Acids, Bases and pH

There are three major frameworks for understanding acids and bases, each broader than the last:

Arrhenius definition: An acid produces H⁠⁺ ions in water; a base produces OH⁠^- ions in water. This is the simplest model but only works for aqueous solutions. HCl is an Arrhenius acid (produces H⁠⁺), and NaOH is an Arrhenius base (produces OH⁠^-).

Brønsted-Lowry definition: An acid is a proton (H⁠⁺) donor; a base is a proton acceptor. This is more general — it explains acid-base behavior in any solvent, not just water. For example, NH⁠₃ is a Brønsted-Lowry base because it accepts a proton from water: NH⁠₃ + H⁠₂O → NH⁠₄⁠⁺ + OH⁠^-.

Lewis definition: An acid is an electron-pair acceptor; a base is an electron-pair donor. This is the broadest definition and includes reactions with no proton transfer at all, such as BF⁠₃ + NH⁠₃ → F⁠₃B-NH⁠₃.

Water is amphoteric — it can act as either an acid or a base depending on what it reacts with. With HF, water acts as a base (accepts H⁠⁺). With NH⁠₃, water acts as an acid (donates H⁠⁺).

The concentration of H⁠₃O⁠⁺ (hydronium) ions in solution determines its acidity. Because these concentrations span many orders of magnitude, we use a logarithmic scale:

pH = −log[H⁠₃O⁠⁺]

pOH = −log[OH⁠^-]

At 25 °C, these two values are always related by:

pH + pOH = 14.00

This comes from the ion-product constant of water: K_w = [H⁠₃O⁠⁺][OH⁠^-] = 1.0 × 10⁠^-14 at 25 °C.

To convert between pH and ion concentrations:

• [H⁠₃O⁠⁺] = 10^−pH
• [OH⁠^-] = 10^−pOH

A neutral solution has pH = 7.00, acidic solutions have pH < 7, and basic solutions have pH > 7.

In every Brønsted-Lowry acid-base reaction, a proton is transferred from the acid to the base. This creates a conjugate pair: the acid loses a proton to become its conjugate base, and the base gains a proton to become its conjugate acid.

HF + H⁠₂O ⇌ F⁠^- + H⁠₃O⁠⁺

In this reaction:

• HF is the acid → F⁠^- is its conjugate base
• H⁠₂O is the base → H⁠₃O⁠⁺ is its conjugate acid

Key relationship: A strong acid has a very weak conjugate base, and a weak acid has a relatively stronger conjugate base. This inverse relationship is fundamental to understanding acid-base equilibrium.

Every acid-base reaction has two conjugate pairs — always identify both when analyzing a reaction.

Strong acids ionize completely in water — every molecule donates its proton. The common strong acids are HCl, HBr, HI, HNO⁠₃, HClO⁠₄, and H⁠₂SO⁠₄ (first proton).

For a strong acid, [H⁠₃O⁠⁺] equals the initial acid concentration:

Example: 0.025 M HCl → [H⁠₃O⁠⁺] = 0.025 M → pH = −log(0.025) = 1.60

Strong bases also dissociate completely. The common ones are Group 1 and 2 hydroxides: NaOH, KOH, Ca(OH)⁠₂, Ba(OH)⁠₂.

Example: 0.010 M NaOH → [OH⁠^-] = 0.010 M → pOH = 2.00 → pH = 12.00

For bases with two OH⁠^- per formula unit (like Ca(OH)⁠₂), remember to double: 0.005 M Ca(OH)⁠₂ → [OH⁠^-] = 0.010 M.

For a mixture of two strong-acid (or two strong-base) solutions: convert each pH back to its ion concentration, add the moles of H⁠₃O⁠⁺ (or OH⁠^-), divide by the total volume, and re-take the log. Never average pH values directly — pH is logarithmic.

Weak acids only partially ionize in water. The extent of ionization is quantified by the acid ionization constant, K_a:

HA + H⁠₂O ⇌ A⁠^- + H⁠₃O⁠⁺

K_a = [H⁠₃O⁠⁺][A⁠^-] / [HA]

A larger K_a means a stronger weak acid (more ionization). To find pH from K_a and initial concentration:

1. Set up an ICE table (Initial, Change, Equilibrium)
2. Let x = [H⁠₃O⁠⁺] at equilibrium
3. Substitute into the K_a expression: K_a = x⁠² / (C⁠₀ − x)
4. If K_a is small relative to C⁠₀, simplify: x ≈ √(K_a × C⁠₀)
5. pH = −log(x)

Always verify the simplification: if x is more than 5% of C⁠₀, solve the full quadratic instead.

The inverse direction works too: given the pH of a solution of known initial concentration, x = 10^−pH gives the equilibrium [H⁠₃O⁠⁺], and K_a = x⁠² / (C⁠₀ − x) recovers the ionization constant. The same inversion with pOH recovers K_b for a weak base.

A weak base only partially accepts protons from water, establishing an equilibrium:

B + H₂O ⇌ BH⁺ + OH⁻

The base-ionisation constant is K_b = [BH⁺][OH⁻] / [B]. A small K_b means the base ionises very little and the solution is only mildly basic.

Solving for pH follows the same ICE-table approach as weak acids, but you solve for [OH⁻] first:

1. Set up the ICE table with initial base concentration C₀, change −x for B and +x for BH⁺ and OH⁻.
2. If the small-x approximation is valid (x < 5% of C₀): x = [OH⁻] ≈ √(K_b · C₀).
3. Calculate pOH = −log[OH⁻], then pH = 14.00 − pOH.

Example: 0.50 M NH₃ with K_b = 1.8 × 10⁻⁵: x = √(1.8 × 10⁻⁵ × 0.50) = 3.0 × 10⁻³ M. Check: 3.0 × 10⁻³/0.50 = 0.6% < 5%, so the approximation holds. pOH = 2.52, pH = 11.48.

For any conjugate acid-base pair, the product of their ionization constants equals the ion-product constant of water:

K_a × K_b = K_w = 1.0 × 10⁠^-14 (at 25 °C)

This means if you know K_a for an acid, you can calculate K_b for its conjugate base, and vice versa:

• K_b = K_w / K_a
• K_a = K_w / K_b

Example: Acetic acid has K_a = 1.8 × 10⁠^-5. Its conjugate base (acetate, CH⁠₃COO⁠^-) has K_b = (1.0 × 10⁠^-14) / (1.8 × 10⁠^-5) = 5.6 × 10⁠^-10.

A useful consequence: the stronger an acid, the weaker its conjugate base — and this is quantitatively precise, not just qualitative.

When a salt dissolves in water, its ions may react with water (hydrolyze) to produce an acidic or basic solution. The outcome depends on the parent acid and base:

• Strong acid + strong base → neutral salt. Example: NaCl from HCl + NaOH. Neither Na⁠⁺ nor Cl⁠^- hydrolyzes.
• Strong acid + weak base → acidic salt. Example: NH⁠₄Cl from HCl + NH⁠₃. The NH⁠₄⁠⁺ ion donates a proton to water.
• Weak acid + strong base → basic salt. Example: CH⁠₃COONa from CH⁠₃COOH + NaOH. The CH⁠₃COO⁠^- ion accepts a proton from water.
• Weak acid + weak base → depends on K_a vs K_b. Compare the ionization constants: if K_a > K_b, the solution is acidic; if K_b > K_a, it's basic.

To predict the pH, use K_a or K_b of the hydrolyzing ion (found via the K_a·K_b = K_w relationship).

Percent ionization measures how much of a weak acid or base actually ionizes in solution:

% ionization = ([H⁠₃O⁠⁺]_eq / [HA]_initial) × 100%

Key observations:

• Percent ionization increases as the solution is diluted. A 0.01 M weak acid is more ionized (percentage-wise) than a 1.0 M solution of the same acid.
• A larger K_a gives a higher percent ionization at any given concentration.
• For the 5% approximation used in ICE tables to be valid, the percent ionization must be ≤ 5%.

Example: If 0.10 M acetic acid gives [H⁠₃O⁠⁺] = 1.34 × 10⁠^-3 M, then % ionization = (1.34 × 10⁠^-3 / 0.10) × 100% = 1.34%.

Polyprotic acids have more than one ionizable proton. They ionize in successive steps, each with its own K_a value:

Diprotic acid (e.g., H⁠₂SO⁠₃):

H⁠₂SO⁠₃ + H⁠₂O ⇌ HSO⁠₃⁠^- + H⁠₃O⁠⁺    K_a1

HSO⁠₃⁠^- + H⁠₂O ⇌ SO⁠₃⁠^2- + H⁠₃O⁠⁺    K_a2

Triprotic acid (e.g., H⁠₃PO⁠₄) has three steps: K_a1 > K_a2 > K_a3.

Each successive ionization is always weaker (smaller K_a) because it's harder to remove a proton from a species that's already negatively charged. In practice, this means:

• The first ionization dominates the pH — you can usually ignore the second and third.
• Treat each step as a separate equilibrium, using the equilibrium concentrations from the previous step as the initial concentrations for the next.

Oxides of elements show a clear pattern in their acid-base behavior that follows the periodic table:

• Metal oxides are typically basic. They react with water to form hydroxides: Na⁠₂O + H⁠₂O → 2NaOH. They react with acids to form salt + water.
• Nonmetal oxides are typically acidic. They react with water to form oxyacids: SO⁠₃ + H⁠₂O → H⁠₂SO⁠₄. They react with bases to form salt + water.
• Amphoteric oxides (like Al⁠₂O⁠₃ and ZnO) can react with both acids and bases.

The trend across a period: oxides become more acidic from left to right as metallic character decreases. Down a group, oxides of the same type become more basic as metallic character increases.

This is directly related to electronegativity: highly electronegative nonmetals form oxides where the O-H bond in the resulting oxyacid is highly polar and breaks easily, releasing H⁠⁺.

Acid-base chapters feel broad because different equations apply to different regimes. Use this selection framework:

1. Strong acid/base only? Use direct stoichiometry, then pH/pOH from concentration.
2. Weak acid/base in water? Use K_a or K_b with ICE-table logic (small-x check).
3. Conjugate-pair mixture? Use Henderson–Hasselbalch (buffer region).
4. Salt solution? Identify parent acid/base strength, then hydrolysis behavior.

Quality checks: pH must be between 0 and 14 for standard aqueous problems, strong-acid solutions must yield pH below 7, and weak-acid pH should be higher than an equal-concentration strong acid. Include units and sig figs consistently when converting between [H⁺], [OH⁻], pH, and pOH.

Just as the wide range of [H₃O⁺] concentrations is compressed onto the logarithmic pH scale, the equally wide range of acid ionization constants is often reported as pK_a:

pK_a = −log₁₀(K_a)

The relationship inverts exactly, so a K_a can always be recovered from a pK_a:

K_a = 10^−pK_a

Because the logarithm is negative, the scale runs opposite to K_a: a smaller pK_a means a larger K_a, and therefore a stronger acid. For example, acetic acid (K_a = 1.8 × 10⁻⁵) has pK_a = −log(1.8 × 10⁻⁵) = 4.74, while the much stronger hydrofluoric acid (K_a = 6.8 × 10⁻⁴) has the smaller pK_a = 3.17.

For a polyprotic acid, each ionization step has its own constant, so each step also has its own pK_a. Since K_a1 > K_a2 > K_a3, the pK_a values increase with each successive step: pK_a1 < pK_a2 < pK_a3. To convert in the other direction, raise 10 to the negative of each value. For phosphoric acid (H₃PO₄) with pK_a1 = 2.12, pK_a2 = 7.21, and pK_a3 = 12.32:

• K_a1 = 10^−2.12 = 7.6 × 10⁻³
• K_a2 = 10^−7.21 = 6.2 × 10⁻⁸
• K_a3 = 10^−12.32 = 4.8 × 10⁻¹³

The first step (smallest pK_a) is by far the strongest and dominates the pH, consistent with the rule that successive ionizations grow progressively weaker.