The acid dissociation constant, symbolized as \(K_a\), is a foundational concept in chemistry used to quantify the strength of an acid in a solution. It functions as an equilibrium constant, providing a quantitative measure of the extent to which an acid ionizes, or dissociates, into its constituent ions when dissolved in water. A higher value for \(K_a\) indicates that the acid dissociates more fully, meaning it is a stronger acid. Conversely, a lower \(K_a\) value signifies a weaker acid that remains largely undissociated in solution. Understanding the \(K_a\) value is necessary for predicting an acid’s behavior and for calculating the pH of its solution.
Defining the Acid Dissociation Equilibrium
The calculation of \(K_a\) begins with defining the chemical reaction for the acid’s dissociation in water. For any generic weak acid, represented as HA, the reaction is a reversible process that establishes a state of chemical equilibrium. In this reaction, the acid donates a proton (\(H^+\)) to a water molecule (\(H_2O\)), producing a hydronium ion (\(H_3O^+\)) and the acid’s conjugate base (\(A^-\)).
The general equilibrium equation is shown as \(\text{HA} + \text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{O}^+ + \text{A}^-\). The concentration of pure liquid water is considered constant and is therefore excluded from the \(K_a\) expression. The resulting mathematical formula for the acid dissociation constant is \(K_a = \frac{[\text{H}_3\text{O}^+][\text{A}^-]}{[\text{HA}]}\).
The square brackets denote the molar concentration of each species at equilibrium. The numerator contains the concentrations of the products, while the denominator contains the concentration of the undissociated acid. Because \(H_3O^+\) is often simplified to \(H^+\) for convenience in calculations, the expression may also be written as \(K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\).
Finding Equilibrium Concentrations Using pH
The most common method for calculating \(K_a\) requires the initial concentration of the acid and the solution’s measured pH. The first step is to use the experimental pH value to determine the concentration of hydrogen ions, \([\text{H}^+]\), present at equilibrium. This concentration is calculated using the inverse logarithmic relationship: \([\text{H}^+] = 10^{-\text{pH}}\).
This calculated \([\text{H}^+]\) concentration is equal to the equilibrium concentration of the conjugate base, \([\text{A}^-]\), because they are produced in a one-to-one molar ratio during the acid’s dissociation. The concept of an Initial, Change, Equilibrium (ICE) table is used to organize the concentrations of all reacting species.
The “Initial” row holds the starting concentration of the acid, \([\text{HA}]_{\text{initial}}\), and zero for the products. The “Change” row accounts for the fact that the acid concentration decreases by an amount, \(x\), while the product concentrations increase by the same amount, \(x\). The value of \(x\) is precisely the \([\text{H}^+]\) concentration calculated from the pH.
The “Equilibrium” row is then filled by combining the initial and change values for each species. For the products, \([\text{H}^+]\) and \([\text{A}^-]\), the equilibrium concentration is simply the value of \(x\) found from the pH. For the reactant acid, the equilibrium concentration, \([\text{HA}]_{\text{equilibrium}}\), is equal to \([\text{HA}]_{\text{initial}} – x\).
For instance, if a \(1.0\) molar solution of a weak acid has a pH of \(2.38\), the \([\text{H}^+]\) and \([\text{A}^-]\) are \(10^{-2.38}\), which is approximately \(0.0042\) molar. The equilibrium concentration of the undissociated acid is then \(1.0 – 0.0042\), or \(0.9958\) molar. These three numerical values represent the equilibrium concentrations needed to solve the \(K_a\) expression.
Solving the \(K_a\) Expression and Interpreting Results
Once the equilibrium concentrations for the acid, the hydrogen ion, and the conjugate base have been determined, the final step is to substitute these values into the \(K_a\) expression. The calculated concentrations are plugged into the formula \(K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]_{\text{equilibrium}}}\). The product of the two ion concentrations in the numerator is then divided by the concentration of the remaining undissociated acid in the denominator.
For the example where \([\text{H}^+] = 0.0042\) M, \([\text{A}^-] = 0.0042\) M, and \([\text{HA}] = 0.9958\) M, the calculation yields a \(K_a\) value of approximately \(1.8 \times 10^{-5}\). This final numerical value is the acid dissociation constant for that specific acid at the temperature of the measurement.
The magnitude of the resulting \(K_a\) value indicates the acid’s strength. A large \(K_a\) value (greater than 1) suggests a strong acid because the equilibrium favors the products, meaning a high degree of dissociation. A small \(K_a\) value (typically between \(10^{-2}\) and \(10^{-14}\)) signifies a weak acid, meaning most of the acid remains undissociated. Comparing the \(K_a\) values of two different weak acids allows for a direct comparison of their relative strengths.
Calculating \(K_a\) Through \(pK_a\)
Chemists frequently use a logarithmic scale to express acid strength, which involves converting the \(K_a\) value into a \(pK_a\) value. The \(pK_a\) is defined as the negative logarithm (base 10) of the \(K_a\) value, a relationship written as \(pK_a = -\log(K_a)\). This conversion is used to manage the very small numbers often associated with \(K_a\) values for weak acids.
Working with \(pK_a\) converts the \(K_a\) values, which are usually expressed in scientific notation with negative exponents, into simpler, more manageable positive numbers. If the \(pK_a\) value is known, the \(K_a\) can be determined by using the inverse operation: \(K_a = 10^{-pK_a}\).
A low \(pK_a\) value corresponds to a high \(K_a\) value, which means a stronger acid. For example, a weak acid with a \(K_a\) of \(1.8 \times 10^{-5}\) has a \(pK_a\) of \(4.74\). This inverse relationship is why acids are considered stronger as their \(pK_a\) decreases.