Consider the reaction 3 CH₄(g) → C₃H₈(g) + 2 H₂(g). (b) Calculate ΔG at 298 K if the reaction mixture consists of 40.0 atm of CH₄, 0.0100 atm of C₃H₈(g), and 0.0180 atm of H₂.

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Identify the reaction: 3 CH_4(g) \rightarrow C_3H_8(g) + 2 H_2(g).

Use the formula for Gibbs free energy change: \( \Delta G = \Delta G^\circ + RT \ln Q \), where \( R = 8.314 \text{ J/mol K} \) and \( T = 298 \text{ K} \).

Calculate the reaction quotient \( Q \) using the partial pressures: \( Q = \frac{P_{C_3H_8} \cdot (P_{H_2})^2}{(P_{CH_4})^3} \).

Substitute the given partial pressures into the expression for \( Q \): \( Q = \frac{0.0100 \cdot (0.0180)^2}{(40.0)^3} \).

Substitute \( \Delta G^\circ \), \( R \), \( T \), and \( Q \) into the Gibbs free energy equation to find \( \Delta G \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gibbs Free Energy (ΔG)

Gibbs Free Energy (ΔG) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. It is a crucial concept in predicting the spontaneity of a reaction; a negative ΔG indicates that a reaction can occur spontaneously, while a positive ΔG suggests non-spontaneity. The relationship between ΔG, enthalpy (ΔH), and entropy (ΔS) is given by the equation ΔG = ΔH - TΔS.

Reaction Quotient (Q)

The reaction quotient (Q) is a measure of the relative concentrations of products and reactants at any point in a reaction, expressed as the ratio of the product concentrations raised to their stoichiometric coefficients to the reactant concentrations raised to their coefficients. It is used to determine the direction in which a reaction will proceed to reach equilibrium. In this case, Q can be calculated using the given partial pressures of CH₄, C₃H₈, and H₂.

Standard Gibbs Free Energy Change (ΔG°)

The standard Gibbs free energy change (ΔG°) is the change in Gibbs free energy for a reaction under standard conditions (1 atm, 298 K). It provides a reference point for calculating ΔG under non-standard conditions using the equation ΔG = ΔG° + RT ln(Q), where R is the universal gas constant and T is the temperature in Kelvin. Understanding ΔG° is essential for evaluating the favorability of a reaction and its equilibrium position.

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Textbook Question

Consider the reaction 3 CH₄(g) → C₃H₈(g) + 2 H₂(g). (a) Using data from Appendix C, calculate ΔG° at 298 K.

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Consider the decomposition of barium carbonate: BaCO₃(s) ⇌ BaO(s) + CO₂(g) Using data from Appendix C, calculate the equilibrium pressure of CO₂ at (a) 298 K.

Consider the reaction 3 CH4(g) → C3H8(g) + 2 H2(g). (b) Calculate ΔG at 298 K if the reaction mixture consists of 40.0 atm of CH4, 0.0100 atm of C3H8(g), and 0.0180 atm of H2.

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Key Concepts

Gibbs Free Energy (ΔG)

Reaction Quotient (Q)

Standard Gibbs Free Energy Change (ΔG°)

Consider the reaction 3 CH₄(g) → C₃H₈(g) + 2 H₂(g). (b) Calculate ΔG at 298 K if the reaction mixture consists of 40.0 atm of CH₄, 0.0100 atm of C₃H₈(g), and 0.0180 atm of H₂.