Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Interdisciplinary paper IDP1 / 1 / b / Solution

In a vertical-beam, effective-wavelength Beer-Lambert law approximation, the optical depth above altitude $z$ is $\tau (z)={\sigma }_{{\mathrm{O}}_2}{N}_{{\mathrm{O}}_2}(z)+{\sigma }_{{\mathrm{O}}_3}{N}_{{\mathrm{O}}_3}(z)$. Using centimetres consistently givesThe transmitted photon flux is proportional to $e^{-\tau }$. If the parent-gas absorption cross section and photodissociation yield are otherwise the same, they cancel in the ratio of photolysis rate coefficients:The corresponding fractions of the unattenuated beam are about $7.52\times 10^{-7}$ and $0.654$. Strong shielding by the overlying oxygen and especially ozone therefore makes destruction vastly faster higher in the stratosphere, even though the molecular absorption properties have not changed.

This numerical ratio uses the intended simplified radiation model. A real photolysis rate coefficient integrates wavelength-dependent actinic flux, absorption cross section and quantum yield. A nonvertical direct beam would use slant optical depth $\tau /\cos \theta$; diffuse scattering and the true solar spectrum would also matter. The supplied effective cross sections do not specify a unique broadband radiative-transfer calculation.