91Ó°ÊÓ

The solubility product constant, often abbreviated as \( K_{sp} \), is a key concept in understanding the solubility of sparingly soluble salts. It is an equilibrium constant that defines the maximum concentration of ions in a solution before precipitation occurs.
For example, in the dissociation of silver benzoate \( \text{AgC}_6\text{H}_5\text{COO(s)} \), the solubility product is given by the expression:\[K_{sp} = [\text{Ag}^+][\text{C}_6\text{H}_5\text{COO}^-]\]This equation signifies that the product of the equilibrium concentrations of the ions \([\text{Ag}^+]\) and \([\text{C}_6\text{H}_5\text{COO}^-]\) in solution must not exceed \( K_{sp} \).

• A small \( K_{sp} \) value implies that the salt is not very soluble in water.
• \( K_{sp} \) values are unique to each compound and vary with temperature.

Understanding \( K_{sp} \) allows chemists to predict whether a precipitate will form in a solution, based on the concentrations of the constituent ions.

The acid dissociation constant, denoted as \( K_a \), measures the strength of an acid in solution. It is the equilibrium constant for the ionization of a weak acid.
Considering our exercise, the benzene acid \( \text{C}_6\text{H}_5\text{COOH} \) partially dissociates in water as follows:\[\text{C}_6\text{H}_5\text{COOH} \rightleftharpoons \text{C}_6\text{H}_5\text{COO}^- + \text{H}^+\]The expression for the \( K_a \) is:\[K_a = \frac{[\text{C}_6\text{H}_5\text{COO}^-][\text{H}^+]}{[\text{C}_6\text{H}_5\text{COOH}]}\]

• A higher \( K_a \) value indicates a stronger acid that ionizes more completely.
• Weak acids, such as benzoic acid, have low \( K_a \) values.

By knowing the \( K_a \), one can calculate the concentration of the acid's ions at equilibrium, which is crucial in buffered solutions where pH is maintained constant.

Equilibrium concentrations refer to the concentrations of reactants and products in a chemical reaction at equilibrium. Using equilibrium constants such as \( K_{sp} \) or \( K_a \), the amounts of various ions present can be determined.
For the given problem, the equilibrium concentration of \([\text{C}_6\text{H}_5\text{COO}^-]\) was calculated using the acid dissociation constant \( K_a \) and the concentration of hydrogen ions \([\text{H}^+]\).

• Given \( K_a = 6.3 \times 10^{-5} \) and \([\text{H}^+] = 10^{-4} \), the concentration \([\text{C}_6\text{H}_5\text{COO}^-] = 0.63 \) M.
• The concentration \([\text{Ag}^+]\) was then calculated to be \( 3.97 \times 10^{-5} \) M using the \( K_{sp} \) expression.

Accurate calculation of these equilibrium concentrations is critical in determining how much of a substance will dissolve and remain in solution.

Buffered solutions play a vital role in controlling pH, which can vastly affect solubility.
Buffers resist changes in pH upon the addition of an acid or base because they contain significant amounts of both a weak acid and its conjugate base.
In this scenario, the solution is buffered at pH 4. This means that the \([\text{H}^+]\) concentration remains constant at \( 10^{-4} \) M, influencing the solubility of silver benzoate.

• The buffering effect ensures consistent ion concentrations, ensuring the ongoing physical and chemical stability of the solution.
• The given solubility of \( 0.0086 \) g/L was calculated by maintaining this buffered condition while considering the \( K_{sp} \) and \( K_a \).

This demonstrates how buffers help maintain desired conditions for controlled solubility and reaction rates, crucial in many chemical and biological applications.