Ionic Equilibrium of Acids and Bases

The position of equilibrium for weak acid-base reactions is determined by the acid's dissociation constant ($K_a$) or the base's dissociation constant ($K_b$). The further the equilibrium lies to the left, the weaker the acid or base. For weak acids and bases, the equilibrium constant expressions are as follows:

For weak acids

$HA \rightleftharpoons H^+ + A^-$

The equilibrium constant for a weak acid is called $K_a$, the dissociation constant of the acid:

$K_a = \frac{[H^+][A^-]}{[HA]}$

For weak bases

$B + H_2O \rightleftharpoons BH^+ + OH^-$

The equilibrium constant for a weak base is called $K_b$, the dissociation constant of the base:

$K_b = \frac{[BH^+][OH^-]}{[B]}$

The $K_a$ and $K_b$ values are used to determine the strength of acids and bases. A small value of $K_a$ or $K_b$ indicates that the acid or base ionizes very little, making it very weak.

Example: comparing the strength of two bases

The $K_b$ value for aniline (C₆H₅NH₂) is $4.17 \times 10^{-10}$, and the $K_b$ value for ammonia (NH₃) is $1.78 \times 10^{-5}$. To determine which base is stronger, compare the $K_b$ values:

Since $K_b$ for NH₃ is greater than that of C₆H₅NH₂, ammonia is a stronger base.

The strength of weak acids and bases can also be determined from their degree of dissociation, which is described by Ostwald's dilution law. This law states that the degree of dissociation of a weak acid or base increases with dilution, and it can be approximated using the following formula:

$\alpha = \sqrt{\frac{K_a}{C}}$

Where $\alpha$ is the degree of dissociation, $K_a$ is the dissociation constant, and $C$ is the concentration of the acid or base.

$K_a$ and $K_b$ values can be difficult to handle, so we often use the $pK_a$ and $pK_b$ values, which are defined as:

$pK_a = -\log{K_a}, \quad pK_b = -\log{K_b}$

The lower the $pK_a$, the stronger the acid, and the lower the $pK_b$, the stronger the base.

Example: formic acid (HCOOH)

Formic acid (HCOOH) has a $K_a$ of $1.78 \times 10^{-4}$ mol/L. We can calculate the concentration of H₃O⁺ ions and the pH of a 0.1 M solution of HCOOH:

Since $K_a$ is small, we can approximate the concentration of H₃O⁺ ions by assuming that the concentration of undissociated HCOOH (HA) is approximately equal to the initial concentration of 0.1 M. Thus, we can use the equation:

$K_a = \frac{[H_3O^+][A^-]}{[HA]}$

Assuming $[H_3O^+] = [A^-]$, we can solve for the concentration of H₃O⁺ and then find the pH:

$[H_3O^+] = \sqrt{K_a \times [HA]}$

Substituting the values:

$[H_3O^+] = \sqrt{(1.78 \times 10^{-4}) \times (0.1)} = 4.22 \times 10^{-3} \text{ M}$

Then, to calculate the pH:

$pH = -\log{[H_3O^+]} = -\log{(4.22 \times 10^{-3})} = 2.375$

For a conjugate acid-base pair, the relationship between $pK_a$ and $pK_b$ is given by:

$pK_a + pK_b = 14$

This relationship holds true at 25°C, as the sum of $pK_a$ and $pK_b$ for a conjugate acid-base pair always equals 14.

Example: formic acid and its conjugate base

If the $pK_a$ of formic acid (HCOOH) is 3.75, we can calculate the $pK_b$ of its conjugate base, formate ion (HCOO⁻):

$pK_a + pK_b = 14$

$pK_b = 14 - pK_a = 14 - 3.75 = 10.25$