A metal salt with the formula MCl₂ crystallizes from water to form a solid with the composition MCl₂ • 6 H₂O. The equilibrium vapor pressure of water above this solid at 298 K is 18.3 mmHg. What is the value of ΔG for the reaction MCl₂ • 6 H₂O(s) ⇌ MCl₂(s) + 6 H₂O(g) when the pressure of water vapor is 18.3 mmHg? When the pressure of water vapor is 760 mmHg?

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Identify the reaction: \( \text{MCl}_2 \cdot 6 \text{H}_2\text{O(s)} \rightleftharpoons \text{MCl}_2\text{(s)} + 6 \text{H}_2\text{O(g)} \).

Use the equation for Gibbs free energy change: \( \Delta G = \Delta G^\circ + RT \ln Q \), where \( Q \) is the reaction quotient.

Calculate \( Q \) for the first scenario (18.3 mmHg): \( Q = \left( \frac{P_{\text{H}_2\text{O}}}{P^\circ} \right)^6 \), where \( P_{\text{H}_2\text{O}} = 18.3 \text{ mmHg} \) and \( P^\circ = 1 \text{ atm} = 760 \text{ mmHg} \).

Calculate \( Q \) for the second scenario (760 mmHg): \( Q = \left( \frac{P_{\text{H}_2\text{O}}}{P^\circ} \right)^6 \), where \( P_{\text{H}_2\text{O}} = 760 \text{ mmHg} \).

Substitute the values of \( Q \) into the Gibbs free energy equation for each scenario 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 crucial for determining the spontaneity of a reaction; a negative ΔG indicates a spontaneous process, while a positive ΔG suggests non-spontaneity. The relationship between ΔG, enthalpy (ΔH), and entropy (ΔS) is given by the equation ΔG = ΔH - TΔS.

Equilibrium and Vapor Pressure

Equilibrium in a chemical reaction occurs when the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products. The vapor pressure of a substance is the pressure exerted by its vapor when it is in equilibrium with its liquid or solid phase. In this question, the equilibrium vapor pressure of water at 298 K is essential for calculating ΔG under specific conditions.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation relates the change in vapor pressure with temperature to the enthalpy of phase changes. It is particularly useful for understanding how the vapor pressure of a substance changes with temperature and can be applied to calculate ΔG for phase transitions. In this context, it helps to determine the relationship between the vapor pressure of water and the Gibbs Free Energy change for the dissolution of MCl2 • 6 H2O.