What a titration measures and why the arithmetic is simple
A titration turns a volume you can read precisely into a concentration you cannot measure directly. You add a solution of exactly known strength — the titrant — from a burette into a measured aliquot of the unknown until an indicator or a pH electrode shows the reaction is complete. The volume it took is the measurement, and everything else is stoichiometry.
The arithmetic rests on a single idea: at the equivalence point, the number of moles of titratable protons supplied by the acid equals the number accepted by the base. Neither the identity of the species nor the shape of the pH curve enters the calculation. Only the mole counts matter.
Moles of a solute equal molarity times volume, so “moles of protons” equals M × V × n, where n is the number of protons per formula unit. Setting the two sides equal gives MₐVₐnₐ = M_bV_bn_b, and solving for whichever quantity you do not know is one division.
Titration remains the reference method for many regulated determinations precisely because it is traceable: your answer depends on a certified primary standard, a calibrated burette and a calibrated pipette, and on nothing else. Water hardness, milk and juice acidity, fatty-acid content in oils, active-ingredient assay in pharmacopoeial monographs and free acid in plating baths are all still titrated.
The proton-count term is the part people get wrong
Write out the neutralisation you are performing before you touch the numbers. For HCl + NaOH → NaCl + H₂O the ratio is one to one, so nₐ = n_b = 1 and the equation reduces to MₐVₐ = M_bV_b. That is the version most textbooks quote, and it is a special case, not the general rule.
Now take H₂SO₄ + 2 NaOH → Na₂SO₄ + 2 H₂O. Each molecule of sulfuric acid delivers two protons, so nₐ = 2. A 0.0500 M sulfuric acid solution neutralises twice as much hydroxide as a 0.0500 M hydrochloric acid solution of the same volume. Omitting the 2 halves your answer. The same applies to Ca(OH)₂, Ba(OH)₂ and Na₂CO₃ titrated to carbonic acid, all of which take two protons.
The proton count also depends on where you stop. Sodium carbonate titrated with HCl has two equivalence points: the first at pH ≈ 8.3, where CO₃²⁻ becomes HCO₃⁻ and n = 1, and the second at pH ≈ 3.8, where HCO₃⁻ becomes H₂CO₃ and n = 2. Phosphoric acid has three. Your indicator choice, not the formula, decides which n applies — so pick the indicator first and set n to match.
Two other terms deserve care. Vₐ is the pipetted aliquot, not the total volume in the flask; water added to make the indicator visible dilutes the solution but does not change the moles present, so it never enters the calculation. And M_b must be the standardised molarity. Sodium hydroxide absorbs carbon dioxide from the air and its concentration drifts, which is why it is standardised against potassium hydrogen phthalate rather than trusted from the bottle.
Normality, reported above, is simply N = M × n: concentration expressed in equivalents per litre rather than moles per litre. In normality terms the equivalence condition is even simpler, NₐVₐ = N_bV_b, which is why older analytical methods are written that way. The pH calculator converts the resulting concentrations into pH if you need the acidity rather than the amount.
Worked example: standardising an unknown HCl against 0.1000 M NaOH
You pipette 25.00 mL of an unknown hydrochloric acid solution into a conical flask, add a few drops of phenolphthalein and titrate with 0.1000 M sodium hydroxide. The endpoint arrives at a burette reading of 23.45 mL.
- Write the reaction. HCl + NaOH → NaCl + H₂O. One proton each way, so nₐ = 1 and n_b = 1.
- Count the moles of titrant. Molarity in mol/L times volume in mL gives millimoles directly: 0.1000 × 23.45 = 2.345 mmol NaOH.
- Convert to moles of analyte. The ratio is one to one, so the flask contained 2.345 mmol HCl.
- Divide by the aliquot volume. Mₐ = 2.345 mmol ÷ 25.00 mL = 0.09380 M.
- Convert to a mass if you want one. HCl has M = 36.46 g/mol, so the aliquot held 2.345 mmol × 36.46 g/mol = 85.50 mg of HCl.
Now check the sensitivity. A burette read to ±0.05 mL means the titre is 23.45 ± 0.05 mL, a relative uncertainty of 0.21%. Because Mₐ is strictly proportional to V_b, the concentration inherits exactly that 0.21%: 0.09380 ± 0.00020 M. This is why analysts aim for a titre between 10 and 40 mL — a 5 mL titre carries five times the relative reading error of a 25 mL one.
Change one thing and watch the proton count bite. Suppose the unknown had been sulfuric acid rather than hydrochloric. The same 23.45 mL of 0.1000 M NaOH still delivers 2.345 mmol of hydroxide, but each H₂SO₄ molecule supplies two protons, so the flask held only 2.345 ÷ 2 = 1.1725 mmol of acid, and Mₐ = 0.04690 M — half the previous answer.
Reading the result and judging whether the titration was any good
Judge a titration by three numbers before you trust its concentration. First, the titre volume: 10 to 40 mL on a 50 mL burette. Below 10 mL the reading error dominates; above 50 mL you have to refill mid-titration and add another reading error. If your titre falls outside that band, change the aliquot volume or the titrant strength, not the arithmetic.
Second, the agreement between replicates. Concordant titres — three readings within 0.10 mL of one another — are the conventional standard for classical volumetric analysis. Discard the rough first titration and average the concordant ones.
Third, the difference between the endpoint and the equivalence point. The equivalence point is where the stoichiometry balances; the endpoint is where your indicator changes colour. They coincide only if the indicator's transition range brackets the pH at equivalence. For a strong acid titrated with a strong base, equivalence sits at pH 7 and almost any common indicator works. For a weak acid titrated with a strong base, equivalence sits above 7 — acetic acid with sodium hydroxide lands near pH 8.7 — so phenolphthalein (8.3–10.0) is right and methyl orange (3.1–4.4) would stop far too early. For a weak base titrated with a strong acid, equivalence sits below 7 and the choice reverses.
The chart above assumes both species are strong. If yours is weak, the region before equivalence is a buffer and the curve is much flatter there; the Henderson-Hasselbalch calculator gives the pH at any point in that region, and the half-equivalence volume is where pH equals pKa. The weak acid pH calculator and weak base pH calculator handle the endpoints themselves.
Proton counts and indicator choice for common titrations
| Analyte | Titrant | pH at equivalence | n (analyte) | Usual indicator |
|---|---|---|---|---|
| HCl (strong acid) | NaOH | 7.0 | 1 | Phenolphthalein or methyl orange |
| H₂SO₄ (diprotic) | NaOH | ≈7 | 2 | Phenolphthalein |
| CH₃COOH (weak acid) | NaOH | ≈8.7 | 1 | Phenolphthalein |
| Citric acid (triprotic) | NaOH | ≈9.4 | 3 | Phenolphthalein |
| NaOH (strong base) | HCl | 7.0 | 1 | Either |
| Ca(OH)₂ (dibasic) | HCl | 7.0 | 2 | Methyl orange |
| NH₃ (weak base) | HCl | ≈5.3 | 1 | Methyl red or methyl orange |
| Na₂CO₃ to HCO₃⁻ | HCl | ≈8.3 | 1 | Phenolphthalein |
| Na₂CO₃ to H₂CO₃ | HCl | ≈3.8 | 2 | Methyl orange |
Equivalence pH values are for roughly 0.1 M solutions and shift with concentration; the two carbonate rows show the same analyte with two different valid answers depending on where you stop.
Sources of error this calculator cannot see
- An unstandardised titrant. Bench sodium hydroxide absorbs atmospheric CO₂ and its molarity falls over weeks. Standardise against a primary standard such as potassium hydrogen phthalate and use that number, not the label.
- The wrong proton count for where you stopped. Polyprotic species have more than one equivalence point. The indicator decides which one you measured; n must match it.
- Indicator error. The endpoint colour change occurs slightly after the equivalence point. Running an indicator blank on the same volume of water and subtracting it removes most of this.
- Carbonate in the sample. A carbonate impurity in the analyte consumes titrant and inflates the apparent concentration, and it produces a two-step curve that is easy to miss with a colour indicator.
- Dilution water counted as aliquot volume. Water added to the flask changes nothing in the calculation. Enter only the pipetted aliquot.
- Temperature effects on volumetric glassware. Class A glassware is calibrated at 20 °C, and water expands by about 0.02% per °C near room temperature, so working 10 °C away shifts the delivered volume by roughly 0.2% — the same size as the burette reading error on a 25 mL titre. Control the temperature when you need that last 0.1%.
- Assuming a 1:1 reaction from the names alone. Write the balanced equation every time. The stoichiometric ratio is the only thing that connects the burette to the answer.
Related techniques and where titration stops being the right tool
Acid-base titration is one member of a family. Redox titrations (permanganate, dichromate, iodine) use electrons rather than protons as the equivalence currency, and the same MVn equation applies with n as the electron count. Complexometric titrations with EDTA determine metal ions and almost always run at n = 1 because EDTA binds one metal ion per molecule. Precipitation titrations such as the Mohr method for chloride use the stoichiometry of the precipitate; the solubility side of that is handled by the solubility product calculator.
Move away from titration when the analyte is very dilute, coloured, or present in a complex matrix. Below roughly 10⁻⁴ M the endpoint break becomes too shallow to see and an instrumental method — ion chromatography, UV-Vis with the Beer-Lambert law, or potentiometry with an ion-selective electrode read through the Nernst equation — is more reliable. Automatic potentiometric titrators extend the usable range by locating the equivalence point from the steepest point of the pH curve rather than from a colour change, which also removes indicator error entirely.
For the reverse problem — you know both concentrations and want the pH at some intermediate point rather than the equivalence volume — the buffer region is described by Henderson-Hasselbalch and the strong-strong region by simple excess-concentration arithmetic. Both are covered by the pH and buffer calculators linked above.