Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 2 / A7 / a / ii / Solution

Resolve the activation free energy into enthalpy and entropy in the Eyring equation:

k_{2} = \frac{k _{B} T}{h c ^{\circ}} exp (\frac{Δ S ^{‡\circ}}{R}) exp (- \frac{Δ H ^{‡\circ}}{RT}) .

Taking logarithms of numerical values in fixed, consistent units gives

ln \frac{k _{2}}{T} = - \frac{Δ H ^{‡\circ}}{R} \frac{1}{T} + ln \frac{k _{B}}{h c ^{\circ}} + \frac{Δ S ^{‡\circ}}{R} .

Thus for slope

s

and intercept

b

,

Δ H^{‡\circ} = - R s, Δ S^{‡\circ} = R [b - ln \frac{k _{B}}{h c ^{\circ}}] .

The logarithms imply reference units, as made explicit in the supplied numerical plot. Approximate linearity requires the activation parameters and transmission coefficient to vary little over the temperature interval; a significant activation heat-capacity difference produces curvature.

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