pH Calculations

pH is a logarithmic scale used to express the concentration of hydrogen ions (H⁺, or more precisely hydronium ions H₃O⁺) in an aqueous solution. Defined by Søren Sørensen in 1909, pH = −log₁₀[H⁺], where [H⁺] is measured in mol L⁻¹. A pH of 7 at 25 °C is neutral, values below 7 are acidic, and values above 7 are basic (alkaline). Because the scale is logarithmic, a change of one pH unit represents a tenfold change in hydrogen ion concentration. Acids and bases are broadly classified as strong or weak depending on their degree of dissociation in water. Strong acids and bases dissociate completely, making pH calculations straightforward from stoichiometry alone. Weak acids and bases only partially dissociate, establishing a dynamic equilibrium described by their dissociation constants K_a (for acids) and K_b (for bases). Ostwald's Dilution Law connects the degree of dissociation to these constants and the initial concentration of the solute. Mastering pH calculations requires understanding both categories. For strong electrolytes, you read [H⁺] or [OH⁻] directly from the molarity. For weak electrolytes, you set up an ICE (Initial–Change–Equilibrium) table, apply the appropriate equilibrium expression, and solve—sometimes using simplifying approximations when K_a or K_b is very small. The relationship pK_a + pK_b = pK_w = 14 (at 25 °C) links conjugate acid–base pairs and is essential for buffer and hydrolysis problems.

Key takeaways
  • Identify the Type of Acid or Base
  • Strong Acid pH Calculation
  • Weak Acid pH Calculation Using K_a
  • Ostwald's Dilution Law and Degree of Dissociation
  • Weak Base pH Calculation Using K_b

Identify the Type of Acid or Base

First, classify the species. Common strong acids: HCl, HBr, HI, HNO₃, H₂SO₄ (first dissociation), HClO₄. Common strong bases: NaOH, KOH, Ca(OH)₂. Everything else—acetic acid (CH₃COOH), ammonia (NH₃), carbonic acid (H₂CO₃)—is treated as weak. This classification determines which calculation pathway you follow.

Strong Acid pH Calculation

For a strong acid at concentration C mol L⁻¹, complete dissociation means [H⁺] = C (monoprotic) or [H⁺] = 2C (diprotic, e.g. H₂SO₄ at low concentrations). Then pH = −log[H⁺]. Example: 0.01 mol L⁻¹ HCl → [H⁺] = 0.01 = 10⁻² → pH = 2. For strong bases, find [OH⁻] the same way, compute pOH = −log[OH⁻], then pH = 14 − pOH at 25 °C.

Weak Acid pH Calculation Using K_a

For a weak acid HA at initial concentration C with dissociation constant K_a, set up an ICE table: [HA] starts at C, [H⁺] and [A⁻] start at ≈ 0. At equilibrium: [HA] = C − x, [H⁺] = x, [A⁻] = x. The equilibrium expression gives K_a = x² / (C − x). If K_a << C (less than 5% dissociation), the approximation C − x ≈ C is valid, so x = √(K_a · C). Then pH = −log(x). If the approximation fails, solve the full quadratic: x² + K_a·x − K_a·C = 0.

Ostwald's Dilution Law and Degree of Dissociation

The degree of dissociation α is defined as the fraction of the original electrolyte that has dissociated. For a weak acid: K_a = Cα² / (1 − α). When α << 1, this simplifies to K_a ≈ Cα², giving α = √(K_a / C). Ostwald's Dilution Law states that α increases as C decreases (dilution promotes dissociation). Since [H⁺] = Cα, substituting gives [H⁺] = √(K_a · C), consistent with the ICE table approach. The degree of dissociation is directly measurable via conductivity experiments.

Weak Base pH Calculation Using K_b

For a weak base B at concentration C: \ce{B + H2O <=> BH+ + OH-}. Similarly, K_b = [BH⁺][OH⁻] / [B] = x² / (C − x) ≈ x² / C. So [OH⁻] = x = √(K_b · C), pOH = −log[OH⁻], and pH = 14 − pOH. For a conjugate acid–base pair (e.g. CH₃COOH / CH₃COO⁻): K_a × K_b = K_w = 1.0 × 10⁻¹⁴ at 25 °C, so pK_a + pK_b = 14.

The Key Equation

Ka=[\ceH+][\ceA−][\ceHA]=x2C−x≈x2CK_a = \frac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]} = \frac{x^2}{C - x} \approx \frac{x^2}{C}
  • K_a — Acid dissociation constant (mol L⁻¹): Equilibrium constant for the dissociation of a weak acid in water; larger K_a means stronger acid.
  • [\ce{H+}] — Hydrogen ion concentration (mol L⁻¹): Concentration of hydronium ions (H₃O⁺) at equilibrium, equal to x in the ICE table.
  • [\ce{A-}] — Conjugate base concentration (mol L⁻¹): Concentration of the anion produced when HA donates a proton; also equal to x.
  • [\ce{HA}] — Undissociated acid concentration (mol L⁻¹): Equilibrium concentration of the weak acid remaining in solution, equal to C − x.
  • C — Initial concentration (mol L⁻¹): The formal (analytical) concentration of the weak acid before any dissociation occurs.
  • x — Extent of dissociation (mol L⁻¹): The concentration of acid that has dissociated; equals [H⁺] = [A⁻] at equilibrium.

Worked Example

Problem. Calculate the pH of a 0.100 mol L⁻¹ solution of ethanoic acid (acetic acid, CH₃COOH) at 25 °C. K_a = 1.8 × 10⁻⁵ mol L⁻¹.

pH=−log⁡(1.342×10−3)=2.87\text{pH} = -\log(1.342 \times 10^{-3}) = 2.87

Answer. pH = 2.87

Figure
Diagram
pH Scale and Dissociation Comparison0123456789101112← ACIDICBASIC →NEUTRALStrong Acid (HCl)100% dissociated → [H⁺] = CWeak Acid (CH₃COOH, K_a=1.8×10⁻⁵)~1.3% dissociated at 0.1 mol L⁻¹Strong Base (NaOH)100% dissociated → [OH⁻] = CWeak Base (NH₃, K_b=1.8×10⁻⁵)~1.3% dissociated at 0.1 mol L⁻¹Ostwald: α = √(K_a/C) | [H⁺] = √(K_a·C) | pH = ½(pK_a − log C)At 25°C: pK_w = 14 → pH + pOH = 14
Origins · 1 minSøren Peter Lauritz Sørensen and how this was discovered
SP
Søren Peter Lauritz Sørensen
1868–1939 · Danish

Sørensen introduced the pH scale in 1909 while working at the Carlsberg Laboratory in Copenhagen. He was studying the effect of ion concentrations on enzyme activity and found that expressing [H⁺] directly was cumbersome, proposing instead the notation p[H] (the 'p' from the Danish/German word for 'power', Potenz). He also developed precise buffer solutions to standardise biochemical experiments, laying the groundwork for modern analytical chemistry and biochemistry. Sørensen's pH work was done at the famous Carlsberg brewery laboratory, which was funded by the Danish beer industry to improve the quality and consistency of lager production—so the pH scale owes its existence, indirectly, to beer!

  • Chemical Equilibrium and Le Chatelier's Principle — Ionic equilibrium is a specific application of chemical equilibrium; understanding how equilibria respond to changes in concentration and dilution (Le Chatelier's Principle) is essential for predicting how pH changes when solutions are diluted or mixed.
  • Buffer Solutions and the Henderson–Hasselbalch Equation — Buffers are mixtures of a weak acid and its conjugate base; calculating their pH uses the same K_a equilibrium expressions developed here, extended by the Henderson–Hasselbalch equation pH = pK_a + log([A⁻]/[HA]).
  • Salt Hydrolysis and pH of Salt Solutions — Salts formed from weak acids or bases undergo hydrolysis to produce acidic or basic solutions. Predicting the pH of these solutions requires K_a, K_b, and the K_w relationship introduced in pH calculations.
  • Acid–Base Titrations and Titration Curves — Titration curves plot pH against volume of titrant added. Calculating pH at different stages—before equivalence, at equivalence, and beyond—requires all the techniques covered in strong and weak acid/base pH calculations.