WEB The pH of a \(2.642 M\) solution of a weak acid, \(\mathrm{HB}\), is \(5.32\). What is \(K_{\mathrm{a}}\) for the weak acid?

Understanding the pH of a solution is crucial for studying acid-base chemistry. The pH is a measure of the acidity or basicity of an aqueous solution. It is defined as the negative logarithm (base 10) of the concentration of hydrogen ions (\text{[H+]}) in the solution.

Mathematically, this is represented by the equation: \[ \text{pH} = -\log_{10}[\text{H}^+] \].
For a solution with a pH of 5.32, the concentration of H+ ions can be calculated using the inverse of the logarithmic function, leading to \[ \text{[H+]} = 10^{-\text{pH}} = 10^{-5.32} \].
This concentration will be an important part of determining the equilibrium constant for the dissociation of a weak acid.

At the heart of understanding weak acid behavior is the equilibrium expression. This mathematical statement quantifies the distribution of species in a reversible reaction at equilibrium. For a weak acid, which we can represent as HB, the dissociation reaction in aqueous solution is: \[ \text{HB} \rightleftharpoons \text{H}^+ + \text{B}^- \].

The equilibrium expression for this dissociation is written as: \[ K_a = \frac{[\text{H}^+][\text{B}^-]}{[\text{HB}]} \],
where \(K_a\) is the acid dissociation constant. It tells us the extent to which the weak acid dissociates into its ions in solution. The square brackets represent the concentration of each species at equilibrium. Subtle changes in these concentrations can significantly impact the acidity, and thus the pH, of the solution.

The acid dissociation constant, \(K_a\), is a fundamental quantity expressing a weak acid's strength. It is determined by the ratio of the concentration of the dissociated ions to the undissociated acid at equilibrium. A larger \(K_a\) value indicates a stronger acid, as it means more acid has dissociated.

In the context of this problem, we express \(K_a\) as: \[ K_a = \frac{([\text{H}^+] - [\text{H}^+]_0)^2}{[\text{HB}]_0 - ([\text{H}^+] - [\text{H}^+]_0)} \],
ignoring the initial concentration of hydrogen ions (\([\text{H}^+]_0\)) when it is significantly smaller compared to the concentration of hydrogen ions from the dissociation of the weak acid. Ultimately, calculating \(K_a\) from the pH and the concentrations of the species involved in the dissociation helps us understand the acid's behavior in solution.

The concentration of H+ ions in a solution is directly tied to the solution's pH and the dissociation of acids within it. Weak acids dissociate partially, and as a result, they contribute to the concentration of hydrogen ions in the solution. The stronger the weak acid, the more it will raise the H+ concentration, leading to a lower pH value.

When the concentration of these ions is calculated from pH, it can be used to find the acid dissociation constant, \(K_a\), by evaluating the equilibrium condition of the weak acid dissociation. In the exercise, we initially calculated the concentration of H+ ions to be \(10^{-5.32}\), and this value directly influences the equilibrium calculation and the final \(K_a\) value. Understanding how the concentration of H+ ions is determined and manipulated is foundational in the study of chemical equilibrium and acid-base reactions.