Table of Contents
In Brønsted acid–base theory, acids and bases were introduced as proton ($\mathrm{H^+}$) donors and acceptors. In this chapter, the focus is on what happens when such proton transfer reaches a balanced, reversible state: the acid–base equilibrium.
Conjugate Acid–Base Pairs and Equilibrium
Any Brønsted acid–base reaction can be written as a reversible process. For a generic acid $\mathrm{HA}$ in water:
$$
\mathrm{HA + H_2O \rightleftharpoons H_3O^+ + A^-}
$$
Here:
- $\mathrm{HA}$ is the acid, $\mathrm{A^-}$ its conjugate base
- $\mathrm{H_2O}$ is the base, $\mathrm{H_3O^+}$ its conjugate acid
Two conjugate pairs appear:
- $\mathrm{HA / A^-}$
- $\mathrm{H_3O^+ / H_2O}$
Acid–base equilibrium is the state where the forward and reverse proton-transfer processes occur at the same rate. Concentrations of all species become constant in time, although both directions continue to occur microscopically.
The position of equilibrium (on the side of reactants or products) depends on the acid and base strengths involved.
Acid Dissociation Constants $K_a$
For the generic acid ionization:
$$
\mathrm{HA + H_2O \rightleftharpoons H_3O^+ + A^-}
$$
The acid dissociation constant $K_a$ is defined as:
$$
K_a = \frac{[\mathrm{H_3O^+}][\mathrm{A^-}]}{[\mathrm{HA}]}
$$
The concentration of water is omitted because it is essentially constant in dilute aqueous solution. $K_a$ is characteristic of a particular acid at a given temperature.
- Large $K_a$: equilibrium strongly on the products’ side → strong acid (in that medium)
- Small $K_a$: equilibrium lies left → weak acid
For a base $\mathrm{B}$ reacting with water:
$$
\mathrm{B + H_2O \rightleftharpoons BH^+ + OH^-}
$$
we define the base dissociation constant $K_b$:
$$
K_b = \frac{[\mathrm{BH^+}][\mathrm{OH^-}]}{[\mathrm{B}]}
$$
Again, larger $K_b$ means a stronger base.
$pK_a$ and $pK_b$
To deal with the wide range of $K_a$ and $K_b$ values, logarithmic scales are used:
$$
pK_a = -\log_{10} K_a
$$
$$
pK_b = -\log_{10} K_b
$$
Interpretation:
- Small $pK_a$ → large $K_a$ → strong acid
- Large $pK_a$ → weak acid
Similarly for bases:
- Small $pK_b$ → strong base
- Large $pK_b$ → weak base
In water at $25^\circ\mathrm{C}$:
$$
K_w = [\mathrm{H_3O^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}
$$
so:
$$
pK_w = 14.00
$$
For a conjugate acid–base pair $\mathrm{BH^+ / B}$ in water:
$$
pK_a(\mathrm{BH^+}) + pK_b(\mathrm{B}) = pK_w = 14.00 \quad (\text{at } 25^\circ\mathrm{C})
$$
Thus, a strong acid corresponds to a very weak conjugate base, and vice versa.
Position of Acid–Base Equilibria and Relative Strengths
For a general proton transfer:
$$
\mathrm{HA_1 + B_2 \rightleftharpoons A_1^- + HB_2^+}
$$
equilibrium favors transfer of the proton from the stronger acid to the stronger base, producing the weaker acid and weaker base. In other words, equilibrium lies on the side with the weaker acid–base pair.
In aqueous solution, a useful rule is:
- Acids stronger than $\mathrm{H_3O^+}$ are “completely” dissociated (equilibrium far to the right)
- Bases stronger than $\mathrm{OH^-}$ are almost completely protonated (equilibrium far to the right in the reverse sense)
Mathematically, for an acid $\mathrm{HA}$ in water:
- If $K_a \gg 1$ (or $pK_a \ll 0$): essentially complete dissociation
- If $K_a \ll 1$ (or $pK_a > 0$): partial dissociation, true equilibrium mixture
Degree of Dissociation and Equilibrium Calculations
For a weak monoprotic acid $\mathrm{HA}$ with initial concentration $c_0$, let the equilibrium concentration of $\mathrm{H_3O^+}$ formed from this acid be $c_0 \alpha$, where $\alpha$ is the degree of dissociation (fraction of $\mathrm{HA}$ that has dissociated):
- Initial: $[\mathrm{HA}] = c_0$, $[\mathrm{H_3O^+}] \approx 0$, $[\mathrm{A^-}] = 0$
- Change: $\mathrm{HA}$ decreases by $c_0\alpha$, $\mathrm{H_3O^+}$ and $\mathrm{A^-}$ increase by $c_0\alpha$
- Equilibrium:
- $[\mathrm{HA}] = c_0(1-\alpha)$
- $[\mathrm{H_3O^+}] \approx c_0\alpha$ (ignoring water autoionization)
- $[\mathrm{A^-}] = c_0\alpha$
Substituting into $K_a$:
$$
K_a = \frac{[\mathrm{H_3O^+}][\mathrm{A^-}]}{[\mathrm{HA}]} = \frac{(c_0\alpha)(c_0\alpha)}{c_0(1-\alpha)} = \frac{c_0 \alpha^2}{1-\alpha}
$$
For weak acids in not-too-dilute solution ($\alpha \ll 1$), $1-\alpha \approx 1$, giving the approximation:
$$
K_a \approx c_0 \alpha^2
$$
and since $[\mathrm{H_3O^+}] \approx c_0\alpha$,
$$
[\mathrm{H_3O^+}] \approx \sqrt{K_a c_0}
$$
A similar treatment applies to weak bases using $K_b$.
These relationships allow estimation of equilibrium concentrations and, combined with the pH concept (treated elsewhere), the acidity of the solution.
Polyprotic Acids and Stepwise Equilibria
Polyprotic acids (e.g., $\mathrm{H_2SO_4}$, $\mathrm{H_2CO_3}$, $\mathrm{H_3PO_4}$) can donate more than one proton in a stepwise manner. Each deprotonation step has its own equilibrium and $K_a$:
Example for a diprotic acid $\mathrm{H_2A}$:
1st dissociation:
$$
\mathrm{H_2A + H_2O \rightleftharpoons H_3O^+ + HA^-}
$$
$$
K_{a1} = \frac{[\mathrm{H_3O^+}][\mathrm{HA^-}]}{[\mathrm{H_2A}]}
$$
2nd dissociation:
$$
\mathrm{HA^- + H_2O \rightleftharpoons H_3O^+ + A^{2-}}
$$
$$
K_{a2} = \frac{[\mathrm{H_3O^+}][\mathrm{A^{2-}}]}{[\mathrm{HA^-}]}
$$
Typically:
$$
K_{a1} > K_{a2} > K_{a3} > \dots
$$
because it becomes progressively harder to remove additional protons from an increasingly negatively charged species. In many practical situations, dissociation can be approximated by considering only the first (or first two) steps, depending on the magnitudes of $K_{a1}$, $K_{a2}$, etc.
Leveling and Differentiating Effects of the Solvent
The acid–base strengths observed in a given solvent depend on the properties of that solvent. In water:
- All strong acids (e.g., $\mathrm{HCl}$, $\mathrm{HBr}$, $\mathrm{HNO_3}$, $\mathrm{HClO_4}$) are essentially fully converted to $\mathrm{H_3O^+}$; water “levels” their strengths — they all appear as equally strong, since they all produce the same strongest acid possible in water.
- Similarly, strong bases that exceed the basicity of $\mathrm{OH^-}$ are “leveled” to $\mathrm{OH^-}$.
In less basic solvents, differences between these acids can become apparent (the “differentiating effect”), because not all are completely ionized. Thus, acid–base equilibria and strength rankings are solvent-dependent.
Buffers as Acid–Base Equilibria
A buffer solution consists essentially of a weak acid $\mathrm{HA}$ and its conjugate base $\mathrm{A^-}$ (or a weak base and its conjugate acid) present in comparable amounts. The characteristic property—resisting changes in $\mathrm{H_3O^+}$ concentration when small amounts of acid or base are added—arises directly from the underlying acid–base equilibrium:
$$
\mathrm{HA \rightleftharpoons H_3O^+ + A^-}
$$
Changes in $\mathrm{H_3O^+}$ are partially compensated by shifts in this equilibrium (via consumption or formation of $\mathrm{HA}$ and $\mathrm{A^-}$). Quantitative buffer behavior and the corresponding equations are treated in more detail elsewhere; here the emphasis is that a buffer is an acid–base equilibrium system designed to minimize $\mathrm{H_3O^+}$ changes.
Competing Acid–Base Equilibria
In many real systems, several acid–base equilibria occur simultaneously. Examples include:
- Solutions of amphoteric substances (discussed separately), where a species can both donate and accept protons, participating in multiple equilibria at once.
- A mixture of several weak acids or bases, which compete for $\mathrm{H_3O^+}$ or $\mathrm{OH^-}$.
- Acid–base indicators, whose color change corresponds to the equilibrium between two differently colored forms.
The dominant equilibrium at a given pH or composition depends on the various $K_a$ and $K_b$ values and the concentrations of the species involved. Qualitatively, the most favorable proton transfers (from stronger to weaker acids, toward weaker bases) will dominate the overall equilibrium composition.
In summary, acid–base equilibria describe the balance between protonated and deprotonated forms of substances in solution. They are quantified by $K_a$, $K_b$, and their $pK$ values, and they underlie phenomena such as buffer action, stepwise deprotonation of polyprotic acids, and solvent-dependent apparent strengths of acids and bases.