Problem 57 - Chemistry: An Atoms-Focused Approach

Question

Each of the following reactions is first order in each reactant and second order overall. Which reaction is fastest if the initial concentrations of all the reactants are the same? a. \(\mathrm{ClO}_{2}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{3}(g)+\mathrm{O}_{2}(g)\) \(k=3.0 \times 10^{-19} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) b. \(\mathrm{ClO}_{2}(g)+\mathrm{NO}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{ClO}(g)\) \(k=3.4 \times 10^{-13} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) c. \(\mathrm{ClO}(g)+\mathrm{NO}(g) \rightarrow \overline{\mathrm{C} 1(g)}+\mathrm{NO}_{2}(g)\) \(k=1.7 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) d. \(\mathrm{ClO}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{2}(g)+\mathrm{O}_{2}(g)\) \(k=1.5 \times 10^{-17} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\)

Step-by-Step Solution

Answer

a. \(\mathrm{H}(g)+\mathrm{O}_{2}(g) \rightarrow \mathrm{HO}_{2}(g) \quad (k_a = 3.0 \times 10^{-19} \mathrm{cm}^{3}/(\text {molecule} \cdot \text{s}))\) b. \(\mathrm{H}(g)+\mathrm{BrO}(g) \rightarrow \mathrm{HBr}(g)+\mathrm{O}(g) \quad (k_b = 3.4 \times 10^{-13} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s}))\) c. \(\mathrm{ClO}(g)+\mathrm{NO}(g) \rightarrow \overline{\mathrm{C} 1(g)}+\mathrm{NO}_{2}(g) \quad (k_c = 1.7 \times 10^{-11} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s}))\) d. \(\mathrm{F}(g)+\mathrm{H}_{2}\mathrm{O}(g) \rightarrow \mathrm{FO}(g)+\mathrm{H}_{2}(g) \quad (k_d = 1.5 \times 10^{-17} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s}))\) Answer: Reaction C is the fastest among the four options, with a rate constant of \(k_c = 1.7 \times 10^{-11} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s})\).

Step 1: Write the general rate equation

Rate = \(k[\mathrm{reactant}_1][\mathrm{reactant}_2]\)

Step 2: Calculate the rates for each reaction

For simplicity, let's assume that the initial concentration of each reactant is represented by \([\mathrm{reactant}]\). The rates for each reaction can be calculated as follows: a. Rate of Reaction A = \(k_a[\mathrm{reactant}_1][\mathrm{reactant}_2] = (3.0 \times 10^{-19} \mathrm{cm}^{3}/(\text {molecule} \cdot \text{s})) [\mathrm{reactant}][\mathrm{reactant}]\) b. Rate of Reaction B = \(k_b[\mathrm{reactant}_1][\mathrm{reactant}_2] = (3.4 \times 10^{-13} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s})) [\mathrm{reactant}][\mathrm{reactant}]\) c. Rate of Reaction C = \(k_c[\mathrm{reactant}_1][\mathrm{reactant}_2] = (1.7 \times 10^{-11} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s})) [\mathrm{reactant}][\mathrm{reactant}]\) d. Rate of Reaction D = \(k_d[\mathrm{reactant}_1][\mathrm{reactant}_2] = (1.5 \times 10^{-17} \mathrm{cm}^{3}/(\text{molecule} \cdot \text{s})) [\mathrm{reactant}][\mathrm{reactant}]\)

Step 3: Compare the reaction rates

Comparing the rates of the reactions, we can observe that the highest rate constant is for Reaction C with \(k_c = 1.7 \times 10^{-11} \mathrm{cm}^{3}/(\mathrm{molecule} \cdot \mathrm{s})\). Since the initial concentrations of all reactants are the same, the reaction with the highest rate constant will be the fastest reaction.

Step 4: Conclusion

Based on the given rate constants and the initial concentrations of the reactants, Reaction C: \(\mathrm{ClO}(g)+\mathrm{NO}(g) \rightarrow \overline{\mathrm{C} 1(g)}+\mathrm{NO}_{2}(g)\) is the fastest reaction among the four options.

Key Concepts

Rate of Reaction

The rate of reaction is a measure of how quickly a chemical reaction proceeds. It helps understand how fast reactants are converted into products over time. In this context, we calculate the rate by using the formula:

• Rate = \( k[\text{reactant}_1][\text{reactant}_2] \).

This rate is affected by several factors such as temperature, concentration of reactants, and presence of a catalyst.
For example, if you have a chemical reaction where the reactants are gases like chlorine dioxide (\( \text{ClO}_2 \)) and ozone (\( \text{O}_3 \)), the rate at which they transform into the products can be quite different under various conditions.

In the exercise scenario, the initial concentrations of all reactants are the same for each reaction. Hence, the speed of each reaction depends heavily on the rate constant associated with each reaction, represented by \( k \). A higher rate constant indicates a faster reaction under identical conditions.

Rate Law

The rate law provides an expression that relates the rate of a reaction to the concentration of the reactants. It is specific to each chemical reaction and needs to be determined experimentally.

• For example, the generalized rate law for a reaction between two reactants \( A \) and \( B \) can be expressed as: \[ Rate = k[A]^x[B]^y \] where \( x \) and \( y \) are the orders of the reaction relative to the reactants \( A \) and \( B \), respectively.

The coefficients \( x \) and \( y \) in the rate law are not always the same as the stoichiometric coefficients of the balanced chemical equation.
In cases such as the one given in the exercise, knowing the rate law helps predict which reaction will be fastest when the initial reactant concentrations are the same by comparing their rate constants (\( k \)).

Reaction Order

Reaction order is a part of the rate law that provides critical insights into the reactant concentration dependency.

• For a reaction \( aA + bB \rightarrow \text{Product(s)} \), if the rate law is \[ ext{Rate} = k[A]^m[B]^n \] then the reaction order with respect to \( A \) is \( m \), and with respect to \( B \) is \( n \). The overall reaction order is the sum \( m + n \).

In the exercise context, each reaction given is first-order in its respective reactants and second-order overall, meaning that the concentration changes of the reactants directly affect the reaction rate. Understanding this concept is key as it allows students to see how changes in concentration might influence the speed of the reaction depending on its order.

Rate Constant

The rate constant, denoted typically as \( k \), is a crucial factor in chemical kinetics. It ties into the rate law and indicates how fast a reaction proceeds when the reactant concentrations are held constant.

• The rate constant has different units depending on the overall reaction order. For a second-order reaction, like those in the exercise, the unit is usually \( \text{cm}^3/(\text{molecule} \, \text{s}) \).

It essentially encapsulates all the variables affecting the reaction speed that are not related to concentration, such as temperature and catalysts present.
By simply comparing the \( k \) values of different reactions with identical reactant concentrations, as in the exercise, one can determine which reaction will occur more quickly since higher \( k \) indicates a faster reaction under the same conditions.