Question

Write the solubility product expression for the ionic compound \(\mathrm{A}_{x} \mathrm{~B}_{y}\)

Short answer

\( K_{sp} = [\text{A}^{y+}]^x [\text{B}^{x-}]^y \)

Step-by-step solution

Understand the decompostion of the compound

The ionic compound \( \text{A}_x \text{B}_y \) dissociates into its ions in solution. The dissociation can be written as: \( \text{A}_x \text{B}_y (s) \rightleftharpoons x\text{A}^{y+} (aq) + y\text{B}^{x-} (aq) \). Here, "s" stands for solid and "aq" stands for aqueous.

Define the solubility product constant, \(K_{sp}\)

The solubility product constant, \( K_{sp} \), is defined based on the equilibrium concentrations of the ions in the aqueous solution. For the general dissolution of an ionic compound \( \text{A}_x \text{B}_y \), \( K_{sp} \) expression is given as: \( K_{sp} = [\text{A}^{y+}]^x [\text{B}^{x-}]^y \).

Write the solubility product expression

Write the solubility product expression by substituting the stoichiometric coefficients and the ions from the dissolution equation. Based on the dissociation of \( \text{A}_x \text{B}_y \), the solubility product expression is \( K_{sp} = [\text{A}^{y+}]^x [\text{B}^{x-}]^y \). This formula accounts for the stoichiometry and the charges on the ions.

Key concepts

Ionic Compounds

Ionic compounds are substances composed of positive and negative ions, which come together due to attractive electrostatic forces. These compounds form a crystal lattice structure where ions are arranged in a repeating pattern. This structure is maintained by the force of attraction between oppositely charged ions, resulting in a stable but rigid formation.

Some basic properties of ionic compounds include:

• High melting and boiling points due to strong ionic bonds.
• Solubility in water; as a polar solvent, water can overcome the electrostatic forces holding the ions together.
• They conduct electricity when dissolved in water or melted, as the ions become free to move.

Ionic compounds often dissolve in water and dissociate into their constituent ions, which is a crucial step in many chemical reactions.

Dissociation

Dissociation refers to the process where ionic compounds separate into ions when they dissolve in a solvent like water. For a compound like \(\text{A}_x\text{B}_y\), dissociation involves breaking its solid structure into its component ions in solution. This process is vital because it allows the ions to interact with other substances in solution.

Consider the dissociation of \(\text{A}_x\text{B}_y\), which can be represented by the equation: \[ \text{A}_x\text{B}_y (s) \rightleftharpoons x\text{A}^{y+} (aq) + y\text{B}^{x-} (aq) \] Here:

• \(x\) is the number of \(\text{A}^{y+}\) ions produced.
• \(y\) is the number of \(\text{B}^{x-}\) ions produced.
• "s" signifies the solid state, and "aq" indicates that ions are in aqueous solution.

The breaking down of ionic compounds into ions is essential for reactions and dictates the behavior of the compound in a solution.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of ions in a solution when a chemical system comes to a dynamic balance in its dissociation process. After an ionic compound dissolves and dissociates, a dynamic equilibrium is established between the dissolved ions and the undissolved solid.

At equilibrium, the rates of dissolving and precipitating are equal, leading to constant concentrations of ions in solution. This means each species' concentration remains unchanged over time although the process is ongoing at the molecular level.

• The concentration of \(\text{A}^{y+}\) ions will be \([\text{A}^{y+}]\).
• The concentration of \(\text{B}^{x-}\) ions will be \([\text{B}^{x-}]\).

These equilibrium concentrations are critical as they are utilized to calculate the solubility product constant, \(K_{sp}\), which helps quantify solubility.

Solubility Product Constant

The solubility product constant, denoted as \(K_{sp}\), is a numerical value that expresses the product of the concentrations of the ions of a dissolved ionic compound at equilibrium. It's an important parameter for understanding how soluble a compound is in a given solvent.

For the ionic compound \(\text{A}_x \text{B}_y\), the \(K_{sp}\) is calculated using the equilibrium concentrations of its dissociated ions:\[K_{sp} = [\text{A}^{y+}]^x [\text{B}^{x-}]^y\]This expression accounts for both the concentration and the stoichiometry of the ions, making it a precise representation of solubility.

• \(x\) and \(y\) are the stoichiometric coefficients indicating the number of ions.
• Brackets \([...]\) signify molarity, which is the concentration of ions in moles per liter.

By knowing \(K_{sp}\), chemists can predict whether a precipitate will form in reactions, and it is fundamental in fields such as analytical chemistry and geochemistry.