The solubility of AgCl(s) in water at 25 °C is 1.33⨉10^-5 mol/L and its ΔH° of solution is 65.7 kJ/mol. What is its solubility at 50.0 °C?

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Identify the given data: solubility at 25 °C (S1 = 1.33 \times 10^{-5} \text{ mol/L}), temperature T1 = 298 \text{ K}, temperature T2 = 323 \text{ K}, and \Delta H^\circ = 65.7 \text{ kJ/mol}.

Use the van 't Hoff equation to relate the solubility at two different temperatures: \ln\left(\frac{S2}{S1}\right) = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T2} - \frac{1}{T1}\right), where R is the gas constant (8.314 J/mol·K).

Convert \Delta H^\circ from kJ/mol to J/mol by multiplying by 1000, so \Delta H^\circ = 65700 \text{ J/mol}.

Substitute the known values into the van 't Hoff equation: \ln\left(\frac{S2}{1.33 \times 10^{-5}}\right) = -\frac{65700}{8.314} \left(\frac{1}{323} - \frac{1}{298}\right).

Solve for S2, the solubility at 50.0 °C, by calculating the right-hand side of the equation and then exponentiating both sides to isolate S2.

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Key Concepts

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

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is an equilibrium constant that applies to the solubility of sparingly soluble ionic compounds. It is defined as the product of the molar concentrations of the ions, each raised to the power of their coefficients in the balanced equation. For AgCl, Ksp can be calculated from its solubility, which is essential for understanding how temperature affects solubility.

Le Chatelier's Principle

Le Chatelier's Principle states that if a system at equilibrium is subjected to a change in concentration, temperature, or pressure, the system will adjust to counteract that change and restore a new equilibrium. In the context of solubility, increasing temperature typically increases the solubility of endothermic dissolution processes, such as that of AgCl, which can be analyzed using this principle.

Van 't Hoff Equation

The Van 't Hoff equation relates the change in the equilibrium constant of a reaction to the change in temperature. It is particularly useful for calculating how the solubility of a compound changes with temperature, given the enthalpy change (ΔH°) of the dissolution process. This equation allows for the prediction of solubility at different temperatures based on the known solubility and ΔH° at a reference temperature.