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For a specified chemical reaction and standard state convention, the dimensionless equilibrium constant is the equilibrium value of the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$, where $a_i$ denotes thermodynamic activity and ${\nu }_i$ a signed stoichiometric vector entry. Thermodynamic activities incorporate the reference state and departures from ideal behavior. See IUPAC Gold Book, equilibrium constant.

In the ideal concentration model for A converting to B, use $a_{\mathrm{A}}=[\mathrm{A}]/c^{\circ }$ and $a_{\mathrm{B}}=[\mathrm{B}]/c^{\circ }$ with the same reference $c^{\circ }=1\mathrm{M}$. Their ratio gives $Q=[\mathrm{B}]/[\mathrm{A}]$ and $K=0.75/0.25=3$. Equality $K=k_f/k_r$ applies to this elementary first-order pair; it is not a universal ratio of two measured rate constants for an arbitrary multistep chemical reaction.

At a specified constant temperature, with consistent standard states, the standard molar reaction Gibbs free energy isHere ${\Delta }_{r}G$ is the local slope per mole of extent of reaction at fixed temperature and pressure, rather than the finite total Gibbs free energy change between two mixtures. At equilibrium ${\Delta }_{r}G=0$, while ${\Delta }_r{G}^{\circ }$ generally remains nonzero. For $K=3$ at $298\mathrm{K}$, it is about $-2.72kJ/mol$. For positive activities, $Q<K$ favors forward conversion and $Q>K$ favors reverse conversion; neither sign supplies a reaction rate. See MIT 5.111, Lecture 18, reaction free energy and equilibrium.

A battery's charge reservoir means a finite amount of electric charge can pass through the external electrical circuit before its usable chemical state is depleted. It does not mean a large net electric charge is sitting inside the casing. For one net cell chemical reaction transferring $n$ electrons per reaction event, an ideal reaction inventory obeyswhere $F$ is the Faraday constant and $\xi$ the extent of reaction in moles, assuming all the electric current serves that reaction. At fixed temperature and pressure, its reversible electromotive force obeys ${\Delta }_{r}G=-nF\mathcal{E}$: a decrease in Gibbs free energy provides the maximum reversible electrical work per mole. The electromotive force also depends on composition; the Daniell cell illustrates this with thermodynamic activity.^[ref]

As an original finite-reservoir exercise, stipulate usable $Q_{\max }=360\mathrm{C}$ and hold the preceding $3\mathrm{V}$, $1\Omega$, $4\Omega$ parameters constant until an abrupt cutoff. Then $t=Q_{\max }/I=600\mathrm{s}$, available electrical work is $\mathcal{E}{Q}_{\max }=1080\mathrm{J}$, load work is $V_L{Q}_{\max }=864\mathrm{J}$, and internal dissipation is $216\mathrm{J}$. This explicitly finite idealization predicts ten minutes, not an experimentally measured runtime. Real accessible charge depends on discharge conditions and the chosen terminal-voltage cutoff. Charge alone is also not an energy: converting it to delivered work requires the terminal electric potential difference throughout discharge.

The dimensionless ratio of a gas species' chemical fugacity to its pressure in the pure-fluid convention. Its thermodynamic activity relative to a gas reference pressure $P^{\circ }$ is $\varphi P/P^{\circ }$. For mixtures the pressure denominator uses the species' partial pressure. The ideal-gas limit has $\varphi =1$.

An aluminium hydroxide mineral with composition $\mathrm{Al}(\mathrm{OH}{)}_3$. In a simplified mineral activity diagram, it is favored over kaolinite at sufficiently low dissolved silica thermodynamic activity.

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.

The three principal aqueous glycine forms differ by loss of one proton at each step:Call these glycine forms $G^{+}$, $G^0$ and $G^{-}$, respectively: a cation, a zwitterion and an anion, with net electric charges $+e$, $0$ and $-e$, where $e$ is the elementary charge. $G^0$ specifically denotes the zwitterion, not an uncharged amino/carboxyl drawing. The first dissociation removes the carboxyl proton; the second removes a proton from the positively charged nitrogen. The chemical equilibrium changes are reversible, not three permanently separate kinds of glycine. ^[ref]

For all calculations and both diagrams here, explicitly stipulate the pKa values $\mathrm{p}{K}_{a1}=2.35$ and $\mathrm{p}{K}_{a2}=9.78$. These are illustrative model inputs, not measurements from the glycine titration experiment or a claim of universal constants. Actual acid dissociation constants depend on temperature and on the convention used for thermodynamic activity; apparent concentration-based values also depend on ionic strength. For comparison, the inspected textbook tabulates $2.34$ and $9.60$ for glycine, which would give a different isoelectric point. ^[ref]

Adding sodium hydroxide to acidified glycine successively removes its two titratable protons. Two broad buffer solution regions appear around the two pKa values, separated by a steep pH rise near the isoelectric point. This is useful evidence for two separated acid-base chemistry steps even though the changing chemical species are invisible. In the approximation that added hydroxide only changes glycine protonation, $b=1-\bar{z}$ is the added base amount of substance per initial amount of substance of glycine. Then $b\approx 0.5$, $1$ and $1.5$ correspond to $\mathrm{p}{K}_{a1}$, $\mathrm{pI}$ and $\mathrm{p}{K}_{a2}$; endpoints are not exactly pure chemical species. ^[ref]

For the solid-curve model, start with $n_G=0.00200\mathrm{mol}$ of glycine in $V_0=0.0200\mathrm{L}$ and exactly $n_G$ of fully dissociated hydrochloric acid; this fixes the chloride inventory, but does not force every glycine molecule to remain $G^{+}$. Let the sodium hydroxide concentration be $C_B=0.100\mathrm{mol}{\mathrm{L}}^{-1}$ and $b=n_{\mathrm{NaOH}}/n_G$. Assume volume is additive and no other ions are present; stipulate $K_w=10^{-14}$ for self-ionization of water at a temperature of $25{}^{\circ }\mathrm{C}$. ThenBalancing all positive and negative electric charges requiresSolving this equation gives the solid curve. It remains finite at $b=0$ and beyond $b=2$, where ignoring free hydronium and hydroxide would fail. At the first pKa, the full model gives $b\approx 0.436$, rather than $0.5$: appreciable free hydronium makes that half-equivalent shortcut inaccurate. At the second pKa, $b\approx 1.502$. Known acid inventory, dilution, and thermodynamic activity assumptions are therefore part of any quantitative interpretation, not optional corrections inferred from the curve's shape.

The Henderson-Hasselbalch equation estimates buffer solution pH using a weak acid's acid dissociation constant and the conjugate base to weak acid concentration ratio.

Its activity form is an exact rearrangement of the chosen chemical equilibrium relation. Write the acid dissociation step as $\mathrm{HA}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{A}}^{-}$, with hydrated hydrogen ions understood in water and the solvent convention incorporated into the acid dissociation constant. For consistent standard states,Here each $a_i$ is a dimensionless thermodynamic activity at chemical equilibrium. Exactness refers to this specified acid dissociation constant relation and the chosen single-ion thermodynamic activity convention, not to identifying the ratio with quantities initially mixed. If water is instead written explicitly as a reactant, its thermodynamic activity must be treated consistently. See Hippler and Metcalfe, Using activities to correct the Henderson-Hasselbalch equation (2020), equations 3–4.

For a toy weak acid with stipulated pKa $5$, use the ideal dilute concentration approximation and put $r=[{\mathrm{A}}^{-}]/[\mathrm{HA}]$. The Henderson-Hasselbalch equation becomes $\mathrm{pH}\approx 5+{\log }_{10}r$:More conjugate base relative to weak acid means higher pH. Each tenfold increase in the ratio contributes exactly one to the calculated pH. The stipulated pKa is not a tabulated value for an identified substance.

The distinction between thermodynamic activity and concentration is explicit when $a_i={\gamma }_i[i]/c^{\circ }$, where ${\gamma }_i$ is an activity coefficient and $c^{\circ }$ is the reference concentration:Neglecting the final term gives the usual approximate Henderson-Hasselbalch equation. It is justified when the ratio of activity coefficients is sufficiently close to one, as in a suitable dilute ideal approximation. Ionic strength, including contributions from other dissolved ions, affects activity coefficients; a nominally dilute weak acid alone does not establish that all corrections are small. The ratio must use chemical equilibrium concentrations. Replacing these by prepared amounts or post-neutralization inventories is a further approximation, suitable only when subsequent redistribution is small. See Hippler and Metcalfe, concentration, activity, and analytical-concentration approximations.

The short derivation is to solve the acid dissociation constant relation for $a_{{\mathrm{H}}^{+}}$, take a negative base-ten logarithm, and use the definitions of pH and pKa. Its useful beauty is that a chemical equilibrium becomes a simple rule about a ratio: select a weak acid with pKa near the desired pH, then adjust the partner ratio. Scaling both partner concentrations equally leaves the ideal pH prediction unchanged, while changing buffer capacity. In practical buffer solutions, temperature, ionic strength, and the available inventories still matter.

In acid-base chemistry, hydrogen ion usually refers to positively charged hydrogen, represented by ${\mathrm{H}}^{+}$. In water, this notation describes solvated species, commonly represented by hydronium, rather than an isolated bare proton. The hydrogen ion thermodynamic activity defines pH.

For ideal independent mixing on crystallographic sublattices, an endmember’s thermodynamic activity is the product of its required site fractions, with multiplicities. If one component occupies $q$ equivalent mixing positions per formula unit and has site fraction $X$, then $a=X^q$, not generally $X$.

For a binary example, the mixing Gibbs free energy per mole of formula units is $qRT[X\ln X+(1-X)\ln (1-X)]$. Differentiating the total free energy with respect to the amount of the all-component endmember gives a mixing chemical-potential term $qRT\ln X$. Comparison with $RT\ln a$ proves $a=X^q$. This derivation assumes ideal mixing and equal site fractions; nonideal interactions and preferential site occupancy require a different model. Mg endmembers of biotite and garnet commonly use $X_{\mathrm{Mg}}^3$ in this simplified three-site approximation.

An ideal solution is a chemical solution whose component chemical potentials depend on composition throughHere $n_i$ is an amount of substance, $x_i$ its mole fraction, $R$ the molar gas constant, and ${\mu }_i^{\ast }$ the pure-component liquid reference at the same temperature $T$ and pressure $P$. This model applies to every component across the composition range; it is stronger than an approximation valid only near a pure solvent.^[ref]

With this pure-liquid standard state convention, the thermodynamic activity satisfies ${\mu }_i={\mu }_i^{\ast }+RT\ln a_i$, so an ideal solution has $a_i=x_i$, or activity coefficient ${\gamma }_i=a_i/x_i=1$.^[ref] This statement concerns the liquid. Assuming that a coexisting vapor is an ideal gas mixture is an additional approximation. Its mole fractions $y_i$ generally differ from the liquid mole fractions $x_i$.

For an original two-component example, stipulate $T=300\mathrm{K}$ and pure-component vapor pressures $p_A^{\ast }=80\mathrm{kPa}$ and $p_B^{\ast }=20\mathrm{kPa}$. A and B are invented distinguishable components; these numbers are chosen model inputs, not material data. Combine an ideal solution with an ideal gas vapor, neglect the pure-liquid reference's pressure correction, and exclude other gases. At liquid mole fractions $x_A=0.4$ and $x_B=0.6$, Raoult's law and addition of partial pressures giveThus the coexisting vapor contains about $72.73\%$ A by amount of substance, although the liquid contains only $40\%$. The total pressure is neither pure-component vapor pressure, and $x_A$ cannot be substituted for $y_A$. This enrichment is the basis of distillation.

The usefulness of the ideal solution is that one composition law connects chemical potentials, vapor-liquid equilibrium, entropy of mixing and Gibbs free energy of mixing. Its liquid components can interact strongly: the defining condition is their composition dependence, rather than the absence of all interactions as in an ideal gas model.

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 product of hydrogen-ion and hydroxide-ion activities at a specified temperature. In dilute aqueous solution at 25 degrees Celsius, its usual concentration approximation is ten to the minus fourteen.

The ion product evaluates the same dissolved-ion thermodynamic activity product as the solubility product, but at the current composition rather than necessarily at chemical equilibrium. For AB, $Q_{\mathrm{sp}}=a_{\mathrm{A}}{a}_{\mathrm{B}}$; its concentration counterpart is $Q_c=c_{\mathrm{A}}{c}_{\mathrm{B}}$. Compare $Q_c$ with $K_{\mathrm{sp},c}$ only under compatible activity coefficient assumptions.^[ref]

For other stoichiometries, use the dissolution coefficients as powers: ${\mathrm{AB}}_2(s)\rightleftharpoons {\mathrm{A}}^{2+}(aq)+2{\mathrm{B}}^{-}(aq)$ gives $Q_{\mathrm{sp}}=a_{\mathrm{A}}{a}_{\mathrm{B}}^2$. A solubility product is not itself the molar concentration dissolved; even in the ideal AB model in pure water, that concentration is $\sqrt{K_{\mathrm{sp},c}}$.

A clay mineral with composition $\mathrm{A}{\mathrm{l}}_2\mathrm{S}{\mathrm{i}}_2{\mathrm{O}}_5(\mathrm{OH}{)}_4$. Relative to gibbsite, its greater silica content allows higher dissolved silica thermodynamic activity to favor its formation, as quantified by the appropriate equilibrium constant.

For equilibrium association of $q$ isolated amphiphiles into one micelle, equality of chemical potentials gives the displayed relation between dimensionless thermodynamic activities. Large aggregation number makes aggregate abundance rise rapidly with monomer activity. A finite-aggregate model gives a concentration crossover, rather than an exactly discontinuous thermodynamic transition.

Membrane transport is the passage of substances across a cell membrane. For the small solutes considered here, distinguish the available route from its driving force: a lipid bilayer, ion channel, or carrier protein determines how passage can occur, while differences in chemical potential and membrane potential determine which direction is energetically downhill. A large driving force cannot make an unavailable route conduct. See Alberts et al.: principles of membrane transport.

For an ideal dilute chemical solution at uniform temperature and pressure, with the same reference molar chemical potential on both sides, the molar electrochemical potential change for transferring a solute from compartment A to B isHere $c_{\mathrm{A}},c_{\mathrm{B}}$ are concentrations, $T$ is absolute temperature, $R$ is the molar gas constant, $z$ is signed ion charge number, $F$ is the Faraday constant, and $V_{\mathrm{B}}-V_{\mathrm{A}}$ is the electric potential difference. The natural logarithm contains a dimensionless ratio. The electrical term vanishes for an uncharged solute. More generally, use thermodynamic activities in place of the ideal concentration ratio. This combines the ideal-solution chemical potential difference with electrical energy per mole; see IUPAC Gold Book: electrochemical potential. For a biological cell, taking B inside and A outside makes the voltage term use the existing membrane potential convention.

A diagram of stable mineral assemblages as functions of specified aqueous thermodynamic activities, at fixed temperature and pressure. The coordinates often use logarithmic activity ratios. Mass-action equations give pairwise coexistence lines; a line segment is stable only when no omitted mineral has a lower relevant thermodynamic potential under the same imposed chemical potentials.

A specified limiting composition used to describe a solid solution in a mineral. Its thermodynamic activity depends on the chosen standard state and activity model. For example, pyrope is a magnesian garnet endmember, while phlogopite is a magnesian biotite endmember.

An activity model in which distinct atomic positions in a crystal structure are occupied independently by specified species. For ideal random mixing, a pure mineral endmember requiring ${\nu }_s$ specified atoms on site type $s$ has thermodynamic activity $a={\prod }_s{X}_s^{{\nu }_s}$ relative to that pure endmember, with other occupations held fixed. Its mixing contribution to the chemical potential is $RT\ln a$. The powers count occupied positions per formula unit; they are not always the mole fraction of whole endmember formulae.