The sum of \(pK_a\) and \(pK_b\) equals 14 is a fundamental concept in acid-base chemistry. This relationship holds true for a conjugate acid-base pair measured in an aqueous solution under standard conditions, typically \(25^\circ C\). The constant value of 14 is a direct consequence of the chemical properties of water itself.
Defining Acid and Base Strength
Chemists use equilibrium constants to quantify the strength of an acid or a base in a solution. For an acid, the extent to which it dissociates is measured by the acid dissociation constant, \(K_a\). A larger \(K_a\) value indicates a stronger acid. Similarly, the base dissociation constant, \(K_b\), measures a base’s ability to accept a hydrogen ion, with a larger \(K_b\) indicating a stronger base.
Because \(K_a\) and \(K_b\) values are often extremely small, they are difficult to compare directly. To simplify these measurements, scientists employ a logarithmic scale. The \(pK_a\) is defined as the negative logarithm (base 10) of the \(K_a\) value: \(pK_a = -\log_{10}(K_a)\). A lower \(pK_a\) value corresponds to a stronger acid. Similarly, \(pK_b\) is defined as \(pK_b = -\log_{10}(K_b)\), where a lower \(pK_b\) indicates a stronger base.
The Ion Product of Water
The constant 14 originates from the self-ionization of water, a process called autoionization. Even in pure water, a small fraction of water molecules react to produce hydronium ions (\(\text{H}_3\text{O}^+\)) and hydroxide ions (\(\text{OH}^-\)). This reaction establishes a chemical equilibrium described by the ion product of water, \(K_w\).
The \(K_w\) constant is the product of the concentrations of the hydronium and hydroxide ions: \([\text{H}_3\text{O}^+][\text{OH}^-]\). At the standard temperature of \(25^\circ C\), this constant is experimentally determined to be \(1.0 \times 10^{-14}\).
Applying the logarithmic p-scale to \(K_w\) yields \(pK_w = -\log_{10}(K_w)\). Substituting the value for \(K_w\) at \(25^\circ C\) results in \(pK_w = -\log_{10}(1.0 \times 10^{-14})\), which equals 14.00.
Proving the Relationship
The mathematical link between acid strength and base strength begins with the fundamental relationship for any conjugate acid-base pair: \(K_a \times K_b = K_w\). This equation is derived by multiplying the equilibrium expressions for the acid and base dissociation reactions. This product must always equal the ion product of water, \(K_w\), since the reactions are occurring in an aqueous environment.
To transform this multiplicative relationship into the additive \(pK\) form, one takes the negative logarithm of both sides: \(-\log(K_a \times K_b) = -\log(K_w)\). Using the property that the logarithm of a product is the sum of the logarithms, this becomes \([-\log(K_a)] + [-\log(K_b)] = -\log(K_w)\).
By definition, this substitution yields the additive form: \(pK_a + pK_b = pK_w\). Since \(pK_w\) equals 14.00 at \(25^\circ C\), the final relationship for a conjugate pair is established as \(pK_a + pK_b = 14.00\).
Applying the Relationship to Conjugate Pairs
The \(pK_a + pK_b = 14\) relationship is a powerful tool for predicting the behavior of chemicals in water, but it applies only to a specific pairing: an acid and its corresponding conjugate base. A conjugate acid-base pair consists of two species that differ by only a single hydrogen ion. For example, acetic acid (\(\text{CH}_3\text{COOH}\)) and the acetate ion (\(\text{CH}_3\text{COO}^-\)) form a conjugate pair.
This mathematical constant clearly demonstrates the inverse relationship between the strength of an acid and its conjugate base. If an acid is very strong, it will have a very low \(pK_a\) value, sometimes even a negative number. Because the sum must equal 14, its conjugate base must necessarily have a very high \(pK_b\).
Consider hydrochloric acid (\(\text{HCl}\)), a very strong acid with an estimated \(pK_a\) of about -7. Its conjugate base is the chloride ion (\(\text{Cl}^-\)). Using the relationship, the chloride ion must have a \(pK_b\) of \(14 – (-7)\), which equals 21. This extremely high \(pK_b\) value confirms the principle that a strong acid yields an exceptionally weak conjugate base.