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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.

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.

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.

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.

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]