Titration Curve Calculator: pH, Volume & Indicator Choice

The equilibrium formulation below is derived here from mass conservation, electroneutrality and mass action. OpenStax Chemistry 2e §§14.3, 14.5 and 14.7 provide the equilibrium relations and independent worked examples.

Let $h=[H_3O^+]/c^\circ$, $c^\circ=1\,\mathrm{mol/L}$, $c_0=C_s/c^\circ$, $c_t=C_t/c^\circ$, and $z=C_tV/(C_sV_s)$. Dilution gives $c_A=c_0/(1+zc_0/c_t)$. The normalized spectator-ion concentration is $zc_A$.

The mean analyte charge $q(h)$ is $-1$ for a strong monoprotic acid, $+1$ for a strong monobasic base, $-K_a/(h+K_a)$ for HA, and $h/(h+K_{a,BH^+})$ for B, with $K_{a,BH^+}=K_w/K_b$. For H₂A:

The solver bisects pH on −2…16 for 46 iterations to satisfy

where $s=+1$ for base titrant and −1 for acid titrant. Here $K_w=10^{-14}$ at 25 °C. Constants and concentrations use the same standard-state normalization. No Henderson–Hasselbalch approximation or excess-reagent shortcut is used for the reported pH.

Stoichiometry gives $V_{eq,j}=jC_sV_s/C_t$ and $V_{half,j}=(j-1/2)C_sV_s/C_t$. In endpoint mode $C_s=C_tV_{endpoint}/(jV_s)$. These are molecular concentrations, with $j=1$ or 2.

For each indicator boundary, the same charge balance is solved algebraically for $z$. Boundaries outside the run are left blank. The comparison reports $100\max|V_{boundary}/V_{eq,j}-1|$. Merck's attached indicator guide supplies the four transition intervals (pp. 6–8). The full curve is cached independently of burette playback.

• Fixed 25 °C; $K_w=10^{-14}$; equilibrium reached after each drop.
• Activity coefficients are ignored: concentrations replace activities. Volumes add.
• One initially unsalted analyte: HA, B, H₂A, or a fully dissociated monoprotic acid/monobasic base. Strong titrant contributes one H⁺ or OH⁻ per mole.
• Spectator ions are inert. No precipitation, complexes, gas exchange or indicator loading.
• The carbonic-acid defaults use OpenStax's effective constants in a closed system. An open carbonate sample exchanges CO₂ and needs a different model.
• Flask colours and liquid levels are original schematic illustrations, not optical or vessel-volume predictions. Colour stages follow pH across the supplier's range; exact hues are not calibrated.
• The 1% and 5% indicator ratings are author-selected screening thresholds for the whole transition interval.

Inputs cover 0.000001–0.5 mol/L; concentrations above 0.2 mol/L trigger a dilute-model warning. Activity effects can matter even below that threshold, particularly for multivalent ions. For accurate analytical work use suitable activities, standardized titrant, a validated endpoint and measured uncertainty.

Diprotic pKa₂ must exceed pKa₁. The coupled equations also handle overlapping dissociations, but separate stoichiometric points may not produce distinct jumps. Half-neutralization pH need not equal pKa when water, dilution or overlapping equilibria matter. pKa ± 1 bands show species ratios, not absolute buffering strength.

Burette playback stops at twice the final stoichiometric equivalence volume. The selected volume is rounded down to complete drops, except for the final partial drop. Indicator comparisons use continuous equilibrium volumes, so their resolution can exceed the chosen drop size. A blank transition boundary means it is outside this run.

Endpoint concentration assumes the measured endpoint is the selected equivalence. This is a stoichiometric inference, not a correction for indicator bias or a titration-curve fit. No weak titrants, mixtures, triprotic samples, initially added salts, variable temperature or gas-transfer kinetics are modeled.