91Ó°ÊÓ

Sulfuric acid is produced in larger amounts by weight than any other chemical. It is used in manufacturing fertilizers, oil refining, and hundreds of other processes. An intermediate step in the industrial process for the synthesis of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) is the catalytic oxidation of sulfur dioxide: $$2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g) \quad \Delta G^{\circ}=-141.8 \mathrm{~kJ}$$ Calculate \(\Delta G\) at \(25^{\circ} \mathrm{C}\), given the following sets of partial pressures: (a) \(100 \mathrm{~atm} \mathrm{SO}_{2}, 100 \mathrm{~atm} \mathrm{O}_{2}, 1.0 \mathrm{~atm} \mathrm{SO}_{3}\) (b) \(2.0 \mathrm{~atm} \mathrm{SO}_{2}, 1.0 \mathrm{~atm} \mathrm{O}_{2}, 10 \mathrm{~atm} \mathrm{SO}_{3}\) (c) Each reactant and product at a partial pressure of \(1.0 \mathrm{~atm}\)

Step by step solution

01

Understand the Reaction

The reaction given is the oxidation of sulfur dioxide: \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{SO}_{3}(g)\). The standard Gibbs free energy change \( \Delta G^{\circ} \) is \(-141.8 \mathrm{~kJ}\).

02

Identify the Formula for \( \Delta G \)

The Gibbs free energy change \( \Delta G \) is calculated using the formula: \( \Delta G = \Delta G^{\circ} + RT \ln Q \), where \( R \) is the ideal gas constant \(8.314 \text{ J/mol K}\), \( T \) is the temperature in Kelvin, and \( Q \) is the reaction quotient.

03

Calculate \( Q \) for Each Scenario

The reaction quotient \( Q \) is given by \( Q = \frac{{(P_{\mathrm{SO}_3})^2}}{{(P_{\mathrm{SO}_2})^2 (P_{\mathrm{O}_2})}} \). Calculate \( Q \) for each set of conditions given.

04

Step 4a: Calculate \( \Delta G \) for Condition (a)

For \((a)\), \( P_{\mathrm{SO}_2} = 100 \text{ atm}, \; P_{\mathrm{O}_2} = 100 \text{ atm}, \; P_{\mathrm{SO}_3} = 1.0 \text{ atm}\). Thus, \( Q = \frac{{1.0^2}}{{100^2 \times 100}} \). First, calculate \( Q \) then \( \Delta G \).

05

Step 4b: Calculate \( \Delta G \) for Condition (b)

For \((b)\), \( P_{\mathrm{SO}_2} = 2.0 \text{ atm}, \; P_{\mathrm{O}_2} = 1.0 \text{ atm}, \; P_{\mathrm{SO}_3} = 10 \text{ atm}\). So, \( Q = \frac{{10^2}}{{2.0^2 \times 1.0}} \). Calculate \( Q \) and then \( \Delta G \).

06

Step 4c: Calculate \( \Delta G \) for Condition (c)

For \((c)\), \( P_{\mathrm{SO}_2} = P_{\mathrm{O}_2} = P_{\mathrm{SO}_3} = 1.0 \text{ atm}\). Hence, \( Q = \frac{{1.0^2}}{{1.0^2 \times 1.0}} = 1 \). Calculate \( \Delta G \).

07

Perform Calculations for \( \Delta G \)

Use the formula \( \Delta G = \Delta G^{\circ} + RT \ln Q \) with \( R = 8.314 \text{ J/mol K} \) and \( T = 298 \text{ K} \) to calculate \( \Delta G \) for each part. Convert \( \Delta G \) from Joules to kilojoules by dividing by 1000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sulfuric Acid Production

Sulfuric acid (\( \text{H}_{2}\text{SO}_{4} \)) is a critical component in numerous industrial processes. It is predominantly used in the manufacture of fertilizers but also finds applications in oil refining, wastewater treatment, and chemical synthesis. The production of sulfuric acid is driven by its demand due to its capability to react with various substances, enhancing reaction efficiency and output.
The process typically involves several steps, with one of the crucial steps being the conversion of sulfur dioxide (\( \text{SO}_{2} \)) to sulfur trioxide (\( \text{SO}_{3} \)) through catalytic oxidation. This serves as a precursor to form sulfuric acid when combined with water. Efficiency and environmental considerations make this step fundamental in the overall production cycle.

Catalytic Oxidation

Catalytic oxidation is vital in transforming sulfur dioxide (\( \text{SO}_{2} \)) into sulfur trioxide (\( \text{SO}_{3} \)), a precursor for sulfuric acid. This reaction uses a catalyst, often vanadium(V) oxide (\( \text{V}_{2}\text{O}_{5} \)), to lower the activation energy and speed up the conversion. Without a catalyst, the reaction would be too slow and inefficient for commercial purposes.

• The catalyst provides a surface for the reaction, enabling more frequent and successful collisions between reactant molecules.
• It allows the reaction to proceed at a lower temperature and pressure than would otherwise be necessary.

Catalytic oxidation is carefully controlled to optimize yield and minimize environmental impact, contributing to the industrial sector's sustainability.

Reaction Quotient

The reaction quotient (\( Q \)) is a measure used to predict the direction and extent of a chemical reaction at any given point during its progress. For the reaction converting sulfur dioxide to sulfur trioxide, \( Q \) is calculated as follows:\[Q = \frac{{(P_{\text{SO}_{3}})^{2}}}{{(P_{\text{SO}_{2}})^{2} \cdot (P_{\text{O}_{2}})}}\]This formula compares the pressure of the products to that of the reactants. When \( Q \) is less than the equilibrium constant (\( K \)), the forward reaction is favored, meaning the reaction will continue producing more products to reach equilibrium.
In industrial setups, maintaining the appropriate pressure conditions as dictated by \( Q \) ensures optimal conversion rates in the production of sulfur trioxide.

Ideal Gas Constant

The ideal gas constant (\( R \)) is a key component in thermodynamics, especially in the calculation of Gibbs Free Energy (\( \Delta G \)). In the formula \( \Delta G = \Delta G^{\circ} + RT \ln Q \), it bridges physical constants with practical conditions under which reactions occur.
The constant \( R \) is valued at 8.314 J/mol K and helps convert energy into understandable terms relating to temperature and reaction conditions.
Using \( R \), calculations consider how temperature and reaction quotient (\( Q \)) influence the Gibbs Free Energy, determining spontaneity and feasibility. This is crucial in real-world applications, where adjustments to temperature or pressure can significantly influence reaction outcomes. Understanding \( R \) ensures control over industrial chemical processes, enhancing efficiency and reducing costs.