The value of Ksp for Mg₃₁(AsO₄)₂₂ is 2.1 * 10⁻²⁰. The AsO₄³⁻ ion is derived from the weak acid H₃AsO₄ with pKa₁ = 2.22, pKa₂ = 6.98, pKa₃ = 11.502. (b) Calculate the pH of a saturated solution of Mg₃₁(AsO₄)₂₂ in water.

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Identify the dissolution reaction for Mg_{31}(AsO_4)_{22} in water: Mg_{31}(AsO_4)_{22} \rightleftharpoons 31Mg^{2+} + 22AsO_4^{3-}.

Write the expression for the solubility product constant (K_{sp}): K_{sp} = [Mg^{2+}]^{31} [AsO_4^{3-}]^{22}.

Assume the molar solubility of Mg_{31}(AsO_4)_{22} is 's'. Then, [Mg^{2+}] = 31s and [AsO_4^{3-}] = 22s.

Substitute the concentrations into the K_{sp} expression: K_{sp} = (31s)^{31} (22s)^{22} = 2.1 \times 10^{-20}.

Use the concentration of AsO_4^{3-} to find the pH. Since AsO_4^{3-} is derived from H_3AsO_4, consider the hydrolysis reaction: AsO_4^{3-} + H_2O \rightleftharpoons HAsO_4^{2-} + OH^{-}. Calculate the pH using the equilibrium expression and the known pK_a values.

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 quantifies the solubility of a sparingly soluble ionic compound. It is defined as the product of the molar concentrations of the ions, each raised to the power of their coefficients in the balanced dissolution equation. For Mg₃₁(AsO₄)₂₂, Ksp helps determine how much of the compound can dissolve in water, influencing the concentration of the AsO₄³⁻ ions in solution.

Acid-Base Equilibrium and pKa

The pKa value is a measure of the strength of an acid, indicating the pH at which half of the acid is dissociated. In this case, H₃AsO₄ has three pKa values, reflecting its ability to donate protons in a stepwise manner. Understanding these pKa values is crucial for determining the predominant species of arsenate ions in solution, which affects the pH and the overall chemistry of the saturated solution.

pH Calculation in Saturated Solutions

The pH of a saturated solution can be calculated by considering the concentration of hydrogen ions produced from the dissociation of the weak acid formed by the dissolved salt. In this scenario, the concentration of AsO₄³⁻ ions will influence the equilibrium of the acid-base reaction, allowing for the calculation of the pH based on the concentration of these ions and their corresponding dissociation constants.

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