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 \(\mathrm{H}_{3} \mathrm{AsO}_{4}\left(\mathrm{p} K_{a 1}=2.22\right.\); \(\left.\mathrm{p} K_{a 2}=6.98 ; \mathrm{p} K_{a 3}=11.50\right) .\) 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's mistake was misunderstanding the stoichiometry of the balanced equation when assuming \([Mg^{2+}] = 1.5 [AsO_4^{3-}]\). The correct relation, obtained from the stoichiometry, is \([Mg^{2+}] = 3x = 1.5 (2x) = 1.5 [AsO_4^{3-}]\). This error occurred because the student didn't consider the stoichiometric coefficients properly and didn't multiply the concentration of the \(AsO_4^{3-}\) ion by 1.5 in the correct relation. It is important to always consider stoichiometry while establishing relations between ion concentrations in dissolution processes.

Step-by-step solution

Write the balanced equation.

The balanced chemical equation for the dissolution of \(Mg_3(AsO_4)_2\) in water can be written as: \[Mg_3(AsO_4)_2(s) \rightleftharpoons 3Mg^{2+}(aq) + 2AsO_4^{3-}(aq)\] #Step 2: Write the expression for the solubility product constant \(K_{sp}\)#

Write the expression for \(K_{sp}\).

For the balanced equation, the solubility product constant (\(K_{sp}\)) is given by: \[K_{sp} = [Mg^{2+}]^3 [AsO_4^{3-}]^2\] #Step 3: Find the correct relation between the concentrations of \(Mg^{2+}\) and \(AsO_4^{3-}\) ions#

Find the correct relation between the concentrations.

Let the molar solubility of \(Mg_3(AsO_4)_2\) be \(x\). The dissolution of one formula unit of the compound will release 3 moles of \(Mg^{2+}\) ions and 2 moles of \(AsO_4^{3-}\) ions, according to the balanced equation. Therefore, \[[Mg^{2+}] = 3x\] \[[AsO_4^{3-}] = 2x\] Now, we can compare this to the student's assumption (\([Mg^{2+}] = 1.5 [AsO_4^{3-}]\)), which is the mistake we are supposed to explain. #Step 4: Explain the student's mistake#

Explain the mistake.

The student's assumption was: \[[Mg^{2+}] = 1.5 [AsO_4^{3-}]\] However, the correct relation obtained in the previous step is: \[[Mg^{2+}] = 3x = 1.5 (2x) = 1.5 [AsO_4^{3-}]\] The student's mistake lies in misunderstanding the stoichiometry of the balanced equation. They didn't consider the stoichiometric coefficients properly and didn't multiply the concentration of the \(AsO_4^{3-}\) ion by 1.5 in the correct relation. The stoichiometric coefficients in a balanced chemical equation must always be considered while establishing the relation between the concentrations of ions involved in the dissolution process.

Key concepts

Solubility Product Constant (Ksp)

Understanding the solubility product constant, also known as the Ksp, is essential when dealing with the solubility of sparingly soluble salts in water. It is a unique value for each salt at a given temperature and provides a quantitative measure of the salt's solubility in a solution. The Ksp expression is derived from the equilibrium state of a dissolved ionic compound, and it is the product of the concentrations of the ions, each raised to the power of its coefficient in the balanced chemical equation.

The key to using this value effectively is to know that it remains constant for a particular compound under constant conditions, which allows us to calculate either the solubility of the salt or the concentrations of individual ions in a solution at equilibrium. For instance, in the provided exercise, the Ksp for magnesium arsenate \(Mg_3(AsO_4)_2\) is \(2.1 \times 10^{-20}\). This very low Ksp value reflects its low solubility in water. Incorrect application or misinterpretation of Ksp, such as failing to consider the correct stoichiometric relation between ions, could lead to errors in calculating molar solubility.

Stoichiometry in Chemical Equations

Stoichiometry plays a pivotal role in chemistry, especially when it comes to understanding chemical reactions. It involves using the coefficients from balanced chemical equations to relate the amounts of reactants and products. It's a bit like a recipe, where the coefficients tell you how much of each ingredient you need for a reaction to occur.

In the context of solubility, stoichiometry is used to determine the molar ratio of ions produced when a compound dissolves. When a compound such as \(Mg_3(AsO_4)_2\) dissociates in water, it does so according to fixed stoichiometric proportions indicated by its chemical formula. For every one formula unit of \(Mg_3(AsO_4)_2\) that dissolves, it forms three magnesium ions (\(Mg^{2+}\)) and two arsenate ions (\(AsO_4^{3-}\)), not in a 1.5:1 ratio as the student wrongly assumed. Understanding this concept is crucial for correctly setting up the mathematical relationships needed to find the molar solubility and to avoid common mistakes.

Concentration Relation of Ions

Once we grasp the stoichiometry of a dissolved ionic compound, we can establish the concentration relations of the ions produced. For a compound that dissociates into more than one of a particular ion, such as \(Mg_3(AsO_4)_2\) releasing three magnesium ions for every two arsenate ions it produces, this relationship must be reflected in the concentration terms used in the calculation of the Ksp.

The molar solubility, symbolized as \(x\) in our example, is the number of moles of \(Mg_3(AsO_4)_2\) that can dissolve in one liter of water. Since the dissolution of one mole of \(Mg_3(AsO_4)_2\) leads to the formation of three moles of \(Mg^{2+}\) ions and two moles of \(AsO_4^{3-}\), the concentration of \(Mg^{2+}\) will be \(3x\) and \(AsO_4^{3-}\) will be \(2x\), a critical detail overlooked by the student in the exercise. These concentration relations are vital in solving for molar solubility and in working with Ksp expressions to understand the extent of a compound's solubility.