Question

Ozone is a reactive gas that has been associated with respiratory illness and decreased lung function. The following reactions are involved in ozone formation [D. Allen and D. Shonnard, Green Engineering (Upper Saddle River, N.J.: Prentice Hall, 2002)]. \(\mathrm{NO}_{2}\) is primarily generated by combustion in the automobile engine. (a) Show that the steady-state concentration of ozone is directly proportional to \(\mathrm{NO}_{2}\) and inversely proportional to NO. (b) Drive an equation for the concentration of ozone in solely in terms of the initial concentrations \(C_{\mathrm{NO}, 0}, C_{\mathrm{VO}, 0},\) and \(\mathrm{O}_{3,0}\) and the rate law parameters. (c) In the absence of \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\), the rate law for ozone generation is $$-r_{0,3}=\frac{k\left(\mathrm{O}_{2}\right)(\mathrm{M})}{\left(\mathrm{O}_{2}\right)(\mathrm{M})+k^{\prime}\left(\mathrm{O}_{3}\right)}$$ Suggest a mechanism. (d) List ways you can work this problem incorrectly. (e) How could you make this problem more difficult?

Short answer

a) Ozone concentration is directly proportional to NO₂ concentration and inversely proportional to NO concentration: \(O_3 = \frac{k_2 [NO_2]}{k_1 [NO]}\). b) The equation for ozone concentration in terms of initial concentrations and rate law parameters is: \[C_{O_3,ss}=\frac{k_2}{k_1}\left[C_{NO_2,0}+(C_{NO,0} - \xi_1) - \left(C_{O_{3,0}} + \xi_1 - \xi_2\right)\right]- \frac{C_{O_{3,0}}}{C_{NO,0}}\] c) A possible mechanism in the absence of NO and NO₂ involves the dissociation of molecular oxygen (O₂) and formation and dissociation of ozone (O₃) with the presence of a third body (M). d) Some ways to work this problem incorrectly include not considering steady-state conditions, not accounting for initial concentrations or rate law parameters, and not applying the correct mechanism in part (c). e) To make this problem more difficult, introduce reaction intermediates, temperature dependence, catalysts effects, non-steady-state behaviors, or other pollutants interacting with NO and NO₂.

Step-by-step solution

a) Proportionality of Ozone Concentration to NO₂ and NO

Begin by writing the reactions involved in ozone formation: Reaction 1: \(NO + O_3 \rightarrow NO_2 + O_2\) (Ozone reacts with nitric oxide) Reaction 2: \(NO_2 + O_3 \rightarrow NO + 2O_2\) (Nitrogen dioxide reacts with ozone) At steady state: \(r_1 = k_1 [NO][O_3] = r_2 = k_2 [NO_2][O_3]\) To find the proportionality we need to get Ozone concentration in terms of NO and NO₂ concentrations: \(O_3 = \frac{k_2 [NO_2]}{k_1 [NO]}\) The concentration of ozone is directly proportional to the concentration of NO₂ and inversely proportional to the concentration of NO.

b) Ozone Concentration in Terms of Initial Concentrations and Rate Law Parameters

Initially, we can write the rate laws for each reaction: Rate law for Reaction 1: \(r_1 = k_1 [NO][O_3]\) Rate law for Reaction 2: \(r_2 = k_2 [NO_2][O_3]\) At the steady state, the rate of each reaction is equal, so \(k_1 [NO][O_3] = k_2 [NO_2][O_3]\) Cancel [O₃] from both sides: \([NO]k_1=[NO_2]k_2\) Now find Ozone concentration when the reaction reaches steady-state using materials balance \(C_{O_3} = C_{O_{3,0}} + \xi_1 - \xi_2\) Where \(\xi_1\) and \(\xi_2\) are the extents of reactions 1 and 2 Divide both sides by [NO] \(C_{O_3}=\frac{k_2}{k_1}[NO_2]- \frac{C_{O_{3,0}}}{C_{NO,0}}\) At steady-state \([NO]_{ss} = C_{NO,0} - \xi_1\), \([NO_2]_{ss}=C_{NO_2,0}+\xi_1-\xi_2\) So \[C_{O_3,ss}=\frac{k_2}{k_1}\left[C_{NO_2,0}+(C_{NO,0} - \xi_1) - \left(C_{O_{3,0}} + \xi_1 - \xi_2\right)\right]- \frac{C_{O_{3,0}}}{C_{NO,0}}\] This equation represents the concentration of ozone in terms of initial concentrations and rate law parameters.

c) Mechanism in the Absence of NO and NO₂

The given rate law for ozone generation in the absence of NO and NO₂ is: \(-r_{0,3}=\frac{k\left(\mathrm{O}_{2}\right)(\mathrm{M})}{\left(\mathrm{O}_{2}\right)(\mathrm{M})+k^{\prime}\left(\mathrm{O}_{3}\right)}\) A possible mechanism can involve the following steps: Step 1: \(O_2 + M \rightarrow 2O + M\) (Dissociation of molecular oxygen) Step 2: \(O + O_2 + M \rightarrow O_3 + M\) (Formation of ozone) Step 3: \(O_3 + M \rightarrow O_2 + O + M\) (Dissociation of ozone) These steps can account for the given rate law and exhibit an autocatalytic behavior.

d) Ways to Work This Problem Incorrectly

1. Not considering the steady-state condition or the initial concentrations in deriving the ozone concentration. 2. Assuming that the rate of each reaction scales linearly with concentration. 3. Ignoring the rate law parameters in the analysis. 4. Not considering the absence of NO and NO₂ in part (c) and trying to apply the same mechanism used in the previous parts.

e) Making the Problem More Difficult

1. Introducing reaction intermediates or another set of reactions to the mechanism, requiring the analysis of the mechanism to be more complex. 2. Adding temperature dependence to the rate law parameters. 3. Exploring the effects of different types of catalysts on the rate of reaction and ozone concentration. 4. Considering non-steady-state behaviors and time-dependent concentration changes for the components involved. 5. Investigating the role of other pollutants in ozone formation and how they interact with NO and NO₂.

Key concepts

Ozone Formation

Ozone formation is a critical topic in chemical reaction engineering, especially when considering its impact on the environment and human health. Ozone (O₃) is a reactive gas that is primarily known for its protective role in the stratosphere. However, at the ground level, ozone becomes a pollutant. Understanding how ozone forms involves examining the chemical reactions that contribute to its creation.

The initial reaction involves nitric oxide (NO) reacting with ozone to form nitrogen dioxide (NO₂) and oxygen (O₂):

• NO + O₃ → NO₂ + O₂

Subsequently, NO₂ can react with ozone in the presence of sunlight, creating more oxygen. This cycling of NO, NO₂, and O₃ is part of what is known as the photochemical smog.

Studying ozone formation not only helps in understanding air pollution but also guides engineers in designing processes that minimize the environmental impacts of industrial emissions.

Steady-State Concentration

The concept of steady-state concentration is crucial in reaction engineering. In simple terms, a steady state occurs when the rate of input and output in a system balances out, making the concentration of reactants and products constant over time. This equilibrium is important for understanding how long a system can sustain production at a given rate without additional changes.

For ozone, the steady-state concentration can be studied through the balance of the involved reactions. Consider the reactions:

• NO + O₃ → NO₂ + O₂ and
• NO₂ + O₃ → NO + 2O₂.

In the steady state, the concentration of ozone is directly proportional to the concentration of NO₂ and inversely proportional to the concentration of NO. This means that for a stable level of ozone to be achieved, these substances must react at equal rates.

This understanding allows us to derive mathematical expressions that predict ozone concentrations based on current levels of NO and NO₂, thereby assisting in the control of air quality and pollution.

Rate Law Parameters

Rate law parameters are essential for quantifying how quickly chemical reactions occur. These parameters include the rate constants, which are specific for each reaction, and they dictate how concentrations of substances affect the reaction rate.

In the context of ozone formation, the rate laws for the reactions are written as:

• For NO + O₃: \( r_1 = k_1 [NO][O_3] \),
• And for NO₂ + O₃: \( r_2 = k_2 [NO_2][O_3] \).

Where \( k_1 \) and \( k_2 \) are the rate constants for reactions involving NO and NO₂, respectively.

The relationship between these parameters not only helps in predicting the rates at which ozone will form but also influences the decisions made in engineering processes to optimize them for minimal environmental impact. Properly understanding rate law parameters is vital for designing effective environmental protection measures.

Reaction Mechanisms

A reaction mechanism is a step-by-step sequence of elementary reactions by which an overall chemical change occurs. Understanding the precise mechanism of a reaction helps us comprehend the intricacies of how substances transform and allows us to design better catalytic systems to control unwanted reactions.

For ozone formation, understanding the steps involved helps in grasping how NO and NO₂ function within the overall photochemical cycle. Without NO and NO₂, ozone generation occurs via alternative mechanisms. Consider the following steps:

• O₂ + M → 2O + M (oxygen dissociates)
• O + O₂ + M → O₃ + M (ozone forms)
• O₃ + M → O₂ + O + M (ozone dissociates)

These reactions highlight an autocatalytic behavior where molecules act as catalysts for their own reaction.

Understanding reaction mechanisms like these inform the development of models that predict how changes in conditions, like the absence of NO or NO₂, affect ozone levels. It can help in designing strategies to manage pollution levels more effectively.