Question

The value of \(K_{s p}\) for \(\mathrm{Mg}_{3}\left(\mathrm{AsO}_{4}\right)_{2}\) is \(2.1 \times 10^{-20}\). The \(\mathrm{AsO}_{4}^{3-}\) ion is derived from the weak acid \(\mathrm{H}_{3} \mathrm{AsO}_{4}\) \(\left(\mathrm{pK}_{a 1}=2.22 ; \mathrm{pK}_{a 2}=6.98 ; \mathrm{pK}_{a 3}=11.50\right) .\) When asked to calculate the molar solubility of \(\mathrm{Mg}_{3}\left(\mathrm{AsO}_{4}\right)_{2}\) in water, a student used the \(K_{s p}\) expression and assumed that \(\left[\mathrm{Mg}^{2+}\right]=1.5\left[\mathrm{AsO}_{4}^{3-}\right]\). Why was this a mistake?

Short answer

The student made a mistake in assuming the concentration of \(\mathrm{Mg}^{2+}\) ions is 1.5 times the concentration of \(\mathrm{AsO}_{4}^{3-}\) ions. The correct relationship should be \([\mathrm{Mg}^{2+}] = \dfrac{3}{2}[\mathrm{AsO}_{4}^{3-}]\), based on the balanced dissociation equation. The mistake led to an incorrect Ksp expression. To solve for molar solubility, the correct relationship should be used in the Ksp equation. The molar solubility of \(\mathrm{Mg}_{3}(\mathrm{AsO}_{4})_{2}\) in water is approximately \(1.10 \times 10^{-4}\) M.

Step-by-step solution

Write the balanced dissociation equation of Mg3(AsO4)2 in water

First, we need to write the balanced equation for the dissociation of Mg3(AsO4)2 in water: \[ Mg_{3}(AsO_{4})_{2}(s) \rightleftharpoons 3Mg^{2+}(aq) + 2AsO_{4}^{3-}(aq) \] This equation shows that when one mole of Mg3(AsO4)2 dissolves in water, it produces 3 moles of Mg2+ ions and 2 moles of AsO4^3- ions.

Identify the mistake in the student's assumption

The student made a mistake in assuming that the concentration of Mg2+ ions is 1.5 times the concentration of AsO4^3- ions. According to the balanced equation, for each mole of Mg3(AsO4)2 that dissolves, 3 moles of Mg2+ ions and 2 moles of AsO4^3- ions are produced. Therefore, the correct relationship between the concentrations of Mg2+ and AsO4^3- ions is: \[ [Mg^{2+}] = 3/2 [AsO_{4}^{3-}] \]

Write the expression for Ksp

To find the molar solubility of Mg3(AsO4)2, we need to write the expression for Ksp: \[ K_{sp} = [Mg^{2+}]^3 [AsO_{4}^{3-}]^2 \] Since \([Mg^{2+}] = 3/2 [AsO_{4}^{3-}]\), we can substitute this relationship into the Ksp equation: \[ K_{sp} = \left(\frac{3}{2}[AsO_{4}^{3-}]\right)^{3} [AsO_{4}^{3-}]^{2} \]

Solve for molar solubility

Now, let x be the molar solubility of Mg3(AsO4)2 in water, then: \[ [AsO_{4}^{3-}] = x \] Substituting this into our Ksp equation gives: \[ 2.1 \times 10^{-20} = \left(\frac{3}{2}x\right)^{3} x^{2} \] Now, we need to solve this equation for x: \(x \approx 1.10 \times 10^{-4}\)

Determine the molar solubility of Mg3(AsO4)2

The molar solubility of Mg3(AsO4)2 in water is approximately 1.10 x 10^-4 M.

Key concepts

Ksp

Ksp, or the solubility product constant, is a special type of equilibrium constant that measures the solubility of a sparingly soluble ionic compound in water. It is determined at a particular temperature and is unique for each compound. The Ksp value is the product of the concentrations of the ions that are produced when the compound dissociates in water, each raised to the power of its stoichiometric coefficient found in the balanced dissociation equation. For instance, with a substance like \( Mg_3(AsO_4)_2 \) dissociating in water, we get:\
\
\[ K_{sp} = [Mg^{2+}]^3 [AsO_{4}^{3-}]^2 \]
\
A higher Ksp value indicates greater solubility of the compound in water. To use the Ksp effectively, one must know the dissociation equation and relate the molar solubility of each ion to the molar solubility of the original compound. Preciseness in utilizing these ratios is crucial for accurate calculation of molar solubility.

Dissociation Equation

The dissociation equation represents how a compound, particularly an ionic compound, breaks down into its constituent ions in a solvent. Writing a balanced dissociation equation is a core step in understanding the solubility and the Ksp calculation. It defines the stoichiometric relationship between the solute and the ion products formed upon dissolving. For example, the dissolution of \( Mg_3(AsO_4)_2 \) in water can be represented as a reversible reaction:\
\
\[ Mg_{3}(AsO_{4})_{2}(s) \rightleftharpoons 3Mg^{2+}(aq) + 2AsO_{4}^{3-}(aq) \]
\
This equation is critical in determining the ratio in which ions are produced and establishes the stoichiometric coefficients needed to calculate the Ksp value. Understanding the ratios of the ions is key in not making the mistake of assuming an incorrect relationship between the ion concentrations, such as the student example provided in the exercise.

Concentration Ratios

Concentration ratios stem from the balanced dissociation equation and give the relative amounts of ions in solution when an ionic compound dissociates. These ratios are essential when solving for the molar solubility from the Ksp value. In our case, the correct concentration ratio comes from the coefficients in the balanced equation:\
\
\[ [Mg^{2+}] = \frac{3}{2} [AsO_{4}^{3-}] \]
\
This indicates that for every mole of \( AsO_{4}^{3-} \) produced, there are one and a half moles (3/2) of \( Mg^{2+} \) ions produced. This ratio should then be used to express the concentrations of ions in terms of a single variable, commonly identified as 'x', for the molar solubility of the compound. Using incorrect concentration ratios, as the student did in the exercise, leads to incorrect computations of molar solubility which can greatly affect understanding of the degree to which a compound will dissolve in a given solvent.