How much and in what direction will each of the following affect the rate of the reaction: \(\mathrm{CO}(g)+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g)\) if the rate law for the reaction is rate \(=k\left[\mathrm{NO}_{2}\right]^{2} ?\) (a) Decreasing the pressure of \(\mathrm{NO}_{2}\) from 0.50 atm to 0.250 atm. (b) Increasing the concentration of CO from \(0.01 M\) to \(0.03 M\).

The reaction rate law is a mathematical equation that expresses the speed of a chemical reaction as a function of the concentration of its reactants. In the context of the given exercise, the rate law for the reaction \(\mathrm{CO}(g)+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g)\) is defined as rate = \(k[\mathrm{NO}_{2}]^{2}\). This equation shows that the reaction rate is directly proportional to the square of the concentration of nitrogen dioxide \(\mathrm{NO}_{2}\).

Understanding the relationship between reaction rate and reactant concentration is fundamental to controlling chemical reactions. For instance, with this rate law, if the concentration of \(\mathrm{NO}_2\) doubles, the reaction rate will increase by a factor of four, because the rate is dependent on \(\mathrm{NO}_2\) concentration squared. In this exercise, it's crucial to note that the concentration of carbon monoxide (CO) is irrelevant to the rate at which the reaction occurs, as it does not appear in the rate law.

The dependence of reaction rate on reactant concentration is highlighted by the rate law equation. In the provided example, only the concentration of \(\mathrm{NO}_2\) appears in the rate equation, indicating that the concentration of carbon monoxide (CO) does not impact the rate. It’s an important point of clarity for students who might mistakenly believe that changes in CO concentration would affect the rate.

To elaborate, the impact on reaction rate is only through the reactants included in the rate law. Learning to identify which reactant's concentration will change the rate is critical in predicting how a reaction proceeds under various conditions. In classroom or laboratory settings, understanding concentration dependence helps in the design of experiments to test reaction kinetics. This concept also has practical implications in industrial processes where controlling reactant concentrations can be crucial for efficiency and safety.

Pressure is another factor that can influence the reaction rate, particularly in reactions involving gases, as in the given exercise. The relationship between pressure and reaction rate can be somewhat indirect, as pressure affects gas concentration and, in turn, concentration influences reaction rate.

In a closed system, an increase in pressure typically leads to an increase in gas concentration, thereby increasing the reaction rate according to the rate law. Conversely, decreasing the pressure reduces the concentration of the gas reactants, slowing the reaction rate. For the reaction between \(\mathrm{CO}\) and \(\mathrm{NO}_2\), decreasing \(\mathrm{NO}_2\) pressure from 0.50 atm to 0.250 atm results in halving its concentration, which then reduces the reaction rate by a factor of four, since rate is proportional to \(\mathrm{NO}_2\) concentration squared. This principle is vital for students to grasp, as it links the physical conditions of a reaction system with the kinetic behavior described by the rate law.