Q.The ionization constant of acetic acid is $K_a = 1.8 \times 10^{-5}$. Calculate the ionization constant $K_b$ of its conjugate base, the acetate ion $\text{CH}_3\text{COO}^-$, given $K_w = 1.0 \times 10^{-14}$ at $298\ \text{K}$.

Weak Acid Ionization: From Intuition to Precision

Imagine you drop a spoonful of sugar into a glass of water. Some sugar dissolves, but a lot just sits at the bottom. Now imagine you drop a spoonful of salt — it all dissolves completely. Acids behave the same way. Some acids, like hydrochloric acid (HCl), dissolve completely in water — every single molecule breaks apart. Others, like acetic acid (vinegar), only partially break apart. Most of the acid molecules stay intact, and only a few actually ionize.

That's the core idea: weak acids are shy about giving away their hydrogen ion. They don't fully commit.

The Precise Statement

A weak acid ($HA$) in water establishes an equilibrium between the intact acid molecule and its ions:

The double arrow ($\rightleftharpoons$) is the key. It tells you the reaction happens in both directions simultaneously. Some $HA$ molecules break apart to form $H_3O^+$ and $A^-$, while some $H_3O^+$ and $A^-$ recombine back into $HA$. At equilibrium, both processes happen at the same rate — so the concentrations stop changing.

The Quantitative Measure: $K_a$

Every weak acid has a number that tells you exactly how "shy" it is — the acid dissociation constant, $K_a$:

The smaller the $K_a$, the weaker the acid. For acetic acid (vinegar), $K_a \approx 1.8 \times 10^{-5}$. That tiny number means the numerator (ions) is very small compared to the denominator (intact acid). For a strong acid like HCl, $K_a$ is effectively infinite — the denominator is essentially zero because all the acid has ionized.

A Concrete Example

Suppose you dissolve 0.10 mol of acetic acid ($CH_3COOH$) in 1 L of water. At equilibrium, you'll find:

• $[CH_3COOH] \approx 0.0998$ M (almost all of it is still intact)
• $[H_3O^+] \approx 0.0013$ M (only about 1.3% has ionized)
• $[CH_3COO^-] \approx 0.0013$ M

Plugging into $K_a = \frac{x^2}{0.10 - x} = 1.8 \times 10^{-5}$ and solving gives $x \approx 0.0013$ M.

Why This Matters …