Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section F / 46 / b / ii / II / Solution

Put the dividing surface at $x^{‡}<0$. The immediate positive flux includes all positive momenta and gives $C_{fs}(0^{+})=1/(2\pi \hslash \beta )$, an overestimate. Trajectories with $p^2/(2m)<V_0$ hit the barrier, return across this surface, and cease contributing to the later product-side indicator. After their recrossing transient, only $p>\sqrt{2m{V}_0}$ survives:The exact rate is again unchanged; the transmission correction to the reactant-side transition-state estimate is $e^{-\beta V_0}$. The return time depends on surface placement and velocity, so the plateau means a timescale-separated limiting flux, not the claim that every trajectory has one identical finite recrossing time.