Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Interdisciplinary paper IDP1 / 2 / d / iii

The change in inverse temperature is $\Delta u=1/290-1/300=1.14943\times 10^{-4}{\mathrm{K}}^{-1}$. Treating the sensitivity as approximately constant over the interval,Thus the predicted concentration increases by about 15.3%. Linearizing the exponential gives a rougher 14.3% estimate; a logarithmic change is not itself the exact fractional change.

An independent finite-change calculation checks this approximation without keeping the loss ratio artificially fixed. Normalize the initial Chapman and nitrogen-cycle loss terms to 1 and 3, so the initial production is 4. If $r=X_{290}/X_{300}$, the new steady balance is $e^{-2610\Delta u}{r}^2+3e^{-1200\Delta u}r=4$. Its positive root is $r=1.15342$, essentially the same result.