For a weak acid, HA: $$ \mathrm{HA}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) $$ a) Write the equilibrium expression for \(K_{\mathrm{a}}\). b) Show that \(\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}-\log \frac{[\mathrm{HA}]}{\left[\mathrm{A}^{-}\right]}\)

The acid dissociation constant, denoted as Ka, is a numerical value that represents the strength of a weak acid in solution. It is defined by the equilibrium concentration of the acid (HA), its conjugate base (A-), and the hydronium ion (H3O+), formed when the acid donates a proton (H+) to water.

The equilibrium expression for a weak acid dissociation can be represented as: $$K_a = \frac{[H_3O^+][A^-]}{[HA]}$$where [H3O+] is the concentration of hydronium ions, [A-] is the concentration of the conjugate base, and [HA] is the concentration of the undissociated acid. A larger Ka value indicates a stronger weak acid, which dissociates more in solution, thereby producing more hydronium ions.

pH is a logarithmic measure of the hydrogen ion concentration, and it is inversely related to the concentration of H3O+ ions in the solution. The pH value provides information about the acidity or basicity of a solution. The formula to calculate pH is given by: $$pH = -\log[H_3O^+]$$For weak acids, the pH is not only influenced by the concentration of H3O+ ions but also by the equilibrium established between the acid and its conjugate base. As such, the concentration of the undissociated acid and its conjugate base also affects the pH of the solution.

The equilibrium expression for a weak acid illustrates the dynamic balance between the forward and reverse reactions of acid dissociation. The expression is derived from the law of mass action, which states that for a reversible chemical reaction at equilibrium and a constant temperature, a certain ratio of reactant and product concentrations has a constant value, known as the equilibrium constant (Ka for acids).

The general form of the equilibrium expression for the dissociation of a weak acid is:$$K_a = \frac{[H_3O^+][A^-]}{[HA]}$$This ratio remains constant for a given acid at a constant temperature, helping in the prediction of the behavior of the acid in various conditions.

The Henderson-Hasselbalch equation is particularly helpful for calculating the pH of buffer solutions, which consist of a mixture of a weak acid and its conjugate base. This equation is derived from the equilibrium expression for an acid and the definition of pH. It states:$$pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right)$$In this equation, pKa is the negative logarithm of the acid dissociation constant, Ka, and it represents the pH at which the concentrations of the acid and its conjugate base are equal.

Rearranging this equation as in the textbook solution, we can express the pH solely in terms of pKa and the ratio of the acid and base concentrations. This is an invaluable tool for chemists and biochemists, as it simplifies the complex processes into a formula that can predict the pH of a solution under certain conditions.