Weak acid

A weak acid establishes a chemical equilibrium between its proton-donating form and its conjugate base in water. For a neutral monoprotic acid $\mathrm{HA}$, the aqueous chemical equation isWater accepts a proton to become hydronium; ${\mathrm{A}}^{-}$ is the conjugate base. The double arrow denotes reversible exchange at chemical equilibrium, not equal populations. The acid dissociation constant and composition determine the hydronium concentration; substituting total chemical acid concentration into the pH formula is generally incorrect.

“Weak” describes the tendency to transfer a proton in a specified solvent, rather than how much chemical acid was dissolved. A weak acid can be concentrated, and a strong acid can be dilute. Even the same weak acid can have a large weak acid dissociation fraction after sufficient dilution, while its acid dissociation constant stays fixed at the same temperature and standard state conventions.^[ref]

A monoprotic acid supplies one proton per acid unit in the acid-base chemistry model under consideration. Acetic acid is an example. Several successive proton transfers require several acid dissociation constants and a larger set of chemical species.

For a monoprotic acid with analytical concentration $C_T=[\mathrm{HA}]+[{\mathrm{A}}^{-}]$, define the dimensionless weak acid dissociation fractionThis counts acid units in the conjugate base form. Identifying it with $[{\mathrm{H}}_3{\mathrm{O}}^{+}]/C_T$ additionally requires negligible self-ionization of water and no other sources of electric charge. In the ideal dilute chemical solution model, with $h=[{\mathrm{H}}_3{\mathrm{O}}^{+}]/c^{\circ }$, the acid dissociation constant gives $\alpha =K_a/(K_a+h)$. Thus lowering hydronium thermodynamic activity favors the conjugate base.

The diagram uses stipulated dimensionless acid dissociation constant $K_a=10^{-5}$ and self-ionization of water ion product $K_w=10^{-14}$ at fixed temperature; these are model inputs, not measurements of a named substance. It solves the weak acid mass and charge balances. At $C_T=0.010\mathrm{mol}{\mathrm{L}}^{-1}$, the calculated weak acid dissociation fraction is about $3.11\%$ and pH about $3.51$; at $C_T=10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$, they are about $27.0\%$ and $4.57$. More acid units are dissociated as a fraction of the total, yet fewer hydronium ions occur per volume.

At extreme dilution, self-ionization of water makes $h$ approach $\sqrt{K_w}$, so pH approaches $7$ for the chosen model. The limiting weak acid dissociation fraction is $K_a/(K_a+\sqrt{K_w})\approx 0.9901$, rather than exactly one. The calculation assumes pure added water, a neutral monoprotic acid, and no competing chemical reactions; actual contamination becomes increasingly important as $C_T$ falls.

Consider only neutral $\mathrm{HA}$ added to pure water, at fixed temperature, with ideal dilute thermodynamic activities, water thermodynamic activity approximated by one, and no added salts or other chemical species. Normalize every concentration by $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$:Conservation of acid units and zero net electric charge, respectively, requireThe second equation balances positive hydronium against negative conjugate base and hydroxide. It would need all additional ions if salts or a chemical base were added. Together with acid dissociation constant $K_a=hx/y$ and the self-ionization of water relation $K_w=ho$, these giveThis is exact within the stated ideal model, including depletion and self-ionization of water. Multiplication by $h(K_a+h)$ yields the cubic equationFor positive $K_a,K_w,c$, the unmultiplied left-hand side increases strictly from negative to positive values as $h$ increases from zero to infinity; its derivative is $1+K_w/h^2+K_{a}c/(K_a+h{)}^2>0$. There is therefore exactly one positive root of a polynomial. A bisection method can find it without choosing an unphysical root of a polynomial.

If $K_w/h^2\ll 1$, neglect hydroxide, so $h\approx x$ and the quadratic equation retaining acid depletion givesThe last form avoids subtracting nearly equal numbers. If also $x/c\ll 1$, neglect depletion to obtainCheck both conditions after computing: $x/c$ measures the fraction lost from the initial undissociated weak acid inventory, while $K_w/h^2$ measures the omitted hydroxide contribution to electric charge balance. A $5\%$ depletion criterion is a chosen tolerance, not a universal boundary. Keeping depletion does not cure the neglect of self-ionization of water. These weak acid mass and charge balances also explain why the square root estimate cannot be extended to arbitrarily dilute chemical solutions.

An analytical concentration records the total conserved inventory of a specified component per volume of chemical solution, summed over its relevant chemical species. Here $C_T=[\mathrm{HA}]+[{\mathrm{A}}^{-}]$; it is not generally the chemical equilibrium concentration $[\mathrm{HA}]$ of the undissociated weak acid.

A pH measurement becomes evidence for a weak acid chemical equilibrium when paired with a known analytical concentration and a calibrated glass electrode. For the pure-weak acid model, measured pH estimates $h=10^{-\mathrm{pH}}$ only in the ideal thermodynamic activity approximation. Then $o=K_w/h$, $x=h-o$, $y=c-x$, and $K_a=hx/y$. Require $0<x<c$; values outside that range signal inconsistent measurements or an inadequate model. Repeating this reconstruction across a dilution series tests whether one acid dissociation constant describes the data, rather than merely fitting one point.

The Analytical Sciences Digital Library's Potentiometric Titration of an Unknown Weak Monoprotic Acid provides a complementary acid-base titration experiment. Its apparatus includes a glass electrode and meter, a stirring magnet, a volumetric flask, pipettes, and a burette containing standardized sodium hydroxide. Prepare an aqueous acid stock of known analytical concentration, transfer a measured portion, and stir gently with the glass electrode clear of the magnet. Record added volume and stable pH after each addition; use a preliminary run to locate the steep region and smaller additions there on a fresh portion. The readout is the pH versus added volume curve, not a visual count of dissociated molecules.

Calibrate with fresh standard buffer solutions bracketing the pH range being interpreted. Use their assigned values at the actual temperature; a pair near pH $4$ and $7$ can cover the midpoint of many weak acid examples, but does not cover an entire alkaline endpoint region. Rinse between standards and samples to limit carryover, keep the sensing bulb and reference electrode junction immersed, and wait for a stable reading. These instrument practices are documented in HI5221 & HI5222 Instruction Manual: pH and mV and ISE and Temperature Bench Meters by Hanna Instruments (pp. 45–47, 49–50 and 73–74: calibration, measurement, rinsing and electrode conditioning).

Locate the acid-base titration equivalence region from the steepest change of pH with added volume. For one-to-one neutralization, standardized sodium hydroxide concentration times equivalence volume estimates the initial amount of substance of acid. Half that volume gives approximately equal $[\mathrm{HA}]$ and $[{\mathrm{A}}^{-}]$ only when subsequent chemical equilibrium redistribution is small. The Henderson-Hasselbalch equation then gives $\mathrm{pH}\approx \mathrm{p}{K}_a$. Unequal activity coefficients or appreciable hydronium or hydroxide inventories can shift this estimate. The full chemical equilibrium curve supplies a stronger consistency check than its midpoint alone.

Useful controls for this model test are a fresh independently prepared acid portion, a water blank, a check standard after measurement to detect instrument drift, and fixed recorded temperature. A blank diagnoses contamination but should not be subtracted directly from pH, which is a logarithm. Keep exposure to atmospheric carbon dioxide comparable, especially at high dilution, and record any background salt that changes ionic strength. These are experimental-design controls suggested by the model's assumptions. The observed curve and its uncertainty determine whether the model is adequate; the diagrams here contain no experimental readouts. An aqueous acid dissociation constant cannot be inferred unchanged from a sample requiring a different solvent mixture, and a nearly neutral dilute sample makes the difference $h-K_w/h$ particularly sensitive to pH error.

A glass electrode is an electrode whose glass membrane responds to hydrogen ion thermodynamic activity. A meter measures its electric potential difference against a reference electrode; calibration with standard buffer solutions converts this response to operational pH.

A reference electrode supplies the comparison electric potential for the glass electrode measurement. A combination probe incorporates both electrodes; changes at its junction can contribute measurement error.

The conventional zero of standard electrode potentials, based on hydrogen oxidation/reduction at specified standard hydrogen fugacity and proton activity.

A silver electrode in contact with silver sulfate and a solution of controlled sulfate activity. The reduction is $\mathrm{A}{\mathrm{g}}_2\mathrm{S}{\mathrm{O}}_4+2{\mathrm{e}}^{-}\rightleftharpoons 2Ag+\mathrm{S}{\mathrm{O}}_4^{2-}$.

An acid-base titration adds measured quantities of a standardized chemical acid or chemical base to determine an initially unknown acid or base inventory through neutralization. Monitoring pH also constrains chemical equilibrium parameters such as the acid dissociation constant.

The added reagent is the titrant; a pipette selects an aliquot of the sample and a burette measures the added volume. Known stoichiometry converts this volume into amount of substance. This separates two questions that a single pH reading cannot answer: how much chemical acid is present, and how readily it donates a proton. A large inventory of weak acid can consume more titrant than a small inventory of strong acid despite having a higher initial pH.

The equivalence point is the composition at which added titrant and the initial sample inventory match the specified chemical reaction's stoichiometry. It is a theoretical inventory condition; the titration endpoint is an observed stopping criterion. For a monoprotic acid consuming one hydroxide per acid unit, initial acid molar concentration $c_A$, sample volume $V_A$, and base titrant molar concentration $c_B$ giveThe proof is amount of substance conservation with one-to-one neutralization; a different stoichiometry requires its coefficients. All volumes must use compatible units. Since $1\mathrm{mL}=10^{-3}\mathrm{L}$, $0.100\mathrm{mol}{\mathrm{L}}^{-1}\times 10.00\mathrm{mL}=0.001000\mathrm{mol}$, not $1\mathrm{mol}$.

An equivalence point does not prescribe pH $7$. A weak acid titrated with a strong base leaves its conjugate base, which reacts with water; the corresponding chemical equilibrium can make the chemical solution alkaline. The neutral strong acid/strong base case additionally depends on temperature, through self-ionization of water.^[ref]

A titration endpoint is the experimentally assigned point indicated by a specified signal: for example, a persistent color of a pH indicator, or a feature extracted from a potentiometric titration. An equivalence point is defined by stoichiometry; an endpoint estimates its location. State the signal rule and its repeatability. A fleeting local color where a drop enters an unmixed sample need not satisfy the rule for the mixed sample.

Define the endpoint offset $\Delta V=V_{\mathrm{end}}-V_{\mathrm{eq}}$. For one-to-one neutralization, using the endpoint without correction gives inferred acid inventory ${\hat{n}}_A=c_B{V}_{\mathrm{end}}$ andThus an illustrative $0.05\mathrm{mL}$ overshoot at $V_{\mathrm{eq}}=10.00\mathrm{mL}$ overestimates the inventory by $0.5\%$. Repeatedly reaching the same color can reduce scatter while retaining this systematic offset. This example stipulates an offset; it is not a measured drop size.

A pH indicator changes color as its protonation distribution changes. In a simple weak acid/conjugate base model, its two forms have different colors and their ratio depends on pH through the acid dissociation constant. The visible transition spans a range of pH; it is not a direct signal that the sample has reached an equivalence point. An indicator is useful when its transition lies in a steep part of the titration curve, so a modest range of pH corresponds to a small added volume.^[ref]

Bromothymol blue is a pH indicator used to assign a color-based titration endpoint in the inspected solubility product measurement by titration.

A titration curve plots a measured or calculated property against added titrant volume or amount of substance. In an acid-base titration, plotting pH exposes gradual changes and a steep endpoint region. A potentiometric titration uses an electrode electric potential difference; a calibrated glass electrode and reference electrode provide operational pH. Finite additions bracket a transition rather than measuring a continuous curve. A chemical equilibrium fit can estimate the equivalence point; identifying the steepest sampled interval alone has a resolution set by the dose spacing and instrument response.

A potentiometric titration monitors an electrode electric potential difference as titrant is added. A calibrated glass electrode can report pH during an acid-base titration.

Consider an original ideal model: $V_A=10.00\mathrm{mL}$ of hydrochloric acid at $c_A=0.100\mathrm{mol}{\mathrm{L}}^{-1}$, titrated with sodium hydroxide at $c_B=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. Treat these numbers as exact model inputs. Assume complete dissociation of an ionic compound, complete strong acid ionization, additive volumes, uniform mixing, and activity coefficients equal to one. Use stipulated $K_w=10^{-14}$, approximately appropriate near $25{}^{\circ }\mathrm{C}$. These are calculated pH values, not measurements of these materials.

The chemical equation for neutralization isInitial acid inventory is $n_A=1.000\mathrm{mmol}$, so the equivalence point is $v=10.00\mathrm{mL}$. Added water does not change this inventory. But the final volume $V_A+v$, including the titrant, determines all concentrations.

Keep self-ionization of water even at the equivalence point. With $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, define the dimensionless acid excess $d$, hydronium level $h$, and hydroxide level $o$ byThe sodium cation and chloride anion inventories fix the electric charge balance $h-o=d$, while chemical equilibrium gives $ho=K_w$. Eliminating $o$ yields a quadratic equation $h^2-dh-K_w=0$ with exactly one positive root. For stable calculation on either side, writeThe second expression follows by multiplying by $s-d$; it avoids subtracting two nearly equal large quantities on the alkaline side. A logarithm acts on the dimensionless $h$, as required by the pH definition.

At $v=5.00\mathrm{mL}$, the excess is $0.500\mathrm{mmol}$ in $15.00\mathrm{mL}$, giving $d=1/30$ and pH $1.477121\dots$. At $v=15.00\mathrm{mL}$, excess hydroxide is $0.500\mathrm{mmol}$ in $25.00\mathrm{mL}$, giving $d=-1/50$ and pH $12.301030\dots$. Neglecting dilution would give the wrong values even far from the equivalence point.

Exactly at $v=10.00\mathrm{mL}$, $d=0$ and $h=o=\sqrt{K_w}=10^{-7}$, giving pH $7$ in this model. At $9.999\mathrm{mL}$ and $10.001\mathrm{mL}$, the same smooth expression gives pH $5.300835\dots$ and $8.699121\dots$. The familiar excess-only formulas would predict zero hydronium or zero hydroxide at the equivalence point; retaining self-ionization of water removes that failure. A steep curve is still continuous.

The sensitivity is especially revealing. Differentiating $h-K_w/h=d$ giveswhen $c_A=c_B$. At this model's equivalence point, the derivative is about $1.086\times 10^4$ pH units per millilitre. This is a local derivative, not a linear prediction over a whole drop. Real endpoint accuracy also depends on dosing, mixing, measurement uncertainty, temperature, extra reacting chemical species, and nonideal thermodynamic activities.

The institutional experiment The Potentiometric Titration of an Acid Mixture, from Massachusetts Institute of Technology's 5.310 Laboratory Chemistry, determines hydrochloric acid and phosphoric acid inventories. Dried potassium hydrogen phthalate standardizes sodium hydroxide. A volumetric pipette selects an acid aliquot, diluted with water in a beaker. A burette feeds a stirred sample beside a glass electrode; the apparatus isolates the sample from stirrer warming. The burette is conditioned with titrant and its tip freed of bubbles. Stable pH and added volume are recorded, with smaller additions near the transition. Controls include replicate standardizations, a consistent pale-pink phenolphthalein endpoint, and a standard buffer solution check between sample runs. The readout supports inventory estimates through known stoichiometry; it does not directly count molecules.^[ref]

The same protocol requires drainage time for burette readings and warns that an undelivered hanging drop spoils the association between volume and pH. Its color-persistence wording differs between the main paragraph and footnote, so a reproducible titration endpoint needs an explicitly chosen criterion. Its acid mixture is not the single-strong acid ideal model above.^[ref]

A distinct control appears in Process Measurement Procedures: Sulfuric Acid Emissions, §1.4.4.2: the phenolphthalein standardization is repeated with water replacing the acid, matching the sample aliquot and indicator quantity. The volume needed for the same pink endpoint is subtracted from the sample reading.^[ref]

The independent bookkeeping for such a matched blank is $n_A=c_B(V_{\mathrm{sample}}-V_{\mathrm{blank}})$, assuming one-to-one neutralization and equal background consumption. A blank cannot correct an unmatched chemical reaction or guarantee exact coincidence of endpoint and equivalence point. Subtracting pH readings would not give this inventory correction, because pH is a logarithm. Repeatability constrains scatter; an independently known check sample also tests systematic bias. Carbon dioxide uptake, incomplete mixing, electrode drift, and uncertain titrant concentration limit the inference.

Titrant standardization determines the actual titrant concentration by reaction with a known inventory, often a primary standard. It is distinct from calibrating the delivered volume or a glass electrode's response. For a known monoprotic acid inventory $n_s$ and a suitable blank-corrected endpoint,An original illustrative standard of $0.980\mathrm{mmol}$ requiring $9.85\mathrm{mL}$, with matched blank $0.05\mathrm{mL}$, gives $c_B=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. The measured quantities here are stipulated examples. Independent solution preparation and volumetric calibration tests the pipette, burette, and volumetric flask; titrant standardization tests the reacting inventory per delivered volume.

The useful chain is known mass, known composition, known amount of substance, and then a measured titre through a specified stoichiometry. If each standard unit consumes $r$ units of titrant, the general inventory equation is $c_{B}V=r{n}_s$, where $V$ is the corrected titre. The one-to-one formula above has $r=1$. A titration endpoint supplies an observable stopping rule; whether it represents the equivalence point accurately remains a separate question. Standardization endpoint and blank bias makes the direction of the resulting error explicit.

A primary standard supplies a reliably known reacting inventory from which a titrant concentration can be determined.

The defining advantage is known high purity: the weighed portion can be assigned a known amount of substance. A titrant whose concentration is established against that inventory is a secondary standard solution in IUPAC terminology.^[ref]

An accurate balance determines total mass, including any unwanted water or other material. To infer reacting inventory, one also needs known composition, stability during storage and weighing, complete transfer and dissolution, and a sufficiently complete chemical reaction of known stoichiometry. A finely resolved balance cannot establish these chemical facts. Potassium hydrogen phthalate is useful because a suitably prepared portion provides a known acid inventory for sodium hydroxide titrant standardization; the KHP standardization protocol below shows how that claim is put into practice.

Preparation belongs to the particular material and method. For example, the MIT handout specifies drying potassium hydrogen phthalate at $110{}^{\circ }\mathrm{C}$ for $1.5$ hours or overnight, cooling in a desiccator, and weighing by difference; its footnote identifies removal of superficial moisture as the purpose. This is a condition of that potassium hydrogen phthalate procedure, not a universal instruction to heat every primary standard in the same way.^[ref]

Let $m$ be the transferred sample mass, $p$ the known mass fraction of the intended chemical compound, and $M$ its molar mass. In a model with complete dissolution and chemically inactive impurities,The purity fraction is dimensionless: $99.95\%$ would mean $p=0.9995$, not $99.95$. These equations assume that the impurity contributes no titrant demand; an impurity that also reacts needs its own inventory equation. The value of $p$ must refer to the same preparation state as $m$, for example the specified dry material. A mass fraction assigned before unaccounted uptake of water cannot simply be used after it.

For an original worked example, stipulate pure potassium hydrogen phthalate, abbreviated KHP, so $p=1$, a selected mass $m=0.51055\mathrm{g}$, and rounded molar mass $M=204.22\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The NIST Chemistry WebBook lists formula ${\mathrm{C}}_8{\mathrm{H}}_5\mathrm{K}{\mathrm{O}}_4$ and molecular weight $204.2212$, consistent with the rounded molar mass used here.^[ref] The selected mass, purity assumption and volumes in this example are chosen inputs, not measured results or a claim about a real batch's purity.

The acid-bearing anion of potassium hydrogen phthalate loses one proton to hydroxide. The net chemical equation isThus sodium hydroxide has one-to-one stoichiometry with potassium hydrogen phthalate in this neutralization, givingChoose a delivered endpoint titre of $24.85\mathrm{mL}$ and matched titration blank of $0.05\mathrm{mL}$. ThenThe extra displayed digits show the arithmetic of the stipulated model; they do not establish experimental accuracy. A compact rounded result is $0.1008\mathrm{mol}{\mathrm{L}}^{-1}$.

When the entire weighed portion is titrated, added water changes concentration and the pH path but preserves the initial acid inventory. The dissolution volume therefore does not appear in $n_s=mp/M$. If instead a volumetric flask prepares a stock and a volumetric pipette selects an aliquot, the selected inventory is $n_s{V}_a/V_f$ for homogeneous stock, with delivered aliquot volume $V_a$ and final stock volume $V_f$. The same bookkeeping explains why complete transfer matters even when the amount of dissolution water is approximate.

Columbia University's institutional laboratory manual, Experiment 1, Glassware Calibration, Primary and Secondary Standards, and Manual Titrations, Part 3, specifies two weighed $0.2$–$0.3\mathrm{g}$ portions of potassium hydrogen phthalate, transferred completely into separate $250\mathrm{mL}$ Erlenmeyer flasks. Dissolve each in approximately $100\mathrm{mL}$ distilled water; any warmed sample cools to room temperature before titration. Add two or three drops of phenolphthalein. Rinse and condition a clamped burette with the diluted sodium hydroxide, then titrate while swirling to faint pink lasting about $20$ seconds. Calculate concentration from each portion's own mass and delivered titre; repeat if results scatter. Part 1 identifies a Mettler balance, and Part 2 includes paired weighings of exposed sodium hydroxide and potassium hydrogen phthalate. These are prescribed procedures, not reported measurements.^[ref]

The same manual directs keeping the diluted sodium hydroxide covered and limiting air exposure after titrant standardization.^[ref] The MIT handout explains the associated carbon dioxide problem: carbonate contamination of sodium hydroxide is controlled during preparation, bottles are promptly capped, and atmospheric carbon dioxide makes the pink phenolphthalein endpoint fade.^[ref] Consequently, a standardized concentration applies to the stored chemical solution under its handling conditions; continued uptake of carbon dioxide can change the reacting inventory per delivered volume.

The color is the readout, and stoichiometry supplies the inference. Phenolphthalein indicates a pH range, so its first reproducible pale pink is an operational titration endpoint, rather than a direct observation of equal reacting inventories. A potentiometric titration can locate an endpoint from a pH versus volume curve, but glass electrode calibration alone still does not determine sodium hydroxide concentration. Repeat portions test scatter; incomplete dissolution, an unmatched titration blank, or a shared endpoint error can bias all repeats together.

Here a titre means the delivered titrant volume needed to reach the specified titration endpoint, obtained from final minus initial burette readings. Report explicitly whether a quoted titre includes or excludes a titration blank. This volume convention is distinct from uses of titre to express a solution's reacting strength.

A titration blank runs the background procedure without the intended reacting sample, matching relevant water, indicator quantity and titration endpoint conditions. Its titrant demand estimates a background contribution to subtract. The inspected EPA procedure uses such a phenolphthalein blank; this is a distinct control, rather than an instruction present in Columbia's Part 3.^[ref] A blank must match the relevant background to justify subtraction; it does not remove sample-specific side reactions or a mismatch between titration endpoint and equivalence point.

Hold the known primary standard inventory fixed, and let $V$ be the correct blank-subtracted titre. In the following model, any remaining endpoint offset has been removed from this reference. If a titration endpoint overshoot adds $\delta V>0$, the inferred titrant concentration is too small:The denominator grows while the numerator stays fixed. For $\delta V\ll V$, the relative effect is approximately $-\delta V/V$, but the displayed fraction is exact for this additive model. An unremoved positive titration blank $b$ has the same mathematical effect: replace $\delta V$ by $b$. If both remain, replace it by $\delta V+b$. This analogy concerns their effects on the denominator; it does not mean a blank measures endpoint overshoot.

For the chosen $24.80\mathrm{mL}$ reference titre, an extra $0.05\mathrm{mL}$ gives $\hat{c}=0.1006036217\dots \mathrm{mol}{\mathrm{L}}^{-1}$ and relative error $-0.05/24.85=-0.201207\dots \%$. Leaving both a $0.05\mathrm{mL}$ blank and a $0.05\mathrm{mL}$ overshoot gives denominator $24.90\mathrm{mL}$ and relative error $-0.10/24.90=-0.401606\dots \%$.

For comparison, the same extra $0.05\mathrm{mL}$ on a $5.00\mathrm{mL}$ reference titre gives $-0.05/5.05=-0.990099\dots \%$. The ratio of these error magnitudes is $24.85/5.05=4.920792\dots$, while the small-error approximation would give $24.80/5.00=4.96$. A larger titre reduces the fractional effect of a fixed extra volume; it does not fix unknown purity, incomplete transfer, or an incorrect stoichiometry.

A titrant is the reagent added in measured quantities during a titration; its known concentration and delivered volume determine the added amount of substance.

A volumetric flask prepares a chemical solution at a specified total volume, using a calibration mark.

Its mark specifies contained volume under its calibration conditions. In contrast, a volumetric pipette specifies a delivered portion. The narrow neck makes a small change of volume visible as a larger change of liquid height. The final chemical solution volume includes the solute; “dilute to $100\mathrm{mL}$” does not mean “add $100\mathrm{mL}$ of water”.^[ref]

For an original dilution example, a fully transferred $10.00\mathrm{mL}$ aliquot of $0.100\mathrm{mol}{\mathrm{L}}^{-1}$ acid contains $1.000\mathrm{mmol}$. Making the final total $100.0\mathrm{mL}$ gives analytical concentration $0.0100\mathrm{mol}{\mathrm{L}}^{-1}$. A subsequent $10.00\mathrm{mL}$ aliquot contains $0.1000\mathrm{mmol}$ and needs $1.000\mathrm{mL}$ of $0.100\mathrm{mol}{\mathrm{L}}^{-1}$ base for one-to-one neutralization. The entire diluted flask still needs $10.00\mathrm{mL}$ of that base: dilution preserves the selected inventory but sampling selects a fraction of it.

Overfilling lowers the intended molar concentration. Removing some well-mixed chemical solution to restore the mark also removes solute, so it does not recover the intended inventory. Full transfer, known final volume, and mixing are separate requirements.

A pipette transfers a measured portion of a chemical solution.

For an acid-base titration, a volumetric pipette selects an aliquot whose amount of substance is $n=cV$ from a homogeneous sample. The delivered volume depends on the instrument's calibration and drainage convention; the residual liquid of a gravity-draining volumetric pipette is allowed for in that calibration. Its mark is read using the liquid meniscus. A volume calibration alone does not establish that the sampled chemical solution has the intended concentration.^[ref]

An aliquot is a measured portion taken from a larger sample. For a homogeneous chemical solution, it contains the corresponding fraction of each conserved component inventory. Representative sampling requires mixing; knowing only the portion's volume is insufficient if the sample is not homogeneous.

A volumetric pipette transfers one nominal volume of chemical solution according to its marked calibration and drainage convention.

A burette delivers an adjustable measured volume of a chemical solution; the change in its reading gives the delivered volume in an acid-base titration.

Its downward-increasing scale measures a change in reading, not the initial liquid inventory. For initial reading $R_i$ and final reading $R_f$ without refilling, the nominal delivered volume is $v=R_f-R_i$. An illustrative $2.35\mathrm{mL}$ to $12.35\mathrm{mL}$ reading therefore represents $10.00\mathrm{mL}$. There is no requirement to start at zero. An air bubble disappearing from the tip uses part of the apparent delivery to fill the tip, and a hanging drop has not yet entered the sample; both break the intended volume account.

For transparent aqueous samples, read the bottom of the liquid meniscus at eye level; viewing from above or below shifts its apparent alignment. The mark is a comparison between a curved liquid surface and a scale, not a direct measure of amount of substance. Eye-level reading and the distinction between contained and delivered volumes are part of solution preparation and volumetric calibration.^[ref]

Both readings contribute measurement uncertainty. If each has independent standard deviation ${\sigma }_R$, their difference has standard deviation $\sqrt{2}{\sigma }_R$; a common additive reading offset cancels, while a scale-factor error remains. For the inventory $n=c_{B}v$, uncertainty in titrant concentration remains even with perfect readings. This is why repeat delivery, independent volume calibration, and titrant standardization answer different questions.