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 \(4.8 \times 10^{3} \mathrm{M}^{-1} \mathrm{~s}^{-1}\). (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 \(\Delta H_{\text {f }}{ }^{\circ}\) values from Appendix \(\mathrm{C}\) to estimate the enthalpy change for this reaction. Would this reaction raise or lower the temperature of the stratosphere?

Short answer

The rate law for this second-order reaction is rate = k[O][O₃]. It is likely that this reaction occurs via a single elementary process, but we cannot be certain without more information. The enthalpy change (∆H) for the reaction is -391.9 kJ/mol, indicating an exothermic reaction that would raise the temperature of the stratosphere.

Step-by-step solution

Determine the order of the reaction

To determine the rate law for this reaction, we need to find the order of the reaction. We can do this by looking at the units of the rate constant. The rate constant has units of M⁻¹ s⁻¹, which indicates a second-order reaction. The general form of a second-order reaction is: rate = k[A]^m[B]^n, where m + n = 2 Since in our reaction we have O(g) and O₃(g) as reactants, we will write the rate law: rate = k[O][O₃] 2. Determine if the reaction occurs via a single elementary process

Reasons why this reaction might not occur via a single elementary process

We cannot be entirely certain if this reaction occurs via a single elementary process, but it is likely since the given reaction involves a direct reaction between O(g) and O₃(g). However, we would need more information (such as experimental data or a reaction mechanism) to conclusively determine if this reaction involves a single elementary process or multiple intermediate steps. 3. Estimate the enthalpy change for this reaction

Calculate the enthalpy change for this reaction

We can estimate the enthalpy change (∆H) for this reaction using the standard heats of formation (∆H_f°) of the reactants and products. The general formula for calculating the enthalpy change is: ∆H = Σ np∆H_f°(products) - Σ nr∆H_f°(reactants) Using the Appendix C values, we have: ∆H_f°(O₃) = 142.7 kJ/mol ∆H_f°(O₂) = 0 kJ/mol (reference compound) ∆H_f°(O) = 249.2 kJ/mol We can now substitute these values into the enthalpy change formula: ∆H = (2 x 0) - (1 x 249.2 + 1 x 142.7) ∆H = -391.9 kJ/mol The negative value for ∆H indicates that this reaction is exothermic and would release heat to the surroundings. Therefore, this reaction would raise the temperature of the stratosphere.

Key concepts

Second-order reaction

A second-order reaction is characterized by a rate constant that has units of \( \text{M}^{-1} \text{s}^{-1} \), which is exactly what we observe in the reaction of oxygen atoms with ozone that contributes to ozone depletion. This type of reaction involves two reactant molecules, which are represented as \([A]^m[B]^n\) in the rate law formula. Here, \(m + n = 2\), indicating that the overall reaction is second-order.

In the given depletion reaction, the reactants are \( \text{O}(g) \) and \( \text{O}_3(g) \). The rate law can therefore be written as:

• \(\text{rate} = k[\text{O}][\text{O}_3]\)

This implies that the rate of reaction depends on the concentration of both oxygen and ozone molecules. Such a reaction is typical in cases involving bimolecular interactions, where the likelihood of collision between the two molecules determines the speed of the reaction.

Rate law

The rate law is a mathematical expression that links the rate of a chemical reaction to the concentration of reactants. For our ozone depletion reaction, the rate law is determined from the reaction order and the units of the rate constant.

The order of a second-order reaction is illustrated by the rate law equation:

• \(\text{rate} = k[A]^m[B]^n\)

This reveals how the rate depends on the concentration of reactants \([A]\) and \([B]\), where \(m + n\) equals the overall order of the reaction.

The rate law sheds light on how variations in the concentrations of \( \text{O} \) and \( \text{O}_3 \) determine the overall reaction speed. Understanding the rate law is crucial for controlling reaction conditions aptly, especially when dealing with delicate balances like those affecting atmospheric gases like ozone.

Reaction enthalpy

Reaction enthalpy (\(\Delta H\)) is an important measure in assessing whether a chemical reaction will absorb or release energy, hence affecting the temperature of the surroundings. To compute this value, we use the formula:

• \[\Delta H = \sum n_p \Delta H_f^\circ(\text{products}) - \sum n_r \Delta H_f^\circ(\text{reactants})\]

In the context of our ozone reaction, the enthalpy change shows the transfer of energy:

• 
  \[ \Delta H = (2 \times 0) - (1 \times 249.2 + 1 \times 142.7) = -391.9 \text{kJ/mol}\]

The resulting negative value communicates that the reaction is exothermic, meaning it releases heat. This can have further implications on the environment, such as increasing the temperature of the stratosphere, contributing to climatic changes.

Elementary process

An elementary process, or elementary step, refers to a reaction that occurs in a single stage, with no intermediate steps. In our examined reaction, which involves oxygen and ozone, the process does occur via a likely elementary process. This conclusion stems from how straightforward the interaction is between these two reactants.

Nonetheless, to be absolutely sure about whether the reaction occurs in a single elementary process, additional experimental data or computational insights would typically be required. These would reveal if intermediate structures might form temporarily, thus breaking the reaction into more than one step.

• Understanding whether a reaction is elementary helps in simplifying kinetic studies.

The basic nature of an elementary reaction offers cleaner models of reaction mechanisms, enhancing both predictability and educational clarity in chemical kinetics.