The acidity of a solution is measured by its pH, a scale indicating the concentration of hydrogen ions present. In chemistry, acids are substances that donate a proton, or hydrogen ion (\(H^+\)), when dissolved in water. The strength of an acid is determined by how readily it releases this proton into the solution. Weak acids only partially dissociate, requiring a specific metric to quantify this limited release of \(H^+\). This metric is the acid dissociation constant, \(K_a\), which is used to accurately determine the pH of weak acid solutions.
Defining Acid Strength and the Equilibrium Constant (\(K_a\))
The acid dissociation constant (\(K_a\)) serves as a quantitative measure of an acid’s strength in an aqueous solution. It is an equilibrium constant that describes the extent of the acid’s ionization, or dissociation, into ions. When a weak acid is placed in water, it reaches a state of chemical equilibrium.
The value of \(K_a\) provides a direct relationship to acid strength. A larger \(K_a\) value indicates that the acid dissociates to a greater extent, resulting in a higher concentration of \(H^+\) ions and therefore a stronger acid. Conversely, a smaller \(K_a\) value signifies that the acid remains mostly undissociated, characterizing it as a weaker acid. Weak acids, like acetic acid, only dissociate partially, which is why their \(K_a\) values are small, generally ranging from \(10^{-2}\) to \(10^{-14}\). Because of this partial dissociation, the \(K_a\) value is necessary to determine the concentration of \(H^+\) ions actually present at equilibrium.
The Mathematical Framework for Weak Acids
To determine the \(pH\) of a weak acid solution, the first step is to establish the chemical equilibrium expression. For a generic weak acid, \(HA\), the dissociation reaction is \(HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)\). The acid dissociation constant is mathematically defined by the equilibrium expression: \(K_a = \frac{[H^+][A^-]}{[HA]}\). The square brackets denote the molar concentrations of the species at equilibrium.
To find the equilibrium concentrations, a systematic approach using an Initial, Change, Equilibrium (ICE) table is necessary. The table organizes the initial concentration of the acid and tracks the change in concentration as it dissociates. The change in concentration, represented by the variable ‘x’, corresponds to the amount of acid that dissociates and the concentration of \(H^+\) ions formed.
A common simplifying assumption is applied if the \(K_a\) value is very small. This suggests that ‘x’ (the amount dissociated) will be negligible compared to the initial concentration of \(HA\). Therefore, the initial concentration of \(HA\) minus ‘x’ is approximated as just the initial concentration of \(HA\). This assumption is generally valid if ‘x’ is less than 5% of the initial concentration, allowing the equation to be solved without the complex quadratic formula.
Step-by-Step pH Calculation Using \(K_a\)
Consider a \(0.10 \text{ M}\) solution of a hypothetical weak acid, \(HA\), with a known \(K_a\) value of \(1.8 \times 10^{-5}\). The goal is to determine the \(pH\) of this solution.
Setting up the ICE Table and Equation
The ICE table is constructed using the initial concentration of \(HA\) as \(0.10 \text{ M}\). The equilibrium concentrations are expressed as \([HA] = 0.10 – x\), \([H^+] = x\), and \([A^-] = x\). Substituting these into the \(K_a\) equation yields: \(1.8 \times 10^{-5} = \frac{(x)(x)}{0.10 – x}\). Because the \(K_a\) is small, we apply the simplifying assumption, setting \(0.10 – x \approx 0.10\).
Solving for \(H^+\) Concentration and pH
The simplified equation becomes \(1.8 \times 10^{-5} = \frac{x^2}{0.10}\). To solve for \(x\), the equation is rearranged: \(x^2 = (1.8 \times 10^{-5}) \times (0.10)\), which yields \(x^2 = 1.8 \times 10^{-6}\). Taking the square root gives \(x = 1.34 \times 10^{-3} \text{ M}\).
This value of \(x\) is the molar concentration of hydrogen ions at equilibrium, \([H^+]\). The final step is to convert this concentration into the \(pH\) using the definition \(pH = -\log[H^+]\). Plugging in the calculated value gives \(pH = -\log(1.34 \times 10^{-3})\), resulting in a final \(pH\) of \(2.87\).
The \(pK_a\) Metric and Its Relationship to \(K_a\)
While \(K_a\) accurately measures acid strength, its values often involve small negative exponents, making them cumbersome to compare. Chemists frequently convert \(K_a\) into a more manageable logarithmic scale called \(pK_a\). The \(pK_a\) value is mathematically defined as the negative logarithm of the acid dissociation constant: \(pK_a = -\log(K_a)\).
Because of the inverse relationship imposed by the negative logarithm, a smaller \(pK_a\) value corresponds to a larger \(K_a\) value, indicating a stronger acid. For instance, an acid with a \(pK_a\) of \(3.0\) is stronger than one with a \(pK_a\) of \(5.0\). The \(pK_a\) metric is particularly useful in applications involving buffer solutions, which resist changes in \(pH\). The \(pK_a\) value of the weak acid component is directly related to the \(pH\) at which the buffer is most effective, allowing for the precise selection of acid-base pairs.