A particular saturated solution of Ca3(PO4)2 has [Ca2+] = [PO43-] = 2.9 x 10^-7 M. (c) What is [PO43-] in a saturated solution that has [Ca2+] = 0.010 M?

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Identify the solubility product expression for calcium phosphate, \( \text{Ca}_3(\text{PO}_4)_2 \). The dissolution can be represented as: \( \text{Ca}_3(\text{PO}_4)_2 (s) \rightleftharpoons 3\text{Ca}^{2+} (aq) + 2\text{PO}_4^{3-} (aq) \).

Write the expression for the solubility product constant, \( K_{sp} \), which is given by: \( K_{sp} = [\text{Ca}^{2+}]^3 [\text{PO}_4^{3-}]^2 \).

Use the initial condition where \([\text{Ca}^{2+}] = [\text{PO}_4^{3-}] = 2.9 \times 10^{-7} \text{ M}\) to calculate \( K_{sp} \). Substitute these values into the \( K_{sp} \) expression to find its value.

Now, consider the new condition where \([\text{Ca}^{2+}] = 0.010 \text{ M}\). Substitute this value into the \( K_{sp} \) expression: \( K_{sp} = (0.010)^3 [\text{PO}_4^{3-}]^2 \).

Solve for \([\text{PO}_4^{3-}]\) by rearranging the equation: \([\text{PO}_4^{3-}] = \sqrt{\frac{K_{sp}}{(0.010)^3}}\).

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주요 개념

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Saturation and Solubility Product (Ksp)

Saturation refers to the point at which a solution can no longer dissolve additional solute at a given temperature. The solubility product constant (Ksp) is an equilibrium constant that quantifies the solubility of a sparingly soluble ionic compound. For calcium phosphate, Ca3(PO4)2, Ksp is determined by the concentrations of its ions in a saturated solution, specifically [Ca2+] and [PO43-].

Stoichiometry of Dissociation

The stoichiometry of dissociation describes how a compound breaks down into its constituent ions in solution. For Ca3(PO4)2, the dissociation can be represented as Ca3(PO4)2 ⇌ 3Ca2+ + 2PO43-. This means that for every mole of Ca3(PO4)2 that dissolves, three moles of Ca2+ and two moles of PO43- are produced, establishing a relationship between their concentrations in a saturated solution.

Concentration Relationships in Saturated Solutions

In a saturated solution, the concentrations of the ions are related through the Ksp expression. For Ca3(PO4)2, Ksp = [Ca2+]^3[PO43-]^2. When the concentration of one ion changes, the concentration of the other must adjust to maintain the equilibrium defined by Ksp. This relationship allows us to calculate the concentration of PO43- when [Ca2+] is known.