The following second–order reaction is important in air pollution. 2 NO2 → 2 NO + O2 — Physical Chemistry — Kinetics Chemistry Question
Kinetics — Atmospheric Chemistry
The following second–order reaction is important in air pollution.
2 NO2 → 2 NO + O2
Derive an integrated relationship between the total pressures in the reaction vessel, originally containing pure NO2, at the time t.
Model Answer
- d p_{NO2} / d t = k p_{NO2}^2
1/p_{NO2} - 1/p_{NO2}^0 = k t
where p_{NO2}^0 denotes the initial pressure of NO2.
By expressing the pressure of NO2 in terms of the total pressure P at time t:
At t = 0:
p_{NO2} = p_{NO2}^0
At time t:
2 NO2 → 2 NO + O2
p_{NO2} = p_{NO2}^0 - 2x
p_{NO} = 2x
p_{O2} = x
Total pressure P = p_{NO2} + p_{NO} + p_{O2} = (p_{NO2}^0 - 2x) + 2x + x = p_{NO2}^0 + x
Thus, x = P - p_{NO2}^0
Substituting x back into the equation for p_{NO2}:
p_{NO2} = p_{NO2}^0 - 2(P - p_{NO2}^0) = 3 p_{NO2}^0 - 2 P
Substituting this into the integrated second-order rate equation gives:
1 / (3 p_{NO2}^0 - 2 P) - 1 / p_{NO2}^0 = k t
It was found that when a 2 liter vessel is filled with NO2 at a pressure of 600 mm of Hg and a temperature of 600 C, the reaction is 50% complete after 3 min. Calculate the rate constant.
Model Answer
At the half-life (t_{1/2} = t = 3 min), the remaining pressure of NO2 is:
p_{NO2} = 1/2 p_{NO2}^0
For a second-order reaction, the half-life is:
t_{1/2} = 1 / (k * p_{NO2}^0)
Therefore, the rate constant k can be calculated as:
k = 1 / (p_{NO2}^0 * t_{1/2})
Convert the initial pressure of NO2 from mm Hg to atm:
p_{NO2}^0 = 600 / 760 atm
Calculate k:
k = 1 / ((600/760 atm) * 3 min) = 0.422 atm^-1 min^-1 (or L atm^-1 min^-1)