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The ion formed in the chemical equilibrium between iodine and iodide in a suitable solution.

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]

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.

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 weak electrolyte yields only a fraction of ions from its dissolved substance under the specified conditions, as illustrated by acetic acid in water.

For a neutral weak acid, unconverted molecules coexist with hydronium and the conjugate base at chemical equilibrium. The weak acid dissociation fraction describes the partition of the dissolved inventory; the acid dissociation constant describes the chemical equilibrium at fixed solvent, temperature and standard states. A single fraction is not a permanent label for every concentration: dilution can increase that fraction while the acid dissociation constant stays fixed. Calling a substance a weak electrolyte does not mean its chemical solution must be dilute.^[ref]

The following ideal weak electrolyte dilution law makes the distinction quantitative for a monoprotic acid. Other weak electrolytes, mixed chemical solutions and additional proton-transfer steps require their own chemical equilibrium and inventory balances.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.

If two thermodynamic systems are each in thermal equilibrium with the same third system in the same equilibrium state, they are in thermal equilibrium with each other. Write $A\sim B$ for this relation. The zeroth law of thermodynamics supplies transitivity:Its symmetric relation property and self-equilibrium (reflexivity), together with transitivity, make thermal equilibrium an equivalence relation. Its equivalence classes can be labelled by a common empirical temperature; choosing an absolute numerical scale requires the further structure of thermodynamic temperature.^[ref]

The comparison assumes internally equilibrated macroscopic bodies, unchanged composition and mechanical constraints, and contact permitting only heat exchange. Use rigid, impermeable diathermal walls so that neither matter transfer nor boundary work spoils the test. The same state of $C$ matters: letting $C$ warm while measuring $A$, then cool while measuring $B$, does not supply the premise. Ordinary nonrelativistic laboratory thermodynamics is assumed; the Tolman temperature law treats gravitational qualifications. Equal temperatures establish the thermal condition, without requiring mechanical equilibrium or chemical equilibrium.

This explains why a thermometer can compare unlike materials: it supplies a reusable third system. A temperature is a property of an equilibrium state, while heat is energy transferred because of a temperature difference. Two bodies at the same temperature can have different internal energies and heat capacities. A thermometer therefore does not measure a body's stored amount of heat; the finite-body disturbance and waiting-time calculations in thermometer describe how making the comparison can change the state being measured.^[ref]

The name “zeroth” reflects logical order rather than the date of the first thermometer: this premise about thermal equilibrium precedes the already numbered first law of thermodynamics, Second law of thermodynamics and third law of thermodynamics. This is the naming explanation given in University Physics Volume 2 by Samuel J. Ling and others (§1.1, “Thermal Equilibrium”).^[ref]

The corresponding Zeroth law of black-hole mechanics states constancy of surface gravity over an equilibrium horizon; Hawking radiation identifies that quantity with physical temperature up to its universal conversion factor.