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) From the magnitude of the rate constant, would you expect the activation energy of this reaction to be large or small? Explain. (d) Use \(\Delta H^{\circ}\), values from Appendix \(C\) 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 reaction is \(\text{Rate} = k [\mathrm{O}] [\mathrm{O_3}]\). It likely occurs via a single elementary process because the coefficients of the reactants in the balanced reaction equation are equal to the exponents in the rate law. The activation energy is relatively small, as the large rate constant indicates a fast reaction. The enthalpy change for this reaction is \(-142.7 \, \text{kJ/mol}\), indicating an exothermic reaction that raises the temperature of the stratosphere.

Step-by-step solution

Understand the reaction and given rate constant

The reaction is given as: \[ \mathrm{O}(g) + \mathrm{O}_3(g) \longrightarrow 2 \mathrm{O}_2(g) \] The rate constant for this reaction is given as \(k = 4.8 \times 10^5 M^{-1}s^{-1}\).

Write the rate law based on the units of the rate constant

The units of the rate constant are \(M^{-1}s^{-1}\), which indicates a second-order reaction. In a second-order reaction, the rate of reaction is proportional to the product of the concentrations of the reactants. The rate law for this reaction can be written as: \[ \text{Rate} = k [\mathrm{O}] [\mathrm{O_3}] \] b) Single elementary process

Observe the coefficients of the reactants

In the given reaction: \[ \mathrm{O}(g) + \mathrm{O}_3(g) \longrightarrow 2 \mathrm{O}_2(g) \] The stoichiometric coefficients are 1 for both reactants, oxygen atoms, and ozone molecules.

Determine if the reaction occurs via a single elementary process

Since the coefficients of the reactants in the balanced reaction equation are equal to the exponents in the rate law, this reaction likely occurs via a single elementary process. c) Activation energy

Analyze the magnitude of the rate constant

The rate constant of this reaction is \(k = 4.8 \times 10^5 M^{-1}s^{-1}\). This value is quite large, indicating that the reaction is fast.

Determine the activation energy based on the rate constant

Since the reaction is fast, we can conclude that the activation energy for this reaction is relatively small. Fast reactions typically have smaller activation energy barriers, allowing more collisions between molecules to result in successful reactions. d) Enthalpy change and stratosphere temperature

Use appendix data to find the enthalpy change, ∆H°

From Appendix C, we get the standard enthalpy change of formation for each species involved: \[ \begin{aligned} \Delta H^{\circ}_{\text{f, O}} &= 0 \, \text{kJ/mol} \\ \Delta H^{\circ}_{\text{f, O}_3} &= 142.7\, \text{kJ/mol} \\ \Delta H^{\circ}_{\text{f, O}_2} &= 0\, \text{kJ/mol} \end{aligned} \]

Calculate the enthalpy change for the reaction

We can calculate the enthalpy change for the reaction as follows: \[ \begin{aligned} \Delta H^{\circ}_{\text{rxn}} &= \sum n_{\text{products}}\Delta H^{\circ}_{\text{f, products}} - \sum n_{\text{reactants}}\Delta H^{\circ}_{\text{f, reactants}} \\ &= 2\Delta H^{\circ}_{\text{f, O}_2} - (\Delta H^{\circ}_{\text{f, O}} + \Delta H^{\circ}_{\text{f, O}_3}) \\ &= 2(0) - (0 + 142.7) \\ &= -142.7 \, \text{kJ/mol} \end{aligned} \]

Determine the effect on the stratosphere temperature

Since the enthalpy change for this reaction is negative, the reaction is exothermic. An exothermic reaction releases heat, which would raise the temperature of the stratosphere.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates of chemical reactions. It's all about understanding how quickly reactions occur and the steps that take place during these reactions. For our stratospheric ozone-depleting reaction, the rate at which oxygen atoms and ozone molecules react to form oxygen molecules is essential in predicting the extent of ozone depletion over time. By analyzing the rate constant and the order of the reaction, we can infer how different conditions, like temperature or concentration of reactants, will impact the reaction rate.

For instance, a high rate constant suggests that the reaction can proceed quickly, contributing significantly to ozone depletion. This is why chemical kinetics is so crucial in environmental chemistry; it helps us predict and potentially control the rate of ozone depletion in the stratosphere.

Second-Order Reaction

A second-order reaction involves the rate of reaction being proportional to the square of the concentration of one reactant or to the product of the concentrations of two different reactants. In our case, the rate law \text{Rate} = k [\mathrm{O}] [\mathrm{O}_3]\results from the given rate constant with units of \(M^{-1}s^{-1}\). The rate law tells us that our ozone depletion reaction rate depends on the concentration of both oxygen atoms and ozone molecules. The simultaneous involvement of these two reactants is what classifies the reaction as second-order.

Second-order reactions can be represented by complex rate laws and often require more intricate integration techniques to solve for concentration as a function of time compared to first-order reactions, which have simpler exponential decay relationships.

Activation Energy

Activation energy is the minimum amount of energy that reacting molecules must possess for a reaction to occur. It's like the initial push needed to start a chemical process. When we talk about a reaction in the stratosphere that has a large rate constant, like \(4.8 \times 10^5 M^{-1}s^{-1}\), this suggests that the activation energy is likely low. This is because a larger number of the colliding molecules will have enough energy to surpass this lower energy barrier, leading to more frequent successful reactions.

Thus, the lower the activation energy, the faster the reaction rate, which in the context of stratospheric ozone depletion can mean a more rapid reduction in ozone levels if no other factors limit the reaction rate.

Enthalpy Change

Enthalpy change, denoted as \(\Delta H\), measures the total heat content during a reaction at constant pressure. This value can be positive or negative, indicating whether a reaction absorbs or releases heat, respectively. In the stratospheric reaction we're exploring, calculating the enthalpy change involves summing the enthalpies of the products and subtracting the enthalpies of the reactants. Since our calculation yields a negative value of -142.7 kJ/mol, this tells us the reaction is releasing heat into the surroundings.

Understanding enthalpy change is vital for predicting not just the direction of heat flow, but also the impact of reactions on their surroundings, which in our case is the temperature of the stratosphere.

Exothermic Reactions

Exothermic reactions are those that release heat, leading to an increase in the temperature of the environment. When we discover that the enthalpy change for the ozone depletion reaction is negative, as calculated to be -142.7 kJ/mol, we can confidently say it's exothermic. This means that as the reaction proceeds, more heat will be emitted into the stratosphere, potentially warming it.

This is an important consideration when studying environmental chemistry, especially in understanding the delicate balance within the Earth's atmosphere. The release of excess heat from such reactions can contribute to changes in atmospheric dynamics, affecting weather patterns and, in large enough quantities, could impact climate.