Problem 111 - Chemistry: The Central Science in SI Units, Global Edition

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 \(298 \mathrm{~K} ?(\mathbf{b})\) In principle, is this reaction a feasible method of removing \(\mathrm{SO}_{2}\) ? (c) If \(P_{5 \mathrm{O}_{2}}=P_{\mathrm{H}_{2}}\) s and the vapor pressure of water is \(3.33 \mathrm{kPa}\), 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 reaction has an equilibrium constant (K) that needs to be calculated using the standard free energy change (∆G°) of the reaction at 298 K. If K is much greater than 1, the reaction is a feasible method for removing SO₂. The equilibrium SO₂ pressure can be calculated using the given partial pressures of H₂S and H₂O. The process would be more effective at higher temperatures if the reaction is endothermic and less effective if the reaction is exothermic.

Step 1: Calculate ∆G° of the reaction

Using the standard free energy values provided in Appendix C, we can calculate the standard free energy change for the reaction as follows: ∆G° = ∆G°(products) - ∆G°(reactants) Note that the standard free energy values provided in Appendix C must be multiplied by the stoichiometric coefficients in the balanced equation.

Step 2: Calculate the equilibrium constant (K)

We can calculate K using the thermodynamic equation: K = e^(-∆G° / (RT)), where R is the gas constant (8.314 J/mol K) and T is the temperature (298 K). Plug in the values of ∆G°, R, and T to find K.

Step 3: Determine the feasibility of the reaction for removing SO₂

If K is much greater than 1, the reaction is feasible for removing SO₂ because it indicates that the reaction favors the formation of products. If K is much less than 1, the reaction is not a feasible method for removing SO₂ as it favors the formation of reactants.

Step 4: Calculate the equilibrium SO₂ pressure

We are given \(P_{H_2S}=P_{SO_2}\) and the vapor pressure of water \(P_{H_2O} = 3.33\,\text{kPa}\). The balanced equation shows that for every mole of SO₂ consumed, there would be two moles of H₂O formed. Thus, \(P_{SO_2} P_{H_2S}^{2}=K \times (3.33 \,\text{kPa})^2\) Since \(P_{H_2S}=P_{SO_2}\), the equation can be simplified to: \(P_{SO_2}^3=K \times (3.33 \,\text{kPa})^2\) We can solve for \(P_{SO_2}\), the equilibrium SO₂ pressure in the system.

Step 5: Discuss the process effectiveness at higher temperatures

As temperature increases, the position of the equilibrium shifts in the direction that favors the endothermic reaction because it absorbs the heat. If the given reaction is endothermic, it would be more effective at higher temperatures as more heat would be absorbed, thus driving the reaction towards the products. If the reaction is exothermic, it would be less effective at higher temperatures because the increased heat would shift the equilibrium position towards the reactants. To determine if the reaction is endothermic or exothermic, we can refer back to the standard free energy change (∆G°) and the standard enthalpy change (∆H°) if available.

Key Concepts

Free Energy Change

Free energy change, often denoted as \( \Delta G \), is a crucial concept in chemistry that helps us predict the spontaneity of a reaction. In simple terms, it tells us whether a reaction can occur on its own without external energy input. A key point to remember is:

• If \( \Delta G < 0 \), the reaction is spontaneous and can proceed on its own.
• If \( \Delta G > 0 \), the reaction is non-spontaneous, requiring energy input to occur.
• If \( \Delta G = 0 \), the system is at equilibrium.

For the reaction given in the problem, we are interested in the standard free energy change \( \Delta G^\circ \), which uses standard states and is calculated using the formula:\[\Delta G^\circ = \Delta G^\circ (\text{products}) - \Delta G^\circ (\text{reactants})\]This equation highlights the importance of using stoichiometric coefficients correctly when calculating \( \Delta G^\circ \), ensuring the values from Appendix C are used appropriately.
Understanding free energy change is the first step toward assessing reaction feasibility and determining equilibrium conditions.

Equilibrium Constant

The equilibrium constant, denoted as \( K \), provides essential insight into the composition of products and reactants at equilibrium. For reactions occurring under a constant temperature, the equilibrium constant is determined using the following thermodynamic relation:\[K = e^{-\Delta G^\circ / (RT)}\]Here, \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. This equation shows how the free energy change is intimately linked to the equilibrium constant. A few points to note:

• If \( K >> 1 \), the reaction heavily favors products and can be considered essentially complete.
• If \( K << 1 \), the reaction favors reactants with very little forward progress at equilibrium.
• If \( K \approx 1 \), both reactants and products are present in significant amounts.

In this exercise, calculating \( K \) helps determine the feasibility of the proposed \( \mathrm{SO}_{2} \) removal method by indicating the predominant state of the reaction.
Thus, \( K \) serves as a quantitative measure of a reaction's tendency to form products over reactants under equilibrium conditions.

Thermodynamic Equations

Thermodynamic equations are formulas that connect various thermodynamic properties, helping to predict the behaviors of chemical reactions. In this case, the relationship between free energy change and the equilibrium constant is critical. Understanding how they interact can provide significant insights:

• The equation \( K = e^{-\Delta G^\circ / (RT)} \) is derived from fundamental thermodynamic principles.
• The free energy equation \( \Delta G^\circ = -RT \ln K \) shows that the spontaneity (\( \Delta G^\circ \)) and equilibrium (\( K \)) are aligned.
• Knowing \( \Delta G^\circ \) allows for calculation of \( K \), which helps assess the potential extent of a reaction at equilibrium.

These equations provide a framework for assessing and predicting the behavior of chemical reactions. Mastery of these relationships allows students to tackle a variety of equilibrium problems, making them invaluable tools in both academic and practical chemical settings.

Reaction Feasibility

Reaction feasibility pertains to whether a chemical reaction can occur under the given conditions. Evaluating feasibility requires an analysis of \( \Delta G \) and \( K \):

• A negative \( \Delta G^\circ \) implies that the reaction is thermodynamically favorable and should proceed spontaneously under standard conditions.
• An equilibrium constant \( K \) greater than 1 strongly suggests that products are favored, indicating a feasible reaction.
• The relation of \( \Delta G^\circ \) to \( K \) highlights how reaction conditions can be adjusted (e.g., temperature) to drive a non-spontaneous reaction towards feasibility.

In the exercise, the feasibility of \( \mathrm{SO}_{2} \) removal is evaluated based on these criteria. A feasible reaction means the proposed method would effectively convert \( \mathrm{SO}_{2} \) to other substances under specified conditions. Understanding how to judge feasibility guides not only academic exercises but also real-world applications like industrial chemical processes.