The gas-phase decomposition of ozone, 2O3 → 3O2, is believed to have the mechanismwhere M...

The gas-phase decomposition of ozone, 2O3 → 3O2, is believed to have the mechanism

where M is any molecule. (a) Verify that d[O2]/dt = 2k2[O][O3] + k1[O3][M] – k−1[O2][O][M]. Write down a similar expression for d[O3]/dt. (b) Use the steady-state approximation for [O] to simplify the expressions in (a) to d[O2]/dt = 3k2[O3] [O] and d[O3]/dt = -2k2[O3][O]. (c) Show that, when the steady-state approximation for [O] is substituted into either d[O2]/dt or d[O3]/dt, one obtains

(d) Assume step 1 is in near equilibrium so that step 2 is rate- determining, and derive an expression for r. Hint: Because O2appears as a product in both the rate-determining step 2 and the preceding step 1, this problem is tricky. From the overall stoichiometry, we have . The O2 production rate in the rate-determining step 2 is (d[O2]/dt)2 = 2k2[O][O3]. However, for each time step 2 occurs, step 1 occurs once and produces one O2 molecule. Hence three O2 molecules are produced each time the rate-determining step occurs, and the total O2 production rate is d[O2]/dt = 3k2[O][O3]. (e) Under what condition does the steady-state approximation reduce to the equilibrium approximation?