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 \(\mathrm{P}_{\mathrm{so}_{2}}=\mathrm{P}_{\mathrm{A}_{2} \mathrm{~s}}\) 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?

Short answer

The short answer based on the given step-by-step solution to the question is as follows: 1. Calculate the standard Gibbs free energy change (∆G°) of the reaction: ∆G° = [3G°(S) + 2G°(H2O)] - [G°(SO2) + 2G°(H2S)] 2. Calculate the equilibrium constant (K) at 298K: K = exp(-∆G°/(8.314 J/(mol·K) × 298 K)) 3. Analyze the feasibility of the reaction based on the value of K. If K >> 1, the reaction will be feasible for removing SO2. 4. Calculate the equilibrium SO2 pressure using the given partial pressures and the equilibrium constant: K = (P(S)^3 × 25^2) / P(SO2)^3 5. Determine the effect of temperature on the process by analyzing the sign of the standard enthalpy change (∆H°). If ∆H° > 0, the process will be more effective at higher temperatures; if ∆H° < 0, the process will be less effective at higher temperatures.

Step-by-step solution

1. Calculating the standard Gibbs free energy change (∆G°)

: From the given exercise, we are provided with the following reaction: \[\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{S}(g) \rightleftharpoons 3 \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)\] Thus, we can calculate the standard Gibbs free energy change (∆G°) with the following formula: \[\Delta G^\circ = \Sigma n_i G_f^\circ (products) - \Sigma n_j G_f^\circ (reactants)\] Using the standard Gibbs free energy values provided in Appendix C: ∆G° = [3G°(S) + 2G°(H2O)] - [G°(SO2) + 2G°(H2S)]

2. Calculating the equilibrium constant (K)

: We can use the following equation to calculate the equilibrium constant K: \[\Delta G^\circ = -RT\ln{K}\] At 298 K, this equation becomes: K = exp(-∆G°/(8.314 J/(mol·K) × 298 K))

3. Analyzing the feasibility of the reaction

: Based on the value of K from step 2, we can analyze the feasibility of the reaction. If K >> 1, then the reaction will lie heavily towards the products, making it a feasible method of removing SO2.

4. Calculating the equilibrium SO2 pressure

: Given that P(SO2) = P(H2S) and the vapor pressure of water is 25 torr, we can use the equilibrium constant to calculate the equilibrium SO2 pressure. From the reaction's stoichiometry: K = (P(S)^3 × P(H2O)^2) / (P(SO2) × P(H2S)^2) Since P(SO2) = P(H2S): K = (P(S)^3 × 25^2) / P(SO2)^3 Solving for P(SO2), we can find the equilibrium SO2 pressure.

5. Determining the effect of temperature on the process

: We can analyze the effect of temperature on the process. In general, higher temperatures favor endothermic reactions, while lower temperatures favor exothermic reactions. To check this for our specific reaction, we can look at the sign of the standard enthalpy change (∆H°). If ∆H° > 0, the reaction is endothermic, and then the process will be more effective at higher temperatures. If ∆H° < 0, the reaction is exothermic, and the process will be less effective at higher temperatures.

Key concepts

Gibbs Free Energy

Understanding Gibbs free energy is crucial for predicting the spontaneity of chemical reactions. It is a thermodynamic quantity represented as \( \Delta G \), and it combines enthalpy, entropy, and temperature to provide useful information about the feasibility of processes. The formula \( \Delta G = \Delta H - T\Delta S \) illustrates the relationship between these variables, where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy. For a reaction at constant temperature and pressure, a negative value of \( \Delta G \) indicates that the reaction can occur spontaneously.

In the context of the textbook problem, calculating the standard Gibbs free energy change (\( \Delta G^\circ \) provides information about the equilibrium state and direction of the reaction. By considering the standard free energies of formation for reactants and products, students can determine whether the reaction is thermodynamically favorable under standard conditions.

Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a dimensionless value that indicates the extent to which a reaction proceeds before reaching a state of equilibrium. It is calculated from the ratio of the concentrations or partial pressures of the products to the reactants, each raised to the power of their stoichiometric coefficients. The relationship between Gibbs free energy and the equilibrium constant is given by the equation \( \Delta G^\circ = -RT\ln{K} \), where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

For the SO2 removal process, calculating \( K \) from the standard Gibbs free energy change gives an indication of whether the reaction favors the formation of the products. A large \( K \) value suggests that the reaction is likely to proceed in the forward direction under equilibrium conditions, which is beneficial for the removal of SO2.

SO2 Removal

The environmental impact of sulfur dioxide (\( SO_2 \) is a significant concern, and its removal from power-plant stack gases is essential to reduce air pollution. One such method is through a chemical reaction that converts \( SO_2 \) to solid sulfur and water vapor. The feasibility of this reaction for \( SO_2 \) removal depends on the equilibrium constant and the reaction conditions, such as temperature and pressure.

As seen in the exercise, the equilibrium pressure of \( SO_2 \) can be calculated under specific conditions, providing insight into the effectiveness of the removal process. The practicality of this method can be deduced from the calculated equilibrium constant and the theoretical equilibrium pressure of \( SO_2 \), which should be low enough to significantly reduce its presence in the output gases.

Reaction Feasibility

Evaluating reaction feasibility is essential for determining whether a chemical process can be used effectively in industrial applications. This entails looking at various factors such as the equilibrium constant, Gibbs free energy, and reaction kinetics. In our example problem, reaction feasibility for \( SO_2 \) removal is analyzed based on the values calculated from equilibrium constants and Gibbs free energy.

The value of the equilibrium constant provides a direct measure of the position of equilibrium. If equilibrium lies far to the right (\( K >> 1 \)), it indicates that the reaction is highly feasible for \( SO_2 \) removal since the products are favored under equilibrium conditions.

Temperature Effects on Equilibrium

Temperature plays a significant role in the behavior of chemical equilibria. According to Le Chatelier's Principle, an increase in temperature shifts the equilibrium of an endothermic reaction to the right, thereby favoring the formation of products. Conversely, an exothermic reaction is favored by a decrease in temperature. This relationship is reflected in the van 't Hoff equation which connects the equilibrium constant \( K \) with temperature \( T \) and the change in standard enthalpy \( \Delta H^\circ \).

In the removal of \( SO_2 \) by reaction with hydrogen sulfide, understanding how the temperature affects the equilibrium is crucial for optimizing the process. If the reaction absorbs heat (\( \Delta H^\circ > 0 \) - endothermic), raising the temperature can make the removal process more efficient, which is often assessed by examining the enthalpy change of the reaction.