot|249x249px]]

==== Inter-nuclear Distance vs. Time plot ====
For the MEP plot (Figure 7), all the bond distances are initially constant, however this quickly changes. the inter-nuclear distance for A-B gradually decreases for four seconds then plateaus. The decrease indicates the formation of the of the A-B and the plateau shows that the bond formed is static. The inter-nuclear distances for B-C and A-C show a gradual increase. The final distances vary from 0.78 to 2.41 Å.

For the dynamic plot (Figure 6), the A-B inter-nuclear distance, constant at 1 Å for over 6 seconds. The distances for B-C and A-C show a an exponential increase in inter-nuclear distance, reaching distances greater than 35 Å.

===Reactive and Unreactive trajectories===

Reactions go into completion if there is enough energy in the system to overcome the activation barrier, which occurs at the transition point. looking at the surface plots produced using Matlab, it is possible to determine whether a trajectory is reactive, as the reaction path would move from the reactant channel to the product channel. It is possible to assume that trajectories stating from the same position but with greater values of momenta would be reactive as the activation barrier would be overcome due to the surplus of kinetic energy provided. This assumption was tested using r1 and r2 value of 0.74 and 2.0 respectively, and the values of p1 and p2 were varied according the the values seen in Table 1.

{| class="wikitable" border="1"
|+ Table 1
!Reaction
! p1 !! p2!! Surface plot!! Reactivity
|-
|a
| -1.25 || -2.5 || [[File:Test1na.PNG|328x328px|thumb]] ||The trajectory is reactive and the reaction path heads towards the product channel. As there is no oscillations of the B-C molecule, it shows that the A atom attacks the diatomic molecule when it is stationary. The new A-B molecule that is formed vibrates.
|-
|b
| -1.5 || -2.5 || [[File:Test2na.PNG|328x328px|thumb]] ||The trajectory is reactive as the reaction path heads towards the product channel. The shape of the reaction path as it moves through the reactant channel, illustrates that the bond length of B-C lengthens at an early stage during the approach of atom A. The A-B molecule formed vibrates.
|-
|c
| -1.5 || -2.0 || [[File:Test3na.PNG|328x328px|thumb]] ||The trajectory is unreactive, as the reaction path does not move towards the product channel. The path corresponds to atom A approaching a vibrating B-C molecule, however the energy and the phase of vibration is not enough to reach the activation barrier and so the reaction does not occur.
|-
|d
| -2.5 || -5.0 || [[File:Test4nan.PNG|328x328px|thumb]] || The trajectory is unreactive, as the reaction path does not move towards the product channel. The path corresponds to atom A approaching an initially non-vibrating B-C molecule. As the A atom gets closer the B-C bond beings to vibrate, an A-B-C complex is formed as the transition state, however the energy of the system is not enough to move from the transtion state to the products; the A atom dissociates from the complex, and moves away from the vibrating B-C molecule.
|-
|e
| -2.5 || -5.2 || [[File:Test5nan.PNG|328x328px|thumb]] || The trajectory is reactive, as the reaction path moves towards the product channel. The path corresponds to an approaching A atom to a non-vibrating B-C molecule. The encounter involves high translational kinetic energy and results in a vibrationally excited A-B product.
|}
From the data seen in table 1, the assumption previously made is not always correct. In some cases, such as when the p1;momentum was increased in value from -2.0 to -2.5 and the p2 momentum remained at -1.5, the reaction became reactive. However, when the p momentum value was greater than -2.0 and -5.0 the reaction was seen to be unreactive.

Transition State theory (TST) is based on three main assumptions<ref>A. C. Lasaga, ''Rev. Miner. States)'', 1981, '''8'''</ref> :

1) The activated complex will be in equilibrium with the reactants and not the products

2) The reactant nuclei behave according to classical mechanics

3) The reaction system will pass through the lowest energy saddle point/transition state on the potential energy surface

These assumptions do not always match what occurs in reality. The second assumption would indicate that whenever atoms collide with enough energy to form a transition state, the reaction will therefore be reactive. Reaction d (table 1) forms a transition state and therefore should be reactive, however this does not occur as a result of barrier re-crossing which is not taken into account by TST, as the activated complex is not in equilibrium with the products.
<center> A + B ⇌ [AB]‡ → C </center>
The system has enough energy to cross the activation barrier and does so, however it recrosses the activation barrier and moves towards the reactants. This TST assumption also ignores the occurrence of tunneling (more prevalent with lower activation barriers) as it is a quantum mechanical property. Tunneling occurs when molecules with a small amount of energy are able to tunnel through the activation barrier resulting in product formation, although the energy produced by the collision of the molecules was not enough to overcome the barrier.

As TST ignores other pathways of reaction such as barrier re-crossing and assumes that energetic collisions will result in reactive reactions, the rates of reaction assumed by TST would be faster than the rates that are obtained experimentally.

== F - H - H system Analysis ==

=== F + H2 and H + HF reaction energetics ===
The reaction of F + H2 is exothermic (energy is released from the system into the surroundings), as the reactants are higher in energy than the products. This is due to the stability if the H-F bond that is formed, as the difference in electronegativity between hydrogen and fluorine is great, so the bond formed is stronger than the H-H bond, and as the enthalpy of formation of the H-F bond is more negative than the positive enthalpy of dissociation for the H-H bond, the reaction is exothermic. However the reaction of H + HF is endothermic, which is due to the strength H-F bond. This strength would result in a very positive enthalpy of dissociation; the enthalpy supplied to the system through the formation of the H-H bond does not compensate the enthalpy required for H-F dissociation, as the H-H bond is weaker, energy is therefore taken up from the surroundings by the system, causing the reaction to be endothermic.

=== The position of the transition state ===
The Hammond's postulate states that the transition state of a reaction will resemble either the reactants or products depending on which one it is closer to. For exothermic reactions it resembles the reactants and for endothermic reactions it resembles the products<ref>J. E. Meany, V. Minderhout and
Y. Pocker, ''J. Chem. Educ'', 2001, '''78''', 204.</ref> Therefore by looking at the H-H bond distance and optimatising for the F + H2 reaction, the transition state position was found. The transition state for the H + HF would also be the same as both reactions form the same [H-H-F]‡ complex. The point at which the transition state occurs for the F + H2 and the HF + H reactions can be seen in figures 8 & 9 where H-F = 0.7463 Å and H-H = 1.810087 Å.

[[File:Interfhh.PNG|300x300px|thumb|Figure 10 - A plot inter-nuclear distance vs. Time for the F-H-H system]][[File:F+h2na.PNG|289x289px|thumb|left|Figure 8 - A Surface plot of the transition state for the F + H2 reaction]][[File:Hf+hna.PNG|298x298px|thumb|center|Figure 9 - A Surface plot of the transition state for the HF + H reaction]]

=== Calculating activation energies ===
The activation energy is given as the difference in energy between the maxima of the transition state and the minima of the reactants. The energy of the transition state was found to be -103.75 Kcal/mol. MEP potential energy vs time plots were carried out for the two reactions, the bond lengths were altered slightly to resemble that of the reactants, which provided the reactant energies (Figures 11 & 12).

Activation energy = Energy of transition state - Energy of reactants

For the F + H2 reaction : -103.75 -(-104) = 0.25 Kcal/mol

For the HF + H reaction : -103.75 -(-134) = 30.25 Kcal/mol

[[File:Pehhna.PNG|381x381px|thumb|left|Figure 11 - A Potential energy vs Time plot for the F + H2 reaction]]
[[File:Pehf.PNG|348x348px|thumb|center|Figure 12 - A Potential energy vs Time plot for the HF + H reaction]]

=== Mechanism for the release of reactive energy ===
The set of conditions used to create a reactive trajectory for the F + H2 reaction were as follows:

'''rF-H = 2.3, rH-H = 0.81, pF-H = -7.0, pH-H = -0.9'''

As F + H2 is an exothermic reaction, it is possible to assume, that its reaction path would result in the release of excess energy. Analysing the inter-nuclear momenta vs time graph produced (figure 13), it is seen that the H-H molecule vibrates with a smaller oscillation than the H-F molecule formed. The excess energy released by the reaction increases the amplitude of the product oscillations. As the law of the conservation of energy states that energy is not lost but conserved (converted to a different form), a decrease in the potential energy would be expected as the kinetic energy increases, which is seen in figures 14 & 15

[[File:Ques41.PNG|296x296px|thumb|left|Figure 13 - A Inter-nuclear momenta vs Time plot for the F + H2 reaction]]
[[File:Ques43.PNG|289x289px|thumb|Figure 15 - A Potential energy vs Time plot for the F + H2 reaction]][[File:Ques42.PNG|287x287px|thumb|center|Figure 14 - A Kinetic energy vs Time plot for the F + H2 reaction]]

This reaction can be analysed experimentally through calorimetry, which would indicate any changes in enthalpy as the reaction progresses; IR spectroscopy which would produce overtone bands for molecules with higher vibrational states and photochemically, by measuring photons released by vibrating molecules, to determine the vibration energies.

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 15:37, 9 June 2017 (BST) Very nice understanding of how the energy is converted using the graphs.

=== Polanyi's empirical rules ===
In chemical reactions, the reactants must overcome an activation barrier to allow product formation. Translational or vibrational energy initially deposited to the reactants can help it overcome the activation barrier. Polanyi's rules states that vibrational energy is more effective than translational when promoting a late transition barrier, where as translational energy is more effective than vibrational when the energy barrier is an early transition.<ref>Z. Zhang, Y. Zhou, D. H. Zhang,
G. Czakó and J. M. Bowman, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416–3419.</ref>

The F + H2 reaction is exothermic, therefore according to Hammond's postulate has an early transition state, and so an increase in translational energy over the vibrational energy would result in a reactive reaction. (p1=vibrational energy, p2 = translational energy)

Figure 16 shows a surface plot of a system that obeys Polanyi's rules, where p1=1.5, p2 = -5.5, producing a system with high translational energy and low vibrational energy. A system of high vibrational energy and low translational energy , where p1=6, p2 = -1.0, was seen not to produce a reactive reaction (Figure 17). This system obeys Polanyi's rules as only systems of high translational energy should cause a successful reaction when it is exothermic.[[File:Pol1.PNG|284x284px|thumb|left|Figure 16 - A surface plot for the F + H2 reaction of High translational energy]]
[[File:Poly2.PNG|289x289px|thumb|right|Figure 17 - A surface plot for the F + H2 reaction of Low translational energy]]

The HF + H reaction is endothermic, therefore according to Hammond's postulate has an late transition state, and so an increase in vibrational energy over the translational energy would result in a reactive reaction. (p1=vibrational energy, p2 = translational energy)

Figure 18 shows a surface plot of a system that obeys Polanyi's rules, where p1=8.6, p2 = -2.0, producing a system with high vibrational energy and low translational energy. A system of low vibrational energy and high translational energy , where p1=1.5, p2 = -6.0, was seen not to produce a reactive reaction (Figure 19). This system obeys Polanyi's rules as only systems of high vibrational energy should cause a successful reaction when it is endothermic.[[File:Pol3.PNG|312x312px|thumb|left|Figure 18 - A surface plot for the H+ HF reaction of High vibrational energy]]
[[File:Pol3.1.PNG|291x291px|thumb|Figure 19 - A surface plot for the H + HF reaction of Low vibrational energy]]

[[User:Nf710|Nf710]] ([[User talk:Nf710|talk]]) 15:44, 9 June 2017 (BST) Welldone for proving polyanis rules. You could have tried to get some examples hihc proove it at lower energy as some of the examples are quite choatic high energy examples. However the rest of the report is excellent well done.

== Bibliography ==
<references />

MRD:POLS

2018-05-24T14:31:39Z

Pg1616: /* Transition state of reactions */

MRD:POLS

2018-05-24T14:31:14Z

Pg1616: /* Transition state of reactions */

MRD:POLS

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Pg1616: /* Transition state of reactions */

File:PG F T.png

2018-05-24T14:27:46Z

Pg1616:

MRD:POLS

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Pg1616: /* Transition state of reactions */

MRD:POLS

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Pg1616: /* F-H-H system */

MRD:POLS

2018-05-23T20:08:12Z

Pg1616: /* Transition State Theory */

MRD:POLS

2018-05-23T20:07:24Z

Pg1616: /* Transition State Theory */

MRD:POLS

2018-05-23T19:54:26Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:45:47Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:43:58Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:38:15Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:35:38Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:32:43Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:23:18Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:22:44Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:21:38Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:20:32Z

Pg1616: /* Reactive and unreactive trajectories */

MRD:POLS

2018-05-23T19:18:56Z

Pg1616: /* Reactive and unreactive trajectories */

File:PG M 5.png

2018-05-23T19:17:34Z

Pg1616:

File:PG M 4.png

2018-05-23T19:17:23Z

Pg1616:

File:PG M 3.png

2018-05-23T19:17:06Z

Pg1616: