Question

As an EPA scientist studying catalytic converters and urban smog, you want to find \(K_{c}\) for the following reaction: $$ 2 \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{N}_{2}(g)+2 \mathrm{O}_{2}(g) \quad K_{\mathrm{c}}=? $$ Use the following data to find the unknown \(K_{\mathrm{c}}\) : $$ \begin{aligned} \frac{1}{2} \mathrm{~N}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) & \rightleftharpoons \mathrm{NO}(g) & & K_{\mathrm{c}}=4.8 \times 10^{-10} \\ 2 \mathrm{NO}_{2}(g) & \Longrightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) & & K_{\mathrm{c}}=1.1 \times 10^{-5} \end{aligned} $$

Short answer

The equilibrium constant, \( K_{c} \), is \( 4.77 \times 10^{13} \)

Step-by-step solution

- Analyze Given Reactions

Identify the given partial reactions and their equilibrium constants.

- Reverse the Direction of the First Reaction and Adjust the Kc Value

Reverse: \( \text{NO} (g) \rightleftharpoons \frac{1}{2} \text{N}_2 (g) + \frac{1}{2} \text{O}_2 (g) \) which yields \( K_{\text{c}} = \frac{1}{4.8 \times 10^{-10}} = 2.083 \times 10^9 \)

- Multiply the First Reaction by 2 and Adjust the Kc Value Again

This gives \( 2 \text{NO} (g) \rightleftharpoons \text{N}_2 (g) + \text{O}_2 (g) \) and \( K_{\text{c}} = (2.083 \times 10^9) ^ 2 = 4.34 \times 10^{18} \)

- Combine Adjusted Reactions

Combine with \( 2 \text{NO}_2 (g) \rightleftharpoons 2 \text{NO} (g) + \text{O}_2 (g) \) and add Kc values by multiplying: \( K_{\text{c-final}} = (1.1 \times 10^{-5} ) \times (4.34 \times 10^{18}) = 4.774 \times 10^{13} \)

- Write the Final Equilibrium Constant Expression

The final expression for \( K_{c} \) is \( 4.77 \times 10^{13} \)

Key concepts

chemistry equilibrium

Understanding chemistry equilibrium is essential to solving problems related to reaction constants. Equilibrium occurs when the rates of the forward and reverse reactions are equal, meaning the concentrations of the reactants and products remain unchanged over time. In the given exercise, the equilibrium constant, or \(K_{c}\), measures the ratio of product to reactant concentrations at equilibrium for the reaction. When calculating \(K_{c}\), we must adjust the equilibrium constants for individual reactions to reflect their contributions to the overall reaction.
Keep in mind the following important points about chemical equilibrium:

• It is dynamic, meaning reactions continue to occur but at equal rates.
• It can be disturbed by changes in concentration, temperature, or pressure, as described by Le Chatelier's Principle.
• The equilibrium constant is temperature-dependent, so it changes with temperature.

By reversing, multiplying, and combining individual reactions as outlined in the problem solution, we can find the equilibrium constant for complex reactions.

catalytic converters

Catalytic converters play a vital role in reducing harmful emissions from vehicle exhausts. They facilitate chemical reactions that convert pollutants into less harmful substances before releasing them into the atmosphere. In catalytic converters, reactions often reach equilibrium, thus having a crucial role in pollution control.
Key functions of catalytic converters include:

• Converting carbon monoxide (CO) into carbon dioxide (CO₂).
• Transforming hydrocarbons (HC) into water (H₂O) and CO₂.
• Reducing nitrogen oxides (NO_x) into nitrogen (N₂) and oxygen (O₂).

The given exercise involves the reduction of nitrogen dioxide, NO₂, which is a crucial part of how catalytic converters mitigate urban smog. By understanding the equilibrium constants and the principles behind these reactions, we can better appreciate how catalytic converters contribute to cleaner air.

urban smog

Urban smog, or photochemical smog, is a type of air pollution that results mainly from vehicle emissions and industrial activities. It is characterized by the presence of ozone (O₃), nitric oxide (NO), and nitrogen dioxide (NO₂). Chemical reactions in the atmosphere, involving sunlight, lead to the formation of smog.
Factors contributing to urban smog include:

• High levels of volatile organic compounds (VOCs) and NO_x from car exhausts.
• Sunlight, which drives the photochemical reactions forming ozone.
• Stagnant air conditions, which prevent the dispersion of pollutants.

The provided exercise illustrates a reaction important for understanding how NO₂ contributes to smog formation. By mastering equilibrium constant calculations, students can better grasp how different pollutants interact and persist in the environment. This knowledge is crucial for developing strategies to combat urban smog and improve air quality in cities.