The annual production of \(\mathrm{HNO}_{3}\) in 2013 was 60 million metric tons Most of that was prepared by the following sequence of reactions, each run in a separate reaction vessel. (a) \(4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\) (b) \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) (c) \(3 \mathrm{NO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{HNO}_{3}(a q)+\mathrm{NO}(g)\) The first reaction is run by burning ammonia in air over a platinum catalyst. This reaction is fast. The reaction in equation (c) is also fast. The second reaction limits the rate at which nitric acid can be prepared from ammonia. If equation (b) is second order in NO and first order in \(\mathrm{O}_{2},\) what is the rate of formation of \(\mathrm{NO}_{2}\) when the oxygen concentration is \(0.50 \mathrm{M}\) and the nitric oxide concentration is \(0.75 \mathrm{M}\) ? The rate constant for the reaction is \(5.8 \times\) \(10^{-6} \mathrm{L}^{2} \mathrm{mol}^{-2} \mathrm{s}^{-1}\).

Step by step solution

01

Understand the rate law

Identify the rate law based on the order of the reaction. For reaction (b), since it is second order in NO and first order in \(\mathrm{O}_{2}\), the rate law is given by \(\text{Rate} = k[NO]^2[O_2]\), where \(k\) is the rate constant, \(\left[NO\right]\) is the concentration of nitric oxide, and \(\left[O_2\right]\) is the concentration of oxygen.

02

Insert the given concentrations

Insert the given concentrations into the rate law. The concentration of \(NO\) is \(0.75 \mathrm{M}\) and the concentration of \(O_2\) is \(0.50 \mathrm{M}\). The rate law becomes \(\text{Rate} = k\cdot(0.75)^2\cdot 0.50\).

03

Calculate the rate of formation of \(\mathrm{NO}_2\)

Using the given rate constant \(k = 5.8 \times 10^{-6} \mathrm{L}^{2}\mathrm{mol}^{-2}\mathrm{s}^{-1}\), calculate the rate: \[\text{Rate} = (5.8 \times 10^{-6}) \cdot (0.75)^2 \cdot 0.50 \]\[= (5.8 \times 10^{-6}) \cdot 0.5625 \cdot 0.50 \]\[= (5.8 \times 10^{-6}) \cdot 0.28125 \]\[= 1.63125 \times 10^{-6} \mathrm{M} \cdot \mathrm{s}^{-1} \]This is the rate of formation of \(\mathrm{NO}_2\) under the given conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chemical Kinetics

Chemical kinetics is a branch of chemistry that deals with the speed or rate at which chemical reactions occur and the factors that affect these rates. It's a study of how fast chemical reactions proceed and how reactants change into products over time. Understanding kinetics can help us to design chemical processes, control their speed, and predict the outcomes under different conditions.

Several factors influence the rates of chemical reactions. These include the concentration of reactants, temperature, presence of catalysts, and the physical state of the reactants. For example, an increase in reactant concentration typically leads to an increase in the reaction rate, while catalysts are substances that can speed up a reaction without being consumed in the process.

Reaction Rate Calculation

To calculate the reaction rate, we use the rate law, which is an equation that relates the rate of a reaction to the concentration of reactants and the rate constant. The general form of a rate law is \(\text{Rate} = k[A]^m[B]^n\), where \(k\) is the rate constant, \(A\) and \(B\) are the concentrations of reactants, and \(m\) and \(n\) are the reaction orders with respect to each reactant.

The rate constant, \(k\), is a proportionality factor that is determined experimentally and is specific to each chemical reaction. It is influenced by factors such as temperature and the presence of a catalyst. The units of \(k\) vary depending on the overall reaction order and must be consistent with the units used for concentration and time in the rate law formula.

Reaction Orders

Reaction orders describe the relationship between the concentration of reactants and the rate of the reaction. They are determined experimentally and indicate how the rate is affected when the concentration of a reactant is changed. Each reactant in a reaction can have a different order.

A reaction is zero order with respect to a reactant if changing its concentration does not affect the rate. It is first order if the rate changes proportionally with the concentration, and second order if the rate changes with the square of the concentration. Higher-order reactions are also possible, though they are less common.

In our example, the reaction is second order in NO and first order in \(O_2\), which means that the rate of formation of \(NO_2\) depends on the square of the concentration of nitric oxide and directly on the concentration of oxygen. If the concentration of NO doubles, the rate of the reaction quadruples; if the concentration of \(O_2\) doubles, the rate also doubles.