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Acid-base chemistry studies how chemical acids and chemical bases exchange protons or electron pairs. Here the focus is transfer of protons in water: a chemical acid donates a proton, and a chemical base accepts it. The resulting chemical equilibrium determines pH, while a buffer solution can resist changes in pH. This connects microscopic chemical reactions with a compact, measurable description of a chemical solution.

The important distinction is between how much chemical acid is present and how readily it transfers a proton. A weak acid and a strong acid at the same total concentration need not have the same pH. Conversely, two buffer solutions can have the same pH while containing very different total quantities of their chemical acids and chemical bases.

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 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.

Buffer capacity measures resistance to a change in pH: how much additional chemical acid or chemical base a chemical solution accommodates per small change in pH. Its differential form is a local derivative, rather than the total inventory available before a buffer solution is exhausted. Two buffer solutions can have identical pH and very different buffer capacities.

Choose a constant volume $V$ and fixed temperature. Let $n_B$ count added moles of fully dissociated, monovalent chemical base supplying hydroxide, and $n_A$ added moles of fully dissociated, monovalent strong acid supplying hydrogen ions. Neither addition supplies the weak acid component itself. The signed addition is $b=(n_B-n_A)/V$, with units $\mathrm{mol}{\mathrm{L}}^{-1}$, andThe last two forms describe separate base-only and acid-only additions. The buffer capacity is positive for the stable chemical equilibrium model below and is reported in $\mathrm{mol}{\mathrm{L}}^{-1}$ per unit of pH. Since pH is dimensionless, the dimensional unit is just $\mathrm{mol}{\mathrm{L}}^{-1}$; stating “per pH” makes the measurement convention clear. Here constant volume is an idealization, realized approximately by small additions of concentrated reagent.

For a sufficiently small addition, $\Delta \mathrm{pH}\approx \Delta b/\beta$. A large buffer capacity means a small local change of pH for a given addition. Over a finite interval, use the changing buffer capacity instead:This integral gives a finite dose in $\mathrm{mol}{\mathrm{L}}^{-1}$; it need not equal the initial buffer capacity times the eventual change in pH.

The conjugate acid of a chemical base is the chemical species formed when it accepts one proton. This is a relationship between partners, rather than a claim that the partner is a strong acid.

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]

Prepare the same ideal monoprotic acid model at initial pH $5$, with appropriate accompanying ions to balance electric charge. To predict the new pH after a finite signed addition $\Delta b$, solve $Q({\mathrm{pH}}_1)-Q(5)=\Delta b$ using the chemical equilibrium expression above. For $\Delta b=0.002\mathrm{mol}{\mathrm{L}}^{-1}$ of added chemical base, the calculated final pH is $5.366798$ at $C=0.010\mathrm{mol}{\mathrm{L}}^{-1}$, but $5.034749$ at $C=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. The initial local approximation gives changes of about $0.34605$ and $0.034730$, respectively; its accuracy decreases as the partner balance changes appreciably.

For the dilute example, additions beyond approximately $0.005\mathrm{mol}{\mathrm{L}}^{-1}$ in either direction exceed the initial inventory of one partner. The full chemical equilibrium calculation still produces a finite pH; an inventory-only Henderson-Hasselbalch equation calculation would attempt to use a negative partner quantity. At $+0.006\mathrm{mol}{\mathrm{L}}^{-1}$ the dilute example reaches pH $10.99564$, whereas the concentrated example reaches only $5.104698$. This demonstrates a finite operating range, not an abrupt cessation of all buffer capacity.

Consider a neutral monoprotic acid $\mathrm{HA}$ dissolved in water, with no added salts or other acids. Assume dilute ideal thermodynamic activities, fixed temperature, water thermodynamic activity one, and negligible self-ionization of water. Write $C_T$ for its analytical concentration, $c=C_T/c^{\circ }$ with $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, and $\alpha$ for its weak acid dissociation fraction. Conservation of acid units and electric charge givesHere ${\mathrm{A}}^{-}$ is the conjugate base, and ${\mathrm{H}}^{+}$ is conventional shorthand for hydrated hydrogen ions, represented by hydronium in this model. Substitution into the dimensionless acid dissociation constant givesSolving this quadratic equation and rejecting its negative root of a polynomial gives an exact result within the stated model:The second form avoids subtracting nearly equal numbers. Unlike the approximation $\alpha \approx \sqrt{K_a/c}$, it retains depletion of neutral weak acid and always satisfies $0<\alpha <1$ for positive $K_a,c$. The broader weak acid mass and charge balances restore self-ionization of water when that omitted contribution matters.

For an original worked example, stipulate $K_a=10^{-5}$ without identifying a real substance. At $C_T=10^{-2}\mathrm{mol}{\mathrm{L}}^{-1}$, the weak acid dissociation fraction is approximately $0.0311267$, or $3.11267\%$, and $[{\mathrm{H}}_3{\mathrm{O}}^{+}]=3.11267\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$. A hundredfold dilution to $C_T=10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$ raises the fraction to $0.270156$, or $27.0156\%$, but lowers $[{\mathrm{H}}_3{\mathrm{O}}^{+}]$ to $2.70156\times 10^{-5}\mathrm{mol}{\mathrm{L}}^{-1}$. A completely ionized strong acid at $C_T=10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$ would have $[{\mathrm{H}}_3{\mathrm{O}}^{+}]\approx 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$: less than the more concentrated weak acid in the first calculation, despite its greater converted fraction.

The directions of both changes follow without relying on the plot. Put $x=[{\mathrm{H}}_3{\mathrm{O}}^{+}]/c^{\circ }=c\alpha$. Rearranging the chemical equilibrium relation givesThus smaller $c$ forces smaller $x$, which in turn makes $\alpha$ larger. The derivative proves that total hydronium concentration and converted fraction move in opposite directions during dilution. In this model, adding solvent also makes additional acid units ionize at fixed total acid amount of substance, even though their ions are spread over more volume. That distinction explains why “more ionized” alone does not predict greater electrical conductivity.

Do not extend the no-background model to unlimited dilution: its prediction $[{\mathrm{H}}_3{\mathrm{O}}^{+}]\to 0$ omits water's own ions. The weak acid mass and charge balances include that background. Added salts, other acids or bases, and nonideal activity coefficients also change the relation between analytical concentration, converted fraction and hydronium concentration.

The pH of a chemical solution is the negative base-ten logarithm of its hydrogen ion thermodynamic activity:The thermodynamic activity is a dimensionless quantity, so the logarithm has a dimensionless argument. The definition uses thermodynamic activity, rather than total chemical acid concentration. Single-ion thermodynamic activities require a convention; practical pH measurements are tied to calibrated standards. See IUPAC Gold Book, pH.

In a dilute ideal chemical solution in water, a useful approximation isSquare brackets denote concentration in moles per litre. In water, $[{\mathrm{H}}^{+}]$ is shorthand for the hydronium concentration, often written $[{\mathrm{H}}_3{\mathrm{O}}^{+}]$; it does not describe isolated bare protons. Departures from ideal behavior need activity coefficients. The concentration formula approximates the thermodynamic activity definition.

For an ideal strong acid supplying one proton per initial unit, stipulate a total concentration of $0.001\mathrm{mol}{\mathrm{L}}^{-1}$ and negligible self-ionization of water. Then $[{\mathrm{H}}^{+}]\approx 10^{-3}\mathrm{mol}{\mathrm{L}}^{-1}$, soReducing that concentration tenfold by dilution gives $10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$ and pH $4$, under the same assumptions. More generally, dividing the hydrogen ion thermodynamic activity by ten raises pH by one. This is why the scale compresses an enormous range: equal steps of pH represent multiplicative changes, rather than equal differences in concentration. These are model calculations, rather than measured samples.

Neutrality depends on temperature. In the dilute ideal water model, define the dimensionless ion product $K_w=([{\mathrm{H}}^{+}]/c^{\circ })([{\mathrm{OH}}^{-}]/c^{\circ })$ and $\mathrm{p}{K}_w=-{\log }_{10}{K}_w$. Equal hydronium and hydroxide concentrations implyAt $25{}^{\circ }\mathrm{C}$, the textbook approximation $K_w\approx 10^{-14}$ gives neutral pH $7$. Since $K_w$ varies with temperature, neutrality need not mean pH $7$. Nor is pH restricted to $0$–$14$: the activity definition permits negative pH when $a_{{\mathrm{H}}^{+}}>1$, and pH above $14$ when $a_{{\mathrm{H}}^{+}}<10^{-14}$. High concentrations can invalidate the dilute ideal approximation. See Flowers et al., Chemistry 2e, pH and pOH.

S. P. L. Sørensen introduced pH at the Carlsberg Laboratory in 1909, as recorded in the Carlsberg Foundation historical timeline.

Two water molecules can exchange a proton, producing hydronium and hydroxide ions: $2{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{H}}_3{\mathrm{O}}^{+}+{\mathrm{OH}}^{-}$. Even pure water therefore supplies ions; extreme dilution eventually invalidates the strong acid assumption that it supplies all the hydronium.

In water, a strong acid transfers its available proton essentially completely to water under the dilute conditions considered here. For the pH example, each initial chemical acid unit supplies one hydronium ion. “Strong” describes this conversion, rather than a high concentration. See 14.3 Relative Strengths of Acids and Bases.

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.

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]