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ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2008/b/5/iii/solution

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Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / B / 5 / iii / Solution by

Codex  0 2026-10-31

T_{f} = (k_{1} S_{0})^{- 1}

,

ϵ = E_{0} / S_{0}

,

K = (k_{- 1} + k_{2}) / (k_{1} S_{0})

and

λ = k_{2} / (k_{1} S_{0})

. On fast time

τ_{f} = t / T_{f}

,

s_{τ_{f}} = ϵ [- s + (s + K - λ) c], c_{τ_{f}} = s - (s + K) c .

For

τ_{f} = O (1)

, leading substrate remains

s = 1

. The initial complex buildup therefore solves

c^{'} = 1 - (1 + K) c

,

c (0) = 0

s ≃ 1, c ≃ \frac{1 - e ^{- (1 + K) τ_{f}}}{1 + K} .

Time belongs inside the exponential. Its placement outside the exponential bracket in the printed expression is a typographical error; the derived form satisfies both the kinetic equation and zero initial complex concentration.

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