Acid-Base Titration: pH Curve, Indicators, Buffers and Equivalence Point

Presets

Live readouts

Graphs

Titration curve: pH against titrant volume (dashed: exact model, orange: your run)

Steepness |dpH/dV|: the equivalence point is the maximum

Species fractions against pH, with the indicator base form

Buffer capacity beta against titrant volume

What is happening now

Start the titration.

Numerical checks

running...

How to use

Choose a preset or build your own: analyte, concentrations, indicator, temperature. Open the stopcock with the slider (or press +1 drop, +0.1 mL, +1 mL) and watch the burette level, the flask colour and the pH meter. The panel under the controls says in words which part of the curve you are in. Fast-forward runs to the end; Auto-slow mimics a careful chemist who slows down at the steep jump. Space pauses, R restarts. Things to try: (1) titrate acetic acid and read the pH at half the equivalence volume - it equals the pKa; (2) repeat with methyl orange and compare the endpoint error in the readout; (3) switch to citric acid and see why a polyprotic buffer has no jump until the last proton; (4) in Blind mode find the unknown concentration from the endpoint, then press Reveal; (5) heat the flask to 60 C and see where neutral pH moves.

The science

In an acid-base titration a solution of known concentration (the titrant, from the burette) is added to a measured volume of the analyte until the reaction is complete. The pH of the flask is found at every step from an exact calculation, not from textbook shortcuts, so the same model gives the strong-acid jump, the weak-acid buffer plateau, the polyprotic staircase and the endpoint error of an indicator.

Exact pH from electroneutrality. The flask always has zero net charge. With analyte family H_nX of total concentration C_a (diluted by the added titrant), titrant ions C_Na or C_Cl, and any spectator ions of the salt, the charge balance is

C_Na − C_Cl + z_c C_a + [H⁺] − K_w/[H⁺] + C_a Σ_j α_j (z₀ − j) = 0

where α_j is the fraction of the family that has lost j protons, from the stepwise acidity constants:

α_j = [H⁺]^n−j K₁…K_j / Σ_i [H⁺]ⁿ⁻ⁱ K₁…K_i, C_a = c_a V_a / (V_a + V_b)

The left-hand side rises monotonically with [H⁺], so it has exactly one root, found by bisection to 10⁻¹⁵ in pH. The same equation covers strong acids and bases (empty pKa list), weak acids and bases, diprotic and triprotic acids and the dibasic salt Na₂CO₃.

Equivalence. The k-th step is complete at V_e,k = k c_a V_a / c_t. The unknown concentration in a real lab follows from c_a = c_t V_end / (k V_a). For a weak acid of pKa the equivalence pH is that of its conjugate base:

pH_eq ≈ 7 + ½ (pK_a + log₁₀ C_eq), half-equivalence: pH = pK_a (Henderson-Hasselbalch)

Buffer capacity measures resistance to pH change, β = dC_titrant/dpH. The closed form of Van Slyke is

β = ln10 [ K_w/[H⁺] + [H⁺] + C_a Var(j) ], Var(j) = Σ α_j j² − (Σ α_j j)²

For a monoprotic acid this peaks at pH = pKa with β_max = ln10 C_a/4 = 0.576 C_a (plus the small water terms). The curve of β (graph 4) is highest in the buffer zones and in the very acidic or basic ends, and lowest at the equivalence point, which is why the pH jumps there.

Indicators are weak acids with a coloured conjugate pair. Their colour is the mixture of both forms, with base-form fraction f = 1/(1 + 10^pK_in−pH); the eye sees a change roughly between f = 0.1 and 0.9, i.e. pK_in ± 1. The endpoint error is the difference between the volume where f = 0.5 and the nearest equivalence volume. A good indicator has its pK_in inside the steep jump; for acetic acid that is phenolphthalein, not methyl orange.

Temperature. Only the ion product of water is changed: pK_w(T) = 4470.99/T − 6.0875 + 0.01706 T (T in K, an empirical fit), so a neutral solution has pH = pK_w/2, which is 6.5 at 60 C. The pK_a values are held at their 25 C values.

Numerical checks run in the page. (1) HCl + NaOH equivalence pH = pK_w/2; (2) acetic acid start pH against the quadratic closed form; (3) half-equivalence pH = pKa; (4) the weak-acid equivalence formula; (5) numerical β from electroneutrality against the Van Slyke closed form; (6) oxalic acid α(H₂A) = α(A²⁻) at (pK₁+pK₂)/2; (7) the charge-balance residual for phosphoric acid; (8) an inverse solve for the volume at pH 7; (9) pK_w at 25 C; (10) the species fractions sum to one.

Simplifications. All activities are taken as 1 (ideal solutions; real strong-electrolyte solutions above about 0.1 mol/L deviate by 0.05 to 0.1 pH units); pK_a values are typical 25 C literature values and are not corrected for temperature or ionic strength; the indicator takes part only as a colour (its own acid-base buffering is ignored; the 0.2 mL of indicator solution is counted only as dilution); CO₂ absorption from air, the burette tolerance (about 0.05 mL), drop size variation and electrode response lag are ignored; the colour mixing delay in the 3D flask is purely visual. It teaches the shape of titration curves and the logic of the method, it is not a substitute for a laboratory measurement.