File:Hz7718 py20.png

2020-05-15T11:41:12Z

Hz7718:

MRD:hz7718

2020-05-15T11:37:02Z

Hz7718: /* Translational energy VS Vibrational Energy */

==Exercise 1: H + H2 system==
===Transition state===
Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal directions are zero.

For finding transition state on graph, the slopes of both r1 and r2 directions should be zero, which means δV/δr1 = 0 and δV/δr2 = 0. Also, if it is the maximum point in one direction, it should be minimum point in the orthognal direction. In math, (δ2V/δr12)*(δ2V/δr22) - δ2V/(δr1*δr2) < 0.

A minium stationary point means in both directions, it is the minimum point with zero gradients (δ2V/δx2 > 0 and (δ2V/δr12)*(δ2V/δr22) -δ2V/(δr1*δr2) > 0). But transition state is the minium point in one direction but a maximum point in the other orthognal direction.

{| class="wikitable" border="1"
!saddle point on a graph !! Minimum stationary point plot !! Maximum stationary point plot
|-
| [[file:hz7718_py1.png|500px|thumb|left]] || [[file:hz7718_py2.png|500px|thumb|right]] || [[file:hz7718_py3.png|500px|thumb|right]]
|}

===locating the transition state===
When the system is at transition state, the potential energies of both p1 and p2 are zero. And because all the three atoms are in equilibrium, the change of the energy is zero, too. It means that the gradients of potential energy surface is zero and the force acting on atoms is zero, which indicates there is no oscillation of the three atoms and r1 and r2 will keep constant. On the graph, r1 and r2 equal to 90.7pm, and two lines are almost perfectly striaght without oscillating, which means it is(or close to) the transition state. So rts=90.7pm.

[[file:hz7718_py4.png|300px]]

===Trajectories from r1=rts+δ, r1=rts===
In MEP type graph, the trajectory follows the valley floor to H1 + H2-H3. But in the dynamic type graph, the shape of trajecrory is wavy. This is because on mep, all points have zero kinetic energy(velocities and momenta are zero), whihc causes that B-C bond has no vibration. However, in dynamic type, kinetic energy is included in the system, so B-C bond will vibrate so the values of r2 will fluctuate.

{| class="wikitable" border="1"
!MEP graph, r1=91.7pm, r2=90.7pm, p1=p2=0!! Dynamic graph, r1=91.7pm, r2=90.7pm, p1=p2=0
|-
| [[file:hz7718_py5.png|300px|thumb|left]] || [[file:hz7718_py6.png|300px|thumb|right]]
|}

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ.mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.28 || YES || HC atom and HA-HB molecule approach to each other at the beginning. When the system passes the transition state region, new bond HB- HC will form and HA-HB bond will break. Then HA atom and HB-HC molecule move away to each other. Before HB-HC forming, the trajectory is a line with very little oscillation. This is because most kinetic energy of HA-HB is in translational energy. After HB-HC bond forming, the trajectory is wavy because most kinetic energy of HB-HC is in vibrational energy. || [[file:hz7718_py7.png|300px|thumb|right]]
|-
| -3.1 || -4.1 || -420.08 || NO || HC atom and HA-HB molecule approach to each other at the beginning. But they do not pass the transition state region, HA-HB bond does not break and HB-HC bond does not form. Then HC atom and HA-HB molecule move away to each other. HA-HB bond keeps vibrating due to the kinetic energy. || [[file:hz7718_py8.png|300px|thumb|right]]
|-
| -3.1 || -5.1 || -413.98 || YES || HC atom and HA-HB molecule approach to each other very slowly becuase most of kinetic ernegy is in vibrational energy, which means the translational ernergy is very little. When HA-HB bond strats to break, HB-HC bond starts to form. After that, HA atom and HB-HC molecule move away to each other. And HB-HC bond keeps vibrating. || [[file:hz7718_py9.png|300px|thumb|right]]
|-
| -5.1 || -10.1 || -357.28 || NO || HC atom and HA-HB molecule approach to each other, when they passes the transition state region, HA-HB bond starts to break and HB-HC bond starts to form. After that, HA atom moves away from HB-HC molecule but HB-HC molecule vibrates strongly. Then HB-HC bond breaks and HA and HB atoms approach to each other to form bond, which means this system recrosses the transition region and reverts to the reactants. In the end, vibrating HA-HB molecule and HC atom move away to each other. || [[file:hz7718_py10.png|300px|thumb|right]]
|-
| -5.1 || -10.6 || -349.48 || YES || The system passes the transition region for three times: At first, HC atom and HA-HB molecule get closer, HB-HC bond forms and HA-HB bond breaks. Then HA atom returns to HB-HC molecule immediately and the system reverts to reactants. After that, HB-HC bond forms again and HA-HB bond breaks again. In the end, vibrating HB-HC molecule and HA atom move away to each. || [[file:hz7718_py11.png|300px|thumb|right]]
|}
From the table above, we can know that even though the system has enough energy to react, it may still be unreactive.

===Transition State Theory===
In TST predictions, if the reactants with enough energy cross the transition state, it will never come back. However, the experimental results show the system can recross the transition state region and reform the reactants. SO, overall, the TST overestimates the reaction rate. And the TST has limitations: 1. High temperature, there are complicated vibrations which may lead the transition state far away from the saddle point of potential energy surface. 2. Quantum tunneling: Molecules and atoms can still tunnel across the barrier even with not enough energy. It will slightly underestimates the reaction rate.

==Exercise 2: F-H-H system==
===PES inspection===
{| class="wikitable" border="1"
! potential energy surface graph !! Internuclear distance vs time plot
|-
| [[file:hz7718_py12.png|300px|thumb|left]] || [[file:hz7718_py13.png|300px|thumb|right]]
|}
The potential energy surface graph show that the system H2 and F has higher potential energy than HF and H. It means H2 + F reaction is exothermic and HF + H reaction is endothermic. It indicates that the bond strength of H-F is stronger than H-H, so from H2 + F to HF + H, the system needs to absorb energy from the environment, and from HF + H to H2 + F, the system releases energy to the environment.
When it comes to the transition state location, it can be analysed by Hammond's postulate. For example, in the reaction H2 + F to HF + H which is an exothermic reaction. So, the transition state is similaar to the structure to the reactants. It means the distance between FA and HB is very large and the distance between HB and HC is smaller. In the Internuclear distance vs time plot, FA-HB is 180pm and HB-HC is 74pm. And from the plot, all atoms only slightly vibrate which means there is almost no force acting on them and the chnange of potential energy is zero.
We can know it is the saddle point.

===Activation Energy===
For finding the actvation energy, the steps are extended to 3500. When FA-HB is 180pm and HB-HC is 74pm, we can find the Contour plot 1 shows that the transition state finally forms HF + H. So, from Graph 1, we can know the activation energy is 121.6 kJ/mol for the reaction HF + H to H2 + F.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py14.png|400px|thumb|left]] || [[file:hz7718_py15.png|300px|thumb|right]]
|}

When FA-HB is 184pm and HB-HC is 74pm, the Contour plot 2 shows that the transition state finally forms H2 + F. So, from Graph 2, we can know the activation energy is 0.03 kJ/mol for the reaction H2 + F to HF + H.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py16.png|400px|thumb|left]] || [[file:hz7718_py17.png|300px|thumb|right]]
|}

===Reaction dynamics===
In reaction H2 + F to HF + H, FA-HB is 184pm and HB-HC is 74pm, p1=-1 and p2=-2. From the contour plot and animation, the product HF keeps vibrating and HC molecule moves away from HF. This is because the reaction is exothermic, the potential energy transfers to kinetic energy which includes vibratioanl energy and translational energy. However, from Momentum vs Time plot, it shows most of potential energy transfers to vibrational energy instead of translational energy because the HBA mometum fluctuates strongly but Hc-HB momentum keeps a relatviely low value.
{| class="wikitable" border="1"
! Contour Plot !! Momentum vs Time
|-
| [[file:hz7718_py18.png|400px|thumb|left]] || [[file:hz7718_py19.png|400px|thumb|right]]
|}

===Translational energy VS Vibrational Energy===
Graph 1 represents an exothermic reaction with an early transition state. And in exothermic reaction, translational energy can make the reaction more efficient because it can help the reactants pass the early transition state region. If most kinetic energy of reactants is vibrational energy, reactants cannot pass the early transition region in exothermic reaction.
Graph 2 represent an endothermic reaction. Endothermic reaction has a late transition state. The reactants need more vibrational energy to cross the late transition state region. And translational energy cannot help reactnats pass the late transition state.

MRD:hz7718

2020-05-15T10:39:04Z

Hz7718: /* PES inspection */

==Exercise 1: H + H2 system==
===Transition state===
Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal directions are zero.

For finding transition state on graph, the slopes of both r1 and r2 directions should be zero, which means δV/δr1 = 0 and δV/δr2 = 0. Also, if it is the maximum point in one direction, it should be minimum point in the orthognal direction. In math, (δ2V/δr12)*(δ2V/δr22) - δ2V/(δr1*δr2) < 0.

A minium stationary point means in both directions, it is the minimum point with zero gradients (δ2V/δx2 > 0 and (δ2V/δr12)*(δ2V/δr22) -δ2V/(δr1*δr2) > 0). But transition state is the minium point in one direction but a maximum point in the other orthognal direction.

{| class="wikitable" border="1"
!saddle point on a graph !! Minimum stationary point plot !! Maximum stationary point plot
|-
| [[file:hz7718_py1.png|500px|thumb|left]] || [[file:hz7718_py2.png|500px|thumb|right]] || [[file:hz7718_py3.png|500px|thumb|right]]
|}

===locating the transition state===
When the system is at transition state, the potential energies of both p1 and p2 are zero. And because all the three atoms are in equilibrium, the change of the energy is zero, too. It means that the gradients of potential energy surface is zero and the force acting on atoms is zero, which indicates there is no oscillation of the three atoms and r1 and r2 will keep constant. On the graph, r1 and r2 equal to 90.7pm, and two lines are almost perfectly striaght without oscillating, which means it is(or close to) the transition state. So rts=90.7pm.

[[file:hz7718_py4.png|300px]]

===Trajectories from r1=rts+δ, r1=rts===
In MEP type graph, the trajectory follows the valley floor to H1 + H2-H3. But in the dynamic type graph, the shape of trajecrory is wavy. This is because on mep, all points have zero kinetic energy(velocities and momenta are zero), whihc causes that B-C bond has no vibration. However, in dynamic type, kinetic energy is included in the system, so B-C bond will vibrate so the values of r2 will fluctuate.

{| class="wikitable" border="1"
!MEP graph, r1=91.7pm, r2=90.7pm, p1=p2=0!! Dynamic graph, r1=91.7pm, r2=90.7pm, p1=p2=0
|-
| [[file:hz7718_py5.png|300px|thumb|left]] || [[file:hz7718_py6.png|300px|thumb|right]]
|}

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ.mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.28 || YES || HC atom and HA-HB molecule approach to each other at the beginning. When the system passes the transition state region, new bond HB- HC will form and HA-HB bond will break. Then HA atom and HB-HC molecule move away to each other. Before HB-HC forming, the trajectory is a line with very little oscillation. This is because most kinetic energy of HA-HB is in translational energy. After HB-HC bond forming, the trajectory is wavy because most kinetic energy of HB-HC is in vibrational energy. || [[file:hz7718_py7.png|300px|thumb|right]]
|-
| -3.1 || -4.1 || -420.08 || NO || HC atom and HA-HB molecule approach to each other at the beginning. But they do not pass the transition state region, HA-HB bond does not break and HB-HC bond does not form. Then HC atom and HA-HB molecule move away to each other. HA-HB bond keeps vibrating due to the kinetic energy. || [[file:hz7718_py8.png|300px|thumb|right]]
|-
| -3.1 || -5.1 || -413.98 || YES || HC atom and HA-HB molecule approach to each other very slowly becuase most of kinetic ernegy is in vibrational energy, which means the translational ernergy is very little. When HA-HB bond strats to break, HB-HC bond starts to form. After that, HA atom and HB-HC molecule move away to each other. And HB-HC bond keeps vibrating. || [[file:hz7718_py9.png|300px|thumb|right]]
|-
| -5.1 || -10.1 || -357.28 || NO || HC atom and HA-HB molecule approach to each other, when they passes the transition state region, HA-HB bond starts to break and HB-HC bond starts to form. After that, HA atom moves away from HB-HC molecule but HB-HC molecule vibrates strongly. Then HB-HC bond breaks and HA and HB atoms approach to each other to form bond, which means this system recrosses the transition region and reverts to the reactants. In the end, vibrating HA-HB molecule and HC atom move away to each other. || [[file:hz7718_py10.png|300px|thumb|right]]
|-
| -5.1 || -10.6 || -349.48 || YES || The system passes the transition region for three times: At first, HC atom and HA-HB molecule get closer, HB-HC bond forms and HA-HB bond breaks. Then HA atom returns to HB-HC molecule immediately and the system reverts to reactants. After that, HB-HC bond forms again and HA-HB bond breaks again. In the end, vibrating HB-HC molecule and HA atom move away to each. || [[file:hz7718_py11.png|300px|thumb|right]]
|}
From the table above, we can know that even though the system has enough energy to react, it may still be unreactive.

===Transition State Theory===
In TST predictions, if the reactants with enough energy cross the transition state, it will never come back. However, the experimental results show the system can recross the transition state region and reform the reactants. SO, overall, the TST overestimates the reaction rate. And the TST has limitations: 1. High temperature, there are complicated vibrations which may lead the transition state far away from the saddle point of potential energy surface. 2. Quantum tunneling: Molecules and atoms can still tunnel across the barrier even with not enough energy. It will slightly underestimates the reaction rate.

==Exercise 2: F-H-H system==
===PES inspection===
{| class="wikitable" border="1"
! potential energy surface graph !! Internuclear distance vs time plot
|-
| [[file:hz7718_py12.png|300px|thumb|left]] || [[file:hz7718_py13.png|300px|thumb|right]]
|}
The potential energy surface graph show that the system H2 and F has higher potential energy than HF and H. It means H2 + F reaction is exothermic and HF + H reaction is endothermic. It indicates that the bond strength of H-F is stronger than H-H, so from H2 + F to HF + H, the system needs to absorb energy from the environment, and from HF + H to H2 + F, the system releases energy to the environment.
When it comes to the transition state location, it can be analysed by Hammond's postulate. For example, in the reaction H2 + F to HF + H which is an exothermic reaction. So, the transition state is similaar to the structure to the reactants. It means the distance between FA and HB is very large and the distance between HB and HC is smaller. In the Internuclear distance vs time plot, FA-HB is 180pm and HB-HC is 74pm. And from the plot, all atoms only slightly vibrate which means there is almost no force acting on them and the chnange of potential energy is zero.
We can know it is the saddle point.

===Activation Energy===
For finding the actvation energy, the steps are extended to 3500. When FA-HB is 180pm and HB-HC is 74pm, we can find the Contour plot 1 shows that the transition state finally forms HF + H. So, from Graph 1, we can know the activation energy is 121.6 kJ/mol for the reaction HF + H to H2 + F.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py14.png|400px|thumb|left]] || [[file:hz7718_py15.png|300px|thumb|right]]
|}

When FA-HB is 184pm and HB-HC is 74pm, the Contour plot 2 shows that the transition state finally forms H2 + F. So, from Graph 2, we can know the activation energy is 0.03 kJ/mol for the reaction H2 + F to HF + H.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py16.png|400px|thumb|left]] || [[file:hz7718_py17.png|300px|thumb|right]]
|}

===Reaction dynamics===
In reaction H2 + F to HF + H, FA-HB is 184pm and HB-HC is 74pm, p1=-1 and p2=-2. From the contour plot and animation, the product HF keeps vibrating and HC molecule moves away from HF. This is because the reaction is exothermic, the potential energy transfers to kinetic energy which includes vibratioanl energy and translational energy. However, from Momentum vs Time plot, it shows most of potential energy transfers to vibrational energy instead of translational energy because the HBA mometum fluctuates strongly but Hc-HB momentum keeps a relatviely low value.
{| class="wikitable" border="1"
! Contour Plot !! Momentum vs Time
|-
| [[file:hz7718_py18.png|400px|thumb|left]] || [[file:hz7718_py19.png|400px|thumb|right]]
|}

===Translational energy VS Vibrational Energy===

MRD:hz7718

2020-05-14T12:39:34Z

Hz7718: /* Transition State Theory */

==Exercise 1: H + H2 system==
===Transition state===
Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal directions are zero.

For finding transition state on graph, the slopes of both r1 and r2 directions should be zero, which means δV/δr1 = 0 and δV/δr2 = 0. Also, if it is the maximum point in one direction, it should be minimum point in the orthognal direction. In math, (δ2V/δr12)*(δ2V/δr22) - δ2V/(δr1*δr2) < 0.

A minium stationary point means in both directions, it is the minimum point with zero gradients (δ2V/δx2 > 0 and (δ2V/δr12)*(δ2V/δr22) -δ2V/(δr1*δr2) > 0). But transition state is the minium point in one direction but a maximum point in the other orthognal direction.

{| class="wikitable" border="1"
!saddle point on a graph !! Minimum stationary point plot !! Maximum stationary point plot
|-
| [[file:hz7718_py1.png|500px|thumb|left]] || [[file:hz7718_py2.png|500px|thumb|right]] || [[file:hz7718_py3.png|500px|thumb|right]]
|}

===locating the transition state===
When the system is at transition state, the potential energies of both p1 and p2 are zero. And because all the three atoms are in equilibrium, the change of the energy is zero, too. It means that the gradients of potential energy surface is zero and the force acting on atoms is zero, which indicates there is no oscillation of the three atoms and r1 and r2 will keep constant. On the graph, r1 and r2 equal to 90.7pm, and two lines are almost perfectly striaght without oscillating, which means it is(or close to) the transition state. So rts=90.7pm.

[[file:hz7718_py4.png|300px]]

===Trajectories from r1=rts+δ, r1=rts===
In MEP type graph, the trajectory follows the valley floor to H1 + H2-H3. But in the dynamic type graph, the shape of trajecrory is wavy. This is because on mep, all points have zero kinetic energy(velocities and momenta are zero), whihc causes that B-C bond has no vibration. However, in dynamic type, kinetic energy is included in the system, so B-C bond will vibrate so the values of r2 will fluctuate.

{| class="wikitable" border="1"
!MEP graph, r1=91.7pm, r2=90.7pm, p1=p2=0!! Dynamic graph, r1=91.7pm, r2=90.7pm, p1=p2=0
|-
| [[file:hz7718_py5.png|300px|thumb|left]] || [[file:hz7718_py6.png|300px|thumb|right]]
|}

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ.mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.28 || YES || HC atom and HA-HB molecule approach to each other at the beginning. When the system passes the transition state region, new bond HB- HC will form and HA-HB bond will break. Then HA atom and HB-HC molecule move away to each other. Before HB-HC forming, the trajectory is a line with very little oscillation. This is because most kinetic energy of HA-HB is in translational energy. After HB-HC bond forming, the trajectory is wavy because most kinetic energy of HB-HC is in vibrational energy. || [[file:hz7718_py7.png|300px|thumb|right]]
|-
| -3.1 || -4.1 || -420.08 || NO || HC atom and HA-HB molecule approach to each other at the beginning. But they do not pass the transition state region, HA-HB bond does not break and HB-HC bond does not form. Then HC atom and HA-HB molecule move away to each other. HA-HB bond keeps vibrating due to the kinetic energy. || [[file:hz7718_py8.png|300px|thumb|right]]
|-
| -3.1 || -5.1 || -413.98 || YES || HC atom and HA-HB molecule approach to each other very slowly becuase most of kinetic ernegy is in vibrational energy, which means the translational ernergy is very little. When HA-HB bond strats to break, HB-HC bond starts to form. After that, HA atom and HB-HC molecule move away to each other. And HB-HC bond keeps vibrating. || [[file:hz7718_py9.png|300px|thumb|right]]
|-
| -5.1 || -10.1 || -357.28 || NO || HC atom and HA-HB molecule approach to each other, when they passes the transition state region, HA-HB bond starts to break and HB-HC bond starts to form. After that, HA atom moves away from HB-HC molecule but HB-HC molecule vibrates strongly. Then HB-HC bond breaks and HA and HB atoms approach to each other to form bond, which means this system recrosses the transition region and reverts to the reactants. In the end, vibrating HA-HB molecule and HC atom move away to each other. || [[file:hz7718_py10.png|300px|thumb|right]]
|-
| -5.1 || -10.6 || -349.48 || YES || The system passes the transition region for three times: At first, HC atom and HA-HB molecule get closer, HB-HC bond forms and HA-HB bond breaks. Then HA atom returns to HB-HC molecule immediately and the system reverts to reactants. After that, HB-HC bond forms again and HA-HB bond breaks again. In the end, vibrating HB-HC molecule and HA atom move away to each. || [[file:hz7718_py11.png|300px|thumb|right]]
|}
From the table above, we can know that even though the system has enough energy to react, it may still be unreactive.

===Transition State Theory===
In TST predictions, if the reactants with enough energy cross the transition state, it will never come back. However, the experimental results show the system can recross the transition state region and reform the reactants. SO, overall, the TST overestimates the reaction rate. And the TST has limitations: 1. High temperature, there are complicated vibrations which may lead the transition state far away from the saddle point of potential energy surface. 2. Quantum tunneling: Molecules and atoms can still tunnel across the barrier even with not enough energy. It will slightly underestimates the reaction rate.

==Exercise 2: F-H-H system==
===PES inspection===
{| class="wikitable" border="1"
! potential energy surface graph !! Internuclear distance vs time plot
|-
| [[file:hz7718_py12.png|300px|thumb|left]] || [[file:hz7718_py13.png|300px|thumb|right]]
|}
The potential energy surface graph show that the system H2 and F has higher potential energy than HF and H. It means H2 + F reaction is exothermic and HF + H reaction is endothermic. It indicates that the bond strength of H-F is stronger than H-H, so from H2 + F to HF + H, the system needs to absorb energy from the environment, and from HF + H to H2 + F, the system releases energy to the environment.
When it comes to the transition state location, it can be analysed by Hammond's postulate. For example, in the reaction H2 + F to HF + H which is an exothermic reaction. So, the transition state is similaar to the structure to the reactants. It means the distance between FA and HB is very large and the distance between HB and HC is smaller. In the Internuclear distance vs time plot, FA-HB is 180pm and HB-HC is 74pm. And from the plot, all atoms only slightly vibrate whihc means there is almost no force acting on them and the chnange of potential energy is zero.
We can know it is the saddle point.

===Activation Energy===
For finding the actvation energy, the steps are extended to 3500. When FA-HB is 180pm and HB-HC is 74pm, we can find the Contour plot 1 shows that the transition state finally forms HF + H. So, from Graph 1, we can know the activation energy is 121.6 kJ/mol for the reaction HF + H to H2 + F.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py14.png|400px|thumb|left]] || [[file:hz7718_py15.png|300px|thumb|right]]
|}

When FA-HB is 184pm and HB-HC is 74pm, the Contour plot 2 shows that the transition state finally forms H2 + F. So, from Graph 2, we can know the activation energy is 0.03 kJ/mol for the reaction H2 + F to HF + H.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py16.png|400px|thumb|left]] || [[file:hz7718_py17.png|300px|thumb|right]]
|}

===Reaction dynamics===
In reaction H2 + F to HF + H, FA-HB is 184pm and HB-HC is 74pm, p1=-1 and p2=-2. From the contour plot and animation, the product HF keeps vibrating and HC molecule moves away from HF. This is because the reaction is exothermic, the potential energy transfers to kinetic energy which includes vibratioanl energy and translational energy. However, from Momentum vs Time plot, it shows most of potential energy transfers to vibrational energy instead of translational energy because the HBA mometum fluctuates strongly but Hc-HB momentum keeps a relatviely low value.
{| class="wikitable" border="1"
! Contour Plot !! Momentum vs Time
|-
| [[file:hz7718_py18.png|400px|thumb|left]] || [[file:hz7718_py19.png|400px|thumb|right]]
|}

===Translational energy VS Vibrational Energy===

MRD:hz7718

2020-05-14T12:28:42Z

Hz7718: /* Exercise 2: F-H-H system */

==Exercise 1: H + H2 system==
===Transition state===
Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal directions are zero.

For finding transition state on graph, the slopes of both r1 and r2 directions should be zero, which means δV/δr1 = 0 and δV/δr2 = 0. Also, if it is the maximum point in one direction, it should be minimum point in the orthognal direction. In math, (δ2V/δr12)*(δ2V/δr22) - δ2V/(δr1*δr2) < 0.

A minium stationary point means in both directions, it is the minimum point with zero gradients (δ2V/δx2 > 0 and (δ2V/δr12)*(δ2V/δr22) -δ2V/(δr1*δr2) > 0). But transition state is the minium point in one direction but a maximum point in the other orthognal direction.

{| class="wikitable" border="1"
!saddle point on a graph !! Minimum stationary point plot !! Maximum stationary point plot
|-
| [[file:hz7718_py1.png|500px|thumb|left]] || [[file:hz7718_py2.png|500px|thumb|right]] || [[file:hz7718_py3.png|500px|thumb|right]]
|}

===locating the transition state===
When the system is at transition state, the potential energies of both p1 and p2 are zero. And because all the three atoms are in equilibrium, the change of the energy is zero, too. It means that the gradients of potential energy surface is zero and the force acting on atoms is zero, which indicates there is no oscillation of the three atoms and r1 and r2 will keep constant. On the graph, r1 and r2 equal to 90.7pm, and two lines are almost perfectly striaght without oscillating, which means it is(or close to) the transition state. So rts=90.7pm.

[[file:hz7718_py4.png|300px]]

===Trajectories from r1=rts+δ, r1=rts===
In MEP type graph, the trajectory follows the valley floor to H1 + H2-H3. But in the dynamic type graph, the shape of trajecrory is wavy. This is because on mep, all points have zero kinetic energy(velocities and momenta are zero), whihc causes that B-C bond has no vibration. However, in dynamic type, kinetic energy is included in the system, so B-C bond will vibrate so the values of r2 will fluctuate.

{| class="wikitable" border="1"
!MEP graph, r1=91.7pm, r2=90.7pm, p1=p2=0!! Dynamic graph, r1=91.7pm, r2=90.7pm, p1=p2=0
|-
| [[file:hz7718_py5.png|300px|thumb|left]] || [[file:hz7718_py6.png|300px|thumb|right]]
|}

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ.mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.28 || YES || HC atom and HA-HB molecule approach to each other at the beginning. When the system passes the transition state region, new bond HB- HC will form and HA-HB bond will break. Then HA atom and HB-HC molecule move away to each other. Before HB-HC forming, the trajectory is a line with very little oscillation. This is because most kinetic energy of HA-HB is in translational energy. After HB-HC bond forming, the trajectory is wavy because most kinetic energy of HB-HC is in vibrational energy. || [[file:hz7718_py7.png|300px|thumb|right]]
|-
| -3.1 || -4.1 || -420.08 || NO || HC atom and HA-HB molecule approach to each other at the beginning. But they do not pass the transition state region, HA-HB bond does not break and HB-HC bond does not form. Then HC atom and HA-HB molecule move away to each other. HA-HB bond keeps vibrating due to the kinetic energy. || [[file:hz7718_py8.png|300px|thumb|right]]
|-
| -3.1 || -5.1 || -413.98 || YES || HC atom and HA-HB molecule approach to each other very slowly becuase most of kinetic ernegy is in vibrational energy, which means the translational ernergy is very little. When HA-HB bond strats to break, HB-HC bond starts to form. After that, HA atom and HB-HC molecule move away to each other. And HB-HC bond keeps vibrating. || [[file:hz7718_py9.png|300px|thumb|right]]
|-
| -5.1 || -10.1 || -357.28 || NO || HC atom and HA-HB molecule approach to each other, when they passes the transition state region, HA-HB bond starts to break and HB-HC bond starts to form. After that, HA atom moves away from HB-HC molecule but HB-HC molecule vibrates strongly. Then HB-HC bond breaks and HA and HB atoms approach to each other to form bond, which means this system recrosses the transition region and reverts to the reactants. In the end, vibrating HA-HB molecule and HC atom move away to each other. || [[file:hz7718_py10.png|300px|thumb|right]]
|-
| -5.1 || -10.6 || -349.48 || YES || The system passes the transition region for three times: At first, HC atom and HA-HB molecule get closer, HB-HC bond forms and HA-HB bond breaks. Then HA atom returns to HB-HC molecule immediately and the system reverts to reactants. After that, HB-HC bond forms again and HA-HB bond breaks again. In the end, vibrating HB-HC molecule and HA atom move away to each. || [[file:hz7718_py11.png|300px|thumb|right]]
|}
From the table above, we can know that even though the system has enough energy to react, it may still be unreactive.

===Transition State Theory===
TST predicts that when the total energy is larger than the activation energy, the trajectories is reeactive. However, the experimnetal results show that trajectories with a total energy greater than the activation energy may still be unreactive. So the TST overestimates the reaction rate. The TST fails in these situations: 1. High Temperature: There will be more complicated vibration modes of molecules which may cause the transition state aeay form the saddle point of the potential energy surface. 2. Quantum tunneling

==Exercise 2: F-H-H system==
===PES inspection===
{| class="wikitable" border="1"
! potential energy surface graph !! Internuclear distance vs time plot
|-
| [[file:hz7718_py12.png|300px|thumb|left]] || [[file:hz7718_py13.png|300px|thumb|right]]
|}
The potential energy surface graph show that the system H2 and F has higher potential energy than HF and H. It means H2 + F reaction is exothermic and HF + H reaction is endothermic. It indicates that the bond strength of H-F is stronger than H-H, so from H2 + F to HF + H, the system needs to absorb energy from the environment, and from HF + H to H2 + F, the system releases energy to the environment.
When it comes to the transition state location, it can be analysed by Hammond's postulate. For example, in the reaction H2 + F to HF + H which is an exothermic reaction. So, the transition state is similaar to the structure to the reactants. It means the distance between FA and HB is very large and the distance between HB and HC is smaller. In the Internuclear distance vs time plot, FA-HB is 180pm and HB-HC is 74pm. And from the plot, all atoms only slightly vibrate whihc means there is almost no force acting on them and the chnange of potential energy is zero.
We can know it is the saddle point.

===Activation Energy===
For finding the actvation energy, the steps are extended to 3500. When FA-HB is 180pm and HB-HC is 74pm, we can find the Contour plot 1 shows that the transition state finally forms HF + H. So, from Graph 1, we can know the activation energy is 121.6 kJ/mol for the reaction HF + H to H2 + F.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py14.png|400px|thumb|left]] || [[file:hz7718_py15.png|300px|thumb|right]]
|}

When FA-HB is 184pm and HB-HC is 74pm, the Contour plot 2 shows that the transition state finally forms H2 + F. So, from Graph 2, we can know the activation energy is 0.03 kJ/mol for the reaction H2 + F to HF + H.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py16.png|400px|thumb|left]] || [[file:hz7718_py17.png|300px|thumb|right]]
|}

===Reaction dynamics===
In reaction H2 + F to HF + H, FA-HB is 184pm and HB-HC is 74pm, p1=-1 and p2=-2. From the contour plot and animation, the product HF keeps vibrating and HC molecule moves away from HF. This is because the reaction is exothermic, the potential energy transfers to kinetic energy which includes vibratioanl energy and translational energy. However, from Momentum vs Time plot, it shows most of potential energy transfers to vibrational energy instead of translational energy because the HBA mometum fluctuates strongly but Hc-HB momentum keeps a relatviely low value.
{| class="wikitable" border="1"
! Contour Plot !! Momentum vs Time
|-
| [[file:hz7718_py18.png|400px|thumb|left]] || [[file:hz7718_py19.png|400px|thumb|right]]
|}

===Translational energy VS Vibrational Energy===

File:Hz7718 py19.png

2020-05-14T12:07:29Z

Hz7718:

MRD:hz7718

2020-05-14T11:59:11Z

Hz7718: /* Exercise 2: F-H-H system */

==Exercise 1: H + H2 system==
===Transition state===
Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal directions are zero.

For finding transition state on graph, the slopes of both r1 and r2 directions should be zero, which means δV/δr1 = 0 and δV/δr2 = 0. Also, if it is the maximum point in one direction, it should be minimum point in the orthognal direction. In math, (δ2V/δr12)*(δ2V/δr22) - δ2V/(δr1*δr2) < 0.

A minium stationary point means in both directions, it is the minimum point with zero gradients (δ2V/δx2 > 0 and (δ2V/δr12)*(δ2V/δr22) -δ2V/(δr1*δr2) > 0). But transition state is the minium point in one direction but a maximum point in the other orthognal direction.

{| class="wikitable" border="1"
!saddle point on a graph !! Minimum stationary point plot !! Maximum stationary point plot
|-
| [[file:hz7718_py1.png|500px|thumb|left]] || [[file:hz7718_py2.png|500px|thumb|right]] || [[file:hz7718_py3.png|500px|thumb|right]]
|}

===locating the transition state===
When the system is at transition state, the potential energies of both p1 and p2 are zero. And because all the three atoms are in equilibrium, the change of the energy is zero, too. It means that the gradients of potential energy surface is zero and the force acting on atoms is zero, which indicates there is no oscillation of the three atoms and r1 and r2 will keep constant. On the graph, r1 and r2 equal to 90.7pm, and two lines are almost perfectly striaght without oscillating, which means it is(or close to) the transition state. So rts=90.7pm.

[[file:hz7718_py4.png|300px]]

===Trajectories from r1=rts+δ, r1=rts===
In MEP type graph, the trajectory follows the valley floor to H1 + H2-H3. But in the dynamic type graph, the shape of trajecrory is wavy. This is because on mep, all points have zero kinetic energy(velocities and momenta are zero), whihc causes that B-C bond has no vibration. However, in dynamic type, kinetic energy is included in the system, so B-C bond will vibrate so the values of r2 will fluctuate.

{| class="wikitable" border="1"
!MEP graph, r1=91.7pm, r2=90.7pm, p1=p2=0!! Dynamic graph, r1=91.7pm, r2=90.7pm, p1=p2=0
|-
| [[file:hz7718_py5.png|300px|thumb|left]] || [[file:hz7718_py6.png|300px|thumb|right]]
|}

===Reactive and unreactive trajectories===
{| class="wikitable" border="1"
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot/ kJ.mol-1 !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.28 || YES || HC atom and HA-HB molecule approach to each other at the beginning. When the system passes the transition state region, new bond HB- HC will form and HA-HB bond will break. Then HA atom and HB-HC molecule move away to each other. Before HB-HC forming, the trajectory is a line with very little oscillation. This is because most kinetic energy of HA-HB is in translational energy. After HB-HC bond forming, the trajectory is wavy because most kinetic energy of HB-HC is in vibrational energy. || [[file:hz7718_py7.png|300px|thumb|right]]
|-
| -3.1 || -4.1 || -420.08 || NO || HC atom and HA-HB molecule approach to each other at the beginning. But they do not pass the transition state region, HA-HB bond does not break and HB-HC bond does not form. Then HC atom and HA-HB molecule move away to each other. HA-HB bond keeps vibrating due to the kinetic energy. || [[file:hz7718_py8.png|300px|thumb|right]]
|-
| -3.1 || -5.1 || -413.98 || YES || HC atom and HA-HB molecule approach to each other very slowly becuase most of kinetic ernegy is in vibrational energy, which means the translational ernergy is very little. When HA-HB bond strats to break, HB-HC bond starts to form. After that, HA atom and HB-HC molecule move away to each other. And HB-HC bond keeps vibrating. || [[file:hz7718_py9.png|300px|thumb|right]]
|-
| -5.1 || -10.1 || -357.28 || NO || HC atom and HA-HB molecule approach to each other, when they passes the transition state region, HA-HB bond starts to break and HB-HC bond starts to form. After that, HA atom moves away from HB-HC molecule but HB-HC molecule vibrates strongly. Then HB-HC bond breaks and HA and HB atoms approach to each other to form bond, which means this system recrosses the transition region and reverts to the reactants. In the end, vibrating HA-HB molecule and HC atom move away to each other. || [[file:hz7718_py10.png|300px|thumb|right]]
|-
| -5.1 || -10.6 || -349.48 || YES || The system passes the transition region for three times: At first, HC atom and HA-HB molecule get closer, HB-HC bond forms and HA-HB bond breaks. Then HA atom returns to HB-HC molecule immediately and the system reverts to reactants. After that, HB-HC bond forms again and HA-HB bond breaks again. In the end, vibrating HB-HC molecule and HA atom move away to each. || [[file:hz7718_py11.png|300px|thumb|right]]
|}
From the table above, we can know that even though the system has enough energy to react, it may still be unreactive.

===Transition State Theory===
TST predicts that when the total energy is larger than the activation energy, the trajectories is reeactive. However, the experimnetal results show that trajectories with a total energy greater than the activation energy may still be unreactive. So the TST overestimates the reaction rate. The TST fails in these situations: 1. High Temperature: There will be more complicated vibration modes of molecules which may cause the transition state aeay form the saddle point of the potential energy surface. 2. Quantum tunneling

==Exercise 2: F-H-H system==
===PES inspection===
{| class="wikitable" border="1"
! potential energy surface graph !! Internuclear distance vs time plot
|-
| [[file:hz7718_py12.png|300px|thumb|left]] || [[file:hz7718_py13.png|300px|thumb|right]]
|}
The potential energy surface graph show that the system H2 and F has higher potential energy than HF and H. It means H2 + F reaction is exothermic and HF + H reaction is endothermic. It indicates that the bond strength of H-F is stronger than H-H, so from H2 + F to HF + H, the system needs to absorb energy from the environment, and from HF + H to H2 + F, the system releases energy to the environment.
When it comes to the transition state location, it can be analysed by Hammond's postulate. For example, in the reaction H2 + F to HF + H which is an exothermic reaction. So, the transition state is similaar to the structure to the reactants. It means the distance between FA and HB is very large and the distance between HB and HC is smaller. In the Internuclear distance vs time plot, FA-HB is 180pm and HB-HC is 74pm. And from the plot, all atoms only slightly vibrate whihc means there is almost no force acting on them and the chnange of potential energy is zero.
We can know it is the saddle point.

===Activation Energy===
For finding the actvation energy, the steps are extended to 3500. When FA-HB is 180pm and HB-HC is 74pm, we can find the Contour plot 1 shows that the transition state finally forms HF + H. So, from Graph 1, we can know the activation energy is 121.6 kJ/mol for the reaction HF + H to H2 + F.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py14.png|400px|thumb|left]] || [[file:hz7718_py15.png|300px|thumb|right]]
|}

When FA-HB is 184pm and HB-HC is 74pm, the Contour plot 2 shows that the transition state finally forms H2 + F. So, from Graph 2, we can know the activation energy is 0.03 kJ/mol for the reaction H2 + F to HF + H.
{| class="wikitable" border="1"
! Contour Plot 1 !! Graph 1
|-
| [[file:hz7718_py16.png|400px|thumb|left]] || [[file:hz7718_py17.png|300px|thumb|right]]
|}

===Reaction dynamics===
In reaction H2 + F to HF + H, FA-HB is 184pm and HB-HC is 74pm, p1=-1 and p2=-2. From the contour plot and animation, the product HF keeps vibrating and HC molecule moves away from HF. This is because the reaction is exothermic, the potential energy transfers to kinetic energy which includes vibratioanl energy and translational energy. However, from Momentum vs Time plot, most potential energy transfers to vibrational energy instead of translational energy because of the strong vibration of HF molecule.
{| class="wikitable" border="1"
! Contour Plot !! Momentum vs Time
|-
| [[file:hz7718_py18.png|400px|thumb|left]] || [[file:hz7718_py19.png|400px|thumb|right]]
|}

File:19.PNG

2020-05-14T11:51:21Z

Hz7718: Hz7718 uploaded a new version of File:19.PNG

MRD:hz7718

2020-05-11T16:03:31Z

Hz7718: Created page with "==Exercise 1: H + H2 system== ===Transition state=== Mathematically, transition state is a saddle point which is on a graph of a function where slopes in orthogonal..."