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 / a / vi / Solution

With $\beta =1/(k_{B}T)$ and a consistently normalized reactant partition function $Q_r$, the one-dimensional channel formula isClassically the integral starts at $V_0$, givingAn infinite free reactant half-line has no finite canonical population, so define a reactant normalization length $L$, for which $Q_r=L\sqrt{m/(2\pi \beta {\hslash }^2)}$. Then $k_{\mathrm{cl}}=L^{-1}\sqrt{k_{B}T/(2\pi m)}{e}^{-V_0/(k_{B}T)}$. Equivalently $kL$ is a one-dimensional thermal flux per unit reactant density, with dimensions of velocity. A rate constant without either this normalization or a specified reactant well is not dimensionally complete.