Question

For the following reactions, predict how the rate of the reaction will change as the concentration of the reactants triple. (a) \(\mathrm{SO}_{2} \mathrm{Cl}_{2} \longrightarrow \mathrm{SO}_{2}+\mathrm{Cl}_{2} \quad\) rate \(=k\left[\mathrm{SO}_{2} \mathrm{Cl}_{2}\right]\) (b) \(2 \mathrm{HI} \longrightarrow \mathrm{H}_{2}+\mathrm{I}_{2}\) rate \(=k[\mathrm{HI}]^{2}\) (c) \(\mathrm{ClOO} \longrightarrow \mathrm{Cl}+\mathrm{O}_{2} \quad\) rate \(=k\) (d) \(\mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{2}^{-}(a q) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}\) rate \(=k\left[\mathrm{NH}_{4}^{+}\right]\left[\mathrm{NO}_{2}^{-}\right]\) (e) \(2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) rate \(=k\left[\mathrm{H}_{2}\right][\mathrm{NO}]^{2}\)

Short answer

For (a) the rate triples, for (b) the rate increases by a factor of 9, for (c) there is no change in the rate, for (d) the rate increases by a factor of 9, and for (e) the rate increases by a factor of 27 when the concentration of the reactants is tripled.

Step-by-step solution

Analyze reaction (a)

For the reaction \(\mathrm{SO}_{2} \mathrm{Cl}_{2} \longrightarrow \mathrm{SO}_{2}+\mathrm{Cl}_{2}\), the rate is directly proportional to the concentration of \(\mathrm{SO}_{2}\mathrm{Cl}_{2}\), as given by the rate law rate = \(k[\mathrm{SO}_{2} \mathrm{Cl}_{2}]\). If the concentration of \(\mathrm{SO}_{2}\mathrm{Cl}_{2}\) is tripled, the rate will also triple because it is a first-order reaction with respect to \(\mathrm{SO}_{2}\mathrm{Cl}_{2}\).

Analyze reaction (b)

For the reaction \(2 \mathrm{HI} \longrightarrow \mathrm{H}_{2}+\mathrm{I}_{2}\), the rate law is rate = \(k[\mathrm{HI}]^{2}\). Since it is a second-order reaction with respect to \(\mathrm{HI}\), tripling the concentration of \(\mathrm{HI}\) will increase the reaction rate by a factor of \(3^2\) or 9.

Analyze reaction (c)

For the reaction \(\mathrm{ClOO} \longrightarrow \mathrm{Cl}+\mathrm{O}_{2}\), the rate is independent of the concentration of reactants, thus it is a zero-order reaction. Tripling the concentration of \(\mathrm{ClOO}\) will have no effect on the rate of the reaction.

Analyze reaction (d)

For the reaction \(\mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{2}^{-}(a q) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}\), the rate law is \(rate = k[\mathrm{NH}_{4}^{+}][\mathrm{NO}_{2}^{-}]\). This is a first-order reaction with respect to \(\mathrm{NH}_{4}^{+}\) and \(\mathrm{NO}_{2}^{-}\), so tripling the concentration of both will mean the rate is increased by \(3 \times 3 = 9\) times.

Analyze reaction (e)

For the reaction \(2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\), the rate law is rate = \(k[\mathrm{H}_{2}][\mathrm{NO}]^{2}\). The reaction is first-order with respect to \(\mathrm{H}_{2}\) and second-order with respect to \(\mathrm{NO}\). Therefore, tripling the concentration of \(\mathrm{H}_{2}\) will triple the rate, and tripling the concentration of \(\mathrm{NO}\) will increase the rate by \(3^2\), so the overall rate will increase by \(3 \times 3^2 = 27\) times when the concentration of both reactants is tripled.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of how quickly chemical reactions occur and the various factors that affect this rate. It's crucial because it helps us understand essential processes, such as how quickly a medicine is metabolized by the body or how long it takes an environmental pollutant to break down. When we triple the concentration of the reactants in a reaction, as seen in the exercise, we're probing how this change influences the speed of the reaction, a key component in the study of chemical kinetics.

One primary goal of chemical kinetics is to delineate the steps by which a reaction proceeds, known as the reaction mechanism. Various factors, such as temperature, pressure, and the presence of a catalyst, can also affect the reaction rate. A detailed knowledge of kinetics is essential for the development of new chemical processes and for understanding and controlling reaction pathways.

Rate Law

The rate law is a mathematical equation that relates the rate of a chemical reaction to the concentration of its reactants. For example, in the given reaction (a), the rate law is rate = \(k[\mathrm{SO}_2 \mathrm{Cl}_2]\). The constant \(k\) is the rate constant, which varies with temperature and the presence of a catalyst. The expression in the brackets represents the concentration of the reactant.

The power to which the concentration term is raised, known as the 'order' of the reaction with respect to that reactant, indicates the dependency of the rate on the reactant's concentration. The rate law is experimentally determined, and it serves as a vital tool to predict how the reaction rate will change as conditions, such as concentrations, are altered.

Order of Reaction

The order of reaction tells us how the concentration of reactants affects the rate of the reaction. It's indicated by the exponent of the concentration term in the rate law. For example, reaction (b) has a rate law rate = \(k[\mathrm{HI}]^{2}\), making it a second-order reaction with respect to \(\mathrm{HI}\) because the exponent is 2.

The overall order of a reaction is the sum of the orders with respect to each reactant. Understanding the order of a reaction is crucial for predicting how changes in concentration will affect the reaction rate. If the concentration of \(\mathrm{HI}\) is tripled, the reaction rate in a second-order reaction will increase by a factor of \(3^2\) or 9, indicating a strong sensitivity to concentration changes.

Concentration-Dependency

Concentration-dependency in chemical kinetics is about understanding how the concentration of one reactant affects the overall rate of the reaction. This is important for controlling reactions, especially in industrial applications where yield and rate are crucial.

When a reaction depends on the concentration of reactants, like the ones in our given exercise, it means that a change in concentration directly affects the reaction rate. In reaction (d), the reaction rate would increase by \(3 \times 3 = 9\) times if the concentrations of both \(\mathrm{NH}_4^{+}\) and \(\mathrm{NO}_2^{-}\) are tripled. This precise control is powerful—imagine fine-tuning reactions to create medications or materials efficiently.