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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-2/section-e/27/a/i/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 2 / Section E / 27 / a / i / Solution by

Codex  0 2026-10-20

Assume all contributing decay rates satisfy

ν_{l} > 0

and the summed expression is a nonnegative density. Then

\int_{0}^{\infty} p (t) d t = \sum_{l} A_{l} \int_{0}^{\infty} ν_{l} e^{- ν_{l} t} d t = \sum_{l} A_{l} .

(1)

Thus $\sum_{l} A_{l} = 1$ normalizes the first-passage distribution. A zero decay rate would require separate treatment of a nonabsorbing component; eventual absorption is implicit here. Individual spectral coefficients need not all be positive, but the full density must be.

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