Question

A.)Use the steady-state approximation

B.)And the rate determining step approximation to derive the rate equation.The gas-phase decomposition of ozone, 2O₃ -> 3O₂, is believed to have the mechanism. M is any molecule.   Compare the the results of the 2 methods

              k₁

O₃ + M ? O₂ + O + M fast equilibrium

  k_-1

           

O + O₃ -> 2O₂                        Slow

k₂

Answer 1

The given equation is O3 + M <-------------> O2 + O + M

O + O3 -----------> 2O2

In these steps, O is the intermediate.

A steady-state approach makes use of the assumption that the rate of production of an intermediate is equal to the rate of its consumption.

Rate of production of O = K1 [O3] [M]

Rate of consumption of O = k2[O][O3]

Thus, we have

K1 [O3] = k2 [O] [O3]

and solving for [O] gives the result,

[O] = k1 [O3] / k2 [O3] . . . (1)

step i is a equilibrium and thus can not give a rate expression.

Step ii leads to the production of some products. The rate expression from step ii as:

           d[O2]
    -1/2  ----- = k2 [O3][O] k1 [O3] / k2 [O3] ....(2)

dt

          d[O2]
    -1/2  ----- = K [O] [O3] 
            dt

Where, K = k1k2 / k2

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B) The rate determining step is the slowest step of the all the possible steps.

Thus, rate = k [O][O3]