Question

The solubility product of \(\mathrm{CaF}_{2}\) is \(1.08\) \(\times 10^{-10}\). What mass of \(\mathrm{CaF}_{2}\) will dissolve in \(500 \mathrm{ml}\) water in order to make a saturated solution? \((\mathrm{Ca}=40, \mathrm{~F}=19)\) (a) \(3 \times 10^{-4} \mathrm{~g}\) (b) \(1.17 \times 10^{-2} \mathrm{~g}\) (c) \(1.17 \mathrm{mg}\) (d) \(3 \times 10^{-3} \mathrm{~g}\)

Short answer

The mass of \(\mathrm{CaF}_{2}\) that will dissolve in \(500\mathrm{ml}\) of water is \(1.17 \times 10^{-2} \mathrm{~g}\) (b).

Step-by-step solution

Write the Solubility Product Expression

For the dissolution of \(\mathrm{CaF}_{2}\), the reaction is \(\mathrm{CaF}_{2}(s) \rightarrow \mathrm{Ca}^{2+}(aq) + 2\mathrm{F}^{-}(aq)\). The solubility product \(K_{sp}\) for \(\mathrm{CaF}_{2}\) is given as \(1.08 \times 10^{-10}\). Hence, \(K_{sp} = [\mathrm{Ca}^{2+}][\mathrm{F}^{-}]^2\).

Key concepts

Understanding CaF2 Dissolution

The process of dissolving calcium fluoride (\textbf{CaF}\(_2\)) in water is an intriguing example of a sparingly soluble salt interacting with a solvent. When \textbf{CaF}\(_2\) is introduced to water, it dissolves to a limited extent, separating into calcium ions (\textbf{Ca}\(^{2+}\)) and fluoride ions (\textbf{F}\(^{-}\)). This is often represented by the chemical equation: \textbf{CaF}\(_2\)(s) \(\rightarrow\) \textbf{Ca}\(^{2+}\)(aq) + 2\textbf{F}\(^{-}\)(aq).

However, not all \textbf{CaF}\(_2\) will dissolve. The amount that does is governed by the solubility product constant, K\(_{sp}\), which quantifies the maximum possible concentration of ions in solution under saturated conditions. Dissolution is a dynamic equilibrium where the rate of dissolving solid is equal to the rate of solid formation out of dissolved ions. Thus, understanding this balance is key to predicting and controlling the behavior of \textbf{CaF}\(_2\) in various chemical processes, including the preparation of saturated solutions.

Ksp Expression and Its Significance

The solubility product constant, K\(_{sp}\), is a crucial concept in chemistry that indicates the extent to which a compound can dissolve in a solvent to form a saturated solution. For calcium fluoride, the K\(_{sp}\) expression is derived from its dissociation in water: K\(_{sp}\) = [\textbf{Ca}\(^{2+}\)][\textbf{F}\(^{-}]^2\).

This mathematical relationship can be interpreted as the product of the molar concentrations of the resulting calcium ions and the square of the fluoride ions' concentration at equilibrium in a saturated solution. The exponent reflects the stoichiometry of fluoride ions in the dissolution reaction. When the product of the ion concentrations exceeds the solubility product, additional solid will form to re-establish equilibrium. The K\(_{sp}\), therefore, provides a quantitative tool for calculating the concentrations of ions in a solution and determining whether a precipitate will form or dissolve.

The Molar Mass Calculation

To calculate the mass of calcium fluoride that can dissolve in water, the molar mass of \textbf{CaF}\(_2\) must be determined. Molar mass is the mass of one mole of a substance and serves as a conversion factor between the mass of a substance and the number of moles.

The atomic masses of calcium (\textbf{Ca}) and fluorine (\textbf{F}) are 40 and 19 units, respectively. Thus, the molar mass of \textbf{CaF}\(_2\) is calculated by adding the mass of one calcium atom to that of two fluorine atoms as follows: Molar mass of \textbf{CaF}\(_2\) = Mass of \textbf{Ca} + 2 \(\times\) Mass of \textbf{F} = 40 + 2 \(\times\) 19 = 78 g/mol.

With this information, one can find the number of moles from the given mass and vice versa, which is an essential step in determining how much \textbf{CaF}\(_2\) will dissolve to form a saturated solution, as stipulated by its K\(_{sp}\) value.

Preparing a Saturated Solution

A saturated solution is one in which the maximum amount of solute has been dissolved at a given temperature and pressure, with no excess solute visible. It represents a state of dynamic equilibrium between the undissolved solute and the dissolved ions. The saturation level is governed by the solubility product (K\(_{sp}\)), which predicts whether a precipitate will form under certain conditions.

To prepare a saturated solution of \textbf{CaF}\(_2\), water is added to the salt until it can no longer dissolve any more \textbf{CaF}\(_2\), signified by excess undissolved solid at the bottom of the container. The process highly depends on maintaining specific conditions, as the solubility of a substance can change with temperature and the presence of other substances. In a laboratory or industrial setting, reaching saturation is a careful process of control and measurement to achieve the desired solubility outcomes.