Problem 107 - Chemistry: The Central Science

Question

The reaction $$ \mathrm{SO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{~S}(g) \rightleftharpoons 3 \mathrm{~S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g) $$ is the basis of a suggested method for removal of \(\mathrm{SO}_{2}\) from power-plant stack gases. The standard free energy of each substance is given in Appendix C. (a) What is the equilibrium constant for the reaction at \(298 \mathrm{~K} ?\) (b) In principle, is this reaction a feasible method of removing \(\mathrm{SO}_{2} ?\) (c) If \(P_{\mathrm{SO}_{2}}=P_{\mathrm{H}_{2} \mathrm{~s}}\) and the vapor pressure of water is 25 torr, calculate the equilibrium \(\mathrm{SO}_{2}\) pressure in the system at \(298 \mathrm{~K}\) (d) Would you expect the process to be more or less effective at higher temperatures?

Step-by-Step Solution

Answer

The equilibrium constant (K) for the reaction at 298 K can be found using the standard reaction Gibbs energy (\(Δ_rG^o\)). A feasible method for removing SO₂ will have K > 1, favoring the formation of products. Under given conditions, we calculate the equilibrium pressure of SO₂ using the equilibrium expression. The effectiveness of the process at higher temperatures depends on the exothermic/endothermic nature of the reaction and Le Chatelier's Principle.

Step 1: (Step 1: Find the standard reaction Gibbs energy)

(To find the equilibrium constant for the reaction, we need to calculate the standard reaction Gibbs energy (\(Δ_rG^o\)). This can be done using the given standard free energy values for each substance and applying the formula: \(Δ_rG^o = Σn_iG_i^o(products) - Σn_jG_j^o(reactants)\), where \(G^o\) is the standard free energy, and \(Δ_rG^o\) is the Gibbs energy change for the reaction.)

Step 2: (Step 2: Calculate the equilibrium constant)

(Next, we will find the equilibrium constant, K, for the reaction at 298 K. To do this, we will use the equation \(Δ_rG^o = -RT \ln K\), where R is the gas constant (8.314 J/mol K) and T is the temperature in Kelvin. This will help us determine the feasibility of the reaction for the removal of SO₂.)

Step 3: (Step 3: Assess reaction feasibility)

(Based on the value of K obtained in Step 2, we can determine whether the reaction is a feasible method of removing SO₂. If K is greater than 1, the reaction will favor the formation of products at 298 K.)

Step 4: (Step 4: Calculate equilibrium SO₂ pressure)

(Under the given conditions, where \(P_{SO₂} = P_{H₂S}\) and the vapor pressure of water is 25 torr, we will use the equilibrium constant expression to calculate the equilibrium pressure of SO₂ in the system at 298 K. The equilibrium expression can be written as: \(K = \frac{[S^3][H₂O^2]}{[SO₂][H₂S]^2}\))

Step 5: (Step 5: Elaborate on the higher temperature effect)

(After calculating the equilibrium SO₂ pressure for the given conditions, we need to discuss if the process would be more or less effective at higher temperatures. To do so, we will consider the exothermic/endothermic nature of the reaction and use Le Chatelier's Principle. If the reaction is exothermic, increasing the temperature will shift the equilibrium toward the reactants, making the process less effective. On the other hand, if the reaction is endothermic, increasing the temperature will shift the equilibrium toward the products, making the process more effective.) By following these steps, we will be able to find and discuss the equilibrium constant and the feasibility of the given reaction for removing SO₂, as well as the effect of temperature on the reaction.

Key Concepts

Standard Reaction Gibbs Energy

The standard reaction Gibbs energy, denoted as \( Δ_rG^o \), is a measure of the change in free energy during a reaction under standard conditions. For a reaction, it indicates whether it is spontaneous (negative \( Δ_rG^o \)) or not (positive \( Δ_rG^o \)). To find \( Δ_rG^o \), we use the formula: \[ Δ_rG^o = Σn_iG_i^o(products) - Σn_jG_j^o(reactants) \] where \( G^o \) represents the standard free energy of formation for each substance, \( n_i \) and \( n_j \) are the stoichiometric coefficients of the products and reactants, respectively.

• Calculate \( Δ_rG^o \) using the values from appendix data.
• Helps determine the direction of the reaction.
• Serves as a crucial parameter in calculating the equilibrium constant, \( K \).

Equilibrium Constant

The equilibrium constant, \( K \), provides insight into the position of equilibrium for a reaction at a given temperature. It is directly related to the standard reaction Gibbs energy with the equation: \[ Δ_rG^o = -RT \ln K \] where \( R \) is the universal gas constant (8.314 J/mol K), and \( T \) is the temperature in Kelvin. This equation shows that if \( Δ_rG^o \) is negative, \( K \) is greater than 1, indicating that products are favored. Conversely, if \( Δ_rG^o \) is positive, \( K \) is less than 1, suggesting reactants are favored.

• A large \( K \) implies the reaction goes largely to completion.
• Provides a quantitative measure of reactant and product concentrations at equilibrium.

Understandably, finding \( K \) allows us to predict how much product will be formed before reaching equilibrium.

Feasibility of Reaction

The feasibility of a reaction is determined by how favorable the formation of products is under given conditions. A fundamental factor to consider is the value of the equilibrium constant, \( K \).

• If \( K > 1 \), the equilibrium favors product formation, suggesting a feasible reaction.
• If \( K < 1 \), reactants are favored, making the reaction less feasible.
• Consider molecular and energetic aspects to assess practicality in real systems.

Furthermore, reactions spontaneous under standard conditions (negative \( Δ_rG^o \)) can be implemented effectively. Feasibility not only depends on \( K \) but also on practical considerations like safety, cost, and available technology. Thus, while a theoretical model may show feasibility, practical experiments and evaluations are essential for real-world applications.

Temperature Effects on Equilibrium

Temperature can significantly impact the position of equilibrium in a chemical reaction. According to Le Chatelier's Principle, if an external condition like temperature changes, the system will adjust itself to counteract that change. Consider the thermal nature of the reaction:

• **Exothermic reactions**: Heat is released. Increasing temperature causes a shift toward reactants, reducing product yield. Thus, higher temperatures make such processes less effective.
• **Endothermic reactions**: Heat is absorbed. Increasing temperature shifts equilibrium toward products, enhancing reaction effectiveness.

Determining whether a reaction is exothermic or endothermic helps predict temperature effects on its efficiency. This understanding aids in optimizing conditions to maintain or improve reaction performance for desired outcomes.