Problem 33 - Chemistry The Molecular Nature of Matter

Question

In water, the solubility of lead(II) chloride is \(0.016 M\). Use this information to calculate the value of \(K_{\mathrm{sp}}\) for \(\mathrm{PbCl}_{2}\).

Step-by-Step Solution

Answer

The value of \(K_{\mathrm{sp}}\) for \(\mathrm{PbCl}_{2}\) is \(1.6384 \times 10^{-5}\).

Step 1: Write the Dissociation Equation

The solubility product (K_{\mathrm{sp}}) refers to the product of the concentrations of the ions each raised to the power of their stoichiometric coefficients. For lead(II) chloride dissolving in water, the equation is: \[\mathrm{PbCl}_{2}(s) \rightarrow \mathrm{Pb}^{2+}(aq) + 2\mathrm{Cl}^- (aq)\].

Step 2: Express the Equilibrium Concentrations

Because the solubility is given as 0.016 M, the concentration of \(\mathrm{Pb}^{2+}\) ions in a saturated solution is 0.016 M. The dissociation produces 2 moles of \(\mathrm{Cl}^-\) ions for every 1 mole of \(\mathrm{PbCl}_{2}\) that dissolves, so the concentration of \(\mathrm{Cl}^-\) ions is 2 \cdot 0.016 M = 0.032 M.

Step 3: Calculate the Solubility Product Constant

The solubility product constant, \(K_{\mathrm{sp}}\), can be calculated by substituting the equilibrium concentrations into the expression for \(K_{\mathrm{sp}}\): \[K_{\mathrm{sp}} = [\mathrm{Pb}^{2+}][\mathrm{Cl}^-]^{2} = (0.016) \cdot (0.032)^{2}\].

Step 4: Solve for the Solubility Product Constant

Perform the calculation: \[K_{\mathrm{sp}} = 0.016 \cdot (0.032)^{2} = 0.016 \cdot 0.001024 = 1.6384 \times 10^{-5}\]. The value of \(K_{\mathrm{sp}}\) for \(\mathrm{PbCl}_{2}\) is therefore \(1.6384 \times 10^{-5}\).

Key Concepts

Solubility in Water

When we discuss the solubility in water of a compound like lead(II) chloride, we're referring to how well it can dissolve in water to form its constituent ions. Solubility is usually expressed in terms of concentration, typically moles per liter (M). The solubility of a substance in water is determined by various factors, including temperature, pressure, and the presence of other substances. Substances that dissolve well in water are termed 'soluble', while those that do not are 'insoluble'.

For sparingly soluble salts such as lead(II) chloride, the solubility is low, and they only dissolve to a small extent. Such compounds reach a point of equilibrium in water, where the rate of dissolution equals the rate of precipitation. This dynamic balance is described by the solubility product constant, or Ksp, which quantifies the solubility at a given temperature.

Lead(II) Chloride Solubility

Lead(II) chloride (PbCl₂) is an example of a sparingly soluble salt, meaning it has a low solubility in water. At a given temperature, the solubility of lead(II) chloride is fixed and can be denoted as a concentration, for instance, 0.016 M. This value represents the maximum amount of lead(II) chloride that can dissolve in water to produce ions.

Understanding the solubility of a compound like lead(II) chloride is crucial in various applications, including environmental science and industrial processes. For instance, knowing its solubility allows for the prediction of how it behaves in natural waters, which is essential for assessing its potential impact on both human health and the environment.

Chemical Equilibrium

Chemical equilibrium is a state in which the forward and reverse reactions occur at the same rate, resulting in no overall change in the concentration of reactants and products. This concept is central to understanding solubility phenomena, including the behavior of lead(II) chloride in water.

At equilibrium, the dissolution of lead(II) chloride in water is balanced by the process of crystallization, where ions in solution reform the solid compound. The solubility product constant (Ksp) is a way to express this equilibrium quantitatively. It is determined by the concentrations of the ions at equilibrium, each raised to the power of its stoichiometric coefficient from the balanced dissolution equation. The Ksp value is unique to each compound and is typically constant at a fixed temperature.

Stoichiometry

Stoichiometry is the area of chemistry that pertains to the quantitative relationships of substances as they participate in chemical reactions. When a compound like lead(II) chloride dissolves in water, stoichiometry allows us to relate the amount of solid that dissolves to the amount of ions produced.

The stoichiometric coefficients from the balanced equation \(\mathrm{PbCl}_{2}(s) \rightarrow \mathrm{Pb}^{2+}(aq) + 2\mathrm{Cl}^- (aq)\) show that one mole of lead(II) chloride produces one mole of lead ions and two moles of chloride ions. Using these ratios, we calculate the concentrations of ions needed to determine the solubility product constant (Ksp), providing insight into the solubility of the compound in water.