The acidity of a solution is measured using the pH scale, which indicates the concentration of hydrogen ions (\(\text{H}^+\)). Calculating the pH of a strong acid is straightforward because it fully dissociates in water. Determining the pH of a weak acid, however, requires a detailed approach involving chemical equilibrium. Weak acids only partially break down into their constituent ions, meaning the final hydrogen ion concentration is not equal to the acid’s initial concentration. This multi-step process involves defining the acid’s dissociation constant, tracking concentration changes, solving the resulting expression for the hydrogen ion concentration, and finally converting that value into the pH.
Defining Weak Acids and the Acid Dissociation Constant
Weak acids are defined by their characteristic of only partially ionizing when dissolved in water, establishing a state of chemical equilibrium. For a generic weak acid (HA), the reaction is reversible and represented as \(\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\). In this equilibrium, the undissociated acid molecules, hydrogen ions, and the conjugate base ions all coexist in a constant ratio.
The degree to which a weak acid dissociates is quantified by the Acid Dissociation Constant (\(\text{K}_a\)). This constant indicates the acid’s strength, where a smaller \(\text{K}_a\) value corresponds to a weaker acid that dissociates less. The \(\text{K}_a\) is mathematically expressed as the ratio of the product concentrations to the reactant concentration at equilibrium: \(\text{K}_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\).
The \(\text{K}_a\) value is a fixed property for any given weak acid at a specific temperature, making it the necessary starting point for any pH calculation. Because weak acids only partially ionize, their \(\text{K}_a\) values are typically very small, reflecting that the reactant (HA) is heavily favored over the products. This small \(\text{K}_a\) necessitates the use of a complex equilibrium calculation method.
Setting Up the Calculation Using the ICE Table
To determine the concentration of hydrogen ions produced by the partial dissociation of the weak acid, the ICE table is used to track changes in concentration. ICE stands for Initial, Change, and Equilibrium, systematically organizing the concentrations of the reactants and products involved in the reaction \(\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\).
The ‘Initial’ row (I) is populated with the starting molar concentration of the acid (HA) and zero for the products (\(\text{H}^+\) and \(\text{A}^-\)). The ‘Change’ row (C) represents the shift in concentration required to reach equilibrium. Since the acid dissociates, its concentration decreases by ‘x’, while the product concentrations increase by ‘x’ due to the 1:1:1 stoichiometric ratio.
The final row, ‘Equilibrium’ (E), combines the initial and change rows. The equilibrium concentration of the undissociated acid is \([\text{HA}] = [\text{HA}]_{\text{initial}} – \text{x}\), and the concentrations of the products are \([\text{H}^+] = \text{x}\) and \([\text{A}^-] = \text{x}\). This systematic setup allows the unknown change ‘x’, which is the equilibrium concentration of the hydrogen ion, to be substituted directly into the \(\text{K}_a\) expression.
Solving the Equilibrium Expression for Hydrogen Ion Concentration
Once the equilibrium concentrations are defined in terms of ‘x’ using the ICE table, they are substituted into the acid dissociation constant expression: \(\text{K}_a = \frac{(\text{x})(\text{x})}{[\text{HA}]_{\text{initial}} – \text{x}}\). This equation must then be solved for ‘x’, which represents the molar concentration of hydrogen ions, \([\text{H}^+]\), at equilibrium. The resulting algebraic equation is a quadratic expression in the form of \(\text{ax}^2 + \text{bx} + \text{c} = 0\), which can be solved using the quadratic formula.
A common simplification, known as the approximation method, is often employed to avoid the complexity of the quadratic formula. This method assumes that because the \(\text{K}_a\) value for a weak acid is very small, the amount of acid that dissociates (‘x’) is negligible compared to the initial concentration of the acid, \([\text{HA}]_{\text{initial}}\). This allows the denominator term \([\text{HA}]_{\text{initial}} – \text{x}\) to be simplified to just \([\text{HA}]_{\text{initial}}\), resulting in the simplified expression \(\text{K}_a \approx \frac{\text{x}^2}{[\text{HA}]_{\text{initial}}}\).
The approximation is considered valid if the calculated value of ‘x’ is less than 5% of the initial acid concentration. If the acid is stronger (higher \(\text{K}_a\)) or the initial concentration is very low, the 5% rule may fail, necessitating the use of the full quadratic formula for greater accuracy. Regardless of the method used, the result of this step is the accurate, calculated value of the equilibrium hydrogen ion concentration, \([\text{H}^+]\).
Converting Hydrogen Ion Concentration to pH
With the equilibrium hydrogen ion concentration, \([\text{H}^+]\), now accurately determined, the final step is a straightforward mathematical conversion to find the pH. The pH is defined as the negative logarithm (base 10) of the molar hydrogen ion concentration, expressed by the formula \(\text{pH} = -\log[\text{H}^+]\).
The logarithmic function compresses the wide range of possible hydrogen ion concentrations into the manageable pH scale, typically running from 0 to 14. Because the concentration is expressed as a negative logarithm, a higher concentration of hydrogen ions corresponds to a lower, more acidic pH value. Calculating the negative log of the \([\text{H}^+]\) value yields the final pH of the weak acid solution.