In the stratosphere, ozone decomposes by the uncatalyzed pathway
2 O3 → 3 O2
with an overall second-order rate law rate = ku[O3]2, where ku = 2.5 × 10−15 cm3 molecule−1 s−1 at 250 K. In the presence of chlorine radicals (Cl·) a catalyzed pathway operates that is effectively first order in [O3]: rate = kc[O3], with kc = 1.8 × 10−4 s−1 at 250 K.
A researcher is modeling a future atmosphere in which chlorine radicals have been virtually eliminated. The researcher asks whether changing only stratospheric temperature and the ozone number density could allow the uncatalyzed route to proceed at the same rate that the catalyzed route currently achieves at 250 K when [O3] = 1.0 × 1012 molecules cm−3. For the uncatalyzed pathway assume an activation energy of 96 kJ mol−1 and an Arrhenius pre-exponential factor that is temperature independent.
(a) Calculate the rate of ozone loss (in molecules cm−3 s−1) via the catalyzed pathway under the current conditions.
(b) At 250 K, determine the ozone number density required for the uncatalyzed pathway to match the rate found in part (a).
(c) The maximum physically plausible stratospheric ozone number density is 5.0 × 1012 molecules cm−3. If [O3] is fixed at this maximum value, calculate the temperature that would be needed for the uncatalyzed pathway to equal the rate found in part (a).
(d) Evaluate the realism of simultaneously achieving the temperature and ozone concentration you identified. Support your evaluation with quantitative comparisons and atmospheric considerations.