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The pH scale, introduced by Søren Sørensen in 1909, quantifies the acidity or basicity of an aqueous solution: pH = -log₁₀[H⁺]. In pure water at 25°C, [H⁺] = [OH⁻] = 1.0×10⁻⁷ M, giving pH = 7.0. For strong acids (e.g., HCl, H₂SO₄) and bases (NaOH, KOH), dissociation is complete. For weak electrolytes (acetic acid, ammonia), equilibrium constants (Ka, Kb) govern [H⁺] via the expression: Ka = [H⁺][A⁻]/[HA]. This calculator rigorously solves the exact quadratic where necessary, and for very dilute strong acids (< 1×10⁻⁶ M) accounts for water's autoprotolysis, ensuring accurate results across all concentration ranges.
? Key equations
• Strong acid: [H⁺] = Cacid | pH = -log₁₀(Cacid)
• Strong base: [OH⁻] = Cbase → pOH = -log₁₀(Cbase), pH = 14 - pOH (at 25°C)
• Weak acid: [H⁺]² + Ka[H⁺] – Ka·Ca = 0 → exact quadratic solution
• Temperature effect: Kw(T) = exp(ΔG°/RT) , e.g., Kw(0°C) ≈ 1.14×10⁻¹⁵ , Kw(50°C) ≈ 5.48×10⁻¹⁴
Our algorithm determines the exact hydrogen ion concentration by considering the type of solute, concentration, and temperature‑dependent Kw. For strong acids (monoprotic), we solve [H⁺] = Ca + [OH⁻] (mass balance + charge) which simplifies to [H⁺] = Ca when Ca >> 1e-6; otherwise we solve the cubic from water autoionization: [H⁺]² – Ca[H⁺] – Kw = 0 → exact analytical root. For weak acids, we solve the full polynomial without approximation using the quadratic formula derived from the equilibrium expression. Weak bases are treated analogously by converting to [OH⁻] via Kb, then pOH. The solver also checks if the 'x is small' approximation is valid and displays an appropriate note.
Water autoprotolysis is endothermic: Kw increases with temperature. At 37°C (body temperature), Kw ≈ 2.5×10⁻¹⁴, making neutral pH ≈ 6.81, not 7.00. This calculator incorporates the temperature‑Kw correlation (validated by IUPAC data) to give accurate results for biological and industrial applications (e.g., swimming pools, aquariums, fermentation).
| Substance / Condition | Concentration (M) | pH (calc) | Type / Note |
|---|---|---|---|
| Hydrochloric acid | 0.100 | 1.00 | Strong acid, complete dissociation |
| Sodium hydroxide | 0.010 | 12.00 | Strong base |
| Acetic acid | 0.100 | 2.88 | Weak acid (Ka = 1.8×10⁻⁵) |
| Ammonia solution | 0.100 | 11.13 | Weak base (Kb = 1.8×10⁻⁵) |
| Ultra-pure water at 25°C | — | 7.00 | Kw=1.0×10⁻¹⁴ |
| Ultra-pure water at 60°C | — | 6.51 | Kw increased → neutral pH drops |
In fish farming, ammonia toxicity depends on pH. Un-ionized ammonia (NH₃) is far more toxic than NH₄⁺. Using our pH calculator, a farmer can input total ammonia concentration (as weak base) and pH to evaluate toxic fraction. The underlying weak base equilibrium (Kb of NH₃ ≈ 1.8×10⁻⁵) determines the [NH₃]/[NH₄⁺] ratio, guiding aeration and water exchange decisions. This tool thus helps prevent fish mortality and ensures sustainable aquaculture.
For a weak acid HA with initial concentration C and dissociation degree α, the equilibrium [H⁺] = x, then x²/(C – x) = Ka. The exact quadratic is x² + Ka·x – Ka·C = 0. The positive root x = [-Ka + sqrt(Ka² + 4KaC)]/2 is used. If x/C < 0.05, the approximation x ≈ sqrt(Ka·C) is acceptable; our calculator shows a note when approximation error is small, enhancing transparency.