MRD:yingtingjiaYJ6018
Contents
Transition State
Define transition state in potential energy surface
Transition state is defined as the first order saddle point in the potential energy surface diagram where the curvature is zero. Also, it should be a maximum in energy in one direction (nuclear configuration) and minimum in its orthogonal directions. It is located in the minimum energy path linking the two minimum which are the reactants and products.
Conditions transition state should satisfy
∂E/∂(rab-rbc)=0 and ∂E2/∂2(rab-rbc)<0 ( trajectory of figure 1)
∂E/∂(rab+rbc)=0 and ∂E2/∂2(rab+rbc)>0 (trajectory of figure 2)
Distinguish transition state from local minimum
Local minimum on the lowest energy surface can be defined as:
∂E/∂(rab-rbc)=0 and ∂E2/∂2(rab-rbc)>0 (trajectory of figure 1)
∂E/∂(rab+rbc)=0 and ∂E2/∂2(rab+rbc)>0 (trajectory of figure 2)
In this symmetric reaction, the transition state occurs when rab=rbc, which can be used to distinguish transition point from other turning points in trajectory shown in figure 1 as points that do not satisfy this condition are local minimum.
Locating transition state position
The transition state in this case is at the point where rab=rbc, from which its coordinate can be located using the intercept of line rab and rbc in Intermolecular distance vs Time plot.(figure 3)
rts=90.775 pm
To prove this result, Intermolecular distance vs Time graph(figure.4)was plotted using the following conditions:
| rab=100pm | pab=0 g.mol-1.pm.fs-1. |
| rbc=100pm | pbc=0 g.mol-1.pm.fs-1. |
The contour graph (figure 5)shows that the atoms oscillate back and forwards there is a straight black line. Also, the Internuclear distance vs Time (figure 6)graphs shows two oscillating waves. At transition state, the gradient of potential energy surface is zero, which means that the there is no force acting on the atoms and the atoms should not be oscillating. This suggests that the real rts should be much smaller than the distance used for testing and the Internuclear distance vs Time should show horizontal lines. Through reducing the length of AB and BC, the force acting on AB and BC reduce, the estimation of rts is then determined to be 90.775 pm at the point where force acting on AB and BC is zero.
Calculation of reaction path
Calculation of reaction path using MEP
| Distance and momentum of reaction | |
|---|---|
| rab=rts=90.775 pm | pab=0 g.mol-1.pm.fs-1 |
| rbc=rts+1=91.775 pm | pbc=0 g.mol-1.pm.fs-1 |
Calculation of reaction path using Dynamics
| Distance and momentum of reaction | |
|---|---|
| rab=rts=90.775 pm | pab=0 g.mol-1.pm.fs-1 |
| rbc=rts+1=91.775 pm | pbc=0 g.mol-1.pm.fs-1 |
Comparison of MEP and Dynamics
Trajectory in contour plot
The trajectory shown by the Dynamics calculation(figure7) shows a fluctuating upward curve towards the right from the transition state whereas that in MEP(figure8) has the same trend but is smooth, so both of them are moving downhill along the reaction. This represents that the trajectory calculated by dynamics illustrates that the new H2 molecule formed by HA and HB is stretching and contracting as they move away from HC, this oscillation leads to the fluctuation in the potential energy surface. MEP does not include this feature, as it does not allow the kinetic energy to be built up from the previous time step, it just allows the atoms to move along the direction which is obtained from the acceleration generated by force calculated.
Internuclear distance vs Time plot
MEP(figure 9) shows a graph with line BC and AC have positive gradient overall and their gradient decreases gradually towards zero as the reaction progresses. In contrast, line BC and AC generated by Dynamics(figure 10) are flat initially, their gradient starts to increase at around 12 fs and remains constant for the rest of the time. In addition, line AB in Dynamic graph fluctuates after 20 fs whereas that in MEP is flat. The differences are mainly due to the fact that MEP does not account for the realistic motion of atoms during reaction whereas the motion of atoms are included in the Dynamics. The vibration of new H2 molecule formed by HA and HB is shown clearly by Dynamic form the continuous fluctuation of line AB which is ignored by MEP.
Momenta vs Time
In Dynamics, the momentum of BC increases as HB and HC move away from each other while that of AB increases after a drop at around 12fs. At around 20fs, AB starts to fluctuate which is because of the oscillation of new H2 molecule formed by HA and HB. However, in the MEP, the momenta is just zero. This also proves that MEP does not take the motion of atoms into account. MEP has infinite friction, which prevents the subsequent kinetic energy being built up overtime.
Energy vs Time
The kinetic energy shown in the MEP keeps at zero, so the total energy in this plot (figure7) is equal to the potential energy. However, the Dynamics plot(figure 8) shows that the kinetic energy increases which is due to the gain of the vibrational energy in AB. The total energy in this plot is constant and is slightly higher than that shown in MEP plot. This indicates that the Dynamics calculates the trajectory base on the conservation of energy, whereas MEP only considers the minimum energy pathway, it includes dispersive force, which transfers the kinetic energy into other forms of energy to overcome friction.
Reactive and unreactive trajectories
| pab/
g.mol-1.pm.fs-1 |
pbc/
g.mol-1.pm.fs-1 |
Etot | Reactive? | Description of the dynamics | illustration of the trajectory |
|---|---|---|---|---|---|
| -5.1 | -2.56 | -414.280 | Yes | HA approaches H2 molecule and interact with it, new HAB molecule formed, HAB molecule vibrates and both HAB and HC move away from each other | the trajectory shows a curve starting from the initial AB and BC distance and reach the transition state where rAB=rBC, after that, it starts to fluctuate where the new HAB molecule is formed. |
| -4.1 | -3.1 | -420.077 | No | HA approaches H2 molecule, but bounce off and move away from it | the trajectory shows that rab decreases by moving towards the right, but the line then fluctuates backwards before reaching the transition state, indicating that the HA are moving away from vibrating HBC molecule. |
| -5.1 | -3.1 | -413.977 | Yes | HA approaches H2 molecule and interact with it, new HAB molecule formed. HAB and HC moves away from each other | the trajectory shows that the rab decrease and reaches the transition point where rab =rbc, after that, the line fluctuates and rab increases, representing new HAB molecule formed stretches and contracts and moves away from the HC. |
| -10.1 | -5.1 | 357.277 | No | HA approaches H2 molecule and collides HB several times, it then bounce off and moves away from theH2 molecule | the trajectory shows a line that is moving towards the right where rab is getting smaller, after reaching the transition state, the lines oscillates significantly for several times indicating HA is colliding with HB. It then fluctuates backwards, showing that HA is moving away from HBC. |
| -10.6 | -5.1 | 349.477 | Yes | HA approaches H2 molecule and collide with HB once, this os followed by HB colliding with HC once. After that, new HAB molecule formed and HC moves away. | The trajectory shows a line approaching she shorter rab with fairly horizontal teen initially. After reaching the transition state, there is a sharp drop in rbc representing that HB collides with HC, the line then fluctuates back to position where rbc is increasing. |
Prediction of rate of reaction using Transition State Theory
Transition state is based on assumptions
- energy of atoms in the reactant state obeys Boltzmann distribution, which should be satisfied for the system with enough time to reach thermal equilibrium.
- Once the system reach the transition state with a velocity towards the product side, it will not go back to retain the initial state again.
- quantum tunneling effect is negligible.
- Born-Oppenheimer approximation is obeyed where the nuclear motion is stationary with respect to electronic motion as their mass are massive.
The rate of reaction predicted by the Transition State Theory should be higher than the experimental values. The assumptions can lead to large deviation from experimental. For multi-step reaction, the intermediate should be long-lived enough to reach Boltzmann distribution. If not, the momentum of the reaction trajectory to the intermediate would affect the formation of product. For reaction with low activation energy, the tunneling effect plays a significant role and will vary the real rate of reaction from the prediction using Transition State Theory.
F-H-H system
F+H2 energetics
reaction chemical equation:
F + H2 = HF + H
Potential Energy Surface
As shown in the surface plot, the reactants F and H2 on the left is lower in energy than the products HF and H on the right, therefore, it can be concluded from the graph that this reaction is exothermic.
Bond strength
This reaction involves the breaking of H-H bond and the formation of H-F bond. To determine whether this reaction is exothermic or endothermic, the bond energy of the two is compared. Since H-F is a stronger bond compared to H-H, energy released from forming H-F bond will be greater then energy absorbed to breaking the H-H bond, therefore, the products are lower in energy than reactant, this reaction is exothermic.
| bond energy | |
|---|---|
| H-F | 565 kJmol-1 |
| H-H | 432 kJmol-1 |
HF+H energetics
reaction chemical equation:
H + H-F = H2 + F
Potential Energy Surface
As shown in the surface pot, the reactants H andH-F are lower in energy than the products F and H2. This reaction needs to absorb energy to be completed. Therefore, it is endothermic.
Bond Strength
The reaction involves the breaking of H-F bond and formation of H-H bond. Since H-F bond is stinger than H-H bond, more reaction needs to be absorbed to break H-F bond than released from forming H-H bond, this reaction is endothermic.
| bond energy | |
|---|---|
| H-F | 565 kJmol-1 |
| H-H | 432 kJmol-1 |
Transition state
The hammond Postulate states that the structure of the transition state will resemble the structure of chem, Therefore, for exothermic reaction, the transition state tends to have similar structure as the reactant whereas for endothermic reaction, the transition state tends to have the structure of the products. When considering the exothermic reaction of F+H2, its transition state will resemble the structure of the reactants which is F and H2. For the endothermic reaction of H+H-F, the transition state has similar structure as the reactant H-F and H. When estimating the transition state, the initial conditions can be set based on the bond length of H-H and H-F.
| bond length | |
|---|---|
| H-F | 92 pm |
| H-H | 74 pm |
Through adjusting rab and rbc until the Internuclear distance vs Time graph shows three horizontal line, the location of transition state can be estimated.
| reactions | transition state | |
|---|---|---|
| F + H2 = HF + H | rab=181.301
rbc=74.482 |
rab=distance between F na dHB
rbc=distance between HB and HC |
| H + HF= H2 + F | rab=74.482
rbc=181.301 |
rab=distance between HA and HB
rbc=distance between HB and F |
Activation energy
F + H2 = HF +H
Activation energy is considered as the difference between the energy of the transition state and the starting material. To determine the energy of the transition state, a graph of Energy vs Time is plotted using MEP, from which a inflection point is observed, by enlarging at that point and locating the lowest point on the expanded curve, the energy of transition state is determined to be -437.772 kJmol-1. Since fluorine is very reactive, so the activation energy is expected to be quite low. Therefore, it is expected that the reactants is fairly close the the transition state. The energy of the reactant is determined from the horizontal region to be -434.162 kJmol-1, giving the activation energy of 3.61kJmol-1.
H + HF = H2 + F
The same method is used for this reaction,
Energy mechanism
Through monitoring the Energy vs Time plot, it is discovered that the kinetic energy shows a pattern which is a reflection of the potential energy pattern when the total energy is conserved. This indicates that as the kinetic energy increases, the potential energy decreases by the same amount. The gain in the kinetic energy is mainly from the heat energy released during the reaction.
The Momenta vs Time plot shows the energy transfer between translational and vibrational energy. Since momenta correspond to bond stretching displacement, when HBC molecule oscillates, its momentum oscillates al well due to the fluctuation in the bond length. The magnitude of the fluctuation shows a slight decrease as the molecule moves towards F, some of the vibrational energy is converted to translational energy. AB shows a reverse pattern, its momentum increases in magnitude when the distance between HB and F decreases, once new H-F bond is formed, the new molecule oscillates, translational energy is converted to vibrational energy.
IR spectrum can be used to prove this energy mechanism, it can show the vibrational modes of the H-F bond formed which is not observable at the start of the reaction. The peak representing H-F shows that translational energy is transferred to vibrational energy of H-F.
