Question

Sparingly soluble \(\mathrm{PbCl}_{2}\) dissolves in water to yield an equilibrium \(\mathrm{Pb}^{2+}(a q)\) concentration of \(0.039 \mathrm{M}\). (a) Write the balanced equilibrium equation for \(\mathrm{PbCl}_{2}(\) s) dissolving in water. (b) Write the \(K_{\text {sp }}\) expression for \(\mathrm{PbCl}_{2}\). (c) What is the equilibrium concentration of chloride ion? (d) Calculate the value of \(K_{\mathrm{sp}}\) for \(\mathrm{PbCl}_{2}\) (show your calculation).

Short answer

The balanced equilibrium equation for PbCl₂ dissolving in water is: PbCl₂ (s) ⇌ Pb²⁺ (aq) + 2Cl⁻ (aq). The \(K_{sp}\) expression for PbCl₂ is: \(K_{sp} = [Pb^{2+}][Cl^{-}]^2\). The equilibrium concentration of chloride ion (Cl⁻) is 0.078 M. The value of \(K_{sp}\) for PbCl₂ is 0.000237.

Step-by-step solution

(a) Balanced equilibrium equation for PbCl₂ dissolving in water

First, we need to write the balanced equation for the dissolution process of PbCl₂ in water. When PbCl₂ dissolves in water, it dissociates into Pb²⁺ and Cl⁻ ions. The balanced equation is: PbCl₂ (s) ⇌ Pb²⁺ (aq) + 2Cl⁻ (aq)

(b) \(K_{sp}\) expression for PbCl₂

Next, we will write the expression for the solubility product constant (\(K_{sp}\)) for PbCl₂, which helps to quantify the solubility of a sparingly soluble salt in water. The \(K_{sp}\) expression for the above equilibrium is: \(K_{sp} = [Pb^{2+}][Cl^{-}]^2\)

(c) Equilibrium concentration of chloride ion

We are given the equilibrium concentration of Pb²⁺ ions as 0.039 M. According to the balanced equation, for each mole of PbCl₂ dissolved, two moles of Cl⁻ ions are produced. Therefore, the equilibrium concentration of Cl⁻ ions will be twice the concentration of Pb²⁺ ions. Equilibrium concentration of Cl⁻ ions: \([Cl^-] = 2 × [Pb^{2+}]\) \([Cl^-] = 2 × 0.039 M\) \([Cl^{-}] = 0.078 M\)

(d) Calculate the value of \(K_{sp}\) for PbCl₂

Now, we will calculate the value of \(K_{sp}\) for PbCl₂ by substituting the equilibrium concentrations of Pb²⁺ and Cl⁻ ions in the \(K_{sp}\) expression. \(K_{sp} = [Pb^{2+}][Cl^{-}]^2\) \(K_{sp} = (0.039)(0.078)^2\) \(K_{sp} = 0.000237\) So, the value of \(K_{sp}\) for PbCl₂ is 0.000237.

Key concepts

Equilibrium Concentration

Understanding equilibrium concentration is essential when studying chemical reactions that reach a state of balance. This equilibrium occurs when the rate of the forward reaction, where reactants turn into products, equals the rate of the backward reaction, where products revert to reactants. At this point, the concentration of the reactants and products remains constant.

For sparingly soluble salts such as lead(II) chloride (PbCl2), the equilibrium concentrations of the ions in solution can be determined. Given the equilibrium concentration of lead(II) ions (Pb2+) in our exercise, one can deduce the concentration of chloride ions (Cl-) using the stoichiometry of the balanced dissolution equation. In this case, for each mole of PbCl2 that dissolves, two moles of Cl- ions are produced, leading to the equilibrium concentration of Cl- being double that of Pb2+. Understanding the equilibrium concentration not only allows for calculation of solubility product constant but is also a fundamental concept in understanding the behavior of solutions at equilibrium.

Dissolution of Ionic Compounds

Ionic compounds like PbCl2 dissolve in water to form ions. This process is called dissolution. The solid salt separates into its constituent ions, becoming surrounded by water molecules. This interaction is facilitated by the polar nature of water molecules, which can stabilize the positive and negative ions in solution.

The extent to which an ionic compound dissolves in water is governed by factors such as the lattice energy of the solid and the hydration energy of the ions. Sparingly soluble salts are those that dissolve to a small extent, resulting in low equilibrium concentrations of ions in solution. These salts reach a dynamic equilibrium where the rate of dissolution equals the rate of precipitation, a necessary concept for understanding the solubility product constant (Ksp).

Sparingly Soluble Salts

Sparingly soluble salts, like lead(II) chloride (PbCl2) mentioned in the exercise, have low solubility in water. Their solubility product constant (Ksp), is a unique value that represents the maximum product of the ionic concentrations at equilibrium. This constant is a critical parameter in predicting the extent of dissolution and whether a precipitate will form under certain conditions.

To illustrate, the value of Ksp can suggest how much of the salt can be dissolved before the solution becomes saturated and excess solid remains undissolved. The exercise effectively demonstrates the calculation of Ksp, which provided us with a clearer picture of how sparingly soluble salts behave in aqueous solutions, an essential concept in both theoretical and practical chemistry applications.

Chemical Equilibrium

Chemical equilibrium is a condition where the concentrations of reactants and products in a reversible chemical reaction remain constant over time because the rates of the forward and reverse reactions are equal. This is a dynamic state, not a static one; molecules continue to react, but there is no net change in concentration.

In the context of our PbCl2 dissolution example, chemical equilibrium occurs when the rate at which PbCl2 dissolves to form Pb2+ and Cl- ions (forward reaction) is equal to the rate at which these ions come together to form solid PbCl2 (reverse reaction). Being able to calculate the Ksp gives a quantitative insight into this dynamic equilibrium, crucial for predicting the solubility and stability of a compound within a solution.