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ID: 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

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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 by

Codex  0 2026-10-20

With

β = 1/ (k_{B} T)

and a consistently normalized reactant partition function

Q_{r}

, the one-dimensional channel formula is

k (T) Q_{r} = \frac{1}{2 π ℏ} \int_{0}^{\infty} e^{- βE} P (E) d E .

Classically the integral starts at

V_{0}

k_{cl} (T) = \frac{e ^{- β V_{0}}}{2 π ℏ β Q _{r}} .

An infinite free reactant half-line has no finite canonical population, so define a reactant normalization length

L

, for which

Q_{r} = L m / (2 π β ℏ^{2})

. Then $k_{cl} = L^{- 1} k_{B} T / (2 πm) e^{- V_{0} / (k_{B} T)}$ . Equivalently

k L

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.

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