Once you choose the reaction A +BC ==> AB + C you also have to choose a case. The options are:
These cases are described in greater detail below.
In this example, the program assumes that A, B and C are confined to a line. The program then picks random initial velocities for A, B and C and integrate the equations of motion and see what occurs. The program plots the trajectories of the atoms on a potential energy surface, a mass weighted potential energy surface, and animates the process. The program also produces a plot of the atomic positions versus time.
When I ask my students to do this problem, I ask them to compare the behavior of each of the plots and look to see which trajectories lead to reaction The first part of the exercise is to get the students used to the idea that a reaction can be represented as a trajectory on a potential energy surface. I set up the program to plot trajectories and the animations side by side so the students can understand what is going on and ask students several questions about the plots. In order to do so, you first select the windows that you want to display. Next you shrink the windows that you do not want to display. Then you click on Window on the top frame and select Tile.
Once the trajectories are displayed correctly, I ask students to
Detailed discussions of each of these issues are included in R. I. Masel, Kinetics and Catalysis, Wiley to appear
Note this is a one dimensional system since we have confined the collision to a line. As a result, the rate constant ends up having units of &Angstrom;/molecule/sec.
Students are often surprised to learn that a significant fraction of the trajectories with more than enough energy to cross the barrier in the potential energy surface do not make it over the barrier. Instead coordinated motion of the atoms is needed in order for the reaction to occur. If you choose random trajectories then only a small fraction of the trajectories with energies between 14 and 18 kcal/mole make it over the top of the barrier even though the barrier height is only 13.86 kcal/mole. The maximum rate occurs when the energy is about 20-25 kcal/mole.
Physically, you need to deposit energy into the B-C bond in order for the bond to break. If there is barely enough energy in the system, then reaction happens only when everything is right so that all of the energy of the collision is deposited in the B-C bond. Those are rare events unless there is extra energy to allow the reaction to happen.
All real reactions show behavior like this. Generally the reaction probability is small at energies just above the height of the barrier and then rises at higher energies. The reaction probability peaks 5-10 kcal/mole above the barrier because if there is too much excess energy, you never form a stable A-B bond.
If the students take enough data, they can integrate the results to calculate a reaction rate. Additional data is given in a file trajectories.csvWhen you integrate the rate equation you find that the activation barrier works out to be about 14. kcal/mole i.e. just over the height of the barrier. Physically, the exp(-E/kT) term in the Boltzman average swamps all of the other variations, so that the actual barrier is just over threshold even though the maximum reaction probability occurs 5-10 kcal/mole above the threshold.
Detailed discussions of each of these issues are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
After the students see run the Choose Random trajectories, the students start to suspect that molecules with barely enough energy to cross the barrier cannot lead to reaction. The objective of this part is to prove that molecules with barely enough energy can cross the barrier if everything is timed right. The program fixes the initial energy, and then adjusts all of the other variables so that all of the energy of the collision is deposited into the B-C bond. The collision energy is raised in small increments and the students are told to look for reaction.
The reaction actually occurs when the energy is just above the barrier, as expected.
When I tell my students about this case, I tell them to examine the trajectories on a potential energy surface, and look very carefully at how the trajectory weaves left to right. I liken the motion to the motion around a race course. During a race, it is critically important for the driver to turn at just the right time to get around a turn in the track. In the same way it is critically important for the trajectory to turn at just the right time to make it over the potential energy surface.
Detailed discussions of each of these issues are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
This case is similar to the one above, except that the parameters have not been optimized to maximize the reaction probability. In this case you need excess energy to make it over the barrier. I ask the students to see how much extra energy is needed. I also ask the students to compare the trajectories which do not lead to reaction to the "optimal" trajectories. How much vibrational energy is in the "optimal" trajectories compared to the trajectories which lead to reaction. How much turning is needed.
Physically, you need to have enough vibrational energy to get around the curve in the potential energy surface and enough translational energy to get over the barrier. If you optimize everything, then the energy is partitioned correctly between vibration and translation, you can get over the barrier as long as the total energy is higher than the saddle point energy. If the energy is not partitioned correctly then you need more total energy, because you still need the minimum energy in vibration and translation.
Another key variable is the timing of the collision. That is explored in the next case below.
Detailed discussions of each of these issues are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
The objective of this case is to fix the total energy and vibrational state of the system, and see how changes in the initial positions of the atoms affect the reaction probability. Generally one finds that some of the trajectories make it, while others do not. Again, I tell my students to examine the trajectories on a potential energy surface, and look very carefully at how the trajectory weaves left to right. I liken the motion to the motion around a race course. During a race, it is critically important for the driver to turn at just the right time to get around a turn in the track. In the same way it is critically important for the trajectory to turn at just the right time to make it over the potential energy surface.
The next question I ask the students to consider is "what does it mean to turn at the right time". Physically, molecules are not turning, they are only translating and vibrating. The "turning" is an illusion created by replotting the trajectories on a potential energy surface.
I usually ask students to look at the animations instead. The key thing to focus on is the motion of C as A hits. If C is moving in toward B, no reaction happens. If C is moving away from B, the reaction happens. I ask the students to see if they can tell when a trajectory leads to reaction by carefully examining the reaction and see if C is moving out or in when A hits.
All of these trends can also be observed in plots of the positions vs time or plots of motion on mass weighted coordinates. Detailed discussions of how to look at the plots are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
This case is used to check the Polanyi rules: i.e. that with an early transition state you want most of the energy in translation, with a middle transition state you want some of the energy in translation and with a late transition state, vibrational energy is more important.
I ask my students to try it by varying the proportion of the energy in translation, vibration and rotation, vary the position of the transition state, and see which trajectories lead to reaction.
Recall from above that the reaction probability varies according to how the trajectory weaves left to right. I liken the motion to the motion around a race course. During a race, it is critically important for the driver needs to turn his wheels to get around a turn in the track. Well, vibrational motion provides the "turning motion". The more the trajectory needs to turn, the more vibrational energy is needed. Still one can overdo it. If you turn too sharply (i.e. provide too much vibrational energy) you crash into the curb instead of going around the racetrack. In the same way, if all of the incident energy is in vibration, the trajectory turns too quickly and never has enough energy to make it over the hill in the potential energy surface.
This is a case that I do not use very much myself, but it is included because people talk about the effects. If one assumes billiard ball collisions, then the rate of the reaction H + ClH ==> HCl + H should be higher than the rate of the reaction D + ClH ==> D Cl + H. However, when you compute the trajectories with a real potential you find that the later reaction has a slightly higher rate.
I included this example so students have a different way to compute reaction probabilities. In this case they can choose a energy and run trajectories and calculate a reaction probability. For example students can fix the total energy at 14 kcal/mole and calculate a reaction probability.
I use this example to show how one can use a MD calculation to estimate a rate constant. Recall, that for a one dimensional system
where
I often tell the students to assume that the cross section for the reaction,is the velocity of A toward BC in center of mass coordinates
is the internal energy of the BC molecule
is the translational energy of A toward BC in center of mass coordinates.
is the reaction probability as a function of
and
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is the distribution of internal energy and translational energy.
I tell the students to do a Laguere integration to calculateis the average velocity in center of mass coordinates.
is Boltzman's constant
T is the absolute temperature
Details are included in Example 8.C of R. I. Masel, Kinetics and Catalysis, Wiley to appear
This example is used to show how changes in the impact parameter change the trajectories which lead to reaction. When you have linear collisions, the only way for reaction to occur is for energy and momentum to be indirectly transferred to the B-C bond. With non-linear collisions, though, A can directly insert into the B-C bond. One observes trajectories where B-C rotates, C vibrates out and then A slips into the B-C bond.
When I use this example I ask my students to compare linear and non-linear collisions and say how they are the same or different.
Detailed discussions of each of these issues are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
This example is used to show how changes in the B-C angle affect the trajectories which lead to reaction. When you have linear collisions, the only way for reaction to occur is for energy and momentum to be indirectly transferred to the B-C bond. With non-linear collisions, though, A can directly insert into the B-C bond. There is another effect too. When A comes in, it can push B out of the way, but then B rotates around and gets in the way of the reaction.
When I use this example I ask my students to compare linear and non-linear collisions and say how they are the same or different.
Detailed discussions of each of these issues are included in Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
This example allows the students to choose the impact parameter and incident energy and calculate a reaction probability. The code is set up to not display the entire trajectory. Rather, only a few points along the trajectory are included to increase the speed.
When I assign this problem, I usually ask students to compute the reaction cross-section using the equation:
See Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley to appear
For further information about applications of this program please look in Chapter 9 of R. I. Masel, Principles of Adsorption and Reaction on Solid Surfaces, Wiley 1996 or Chapter 8 of R. I. Masel, Kinetics and Catalysis, Wiley 2000