Problem 107 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The student's mistake was in assuming that the relationship between the concentrations of Mg²⁺ and AsO₄³⁻ ions was \([Mg^{2+}] = 1.5[AsO_4^{3-}]\) without considering the presence of weak acid H₃AsO₄, which forms AsO₄³⁻ ions. The correct relationship, according to the balanced chemical equation, is \([Mg^{2+}] = 3s\) and \([AsO_4^{3-}] = 2s\), where s represents the molar solubility of the salt. Using this relationship, the molar solubility of Mg₃(AsO₄)₂ in water is found to be approximately \(2.714 \times 10^{-5} M\).

Step 1: Write the balanced chemical equation for the dissolution of Mg₃(AsO₄)₂ in water

The balanced chemical equation for the dissolution of Mg₃(AsO₄)₂ in water is given by: \[Mg_3(AsO_4)_2(s) \longleftrightarrow 3Mg^{2+}(aq) + 2AsO_4^{3-}(aq)\]

Step 2: Write the Ksp expression for the dissolution process

For the given chemical equation, Ksp can be written as: \[K_{sp} = [Mg^{2+}]^3 [AsO_4^{3-}]^2\]

Step 3: Understand the mistake made by the student

The student assumed that the relationship between the concentrations of Mg²⁺ and AsO₄³⁻ ions was: \([Mg^{2+}] = 1.5[AsO_4^{3-}]\). However, this does not take into account the formation of H₃AsO₄, which can donate protons to form the AsO₄³⁻ ion. Since H₃AsO₄ is a weak acid, we cannot ignore its influence on the equilibrium concentrations of the involved ions. This is the main mistake made by the student.

Step 4: Write the correct relationship between the concentrations of Mg²⁺ and AsO₄³⁻ ions

According to the balanced chemical equation, \[Mg_3(AsO_4)_2(s) \longleftrightarrow 3Mg^{2+}(aq) + 2AsO_4^{3-}(aq)\] Let the molar solubility of the salt be represented by "s," then the equilibrium concentrations can be represented as: \[[Mg^{2+}] = 3s \quad\text{and}\quad [AsO_4^{3-}] = 2s\]

Step 5: Re-write the Ksp expression using the correct relationship

Now we can rewrite the Ksp expression using the correct relationship between the concentrations: \[K_{sp} = (3s)^3 (2s)^2\] Expanding this expression yields: \[K_{sp} = 108s^5\]

Step 6: Solving for molar solubility

Since we know the value of Ksp (2.1 x 10⁻²⁰) for Mg₃(AsO₄)₂, we can solve for the molar solubility "s": \[2.1 \times 10^{-20} = 108s^5\] \[s^5 = \frac{2.1 \times 10^{-20}}{108}\] \[s^5 \approx 1.944 \times 10^{-22}\] \[s \approx 2.714 \times 10^{-5}\] Hence, the molar solubility of Mg₃(AsO₄)₂ in water is approximately \(2.714 \times 10^{-5} M\).

Key Concepts

Molar Solubility

Molar solubility refers to the number of moles of a solute that can dissolve in a liter of solution until the solution becomes saturated. It's a crucial concept when dealing with the solubility product constant (\(K_{sp}\)), which determines the point at which the dissolved ions in a solution will begin to precipitate.

In the example of \(\mathrm{Mg}_3(\mathrm{AsO}_4)_2\), we start with the equation for its dissolution:\[\mathrm{Mg}_3(\mathrm{AsO}_4)_2(s) \leftrightarrow 3\mathrm{Mg}^{2+}(aq) + 2\mathrm{AsO}_4^{3-}(aq)\]From this, we derive the relationship between ion concentrations and the molar solubility "s."

• \([\mathrm{Mg}^{2+}] = 3s\)
• \([\mathrm{AsO}_4^{3-}] = 2s\)

Replacing these into the \(K_{sp}\) expression, we have:\[K_{sp} = (3s)^3 (2s)^2 = 108s^5\]Solving for "s," given \(K_{sp} = 2.1 \times 10^{-20}\), gives us a molar solubility of approximately \(2.714 \times 10^{-5} M\).

Understanding this concept is key for analyzing the solubility behavior of compounds in various chemical reactions.

Weak Acid

A weak acid, like \(\mathrm{H}_3\mathrm{AsO}_4\), doesn't fully dissociate in water. Instead, only a small portion of its molecules release hydrogen ions.

For \(\mathrm{H}_3\mathrm{AsO}_4\), the dissociation equilibria are defined by \(pK_{a1}=2.22\), \(pK_{a2}=6.98\), and \(pK_{a3}=11.50\). Each \(pK_a\) value represents a different stage of dissociation:

• Stage 1: \(\mathrm{H}_3\mathrm{AsO}_4 \leftrightarrow \mathrm{H}^+ + \mathrm{H}_2\mathrm{AsO}_4^{-}\)
• Stage 2: \(\mathrm{H}_2\mathrm{AsO}_4^{-} \leftrightarrow \mathrm{H}^+ + \mathrm{HAsO}_4^{2-}\)
• Stage 3: \(\mathrm{HAsO}_4^{2-} \leftrightarrow \mathrm{H}^+ + \mathrm{AsO}_4^{3-}\)

The presence of multiple dissociation stages affects the equilibrium concentration of \(\mathrm{AsO}_4^{3-}\) ions. These ions are only partly available from \(\mathrm{Mg}_3(\mathrm{AsO}_4)_2\) dissolution, as some originate from the dissociation of the acid.

Awareness of weak acid behavior is essential when calculating solubility, because ignoring it can lead to inaccurate assumptions about ion concentrations.

Ion Concentration

Ion concentration is a vital factor determining the behavior of a solution. It directly connects to the solubility and the dissolution of compounds.

In dissolving \(\mathrm{Mg}_3(\mathrm{AsO}_4)_2\), we assess the concentrations of \(\mathrm{Mg}^{2+}\) and \(\mathrm{AsO}_4^{3-}\) ions. These concentrations help shape the \(K_{sp}\) expression:\[K_{sp} = [\mathrm{Mg}^{2+}]^3 [\mathrm{AsO}_4^{3-}]^2\]
Correct relationships are established as \([\mathrm{Mg}^{2+}] = 3s\) and \([\mathrm{AsO}_4^{3-}] = 2s\), not \([\mathrm{Mg}^{2+}] = 1.5[\mathrm{AsO}_4^{3-}]\), as initially assumed.

Calculating correct ion concentrations ensures accurate determination of molar solubility. Misunderstanding or miscalculating these can lead to incorrect conclusions about the solubility of the compound. Properly managing ion concentrations in calculations improves the efficiency and reliability of chemical processes.