Question

The solubility products of \(\mathrm{PbSO}_{4}\) and \(\mathrm{Sr} \mathrm{SO}_{4}\) are \(6.3 \times 10^{-7}\) and \(3.2 \times 10^{-7}\), respectively. What are the values of \(\left[\mathrm{SO}_{4}{ }^{2-}\right],\left[\mathrm{Pb}^{2+}\right]\), and \(\left[\mathrm{Sr}^{2+}\right]\) in a solution at equilibrium with both substances?

Short answer

In a solution at equilibrium with both PbSO4 and SrSO4, we have the following solubility product constant expressions: \[K_{sp} (\mathrm{PbSO_{4}}) = [\mathrm{Pb}^{2+}] [\mathrm{SO}_{4}^{2-}]\] \[K_{sp} (\mathrm{SrSO_{4}}) = [\mathrm{Sr}^{2+}] [\mathrm{SO}_{4}^{2-}]\] Given the solubility products, we can set up a system of equations relating the ion concentrations: \[6.3 \times 10^{-7} = [\mathrm{Pb}^{2+}] [\mathrm{SO}_{4}^{2-}]\] \[3.2 \times 10^{-7} = [\mathrm{Sr}^{2+}] [\mathrm{SO}_{4}^{2-}]\] However, without additional information such as initial concentrations, this system of equations cannot be fully solved as given.

Step-by-step solution

1. Write the solubility product constant expressions for PbSO4 and SrSO4

For any sparingly soluble compound, we can write the solubility product constant expression as the product of the concentrations of its constituent ions raised to their stoichiometric coefficients. In our case, for \(PbSO_{4}\) and \(SrSO_{4}\), the expressions are: \[K_{sp} (\mathrm{PbSO_{4}}) = [\mathrm{Pb}^{2+}] [\mathrm{SO}_{4}^{2-}]\] \[K_{sp} (\mathrm{SrSO_{4}}) = [\mathrm{Sr}^{2+}] [\mathrm{SO}_{4}^{2-}]\]

2. Insert the given values of \(K_{sp}\) for PbSO4 and SrSO4 in the expressions

We have the following values for the solubility products of the two salts: \(K_{sp} (\mathrm{PbSO_{4}}) = 6.3 \times 10^{-7}\) \(K_{sp} (\mathrm{SrSO_{4}}) = 3.2 \times 10^{-7}\) Substitute these values into the expressions: \[6.3 \times 10^{-7} = [\mathrm{Pb}^{2+}] [\mathrm{SO}_{4}^{2-}]\] \[3.2 \times 10^{-7} = [\mathrm{Sr}^{2+}] [\mathrm{SO}_{4}^{2-}]\]

3. Solve the system of equations for the ion concentrations

We have two equations with three unknowns, so let's solve for one ion (e.g., the sulfate ion) in terms of the other ion and then substitute into the remaining equation to find their values: \[[\mathrm{SO}_{4}^{2-}] = \frac{6.3 \times 10^{-7}}{[\mathrm{Pb}^{2+}]}\] \[[\mathrm{SO}_{4}^{2-}] = \frac{3.2 \times 10^{-7}}{[\mathrm{Sr}^{2+}]}\] Now, we can equate the above expressions since both represent the sulfate ion concentration: \[\frac{6.3 \times 10^{-7}}{[\mathrm{Pb}^{2+}]} = \frac{3.2 \times 10^{-7}}{[\mathrm{Sr}^{2+}]}\] We will now rearrange the equation to isolate the unknowns: \[[\mathrm{Pb}^{2+}][3.2 \times 10^{-7}] = [\mathrm{Sr}^{2+}][6.3 \times 10^{-7}]\] Now, let's divide both sides by \(6.3 \times 10^{-7}\) and simplify the equation: \[[\mathrm{Pb}^{2+}] = \frac{[\mathrm{Sr}^{2+}]}{2}\] Now let's express \([\mathrm{SO}_{4}^{2-}]\) in terms of \([\mathrm{Pb}^{2+}]\), using one of the expressions above: \[[\mathrm{SO}_{4}^{2-}] = \frac{6.3 \times 10^{-7}}{[\mathrm{Pb}^{2+}]}\] Assuming the sulfates come only from PbSO4 and SrSO4, we write the total concentration of sulfate: \[[\mathrm{SO}_{4}^{2-}]_{total} = [\mathrm{SO}_{4}^{2-}]_{Pb} + [\mathrm{SO}_{4}^{2-}]_{Sr}\] \underline{\textbf{Important Note:}} This exercise cannot be solved without additional information, such as the initial concentrations of the ions, or a solubility constraint. For example, if we know the initial concentrations of either Pb²⁺ or Sr²⁺, we can solve the problem. In this context, the exercise as given cannot be fully solved.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is crucial when studying solubility in chemistry. It refers to the state of a reaction where the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of reactants and products remain constant over time. At equilibrium, although both reactions occur, there is no net change in the concentrations of the chemical species.

When we dissolve a sparingly soluble salt, like PbSO4 or SrSO4, in water, it dissociates into its ions. However, not all solid dissolves; some amount remains intact. At this point, the dissolution of additional solid and the reformation of solid from ions in solution occur at equal rates, creating a dynamic equilibrium. The solubility product constant (Ksp) quantifies this exact balance point, correlating to the maximum product of the ion concentrations that can coexist in a saturated solution without additional solid precipitating out.

Solubility Calculations

Solubility calculations involve determining the amount of a substance that can dissolve in a solvent at a given temperature, which is crucial for understanding how various compounds behave in solution. In the case of sparingly soluble ionic solids, this means calculating the concentrations of the ions in solution at equilibrium.

In most cases, the solubility of a substance in water can be directly calculated if its Ksp is known. For example, if you have a simple ionic compound that dissociates into one cation and one anion, the solubility can be expressed as the concentration of either ion present in a saturated solution. Since the stoichiometry of dissolution is known, by establishing the relationship between the ion concentrations and Ksp, we can calculate the solubility. However, these calculations become more complex when multiple solutes affect the equilibrium concentrations of shared ions, as is the case with coexisting PbSO4 and SrSO4.

Ksp Expressions

Ksp, or the solubility product constant, provides insight into the solubility of a compound. It is the product of the concentrations of the ions resulting from a salt in a saturated solution, each raised to the power of their coefficient in the balanced dissolution equation. The general expression for a salt AX By is given by:

Ksp= [A^+]^m [B^-]^n. Here m and n represent the stoichiometric coefficients from the balanced equation for the dissolution of the salt in water.

For instance, the Ksp expression for PbSO4 is written as Ksp = [Pb²⁺][SO₄²⁻]. In a solution with both PbSO4 and SrSO4, the sulfate concentration is a common ion affecting the solubility of both salts. The calculations for ion concentrations in these solutions require setting up a system of equations using the individual Ksp values for each salt and then solving the system considering the contribution to the sulfate concentration from both salts. Nevertheless, as per the exercise note, additional information such as initial concentrations is often needed to uniquely determine the values of all ionic concentrations in such complex equilibria.