 Solution

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/ii/solution

For an unnormalized representation, divide the first moment by its total mass. Integration by parts gives

\int_{0}^{\infty} t ν_{l} e^{- ν_{l} t} d t = \frac{1}{ν _{l}} .

Consequently

T = \frac{\sum _{l} A _{l} / ν _{l}}{\sum _{l} A _{l}} .

For normalized amplitudes the denominator is one. A mode with small weight can still dominate the mean if its lifetime

1/ ν_{l}

is sufficiently large; this distinction matters when interpreting the later graph.

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