When acids and bases dissolve in water, they alter the concentration of hydrogen ions (\(\text{H}^+\)) and hydroxide ions (\(\text{OH}^-\)). Strong acids and bases dissociate almost completely, but weak acids and bases only partially ionize in a reversible process. This partial ionization establishes chemical equilibrium. To quantify the extent of this partial dissociation, chemists use the acid dissociation constant (\(\text{K}_a\)) and the base dissociation constant (\(\text{K}_b\)). These constants directly measure a substance’s strength in an aqueous solution.
Defining the Acid Dissociation Constant (\(\text{K}_a\))
The acid dissociation constant, \(\text{K}_a\), quantitatively measures the strength of a weak acid in water. When a generic weak acid (HA) is added to water, it donates a proton (\(\text{H}^+\)) to a water molecule. This reaction produces the hydronium ion (\(\text{H}_3\text{O}^+\)) and the acid’s conjugate base (\(\text{A}^-\)).
This reversible reaction is written as: \(\text{HA}(\text{aq}) + \text{H}_2\text{O}(\text{l}) \rightleftharpoons \text{H}_3\text{O}^+(\text{aq}) + \text{A}^-(\text{aq})\). The \(\text{K}_a\) value is derived from the Law of Mass Action, which expresses the ratio of product concentrations to reactant concentrations at equilibrium. The equilibrium expression is \(\text{K}_a = [\text{H}_3\text{O}^+][\text{A}^-] / [\text{HA}]\).
The square brackets indicate molar concentration. Water (\(\text{H}_2\text{O}\)) is excluded from the denominator because its concentration remains constant as the solvent. The numerator contains the products (\(\text{H}_3\text{O}^+\) and \(\text{A}^-\)), while the denominator contains the undissociated weak acid. A larger \(\text{K}_a\) value indicates that equilibrium favors the products, meaning a greater fraction of the acid has dissociated, signifying a stronger weak acid.
Defining the Base Dissociation Constant (\(\text{K}_b\))
The base dissociation constant, \(\text{K}_b\), measures the strength of a weak base in solution. A weak base (B) reacts with water by accepting a proton (\(\text{H}^+\)). This creates the base’s conjugate acid (\(\text{BH}^+\)) and the hydroxide ion (\(\text{OH}^-\)).
The general reaction is: \(\text{B}(\text{aq}) + \text{H}_2\text{O}(\text{l}) \rightleftharpoons \text{BH}^+(\text{aq}) + \text{OH}^-(\text{aq})\). The \(\text{K}_b\) expression is \(\text{K}_b = [\text{BH}^+][\text{OH}^-] / [\text{B}]\). This equation reflects the base’s ability to produce hydroxide ions, which are responsible for basic properties.
As with the \(\text{K}_a\) expression, water is omitted from the \(\text{K}_b\) expression. The numerator includes the conjugate acid and hydroxide ion concentrations, while the denominator holds the concentration of the unreacted base. A higher numerical value for \(\text{K}_b\) indicates a greater extent of dissociation, meaning the base is stronger.
Interpreting Acid and Base Strength (\(\text{pK}_a\) and \(\text{pK}_b\))
The numerical values for \(\text{K}_a\) and \(\text{K}_b\) often span a wide range, frequently involving small exponents, making comparison cumbersome. To simplify these numbers, chemists convert them to a logarithmic scale called \(\text{pK}_a\) and \(\text{pK}_b\). The \(\text{p}\) function is defined as the negative logarithm (base 10) of the constant: \(\text{pK}_a = -\text{log}(\text{K}_a)\) and \(\text{pK}_b = -\text{log}(\text{K}_b)\).
A larger \(\text{K}_a\) value corresponds to a stronger acid, but the relationship is reversed on the \(\text{pK}_a\) scale. For example, an acid with a \(\text{K}_a\) of \(10^{-3}\) (\(\text{pK}_a = 3\)) is stronger than one with a \(\text{K}_a\) of \(10^{-5}\) (\(\text{pK}_a = 5\)). Consequently, a smaller \(\text{pK}_a\) signifies a stronger acid.
The same inverse relationship applies to \(\text{pK}_b\) values. A smaller \(\text{pK}_b\) value corresponds to a higher \(\text{K}_b\) value, indicating a stronger base. This logarithmic transformation allows for easier comparison of acid and base strengths across many orders of magnitude.
The Conjugate Relationship Between \(\text{K}_a\) and \(\text{K}_b\)
Acids and bases are linked through conjugate pairs, differing only by a single proton (\(\text{H}^+\)). For example, the weak acid \(\text{HA}\) and its conjugate base \(\text{A}^-\) form a pair. The acid’s strength is directly related to the strength of its conjugate base through the ion-product constant for water, \(\text{K}_w\).
The ion-product constant for water, \(\text{K}_w\), is the equilibrium constant for water’s self-ionization, where two water molecules form \(\text{H}_3\text{O}^+\) and \(\text{OH}^-\). At \(25^\circ\text{C}\), \(\text{K}_w\) is fixed at \(1.0 \times 10^{-14}\). This constant links the dissociation constants of a conjugate acid-base pair.
The fundamental relationship is \(\text{K}_a \cdot \text{K}_b = \text{K}_w\). This means that if you multiply the acid dissociation constant of a weak acid by the base dissociation constant of its conjugate base, the product will always equal \(\text{K}_w\). This equation shows that a strong acid (large \(\text{K}_a\)) must have a conjugate base with a small \(\text{K}_b\), making it a weak base.
This inverse strength relationship can also be expressed on the logarithmic scale as \(\text{pK}_a + \text{pK}_b = 14\) at \(25^\circ\text{C}\). If the \(\text{pK}_a\) of an acid is known, this simple equation allows for the immediate calculation of the \(\text{pK}_b\) of its conjugate base, providing a complete picture of the pair’s relative strengths. This interconnectedness is a direct consequence of the equilibrium of water.