## Chapman ozone

Sydney Chapman proposed a kinetic model for the production and destruction of stratospheric ozone. His model turned out to be unable to account quantitatively for ozone abundances. Still, the Chapman model remains at the core of the much more extensive models of ozone kinetics used by current atmospheric scientists. This set of exercises deals with his proposed mechanism.

Actually Chapman proposed a set of 6 reactions, shown below:
 (1) O + O -- > O2 (2) O + O2 -- > O3 (3) O + O3 -- > 2 O2 (4) O3 -- > O + O2 (5) 2 O3 -- > 3 O2 (6) O2 -- > O + O
He reasoned that reactions 1 and 5 would have negligible effect, so we will not consider them further. Assume that each step in the proposed mechanism is elementary, so that, for example, the rate of reaction (1) is

rate1 = k1[O]2 .
1) Write expressions for the rates of change of the concentrations of O, O2, and O3. For example, noting that oxygen atoms are consumed in steps 2 and 3 and produced in steps 4 and 6. So the rate of change of oxygen atom concentration, d[O]/dt, is the rate of step 4 plus twice that of step 6 minus the rates of steps 2 and 3. Express these rates of change in equations.

2) Chapman examined the equilibrium state of this model, that is, a state in which the rates of change of all species are zero. Take the expressions obtained in exercise 1, set them equal to zero, and solve for [O] in terms of the rate constants. (Hint: take two of the rate equations and solve for [O3] in terms of the rate constants and the other concentrations. Set these two expressions equal to each other, and solve for [O]. You should find that [O2] drops out, and you can solve for [O] in terms of rate constants only.)

3) It turns out not to be possible to solve for [O3] and [O2] separately, but it is possible to solve for the ratio [O3]/[O2] in terms of rate constants only. Do so.

### Reference

Sydney Chapman, "A Theory of Upper-Atmospheric Ozone," Memoirs of the Royal Meteorological Society 3(26), 103-25 (1930).
Copyright 2003 by Carmen Giunta. Permission is granted to reproduce for non-commercial educational purposes.