Problem 86 - Chemistry: The Central Science in SI Units, Global Edition

Question

A reaction that contributes to the depletion of ozone in the stratosphere is the direct reaction of oxygen atoms with ozone: $$ \mathrm{O}(g)+\mathrm{O}_{3}(g) \longrightarrow 2 \mathrm{O}_{2}(g) $$ At \(298 \mathrm{~K}\) the rate constant for this reaction is \(4.8 \times 10^{5}\) \(M^{-1} \mathrm{~s}^{-1}\). (a) Based on the units of the rate constant, write the likely rate law for this reaction. (b) Would you expect this reaction to occur via a single elementary process? Explain why or why not. (c) Use \(\Delta H_{f}^{\circ}\) values from Appendix \(\mathrm{C}\) to estimate the enthalpy change for this reaction. Would this reaction raise or lower the temperature of the stratosphere?

Step-by-Step Solution

Answer

(a) The rate law for this reaction is \(Rate = k[\mathrm{O(g)}][\mathrm{O}_{3}(g)]\) based on the rate constant's units of \(M^{-1}s^{-1}\), indicating a second-order reaction. (b) This reaction likely occurs via a single elementary process because its order equals its molecularity (number of molecules directly involved). (c) The enthalpy change for this reaction is \(-391.9 \mathrm{kJ/mol}\), an exothermic reaction that would raise the temperature of the stratosphere.

Step 1: (a) Determine the rate law of the reaction given the units of the rate constant

The rate constant for the reaction is given as \(4.8 \times 10^5 M^{-1}s^{-1}\). The units of the rate constant can help us determine the order of the reaction. Since the units are \(M^{-1}s^{-1}\), it suggests that the reaction is second order, i.e., the overall order is 2, with one reactant molecule participating in the reaction. The rate law for this reaction can be written as: $$ Rate = k[\mathrm{O(g)}][\mathrm{O}_{3}(g)] $$ Where \(k\) is the rate constant and \([\mathrm{O(g)}]\) and \([\mathrm{O}_{3}(g)]\) are the concentrations of the reactants.

Step 2: (b) Determine if the reaction occurs via a single elementary process

The reaction has a rate law of order 2. An elementary process has a reaction order equal to its molecularity (the number of molecules directly involved in the elementary step). In this reaction, there are two molecules directly involved (O and O3). As the reaction has an order equal to its molecularity, it is an elementary process. Therefore, this reaction likely occurs via a single elementary process.

Step 3: (c) Calculate the enthalpy change for the reaction

To calculate the enthalpy change for the reaction, we need to use the given ΔH°f values from Appendix C. We can calculate the enthalpy change for the reaction (ΔH°rxn) by taking the sum of the enthalpies of formation of the products and subtracting the sum of the enthalpies of formation of the reactants: $$ \Delta H_{rxn}^{\circ} = \Sigma \Delta H_{f, products}^{\circ} - \Sigma \Delta H_{f, reactants}^{\circ} $$ From Appendix C, we get the following values: ΔH°f (O(g)) = 249.2 kJ/mol ΔH°f (O3(g)) = 142.7 kJ/mol ΔH°f (O2(g)) = 0 kJ/mol Now we can calculate the enthalpy change: $$ \Delta H_{rxn}^{\circ} = (2 \times \Delta H_{f, O_{2}(g)}^{\circ}) - (\Delta H_{f, O_{3}(g)}^{\circ} + \Delta H_{f, O(g)}^{\circ}) $$ $$ \Delta H_{rxn}^{\circ} = (2 \times 0) - (142.7 + 249.2) = -391.9 \mathrm{kJ/mol} $$ Since the enthalpy change is negative, the reaction is exothermic, meaning it releases heat. As a result, this reaction would raise the temperature of the stratosphere.

Key Concepts

Reaction Rate Law

In chemical kinetics, the reaction rate law describes how the rate of a chemical reaction depends on the concentration of its reactants. For the ozone-depleting reaction \[\mathrm{O}(g) + \mathrm{O}_{3}(g) \longrightarrow 2 \mathrm{O}_{2}(g),\]the rate law is an expression that details how the concentration of \(\mathrm{O}(g)\) and \(\mathrm{O}_{3}(g)\) influence the reaction rate.

Given the rate constant unit \(M^{-1} s^{-1}\), the reaction is determined to be second order. This means the rate of reaction is proportional to the product of the concentrations of \(\mathrm{O}(g)\) and \(\mathrm{O}_{3}(g)\). The rate law expression can be written as:

• \(Rate = k[\mathrm{O(g)}][\mathrm{O}_{3}(g)],\)
• where \(k\) is the rate constant.

Understanding the order of the reaction in terms of reactants' concentrations is crucial in predicting and controlling reaction behavior. Second-order rate laws indicate that the reaction rate doubles when the concentration of either reactant is doubled.

Enthalpy Change Calculation

Enthalpy change, denoted as \(\Delta H\), is a key thermodynamic quantity representing heat absorbed or released during a chemical reaction. To calculate the enthalpy change for the ozone reaction, use the enthalpies of formation (\(\Delta H_f^{\circ}\)) data, which is available in standard reference tables like Appendix C.

The enthalpy change for the reaction is calculated using the formula:\[\Delta H_{rxn}^{\circ} = \Sigma \Delta H_{f, \text{products}}^{\circ} - \Sigma \Delta H_{f, \text{reactants}}^{\circ}.\]From the given data, the enthalpy of formation values are:

• \(\Delta H^\circ_f\) for \(\mathrm{O}(g)\) is 249.2 kJ/mol
• \(\Delta H^\circ_f\) for \(\mathrm{O}_{3}(g)\) is 142.7 kJ/mol
• \(\Delta H^\circ_f\) for \(\mathrm{O_{2}}(g)\) is 0 kJ/mol

The overall enthalpy change becomes:\[\Delta H_{rxn}^{\circ} = (2 \times 0) - (142.7 + 249.2) = -391.9 \text{kJ/mol}.\]This negative value indicates that the reaction is exothermic, releasing heat into the stratosphere. Exothermic reactions contribute to increased temperatures in their environment, such as the stratosphere in this example.

Elementary Reaction Processes

Elementary reaction processes are reactions that occur in one step and match their molecularity to the reaction order. In other words, the molecules involved in the reaction directly interact in a single event, rather than via a complex sequence of steps.

In the context of the reaction for ozone depletion:\[\mathrm{O}(g) + \mathrm{O}_{3}(g) \longrightarrow 2 \mathrm{O}_{2}(g),\]there are two reactant molecules of oxygen involved. Since the study indicates a reaction order based on these two molecules, the reaction likely occurs as an elementary process.

This one-step mechanism implies simplicity in the reaction pathway, contributing to more effective rate prediction and control. Elementary reactions are crucial in understanding how complex reactions can be dissected into simpler components. By understanding these core processes, scientists can better predict the behavior of larger, more complex reactions and improve atmospheric modeling to assess phenomena like ozone depletion.