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Chemistry · Ch 7 — Equilibrium

Ionization Constants of Weak Acids

7.11.3

Ionization Constants of Weak Acids

The Concept of a Weak Acid and Its Ionization Constant

A weak acid, unlike a strong acid, does not completely dissociate into ions when dissolved in water. Instead, it establishes an equilibrium between the undissociated acid molecules and the ions produced. Consider a general weak acid, which we will denote as HX. In aqueous solution, it undergoes partial ionization according to the following reversible reaction:

HX(aq)+H2O(l)⇌H3O+(aq)+X−(aq)\text{HX(aq)} + \text{H}_2\text{O(l)} \rightleftharpoons \text{H}_3\text{O}^+\text{(aq)} + \text{X}^-\text{(aq)}

To describe this equilibrium quantitatively, we define two key quantities:

  • cc: the initial concentration of the acid HX (in mol/L) before any dissociation occurs.
  • α\alpha: the degree of ionization, which represents the fraction of the initial acid molecules that have dissociated into ions at equilibrium.

Using these, we can build an ICE (Initial, Change, Equilibrium) table to track the concentrations of all species.

SpeciesInitial Concentration (M)Change in Concentration (M)Equilibrium Concentration (M)
HXcc−cα-c\alphac−cα=c(1−α)c - c\alpha = c(1-\alpha)
H3O+\text{H}_3\text{O}^+00+cα+c\alphacαc\alpha
X−\text{X}^-00+cα+c\alphacαc\alpha

The equilibrium constant for this acid-dissociation reaction is called the acid ionization constant, denoted by KaK_a. It is defined by the law of mass action. Substituting the equilibrium concentrations from the table gives:

Ka=[H3O+][X−][HX]=(cα)(cα)c(1−α)=c2α2c(1−α)K_a = \frac{[\text{H}_3\text{O}^+][\text{X}^-]}{[\text{HX}]} = \frac{(c\alpha)(c\alpha)}{c(1-\alpha)} = \frac{c^2 \alpha^2}{c(1-\alpha)}

This simplifies to the fundamental expression for the ionization constant of a weak acid:

Ka=cα21−αK_a = \frac{c \alpha^2}{1-\alpha}

A more general form, which is often more convenient for calculations, expresses KaK_a directly in terms of the equilibrium molar concentrations of the species:

Ka=[H+][X−][HX]K_a = \frac{[\text{H}^+][\text{X}^-]}{[\text{HX}]}

Important

The value of KaK_a is a direct measure of the strength of a weak acid. At a given temperature, a larger KaK_a value indicates a stronger acid, meaning it dissociates to a greater extent.

Table 6.6The Ionization Constants of Some Selected Weak Acids (at 298 K)
AcidIonization Constant, KaK_a
Hydrofluoric Acid (HF)3.5×10−43.5 \times 10^{-4}
Nitrous Acid (HNO2HNO_2)4.5×10−44.5 \times 10^{-4}
Formic Acid (HCOOH)1.8×10−41.8 \times 10^{-4}
Niacin (C5H4NCOOHC_5H_4NCOOH)1.5×10−51.5 \times 10^{-5}
Acetic Acid (CH3COOHCH_3COOH)1.74×10−51.74 \times 10^{-5}
Benzoic Acid (C6H5COOHC_6H_5COOH)6.5×10−56.5 \times 10^{-5}
Hypochlorous Acid (HClO)3.0×10−83.0 \times 10^{-8}

KaK_a is a dimensionless quantity, with the understanding that the standard state concentration of all species is 1 M.

The pKa Scale

Just as the pH scale was introduced for hydrogen ion concentration, a similar logarithmic scale is used for acid ionization constants. The pKa of an acid is defined as:

pKa=−log⁡(Ka)\text{p}K_a = -\log(K_a)

This provides a more manageable scale for the very small KaK_a values typical of weak acids. A smaller pKa corresponds to a larger KaK_a and therefore a stronger acid.

A Step-by-Step Methodology for pH Calculation

The textbook outlines a systematic, general approach for calculating the equilibrium concentrations, degree of ionization, and pH of a solution of a weak acid. This method is crucial for solving problems.

  1. Identify all species present before any reaction occurs. Recognize which can act as Brønsted-Lowry acids or bases.
  2. Write balanced equations for all possible proton-transfer reactions that can occur between these species.
  3. Identify the principal reaction. Compare the KaK_a (or KbK_b) values of the possible reactions. The reaction with the larger equilibrium constant is the primary reaction and will dominate the equilibrium. The other reactions are subsidiary and can often be ignored.
  4. Construct an ICE table for the principal reaction. Express the change in concentration in terms of α\alpha, the degree of ionization.
  5. Substitute the equilibrium concentrations from the ICE table into the KaK_a expression for the principal reaction. Solve the resulting equation for α\alpha.
  6. Calculate the equilibrium concentrations of all species in the principal reaction using the value of α\alpha found in step 5.
  7. Calculate the pH of the solution using the concentration of hydronium ions: pH=−log⁡[H3O+]\text{pH} = -\log[\text{H}_3\text{O}^+]. …
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