Reaction Dynamics: Trajectories Across a Potential Energy Surface
Rate theory treats a reaction as a population crossing a barrier. Dynamics follows individual trajectories instead, and some of them do not behave: they recross the barrier, or they skip the expected intermediate and land somewhere else.
What a learner can do afterwards
- Reads a two-dimensional surface and traces the path a trajectory would take across it
- Explains what recrossing does to a rate constant predicted from a barrier height
- Describes an outcome that a statistical theory predicts wrongly and says why
- Names an experiment whose product ratio is evidence for dynamic rather than statistical control
1 · Read
A potential energy surface maps a reaction. For A + BC -> AB + C, the two axes are the B-C distance and the A-B distance, and height is energy. The reactant valley lies where A is far away and B-C sits near its bond length. The product valley is the reverse. The lowest route between them crosses a saddle point, the transition state. If the saddle sits early, near the reactant valley, energy in the approach (translation) helps most. If it sits late, near the product valley, vibration of the B-C bond helps most. These are Polanyi's rules.
Transition state theory treats the saddle as a dividing surface between reactants and products. It assumes that every system crossing it toward products goes on to form products and never comes back, and that the reactants stay in thermal equilibrium. From those assumptions it gives a rate constant from the barrier height and the temperature, the Arrhenius-style dependence you have met before.
Dynamics tests the no-return assumption. Start many trajectories at the dividing surface, with positions and momenta sampled at the reaction temperature, and run each one forward and backward in time. Some reach the top and fall back, which is recrossing. The share of crossings that really join reactants to products gives the transmission coefficient, kappa, between 0 and 1. The true rate constant is kappa times the transition state theory rate. If 70 of 100 crossings are reactive, kappa is 0.7.
Watch for product ratios that refuse to match the computed barriers. On some surfaces one transition state leads to a ridge that splits into two product valleys. Trajectories then choose a valley by the downhill shape, not by any barrier. The test is to compare the measured product ratio with both predictions: if it matches the share of trajectories that reach each valley, and not the barrier heights, that is evidence for dynamic rather than statistical control.
Read the surface to see where the barrier sits, use transition state theory for the rate, correct it by kappa for recrossing, and suspect dynamic control when product ratios ignore the barriers.
Reading 'Early and late barriers' is original to this dataset.
2 · Watch
En anglaisTake it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.