Question

What is the relationship between \(K_{\mathrm{sp}}\) and the product of the ion concentrations in terms of determining whether a solution of those ions is saturated?

Short answer

Compare \(Q\) (product of ion concentrations) to \(K_{\text{sp}}\) to determine saturation: \(Q < K_{\text{sp}}\) = unsaturated, \(Q = K_{\text{sp}}\) = saturated, \(Q > K_{\text{sp}}\) = supersaturated.

Step-by-step solution

- Understand the Concept of Solubility Product Constant (\(K_{\text{sp}}\text{)}

The solubility product constant, represented as \(K_{\text{sp}}\), essentially reflects the extent to which a compound can dissolve in water. It is used for sparingly soluble salts. The \(K_{\text{sp}}\) is determined by the equilibrium concentrations of the ions in a saturated solution of the salt.

- Define the Relationship

\(K_{\text{sp}}\) of a compound is calculated from the concentrations of its constituent ions at equilibrium. For a generic salt \AB\ that dissociates into \A^+\ and \B^-\, the expression is: \K_{\text{sp}} = [A^+][B^-]\.

- Compare Product of Ion Concentrations to \(K_{\text{sp}}\)

To determine if a solution is saturated, unsaturated, or supersaturated, compare the product of actual ion concentrations \Q = [A^+][B^-]\ to \(K_{\text{sp}}\). 1) \(Q < K_{\text{sp}}\) indicates an unsaturated solution. 2) \(Q = K_{\text{sp}}\) means the solution is saturated. 3) \(Q > K_{\text{sp}}\) signifies a supersaturated solution.

Key concepts

Understanding Saturation and Ksp

Saturation refers to the point at which a solution cannot dissolve any more solute at a given temperature. When a solution is saturated, the rate at which the solute dissolves is equal to the rate at which it precipitates. This means that the solution has reached an equilibrium.

Saturation is determined by comparing the solubility product constant, or \(K_{\text{sp}}\), to the product of the ion concentrations in the solution. \(K_{\text{sp}}\) is a specific value for each compound that represents the product of the concentrations of its ions in a saturated solution at a given temperature. For instance, if you have a salt \(AB\) that dissociates into \(A^+\) and \(B^-\), the \(K_{\text{sp}}\) expression would be: \[K_{\text{sp}} = [A^+][B^-].\]

When the actual ion product, denoted as \(Q\), which is \[Q = [A^+][B^-].\] , differs from \(K_{\text{sp}}\), the saturation status of the solution can be determined.

Ion Concentration and Its Role

Ion concentration is crucial in determining whether a solution is saturated. The concentration of ions is influenced by how much of the compound has dissolved. When a solution of a salt (e.g., \(AB\)) is prepared, the ions \(A^+\) and \(B^-\) will dissolve into the solution until the solution reaches saturation.

The relationship between ion concentration and \(K_{\text{sp}}\) is described by the equation \[K_{\text{sp}} = [A^+][B^-].\] If the calculated ion product, \(Q\), is less than \(K_{\text{sp}}\), then more salt can dissolve, indicating an unsaturated solution. If \(Q\) equals \(K_{\text{sp}}\), the solution is exactly saturated. Lastly, if \(Q\) exceeds \(K_{\text{sp}}\), the solution is supersaturated, meaning excess salt will start to precipitate out because the solution contains more dissolved ions than can be maintained in equilibrium.

The Concept of Equilibrium in Solubility

Equilibrium in the context of solubility refers to the state in which the rate of dissolution and the rate of precipitation of a solute are equal, leading to a dynamic balance. At this point, the concentrations of the dissolved ions remain constant over time.

When dealing with a sparingly soluble salt, equilibrium is described by the solubility product constant, \(K_{\text{sp}}\). This constant value is a reflection of the maximum amount of ions that can exist in solution without causing precipitation. The mathematical representation is \[K_{\text{sp}} = [A^+][B^-].\]

To determine whether a mixture of ions in a solution has reached equilibrium, you compare the ion product \(Q\) to \(K_{\text{sp}}\). If \(Q = K_{\text{sp}}\), the system is at equilibrium, and the solution is saturated with the dissolved ions. Understanding these concepts is key to mastering problems involving solubility and \(K_{\text{sp}}\).