Question

The balance between \(\mathrm{SO}_{2}\) and \(\mathrm{SO}_{3}\) is important in understanding acid rain formation in the troposphere. From the following information at \(25^{\circ} \mathrm{C}\) : $$ \begin{aligned} \mathrm{S}(s)+\mathrm{O}_{2}(g) & \rightleftarrows \mathrm{SO}_{2}(g) & & K_{1}=4.2 \times 10^{52} \\ 2 \mathrm{~S}(s)+3 \mathrm{O}_{2}(g) & \rightleftarrows 2 \mathrm{SO}_{3}(g) & & K_{2}=9.8 \times 10^{128} \end{aligned} $$ calculate the equilibrium constant for the reaction: $$ 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightleftarrows 2 \mathrm{SO}_{3}(g) $$

Short answer

The equilibrium constant \( K_3 = 5.5 \times 10^{24} \).

Step-by-step solution

Write the Given Reactions and Constants

First, note down the given chemical reactions and their respective equilibrium constants. The reactions are:1. \( \mathrm{S}(s) + \mathrm{O}_{2}(g) \rightleftarrows \mathrm{SO}_{2}(g) \) with \( K_1 = 4.2 \times 10^{52} \).2. \( 2\mathrm{~S}(s) + 3\mathrm{O}_{2}(g) \rightleftarrows 2\mathrm{SO}_{3}(g) \) with \( K_2 = 9.8 \times 10^{128} \).

Identify the Target Reaction

The target reaction for which we need to find the equilibrium constant \( K_3 \) is:\[ 2\mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightleftarrows 2\mathrm{SO}_{3}(g) \]

Derive the Target Reaction Using Given Reactions

To derive the target reaction, observe that the target reaction can be formed by reversing the first reaction and combining it with the second reaction. Reverse the first reaction:\[ \mathrm{SO}_{2}(g) \rightleftarrows \mathrm{S}(s) + \mathrm{O}_{2}(g) \]This has an equilibrium constant \( K_1' = \frac{1}{K_1} = \frac{1}{4.2 \times 10^{52}} \).

Write the Combined Reaction

When you reverse the first reaction and add it to the second reaction, you should gain the following reaction:Adding \( 2 \times [\mathrm{SO}_{2}(g) \rightleftarrows \mathrm{S}(s) + \mathrm{O}_{2}(g)] \) and \( 2\mathrm{~S}(s)+3\mathrm{O}_{2}(g) \rightleftarrows 2\mathrm{SO}_{3}(g) \), we have:\[ 2\mathrm{SO}_{2}(g) + \mathrm{O}_{2}(g) \rightleftarrows 2\mathrm{SO}_{3}(g) \]

Calculate the Equilibrium Constant for the Derived Reaction

Since the target reaction is obtained by reversing the first one and then adding, the equilibrium constant \( K_3 \) for the target reaction is:\[ K_3 = \left(K_1'\right)^2 \times K_2 = \left(\frac{1}{4.2 \times 10^{52}}\right)^2 \times 9.8 \times 10^{128} \]And after calculating, we find:\[ K_3 = 5.5 \times 10^{24} \]

Key concepts

Acid Rain Formation

Acid rain is a type of precipitation with high levels of hydrogen ions, making it acidic. It is primarily caused by emissions of sulfur dioxide (\(\text{SO}_2\)) and nitrogen oxides which react with water molecules in the atmosphere to produce acids. Understanding the chemical reactions leading to acid rain is essential for environmental science.
One of the pivotal reactions involved in the creation of acid rain is the conversion of \(\text{SO}_2\) into \(\text{SO}_3\), which further reacts with water to form sulfuric acid (\(\text{H}_2\text{SO}_4\)). The presence of sulfuric acid in rainwater results in a lowered pH, characterizing what we know as acid rain. This is a significant environmental issue as it affects water bodies, soil, and wildlife.
To address acid rain, it's crucial to study these chemical reactions and their equilibrium constants, to control and minimize emissions effectively.

Sulfur Dioxide

Sulfur dioxide (\(\text{SO}_2\)) is a colorless gas with a pungent odor, produced by the combustion of fossil fuels and volcanic eruptions. It is a primary pollutant contributing to the formation of acid rain. When \(\text{SO}_2\) enters the atmosphere, it can undergo several reactions:

• It reacts with water to form sulfurous acid (\(\text{H}_2\text{SO}_3\)).
• It can be oxidized to sulfur trioxide (\(\text{SO}_3\)).

The latter reaction is critical because \(\text{SO}_3\) is more reactive towards water, producing sulfuric acid (\(\text{H}_2\text{SO}_4\)).
Managing \(\text{SO}_2\) emissions is vital in reducing acid rain. Industrial processes can employ technologies such as flue gas desulfurization to capture \(\text{SO}_2\) before it reaches the atmosphere. Understanding its behavior through equilibrium reactions aids in developing strategies for emission control.

Sulfur Trioxide

Sulfur trioxide (\(\text{SO}_3\)) plays a critical role in forming the acids that contribute to acid rain. It is formed from sulfur dioxide through oxidation, often aided by catalysts in the atmosphere or industrial processes. The conversion of \(\text{SO}_2\) to \(\text{SO}_3\) is a crucial step:
This reaction is represented as \(2\text{SO}_2 + \text{O}_2 \rightleftarrows 2\text{SO}_3\). The equilibrium constant \(K_3\) determined in the exercise indicates the favorability of \(\text{SO}_3\) formation under specific conditions.
\(\text{SO}_3\) readily reacts with water vapor to form sulfuric acid (\(\text{H}_2\text{SO}_4\)), a major component of acid rain. This reaction takes place as \(\text{SO}_3 + \text{H}_2\text{O} \rightarrow \text{H}_2\text{SO}_4\). Controlling the second reaction is just as significant as managing \(\text{SO}_2\) emissions, as it directly affects acid rain's impact on the environment.

Equilibrium Reactions

Equilibrium reactions are chemical reactions where the reactants and products are in a dynamic balance. This means the rate of the forward reaction is equal to the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.
The equilibrium constant, \(K\), reflects the abundance of products relative to reactants at equilibrium. A large \(K\) value indicates the products are favored at equilibrium, whereas a small \(K\) suggests reactants are favored. In the case of sulfur dioxide converting to sulfur trioxide, the equilibrium constant \(K_3 = 5.5 \times 10^{24}\) suggests that sulfur trioxide formation is highly favored, explaining why \(\text{SO}_3\) is prevalent in atmospheric reactions leading to acid rain.
Understanding these equilibrium processes is key to predictive modeling and environmental chemistry, allowing for better interventions in pollution management and control strategies.