Titration Calculator with Steps (Free) | StatsUnlock

A titration calculator uses moles = concentration × volume and the mole ratio from the balanced equation: c(analyte) = c(titrant) × V(titrant) × a/b ÷ V(analyte). For 25.00 mL of HCl needing a mean concordant titre of 24.55 mL of 0.1000 M NaOH: 0.1000 × 24.55 ÷ 25.00 = 0.09820 mol/L HCl (± 0.00042, k = 2).

Rule of thumb: ignore the rough titre, average only concordant titres (within 0.10 mL), convert mL to L, and use the mole ratio: H₂SO₄ needs twice as much NaOH as HCl.

A titration measures an unknown amount of a substance (the analyte) by adding a solution of known concentration (the titrant) from a burette until the reaction is just complete: the equivalence point, seen as the endpoint by an indicator color change or a pH jump. Because moles = concentration × volume and the balanced equation fixes the mole ratio, one volume reading gives the unknown concentration. Students often find the mole ratio and the meaning of the equivalence point the hardest parts (Sheppard, 2006). A titration calculator handles the arithmetic, the concordant titres, the uncertainty and, for acid–base titrations, the full pH curve from an exact charge balance (de Levie, 1999).

A worked one-liner: n(NaOH) = 0.1000 × 24.55 ÷ 1000 = 2.455 × 10⁻³ mol = n(HCl); c(HCl) = 2.455 × 10⁻³ ÷ 0.02500 = 0.09820 mol/L.

Who uses it: GCSE, A level, IB and AP chemistry students writing up titration practicals; university analytical chemistry labs; quality control in food, pharmaceutical and water labs; and anyone checking an acid–base titration lab report, choosing an indicator or predicting the endpoint before running a titration.

Quick answers to the questions people ask most

4.1 What the result means

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4.2 The formula

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4.3 The detail that changes the answer

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4.4 Accuracy and uncertainty

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4.5 What to do next

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4.6 Endpoint versus equivalence point

The equivalence point is where the moles of titrant exactly match the analyte according to the equation; the endpoint is where you see the change. A well-chosen indicator changes color on the steep part of the curve, so the two differ by less than a drop. For weak endpoints, a pH meter and a first-derivative or Gran plot locate the equivalence point more precisely (Gran, 1952; Rossotti & Rossotti, 1965).

4.7 Why exact pH curves

Textbook pH calculations split the curve into regions (strong acid, buffer, salt, excess base) with a different approximation for each. Solving the full charge balance at every volume gives one exact expression that works for strong, weak and polyprotic systems and mixtures (de Levie, 1993; Asuero & Michałowski, 2011). It also shows when an equivalence point is too flat to detect, as for the third proton of phosphoric acid.

4.8 Choosing the titrant

Titrating a weak base with a weak acid gives no sharp endpoint, so one partner is always strong. The choice of titrant and receiver solution still matters in methods such as Kjeldahl nitrogen, where the theory of the titration curve guides it (Michałowski et al., 2013).

A titration result in a report needs the concentration with units, the mean concordant titre and how many titres it came from, the titrant concentration, the indicator or endpoint method, and the uncertainty. The seven write-ups below are filled from the titration calculator; they show the headline example until you press Calculate.

Showing the headline example (HCl with 0.1000 M NaOH). Press Calculate above to fill all seven write-ups with your own result.

5.1 Style rules

5.2 Seven ready-to-copy write-ups

The same worked problem as Sections 0, 9, 10, 11 and 14: 25.00 mL HCl, 0.1000 M NaOH, titres 24.80 (rough), 24.55, 24.60, 24.50 mL.

1. Pick what you want to calculate. Choose a mode in Section 2: concentration, pH titration curve, molar mass of an unknown acid, back titration, double titration or titratable acidity.
2. Load a sample or type your data. Twelve worked problems are in the sample menu. Sample 1 is the headline HCl and NaOH titration. Nothing is calculated until you press Calculate Titration.
3. Say what you are solving for. In the concentration mode, find the analyte concentration, standardize the titrant, or predict the endpoint volume before titrating.
4. Enter the mole ratio. Type the coefficients of the analyte (a) and titrant (b) from the balanced equation: 1 and 2 for H2SO4 with NaOH, 5 and 1 for Fe2+ with KMnO4.
5. Enter all titres in order. List every titre separated by commas. Choose whether the first is a rough run; the calculator keeps the largest concordant set within 0.10 mL.
6. Add tolerances for the uncertainty. Optional: burette, pipette and titrant tolerances give an expanded uncertainty and a budget showing the biggest source.
7. Press Calculate Titration. Press the button or Enter. The concentration, moles, a Verified badge, the formula and every step appear in Section 3.
8. Read the curve and charts. The figure draws the burette or the pH curve; four charts show the titres, the derivative, moles and sensitivity.
9. Copy the write-up. Section 5 fills seven ready texts for a lab report, a log, code, an exam answer, chat, a results paragraph and a table.
10. Download or run a batch. Download Doc or PDF, or calculate many titrations at once from a CSV of titrant molarity, titre, aliquot and ratio in the Batch tab.

A titration sheet needs one formula for the mean titre and one for the concentration. The sheet repeats the headline problem, so the numbers match Sections 0, 7, 10, 11 and 14.

9.1 The formulas you need

9.2 Step by step

9.3 Many groups or samples

Put one titration per row and fill the formula down. Add a column with the concordance check so non-concordant rows stand out.

9.4 Why Excel may differ from R or Python

9.5 Charting a pH curve

Enter volume and measured pH in two columns and insert a Scatter chart with smooth lines. Add a column of ΔpH/ΔV to plot the first derivative; its peak marks the equivalence point.

9.6 Errors you may see

• #DIV/0!: aliquot cell empty.
• An answer 1000 times too large: mL divided by L, or vice versa.
• A result double or half the expected: mole ratio missing.

A complete, copy-and-run script that reproduces the titration calculator for the headline problem. It excludes the rough titre, averages concordant titres, calculates the HCl concentration with its expanded uncertainty, solves the exact pH curve for acetic acid with NaOH, finds the molar mass of an unknown acid and saves one 300 dpi figure.

10.1 The complete script

R, base only, saves titration_curve.png

# Titration calculator in base R (StatsUnlock)
concordant <- function(v, rough = TRUE, tol = 0.10) {
  if (rough) v <- v[-1]                           # drop the rough (first) titre
  s <- sort(v); best <- s[1]
  for (i in seq_along(s)) for (j in i:length(s))
    if (s[j] - s[i] <= tol + 1e-9 && (j - i + 1) > length(best)) best <- s[i:j]
  best
}
titres <- c(24.80, 24.55, 24.60, 24.50)            # mL; first is the rough run
used <- concordant(titres); vt <- mean(used)
ct <- 0.1000; va <- 25.00; ratio <- 1             # NaOH M, HCl aliquot mL, a/b
ca <- ct * vt / 1000 * ratio / (va / 1000)
cat(sprintf("concordant: %s; mean %.2f mL; c(HCl) = %.5f mol/L\n", paste(used, collapse = ", "), vt, ca))
# uncertainty: burette 0.05, repeatability, pipette 0.03, NaOH 0.0002 (rectangular)
urel <- sqrt((0.05 / sqrt(3) / vt)^2 + (sd(used) / sqrt(length(used)) / vt)^2 +
             (0.03 / sqrt(3) / va)^2 + (0.0002 / sqrt(3) / ct)^2)
cat(sprintf("U (k = 2) = %.5f mol/L\n", 2 * urel * ca))
# exact pH: 25 mL 0.1 M acetic acid (pKa 4.76) with 0.1 M NaOH
ph_at <- function(v, va = 25, ca = 0.1, cb = 0.1, Ka = 10^-4.76, Kw = 1e-14) {
  V <- va + v; CA <- ca * va / V; Na <- cb * v / V
  f <- function(lh) { h <- 10^lh; h + Na - Kw / h - CA * Ka / (Ka + h) }
  -uniroot(f, c(-16, 2), tol = 1e-12)$root
}
cat(sprintf("pH: start %.2f, half-equivalence %.2f, equivalence %.2f\n", ph_at(0), ph_at(12.5), ph_at(25)))
# molar mass: 0.1500 g diprotic acid, 25.40 mL of 0.1000 M NaOH
cat(sprintf("molar mass = %.1f g/mol\n", 0.1500 / (0.1000 * 25.40 / 1000 / 2)))
png("titration_curve.png", width = 7, height = 4, units = "in", res = 300)
v <- seq(0, 40, 0.2)
plot(v, sapply(v, ph_at), type = "l", lwd = 2, xlab = "NaOH added, mL", ylab = "pH", ylim = c(0, 14))
abline(v = 25, lty = 2); points(25, ph_at(25), pch = 19)
invisible(dev.off())

10.2 Output

concordant: 24.5, 24.55, 24.6; mean 24.55 mL; c(HCl) = 0.09820 mol/L
U (k = 2) = 0.00042 mol/L
pH: start 2.88, half-equivalence 4.76, equivalence 8.73
molar mass = 118.1 g/mol

10.3 Traps

• sd() is the sample SD, which is right for repeatability.
• uniroot() needs a bracket where the charge balance changes sign; −16 to 2 in log[H⁺] covers pH −2 to 16.
• Keep volumes in mL in the formula but divide by 1000 for moles.

10.4 What each part does

10.5 Adapting it

Change the titres, concentrations and pKa for your titration; ph_at() works for any monoprotic acid and can be extended to polyprotic systems.

A complete, copy-and-run script that reproduces the titration calculator for the headline problem. It excludes the rough titre, averages concordant titres, calculates the HCl concentration with its expanded uncertainty, solves the exact pH curve for acetic acid with NaOH, finds the molar mass of an unknown acid and saves one 300 dpi figure.

Install once: pip install numpy scipy matplotlib

11.1 The complete script

Python with NumPy, SciPy and Matplotlib, saves titration_curve.png

# Titration calculator in Python (StatsUnlock)
import numpy as np
from scipy.optimize import brentq
import matplotlib.pyplot as plt

def concordant(v, rough=True, tol=0.10):
    """Drop the rough titre, return the largest set of titres within tol mL."""
    v = sorted(v[1:] if rough else v); best = v[:1]
    for i in range(len(v)):
        for j in range(i, len(v)):
            if v[j] - v[i] <= tol + 1e-9 and j - i + 1 > len(best):
                best = v[i:j + 1]
    return best

used = concordant([24.80, 24.55, 24.60, 24.50])      # mL, first is rough
vt = np.mean(used); ct, va, ratio = 0.1000, 25.00, 1.0
ca = ct * vt / 1000 * ratio / (va / 1000)
print(f"concordant {used}; mean {vt:.2f} mL; c(HCl) = {ca:.5f} mol/L")

urel = np.sqrt((0.05 / np.sqrt(3) / vt) ** 2 + (np.std(used, ddof=1) / np.sqrt(len(used)) / vt) ** 2
               + (0.03 / np.sqrt(3) / va) ** 2 + (0.0002 / np.sqrt(3) / ct) ** 2)
print(f"U (k = 2) = {2 * urel * ca:.5f} mol/L")

def ph_at(v, va=25, ca=0.1, cb=0.1, Ka=10 ** -4.76, Kw=1e-14):
    """Exact pH of a weak acid titrated with NaOH (charge balance)."""
    V = va + v; CA = ca * va / V; Na = cb * v / V
    f = lambda lh: 10 ** lh + Na - Kw / 10 ** lh - CA * Ka / (Ka + 10 ** lh)
    return -brentq(f, -16, 2, xtol=1e-12)

print(f"pH: start {ph_at(0):.2f}, half-equivalence {ph_at(12.5):.2f}, equivalence {ph_at(25):.2f}")
print(f"molar mass = {0.1500 / (0.1000 * 25.40 / 1000 / 2):.1f} g/mol")

v = np.linspace(0, 40, 201)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(v, [ph_at(x) for x in v], lw=2); ax.axvline(25, ls="--"); ax.plot(25, ph_at(25), "o")
ax.set_xlabel("NaOH added, mL"); ax.set_ylabel("pH"); ax.set_ylim(0, 14)
fig.tight_layout(); fig.savefig("titration_curve.png", dpi=300)

11.2 Output

concordant [24.5, 24.55, 24.6]; mean 24.55 mL; c(HCl) = 0.09820 mol/L
U (k = 2) = 0.00042 mol/L
pH: start 2.88, half-equivalence 4.76, equivalence 8.73
molar mass = 118.1 g/mol

11.3 Traps

• np.std needs ddof=1 for the sample SD.
• brentq() needs opposite signs at the two ends of the bracket.
• Solve in log[H⁺], not [H⁺], so the root finder covers 16 orders of magnitude.

11.4 What each part does

11.5 Adapting it

Change the titres, concentrations and pKa for your titration; ph_at() works for any monoprotic acid and can be extended to polyprotic systems.

12.1 Acid–base indicators

Transition ranges are approximate and vary slightly between sources. Last checked: October 2026.

12.2 pKa values used by the calculator (25 °C)

Typical literature values at 25 °C and low ionic strength; see Goldberg et al. (2002) for buffer acids. Enter your own values in custom mode. Last checked: October 2026.

12.3 Concentration from titre (0.1000 M titrant, 25.00 mL aliquot)

c = 0.1000 × titre × a/b ÷ 25.00. Exact. Last checked: October 2026.

12.4 Titratable acidity factors

meq weight = molar mass ÷ acidic protons ÷ 1000 (Tyl & Sadler, 2017). Last checked: October 2026.

12.5 Class A glassware tolerances

Typical Class A tolerances for burettes (ISO 385) and one-mark pipettes (ISO 648); check the marking on your glassware. Last checked: October 2026.

12.6 pH during the titration of 25 mL of 0.1 M acetic acid with 0.1 M NaOH

Exact charge balance, pKa 4.76, Kw 1.0 × 10⁻¹⁴; the jump from 24.9 to 25.1 mL is about 3 pH units. Last checked: October 2026.

Eight worked problems that cover what a titration calculator meets: a carbonate standardization, vinegar, a redox titration, two pH curves, an unknown acid, an antacid back titration and the classic mistake. Each number was checked in Python.

The written method, calculator keys and the titration procedure, for the headline problem: 25.00 mL HCl, 0.1000 M NaOH, titres 24.80 (rough), 24.55, 24.60, 24.50 mL.

14.1 The written method

1. Equation. HCl + NaOH → NaCl + H2O (1 : 1).
2. Mean titre. Exclude 24.80; (24.55 + 24.60 + 24.50) ÷ 3 = 24.55 mL.
3. Moles of NaOH. 0.1000 × 0.02455 = 2.455 × 10⁻³ mol.
4. Moles of HCl. 2.455 × 10⁻³ mol (1 : 1).
5. Concentration. 2.455 × 10⁻³ ÷ 0.02500 = 0.09820 mol/L.
6. Check. 0.09820 × 25.00 = 0.1000 × 24.55 ✓

14.2 On a calculator

14.3 Lab SOP: acid–base titration

14.4 Practice problems

14.5 Mistakes that lose marks

• Averaging the rough titre.
• Forgetting to convert mL to L.
• Ignoring the mole ratio.
• Recording burette readings to 0.1 mL instead of 0.05 mL.
• Too many significant figures.
• Wrong indicator for a weak acid or base.

14.6 Check-yourself and lab questions

Almost every wrong titration answer comes from volumes, ratios or the endpoint. The titration calculator guards against each one below.

Ionic equilibria at higher concentrations, activity corrections and mixed solvents are covered in specialist texts (Burgot, 2012).

Your result

The titration calculator gives 0.09820 mol/L from a mean titre of 24.55 mL.

What the titration calculator does

Every titration rests on moles = concentration × volume and a mole ratio. This titration calculator applies that with concordant titres and uncertainty, standardizes titrants, predicts endpoints, draws exact pH curves with equivalence points and indicators, and handles molar mass, back titration, double titration and titratable acidity, with every step shown.

Watch out for

• Exclude the rough titre.
• Convert mL to L.
• Use the mole ratio.
• Match the indicator to the pH jump.
• State which acid titratable acidity is expressed as.

Next steps

To make up titrant and standard solutions, use the molarity calculator and the dilution calculator; for EDTA titrations of calcium and magnesium, the water hardness calculator. Whatever the reaction, start and finish with the titration calculator.

More lab, chemistry and measurement tools from StatsUnlock, with worked steps, code and references.

Peer-reviewed sources behind this titration calculator: exact acid–base titration curves, Gran plots and equivalence-point uncertainty, pKa data, carbonate alkalinity, the Kjeldahl titration, EDTA titrations, titratable acidity of foods and wine, antacid back titration, student understanding of titrations, atomic weights and measurement uncertainty. Standards (JCGM 100:2008, ISO 385 and ISO 648 glassware classes) are named where they are used. Every DOI checked October 2026.

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2. Ayensu, I., Bekoe, S. O., Adu, J. K., Brobbey, A. A., & Appiah, E. (2020). Evaluation of acid neutralizing and buffering capacities of selected antacids in Ghana. Scientific African, 8, e00347. https://doi.org/10.1016/j.sciaf.2020.e00347
3. Boulton, R. (1980). The relationships between total acidity, titratable acidity and pH in wine. American Journal of Enology and Viticulture, 31(1), 76–80. https://doi.org/10.5344/ajev.1980.31.1.76
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