Problem 44 - Chemistry and Chemical Reactivity

Question

What is the rate law for each of the following elementary reactions? (a) \(\mathrm{Cl}(\mathrm{g})+\mathrm{ICl}(\mathrm{g}) \rightarrow \mathrm{I}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})\) (b) \(\mathrm{O}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (c) \(2 \mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g})\)

Step-by-Step Solution

Answer

(a) \( \text{Rate} = k[\mathrm{Cl}][\mathrm{ICl}] \), (b) \( \text{Rate} = k[\mathrm{O}][\mathrm{O}_3] \), (c) \( \text{Rate} = k[\mathrm{NO}_2]^2 \).

Step 1: Understanding Elementary Reactions

An elementary reaction is a single step reaction where reactants convert directly to products. For such reactions, the stoichiometric coefficients in the balanced equation provide the exponents in the rate law expression.

Step 2: Determine the Rate Law for Reaction (a)

In the reaction \( \mathrm{Cl} (\mathrm{g}) + \mathrm{ICl} (\mathrm{g}) \rightarrow \mathrm{I} (\mathrm{g}) + \mathrm{Cl}_2 (\mathrm{g}) \), there is one molecule each of \( \mathrm{Cl} \) and \( \mathrm{ICl} \). This implies that the reaction rate law is first order in each reactant. Thus, the rate law is \( \text{Rate} = k[\mathrm{Cl}][\mathrm{ICl}] \).

Step 3: Determine the Rate Law for Reaction (b)

In the reaction \( \mathrm{O} (\mathrm{g}) + \mathrm{O}_3 (\mathrm{g}) \rightarrow 2 \mathrm{O}_2 (\mathrm{g}) \), there is one molecule each of \( \mathrm{O} \) and \( \mathrm{O}_3 \). This indicates the reaction rate is first order with respect to each reactant. Hence, the rate law is \( \text{Rate} = k[\mathrm{O}][\mathrm{O}_3] \).

Step 4: Determine the Rate Law for Reaction (c)

For the reaction \( 2 \mathrm{NO}_2 (\mathrm{g}) \rightarrow \mathrm{N}_2 \mathrm{O}_4 (\mathrm{g}) \), there are two molecules of \( \mathrm{NO}_2 \), making the reaction second order with respect to \( \mathrm{NO}_2 \). Thus, the rate law is \( \text{Rate} = k[\mathrm{NO}_2]^2 \).

Key Concepts

Elementary Reactions

Elementary reactions are the simplest type of chemical reactions. They occur in a single step, with reactants directly forming products without any intermediates. An elementary reaction represents a fundamental event at the molecular level. Because they happen in one step, the rate law for an elementary reaction can be directly derived from the stoichiometric coefficients of the balanced chemical equation.
For instance, if you have a reaction like \( A + B \rightarrow C \), the rate law is expressed as \( \text{Rate} = k[A][B] \) since there is one molecule each of \( A \) and \( B \) involved in the reaction.
In practice, understanding elementary reactions helps simplify the determination of the reaction's rate law, making it directly proportional to the product of the reactants' concentrations each raised to their respective stoichiometric coefficients.

Reaction Order

The reaction order tells us how the rate of a reaction depends on the concentration of the reactants. It is determined by the sum of the exponents of the concentration terms in the rate equation. For elementary reactions, the reaction order is simply the sum of the stoichiometric coefficients of the reactants.
For example, in the reaction \( A + B \rightarrow C \), if the rate law is given by \( \text{Rate} = k[A][B] \), then the reaction is first order with respect to \( A \) and first order with respect to \( B \). Overall, this makes it a second order reaction.
Understanding the reaction order is crucial for predicting how changes in concentration affect the reaction rate, which is essential for controlling and optimizing chemical processes.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. In the context of rate laws for elementary reactions, stoichiometry helps us determine the powers to which the concentration of each reactant is raised in the rate law.
For example, in the reaction \( 2NO_2 \rightarrow N_2O_4 \), the stoichiometric coefficient for \( NO_2 \) is 2. Hence, in the rate law \( \text{Rate} = k[NO_2]^2 \), the concentration of \( NO_2 \) is raised to the power of 2.
This straightforward relationship between stoichiometry and rate laws is a characteristic feature of elementary reactions, providing a clear understanding of how reactant quantities affect the rate of reaction.

Kinetics

Kinetics is the study of the rate of chemical processes and the factors that affect these rates. It explores how different conditions, such as concentration, temperature, and the presence of catalysts, influence the speed of a reaction.
The rate law, a key component of chemical kinetics, provides a mathematical expression that links the reaction rate to the concentrations of reactants, often incorporating a rate constant \( k \) which is specific to the reactions and conditions involved.
By studying kinetics, chemists can gain insights into the mechanisms of reactions, predict how changes in conditions will impact reaction rates, and design processes that can either speed up or slow down reactions based on practical needs.