Wait, What? A titration curve does not have to show a dramatic vertical jump for you to estimate where chemical equivalence lies. You can transform part of the curve into a nearly straight line and let its intercept do the work.
That is the core idea of a Gran plot. Instead of locating an equivalence point by eye, a Gran function combines titrant volume with a measured potential or pH so that an appropriate region becomes approximately linear. Extrapolating that line to its intercept estimates the equivalence volume.
The direct answer
A Gran plot is a model-based linearisation of titration data. For a strong monoprotic acid titrated with a strong base before equivalence, an idealised Gran function is:
F = (V0 + V)10−pH
If activity coefficients and electrode response are sufficiently stable, F is proportional to (Ve − V). Plot F against titrant volume V and the x-intercept gives an estimate of the equivalence volume Ve.
The power comes from chemistry, not graph magic. The linear relation only holds because stoichiometry, mass balance, acid–base equilibrium and electrode response have been simplified into a regime where the transformed equation is approximately linear.
Equivalence point is not the same as endpoint
IUPAC defines the equivalence point as the stage of a titration at which titrant has reacted with titrand according to the reaction stoichiometry. An endpoint is the experimentally recognised signal used to indicate that region—such as a colour change, potential criterion or mathematical estimate.
A Gran plot estimates the stoichiometric equivalence volume from measured data and a chemical/electrode model. It does not make the experimental endpoint and theoretical equivalence point conceptually identical.
Build the strong-acid case from a mole balance
Suppose an initial volume V0 of a strong monoprotic acid has formal concentration CA. A strong base of concentration CB is added in volume V. Before equivalence, acid remains in excess:
n(H+)excess = CAV0 − CBV
At equivalence, CAV0 = CBVe. Substitute that into the excess expression:
n(H+)excess = CB(Ve − V)
After accounting for dilution, the hydrogen-ion concentration is approximately:
[H+] ≈ CB(Ve − V)/(V0 + V)
If pH tracks hydrogen-ion activity and the activity coefficient remains sufficiently constant over the chosen points, 10−pH is proportional to [H+]. Multiplying by total solution volume removes the dilution denominator:
(V0 + V)10−pH ∝ (Ve − V)
That is why the transformed points can approach a line whose x-intercept is Ve.
Why the volume correction matters
Every titrant addition changes two things at once: it reacts chemically and increases the total volume. If you plotted 10−pH alone, dilution would remain entangled with stoichiometric consumption. Multiplying by V0 + V approximately converts concentration-like information back toward an amount-like quantity.
This is a general analytical lesson: transformations should have a chemical reason. A straight line is only useful if you know which physical term was removed or isolated.
A weak acid needs a different Gran function
For a weak monoprotic acid HA titrated with strong base before equivalence, most of the curve is a buffer mixture containing HA and A−. Under simplifying activity assumptions:
Ka ≈ a(H+)[A−]/[HA]
The stoichiometric ratio of conjugate base formed to acid remaining is approximately V/(Ve − V), so a useful transformed relationship becomes:
V·10−pH ∝ Ka(Ve − V)
The exact form depends on the activity convention and model. The important point is that one should not copy the strong-acid Gran function blindly into a weak-acid system. The chemistry determines the transform.
After equivalence, the dominant ion changes
Beyond the equivalence point of a strong-acid/strong-base titration, excess OH− rather than H+ controls the acid–base signal. A corresponding post-equivalence Gran function can be built from hydroxide activity, often expressed through pH and pKw. In the ideal case, pre- and post-equivalence extrapolations should point toward the same Ve.
If they do not, the disagreement is useful information. Electrode drift, changing activity coefficients, CO2 uptake, additional acid–base species, insufficient equilibration or an incomplete equilibrium model may be present.
pH is an activity measurement, not a direct concentration readout
One of the most important upgrades from school titration to analytical chemistry is recognising that pH is linked to hydrogen-ion activity, not simply numerical molar concentration:
pH = −log10a(H+)
At low and controlled ionic strength, concentration can be a useful approximation. As ionic strength changes, activity coefficients can change too. A Gran line can therefore curve even when stoichiometry is correct if the activity model is too simple.
The glass electrode adds another model layer
An ideal hydrogen-ion-responsive electrode follows a Nernst-type relation. Around 25 °C, the ideal magnitude is close to 59.16 mV per decade change in hydrogen-ion activity for a monovalent response. Real electrodes can show offset, slope deviation, drift, junction-potential changes, alkaline error or acid error depending on regime and instrument.
So a Gran plot built from pH contains at least three layers of reasoning: chemical stoichiometry, solution thermodynamics and electrode response. A beautiful straight line does not remove the need to understand all three.
A numerical thought experiment
Imagine 25.00 mL of 0.0100 mol dm−3 strong monoprotic acid titrated conceptually with 0.0100 mol dm−3 strong base. The stoichiometric equivalence volume is 25.00 mL because the reacting mole ratio is 1:1 and the formal concentrations are equal.
At V = 20.00 mL, the remaining acid amount is proportional to 5.00 mL of titrant-equivalent capacity. At V = 24.00 mL, it is proportional to 1.00 mL. The raw pH curve becomes increasingly steep, but the properly transformed Gran function should decrease approximately linearly toward zero at V = 25.00 mL under ideal assumptions.
This example is a calculation model, not a laboratory instruction. In real analysis, concentration standardisation, temperature, electrode behaviour and uncertainty must also be controlled and reported.
Why not simply use the steepest part of the titration curve?
Derivative methods can work very well when the potential change near equivalence is large and densely sampled. Gran’s original motivation included cases where the endpoint signal was small or awkward. Linearisation can use information from a broader region and extrapolate toward equivalence rather than depending entirely on a local maximum in slope.
But linearisation has its own risks. Transforming measurements changes the error structure, and the intercept can be sensitive to which points are selected. A method is not automatically more accurate because the graph looks straighter.
Choosing the linear region is a chemical decision
Points too far from equivalence may be influenced by species neglected in the simplified function. Points extremely close to equivalence can show curvature because neither excess H+ nor excess OH− overwhelmingly dominates, water autoionisation matters more, and the mathematical approximation underlying the chosen form weakens.
Therefore, one should look for a region that is both statistically linear and chemically compatible with the assumptions. “Use the points with the best R²” is not enough if the selected points correspond to the wrong equilibrium regime.
What the residuals can tell you
A fitted line produces residuals: differences between observed and predicted Gran-function values. Random scatter around zero is more reassuring than systematic curvature. Curvature can signal changing activity coefficients, additional proton-active species, electrode non-linearity, or a transformation whose assumptions do not hold over the full chosen range.
This is a key professional habit: inspect the failure pattern, not just the fitted intercept.
Gran plots beyond simple acid–base titration
Gran’s method is broader than the classroom strong-acid example. Linearised functions have been developed for precipitation, complexometric and redox titrations when an electrode signal can be related to the concentration or activity of a reacting species. Gran-function approaches are also important in natural-water alkalinity and acidity analysis.
A 2026 Ocean Science metrology study, for example, describes using Gran’s method as an initial estimate in seawater total-alkalinity measurement before fuller carbonate-system treatment. That application also illustrates the limitation: real seawater contains multiple acid–base systems and requires a model richer than “one strong acid plus one strong base”.
Observation versus inference
The instrument directly gives potential or a processed pH value at each titrant volume. Those are observations. The equivalence volume from a Gran intercept is an inference produced by a chemical and electrode model. Its uncertainty includes both measurement scatter and model adequacy.
Uncertainty belongs in the answer
At research or metrology level, a reported Ve should not be treated as exact. Important contributors can include titrant concentration, delivered-volume calibration, initial sample amount, electrode calibration and slope, temperature, regression uncertainty, blank corrections, drift, ionic-strength effects and model selection.
The x-intercept may look like one number, but it is the end of an uncertainty chain.
A learning progression from Secondary to professional analytical chemistry
- Foundation: understand neutralisation and stoichiometric mole ratios.
- Secondary: read titration curves and distinguish acid, base and equivalence regions.
- JC: connect pH, Ka, buffers, strong/weak acid behaviour and mole balance.
- Undergraduate: derive Gran functions from mass balance, activity approximations and Nernstian electrode response; use regression rather than graphical guessing.
- Advanced: test activity-coefficient, multi-equilibrium, junction-potential and nonlinear-response effects.
- Professional/research: validate the chosen function over its operating range and include regression plus calibration/model contributions in the uncertainty budget.
Misconceptions worth removing
- “The x-intercept is the experimentally observed endpoint.” It is a model-based estimate of equivalence volume.
- “pH equals −log concentration.” Strictly, pH is defined through hydrogen-ion activity.
- “Any titration can use the same Gran formula.” The function depends on reaction stoichiometry, equilibrium and electrode response.
- “The points closest to equivalence are always best.” They can show curvature when simplifying assumptions break down.
- “A high R² proves the chemistry is correct.” A narrow curved function can appear linear; residuals and chemical assumptions still need inspection.
Counterexamples and limits
A sample containing several weak acids can produce a Gran region that reflects multiple overlapping equilibria. Carbonate contamination can shift acid–base balances. A highly variable ionic strength can change activity coefficients during the titration. A drifting glass electrode can create an apparently chemical curvature. Metal ions that hydrolyse can consume or release protons and break the single-reaction model. Published analytical studies have documented such interference and systematic errors.
The correct response is not to force a straight line. It is to improve the chemical model or choose another measurement strategy.
Transfer checks
Check 1. Why does the pre-equivalence strong-acid Gran function multiply 10−pH by total volume?
Check 2. A pre-equivalence Gran plot is visibly curved although the titrant volumes are precise. Name three chemical or electrochemical assumptions that might be failing.
Check 3. The pre- and post-equivalence extrapolations give noticeably different Ve. Why is this disagreement useful rather than merely inconvenient?
Delayed reasoning check. Tomorrow, reconstruct the strong-acid Gran relation from only three ideas: mole balance, dilution, pH activity. If you can reach a function proportional to Ve − V without memorising the final formula, the method is understood.
How we know: selected evidence and standards
- Gunnar Gran, Analyst 1952, 77, 661–671, DOI 10.1039/AN9527700661: foundational linearisation treatment for potentiometric equivalence-point determination.
- IUPAC Gold Book, equivalence point, current 2025 terminology: the stoichiometric stage at which titrant has reacted with titrand.
- IUPAC Gold Book, titration: analytical terminology distinguishing titration, endpoint and equivalence.
- Analytical Chemistry 1975, DOI 10.1021/ac60356a024: documented errors inherent in Gran-plot use.
- Ocean Science 2026, metrological treatment of seawater total alkalinity, showing Gran estimation within a modern uncertainty-aware analytical workflow.
The quiet return
A Gran plot is elegant because it converts a curved experimental story into a linear question. But the intercept is trustworthy only when the chemistry underneath the line is trustworthy. Stoichiometry tells us where equivalence should exist; equilibria and activities connect composition to pH; the electrode turns activity into signal; the regression turns signal into an estimate. The straight line is the final surface of that reasoning, not a substitute for it.
