14.3 Relative Strengths of Acids and Bases

Acid-Base Strength and Equilibrium

Acid vs base strength comparisons

Not all acids and bases ionize to the same extent in water. The ionization constant tells you exactly how far the reaction goes, which is how we compare strengths quantitatively.

• The acid ionization constant ($K_a$) measures how completely an acid donates protons in water. A higher $K_a$ means a stronger acid. For example, HCl has an extremely large $K_a$ (essentially complete ionization), while acetic acid ($CH_3COOH$) has a $K_a$ of only $1.8 \times 10^{-5}$, making it a weak acid.
• The base ionization constant ($K_b$) works the same way for bases. A higher $K_b$ means a stronger base. NaOH dissociates completely (strong base), while ammonia ($NH_3$) has a $K_b$ of $1.8 \times 10^{-5}$ (weak base).
• $pK_a$ and $pK_b$ are just the negative logarithms of $K_a$ and $K_b$. These flip the scale: a lower $pK_a$ means a stronger acid, and a lower $pK_b$ means a stronger base. This is the same log relationship you've seen with pH.
• Every acid has a conjugate base, and every base has a conjugate acid. These pairs are linked by the relationship $K_a \times K_b = K_w$, where $K_w = 1.0 \times 10^{-14}$ at 25°C. This means a strong acid always has a weak conjugate base, and vice versa.

Molecular structure and acid-base strength

Why is one acid stronger than another? It comes down to how stable the conjugate base is after the proton leaves. The more stable the conjugate base, the more the equilibrium favors ionization, and the stronger the acid.

• Inductive effects: Electron-withdrawing groups (like halogens or nitro groups) pull electron density away from the conjugate base, stabilizing the negative charge and making the acid stronger. Electron-donating groups (like alkyl groups) do the opposite, pushing electron density toward the negative charge and making the acid weaker. For example, trichloroacetic acid ($CCl_3COOH$) is much stronger than acetic acid because the three chlorine atoms withdraw electron density.
• Resonance effects: If the conjugate base can spread its negative charge across multiple atoms through resonance, it's more stable. This is why carboxylic acids ($RCOOH$) are much more acidic than alcohols ($ROH$). The carboxylate ion delocalizes the charge over two oxygen atoms.
• Electronegativity: A more electronegative atom bonded to the acidic hydrogen holds the bonding electrons more tightly, making it easier for the proton to leave. Oxygen-hydrogen bonds are more acidic than carbon-hydrogen bonds for this reason.
• Hybridization: Atoms with more s-character in their bonding orbital hold electrons closer to the nucleus, stabilizing the conjugate base. So an $sp$ hybridized carbon (50% s-character) is more acidic than $sp^3$ (25% s-character).
• Leveling effect: A solvent can limit the observable strength of acids or bases. In water, all strong acids (HCl, HBr, $HClO_4$) appear equally strong because water is a strong enough base to fully deprotonate all of them. To distinguish their strengths, you'd need a weaker base as the solvent.

Weak acid-base equilibrium problems

These problems follow a consistent process. Here's how to work through them:

1. Write the equilibrium expression. Set up the balanced equation and the corresponding $K$ expression.

   • Weak acid HA: $HA + H_2O \rightleftharpoons H_3O^+ + A^-$, so $K_a = \frac{[H_3O^+][A^-]}{[HA]}$
   • Weak base B: $B + H_2O \rightleftharpoons BH^+ + OH^-$, so $K_b = \frac{[BH^+][OH^-]}{[B]}$

2. Build an ICE table. ICE stands for Initial, Change, Equilibrium.

   • Fill in the initial concentrations from the problem (products usually start at 0).
   • Define the change using a variable $x$ based on stoichiometry (reactant decreases by $x$, products each increase by $x$).
   • Equilibrium concentrations equal initial + change.

3. Substitute into the $K$ expression and solve for $x$. This often gives a quadratic equation, but you can simplify with the small-x approximation: if $x$ is very small compared to the initial concentration (typically less than 5%), you can drop it from the denominator. Always check this assumption afterward by verifying that $\frac{x}{[HA]_0} \times 100\% < 5\%$.

4. Calculate pH.

   • For a weak acid: $x = [H_3O^+]$, so $pH = -\log{[H_3O^+]}$
   • For a weak base: $x = [OH^-]$, so $pOH = -\log{[OH^-]}$, then use $pH + pOH = 14$ (at 25°C)

Advanced Acid-Base Concepts

• Buffer solutions resist changes in pH when small amounts of acid or base are added. They work because they contain both a weak acid and its conjugate base (or a weak base and its conjugate acid), so they can neutralize added $H_3O^+$ or $OH^-$.
• The Henderson-Hasselbalch equation provides a shortcut for finding the pH of a buffer: $pH = pK_a + \log{\frac{[A^-]}{[HA]}}$. This is especially useful when you know the ratio of conjugate base to weak acid.
• Autoionization of water is the reaction where two water molecules produce a hydronium ion and a hydroxide ion: $2H_2O \rightleftharpoons H_3O^+ + OH^-$. The equilibrium constant for this process is $K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14}$ at 25°C. This is why pure water has a pH of 7.