Problem 47 - Chemistry and Chemical Reactivity

Question

What is the rate law for each of the following elementary reactions? (a) \(\mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) (b) \(\mathrm{Cl}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \rightarrow \mathrm{HCl}(\mathrm{g})+\mathrm{H}(\mathrm{g})\) (c) \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}(\mathrm{aq}) \rightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}(\mathrm{aq})+\mathrm{Br}^{-}(\mathrm{aq})\)

Step-by-Step Solution

Answer

(a) Rate = k [NO][NO3]; (b) Rate = k [Cl][H2]; (c) Rate = k [(CH3)3CBr].

Step 1: Understanding the concept of rate law

The rate law of an elementary reaction can be determined from its stoichiometry. For an elementary reaction of form \(aA + bB \rightarrow cC\), the rate law is \(Rate = k [A]^a[B]^b\) where \(a\) and \(b\) are the stoichiometric coefficients of the reactants \(A\) and \(B\), respectively.

Step 2: Analyzing reaction (a)

For the reaction \(\mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g}) \rightarrow 2\mathrm{NO}_{2}(\mathrm{g})\), the stoichiometric coefficients are 1 for \(\mathrm{NO}\) and 1 for \(\mathrm{NO}_{3}\). Therefore, the rate law is \(Rate = k [\mathrm{NO}][\mathrm{NO}_{3}]\).

Step 3: Analyzing reaction (b)

For the reaction \(\mathrm{Cl}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \rightarrow \mathrm{HCl}(\mathrm{g})+\mathrm{H}(\mathrm{g})\), the stoichiometric coefficients are 1 for \(\mathrm{Cl}\) and 1 for \(\mathrm{H}_{2}\). Thus, the rate law is \(Rate = k [\mathrm{Cl}][\mathrm{H}_{2}]\).

Step 4: Analyzing reaction (c)

For the reaction \((\mathrm{CH}_{3})_{3} \mathrm{CBr}(\mathrm{aq}) \rightarrow (\mathrm{CH}_{3})_{3} \mathrm{C}^{+}(\mathrm{aq}) + \mathrm{Br}^{-}(\mathrm{aq})\), the stoichiometric coefficient is 1 for \((\mathrm{CH}_{3})_{3}\mathrm{CBr}\). Hence, the rate law is \(Rate = k [(\mathrm{CH}_{3})_{3} \mathrm{CBr}]\).

Key Concepts

Elementary Reactions

Elementary reactions are the simplest types of chemical reactions. They take place in a single step and involve either one or a few molecular collisions that lead directly from reactants to products. In an elementary reaction, the way molecules come together can be easily predicted using their chemical equations.

Because these reactions occur in a single step, the rate law for an elementary reaction is directly determined by its stoichiometry. This means that the rate at which the reaction proceeds is proportionate to the concentrations of the reactants raised to the power of their respective stoichiometric coefficients in the balanced equation.

• The reaction \( ext{NO(g) + NO}_3 ext{(g) } \rightarrow \text{2 NO}_2 ext{(g)}\) being elementary, its rate law can be expressed as \(\text{Rate} = k [\text{NO}][\text{NO}_3]\) because each reactant is involved once in the reaction.
• Similarly, for \(\text{Cl(g) + H}_2\text{(g) } \rightarrow \text{HCl(g) + H(g)}\), the rate law is \(\text{Rate} = k [\text{Cl}][\text{H}_2]\).
• The simpler decomposition reaction \(\text{(CH}_3)_3\text{CBr(aq) } \rightarrow \text{(CH}_3)_3\text{C}^+\text{(aq) } + \text{Br}^-\text{(aq)}\) leads to a straightforward rate law of \(\text{Rate} = k [(\text{CH}_3)_3\text{CBr}]\).

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the relative quantities of reactants and products in chemical reactions. It uses quantitative relationships to understand how much of each substance is involved in a reaction.

In the context of rate laws for elementary reactions, stoichiometry helps in determining how the concentration of reactants affects the reaction rate. The stoichiometric coefficients in a balanced chemical equation serve as the exponents in the rate law equation for elementary reactions.

This is because, in these direct one-step reactions, each molecule of reactant collides and reacts in a manner proportional to its presence in the reaction mixture. Thus, stoichiometry gives us a clear insight into the dynamic nature of chemical transformations in elementary steps.

• In a reaction \(aA + bB \rightarrow \text{Products}\), the exponents of \( [A]^a[B]^b\) in the rate law represent the stoichiometric coefficients \(a\) and \(b\).
• For example, in \(\text{NO(g) + NO}_3\text{(g) } \rightarrow \text{2 NO}_2\text{(g)}\), the coefficients of NO and NO₃ are both 1, leading directly to a rate law where each concentration term is raised to the first power.

Reaction Kinetics

Reaction kinetics is the study of the speed at which chemical reactions occur and the factors that influence these rates. The central element of kinetics is the reaction rate, which depends on such variables as reactant concentrations, temperature, presence of catalysts, and the physical state of the reactants.

In elementary reactions, the reaction rate is determined by the rate law, which is derived directly from the stoichiometry of the reaction. This direct link simplifies the kinetic analysis of elementary reactions compared to more complex reactions involving multiple steps.

• The rate constant \(k\) is a crucial parameter in the rate law, influencing how fast a reaction proceeds under given conditions.
• Temperature usually increases the reaction rate as it furnishes the reactant molecules with more energy, leading to more effective collisions.

Understanding these factors helps chemists control and optimize reaction conditions, making kinetic analysis an essential tool in fields ranging from industrial chemistry to pharmaceuticals.

Chemical Reactions

Chemical reactions are processes where substances, the reactants, are transformed into new substances, the products. They involve the making and breaking of chemical bonds, leading to changes in the arrangement of atoms among the reacting species.

In any chemical reaction, understanding the mechanism—the step-by-step sequence—through which reactants turn into products is crucial. This is where the concept of rate law becomes important, especially in elementary reactions. The simplicity of these reactions makes them an ideal model for understanding more complex reactions.

• For example, in the reactions presented, each transforms reactants directly to products without intermediate steps, making the study of their rate laws particularly straightforward.
• Identifying a reaction as elementary can simplify predictions about how changes in conditions will affect the reaction rate.

Thus, insights gained from elementary reaction studies can be invaluable guides to comprehending and manipulating chemical behaviors in more complicated reaction systems.