The gas-phase oxidation of nitric oxide (nitrogen monoxide, NO) has an overall r
ID: 921648 • Letter: T
Question
The gas-phase oxidation of nitric oxide (nitrogen monoxide, NO) has an overall reaction written as 2NO (g) + O2 (g) --> 2NO2 (g). The reaction has an overall third order with a rate law rate = kr[NO]^2[O2]. The reaction goes through the following elementary steps:
NO + NO <-ka-> N2O2
N2O2 + O2 -kb-> 2NO2
A.) Derive the above rate law for NO2 formation using the steady-state approximation
B.) Assuming all the rate constants obey Arrhenius equation, drive a relationship that relates the "apparent activation energy in kr to the activation energies of the elementary steps.
Explanation / Answer
which has an experimentally determined third order rate law:
rate of reaction=k [NO2]2 [O2]
Because a termolecular mechanism (a three-body reaction occurring in a single step) is extremely rare, a mechanism with two bimolecular elementary steps can be suggested:
2NO(g)N2O2(g) (Step 1 fast)
N2O2(g)+O2(g)2NO2(g) (Step 2 slow)
The proposed mechanism has the same stoichiometry as the overall reaction. Step 1 is a reversible process in which equilibrium is reached quickly. Step 2 is therefore the rate-determining step. The relative rate of the Step 2 is k3[N2O2][O2], which does not seem to match with the experimental rate of k[NO2]2[O2]. However, N2O2 is actually a reaction intermediate, because it appears in both elementary steps but not in the equation of the overall reaction. Because an intermediate cannot appear in the rate law for the overall reaction, a value equivalent to N2O2 is required. Because Step 1 is a fast, reversible reaction, it can be assumed that:
k1 [NO]2 = k2 [N2O2]
Therefore,
[N2O2]=k1k2[NO]2
Through substitution,
k3 [N2O2] [O2] = k3 (k1k2) [NO]2 [O2]
The rate constant is k=k3k1k2, so the rate of reaction is k[NO]2[O2], matching the experimental rate law. This proposed mechanism fulfills both requirements for a possible mechanism.
b.
The Arrhenius equation gives the quantitative basis of the relationship between the activation energy and the rate at which a reaction proceeds. From the Arrhenius equation, the activation energy can be found through the relation
K = A e Ea / RT
where A is the frequency factor for the reaction, R is the universal gas constant, T is the temperature (in kelvin), and k is the reaction rate coefficient. Even without knowing A, Ea can be evaluated from the variation in reaction rate coefficients as a function of temperature (within the validity of the Arrhenius equation).
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