PH of weak acids and bases Knowledge-Based Study

Study notes

1. Incomplete Ionization in Aqueous Solution

   Unlike strong acids and bases, which ionize completely in water, weak acids and bases undergo partial dissociation to establish a dynamic equilibrium.

   For a general weak monoprotic acid $\text{HA}$, the ionization in water is represented by the equilibrium expression:

   $$\text{HA}(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{A}^-(aq)$$

   The extent of dissociation is quantified by the acid ionization constant, $K_a$:

   $$K_a = \frac{[\text{H}_3\text{O}^+][\text{A}^-]}{[\text{HA}]}$$

   Similarly, for a weak base $\text{B}$, the equilibrium reaction and base ionization constant, $K_b$, are defined as:

   $$\text{B}(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{BH}^+(aq) + \text{OH}^-(aq)$$$$K_b = \frac{[\text{BH}^+][\text{OH}^-]}{[\text{B}]}$$

   Smaller values of $K_a$ and $K_b$ indicate weaker acids and bases, reflecting equilibria that lie predominantly to the reactant side.

   Systematic Approach Using an ICE Table

   To determine the pH of a weak acid solution with an initial concentration $C_0$, set up an Initial-Change-Equilibrium (ICE) table:

   • Initial: $[\text{HA}] = C_0$, $[\text{H}_3\text{O}^+] \approx 0$, $[\text{A}^-] = 0$
   • Change: $[\text{HA}] = -x$, $[\text{H}_3\text{O}^+] = +x$, $[\text{A}^-] = +x$
   • Equilibrium: $[\text{HA}] = C_0 - x$, $[\text{H}_3\text{O}^+] = x$, $[\text{A}^-] = x$

   Substituting these terms into the equilibrium constant expression yields:

   $$K_a = \frac{x^2}{C_0 - x}$$

   The 5% Approximation Rule

   If $K_a$ is sufficiently small such that $C_0 / K_a > 400$, the extent of ionization is negligible compared to $C_0$. We can assume $C_0 - x \approx C_0$:

   $$x = [\text{H}_3\text{O}^+] \approx \sqrt{K_a \cdot C_0}$$

   Once $x$ is determined, the pH is calculated directly:

   $$\text{pH} = -\log[\text{H}_3\text{O}^+]$$

   If the calculated value of $x$ exceeds 5% of $C_0$, the assumption is invalid, and the exact quadratic equation must be solved.

   Determining Hydroxide Concentration and pH

   For a weak base solution with initial concentration $C_0$, ionization produces hydroxide ions rather than hydronium ions. Using an ICE table, the equilibrium expression simplifies under the negligible ionization assumption to:

   $$K_b = \frac{[\text{BH}^+][\text{OH}^-]}{[\text{B}]} = \frac{x^2}{C_0 - x} \approx \frac{x^2}{C_0}$$

   Solving for the hydroxide ion concentration gives:

   $$x = [\text{OH}^-] \approx \sqrt{K_b \cdot C_0}$$

   To find the pH from $[\text{OH}^-]$, use the following sequential relations at 25 °C:

   1. Calculate $\text{pOH} = -\log[\text{OH}^-]$
   2. Determine $\text{pH} = 14.00 - \text{pOH}$

   Conjugate Acid-Base Relationship

   For any conjugate acid-base pair in aqueous solution at 25 °C:

   $$K_a \cdot K_b = K_w = 1.0 \times 10^{-14}$$$$\text{p}K_a + \text{p}K_b = 14.00$$

   This relationship allows the calculation of $K_b$ when only the $K_a$ of the conjugate acid is provided.

   Degree of Dissociation

   The degree of ionization provides a measure of acid or base strength under specific conditions. Percent ionization for a weak acid is defined as:

   $$\text{Percent Ionization} = \frac{[\text{H}_3\text{O}^+]_{\text{eq}}}{[\text{HA}]_0} \times 100\%$$

   For a weak base, it is formulated identically using the equilibrium hydroxide concentration relative to the initial base concentration:

   $$\text{Percent Ionization} = \frac{[\text{OH}^-]_{\text{eq}}}{[\text{B}]_0} \times 100\%$$

   Effect of Dilution (Le Chatelier's Principle)

   As a weak acid or weak base solution is diluted with water:

   • The total concentrations of all dissolved species decrease.
   • According to Le Chatelier's principle, the equilibrium shifts toward the side with more aqueous ions (the products), thereby increasing the percent ionization.
   • However, despite the higher percentage of dissociated molecules, the absolute concentration of $[\text{H}_3\text{O}^+]$ decreases, causing the solution pH to move closer to 7.00.