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An activity coefficient expresses departure from a chosen ideal-mixture activity convention. In a concentration-based convention,The thermodynamic activity $a_i$ and coefficient ${\gamma }_i$ are dimensionless; $c^{\circ }$ is the standard concentration. The reference convention fixes which ideal or limiting state has ${\gamma }_i=1$. A mole-fraction convention uses a different ideal activity and must not be mixed silently with the concentration convention.^[ref]

Substituting activities into an equilibrium constant separates concentration ratios from interaction corrections. For $AB\rightleftharpoons A+B$, the correction contains ${\gamma }_A{\gamma }_B/{\gamma }_{AB}$. Changing composition can therefore change an observed concentration quotient even when the thermodynamic constant at fixed temperature and reference state is unchanged. An activity coefficient can be greater or less than one; it is not a fraction of molecules assigned to an active population.

For ideal concentration activities and signed stoichiometric coefficients ${\nu }_i$, $K_c={\prod }_i(c_i/c^{\circ }{)}^{{\nu }_i}$ at equilibrium. Its numerical value depends on the stated standard concentration when ${\sum }_i{\nu }_i\ne 0$. Number concentrations require a number-concentration reference; molar concentrations require the corresponding molar reference.

A dissociation constant is an equilibrium constant for a reaction in which a combined species separates into components. For $AB\rightleftharpoons A+B$, a concentration convention writesIt has units of concentration. The dimensionless thermodynamic form instead uses thermodynamic activities, $K_d^{\circ }=a_A{a}_B/a_{AB}$. For ideal concentration activities, $K_d^{\circ }=K_d^{(c)}/c^{\circ }$, with specified standard concentration $c^{\circ }$. The choice of quantity and reference is therefore part of quoting the constant.^[ref]

In the simple dilute $1:1$ model with free $c_A$ held fixed, the fraction of B present as AB is $c_A/(K_d^{(c)}+c_A)$. A smaller $K_d^{(c)}$ therefore means more association at the same free concentration. The inverse association constant describes the reversed reaction only when conventions, solvent conditions and temperature match. A dissociation rate is a kinetic quantity and is not specified by this equilibrium ratio alone.

At chemical equilibrium, ${\mu }_{\mathrm{AB}}={\mu }_{\mathrm{A}}+{\mu }_{\mathrm{B}}$. Using the number-concentration form givesHere $\Delta {\epsilon }_0$ is the reaction energy between the chosen molecular reference states. Rearranging,Define the dimensionless concentration equilibrium constant byThusThe standard concentration $c^{\circ }$ has the same units as $c_i$. It converts concentration ratios to ideal thermodynamic activities; changing it changes the numerical dimensionless $K_c$ for this dissociation, which creates an extra particle. The unnormalized quotient $c_{\mathrm{A}}{c}_{\mathrm{B}}/c_{\mathrm{AB}}$ instead has concentration units.^[ref]

Use ground-state references and the supplied ideal-gas, rigid-rotor and harmonic-oscillator approximations. The two ions have closed electronic shells; take their electronic ground-state degeneracies and the molecular electronic ground-state degeneracy as one, and neglect higher electronic levels. Consistently treated nuclear-spin factors cancel between the species. Since the molecule is heteronuclear, its rotational symmetry number is one.

Let ${\mu }_r=m_{\mathrm{Cs}}{m}_{\mathrm{I}}/m_{\mathrm{CsI}}$ be the reduced mass, neglecting the negligible mass correction from electron transfer. The ratio of translational partition functions per volume reduces toThe molecule has additional rotational and vibrational states, whereas the monatomic ions do not. At $3000\mathrm{K}$,Hence the dimensional equilibrium quotient isThe last form makes the unassigned standard concentration explicit. For the common choice of $1\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$, the corresponding microscopic reference is $c^{\circ }=6.02214\times 10^{26}{\mathrm{m}}^{-3}$, givingEquivalently, the dimensional molar quotient is $8.27\times 10^{-7}\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$. A pressure-based gas standard is a different convention: taking $c^{\circ }=p^{\circ }/(k_{B}T)$ with $p^{\circ }=1\mathrm{bar}$ would give $2.06\times 10^{-4}$, rather than the same numerical constant.

The well-minimum convention gives the identical answer becauseConstants used here are $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, $m_u\simeq 1.66053907\times 10^{-27}\mathrm{kg}$ and $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$.^[ref]

Let ${\rho }_i=N_i/V$ be a molecular number density and define the volume-independent molecular partition function density $f_i=q_i/V$. Then $q_i/N_i=f_i/{\rho }_i$. Also define the reaction reference-energy differenceSubstitution into the equilibrium chemical-potential relation givesAn equilibrium constant expressed using concentration activities $a_i={\rho }_i/c^{\circ }$ is dimensionless:Here every $f_i$ has units ${\mathrm{m}}^{-3}$, and $c^{\circ }$ is the chosen standard concentration, initially one molecule per cubic metre. The factor $c^{\circ }$ is essential: without it the association quotient has units of volume.

Convert the new standard concentration to the number-density convention:For this association, $K_c={\rho }_{{\mathrm{H}}_2\mathrm{CO}}{c}^{\circ }/({\rho }_{{\mathrm{H}}_2}{\rho }_{\mathrm{CO}})$, so it is proportional to $c^{\circ }$. ThereforeBoth dimensionless constants describe the same physical equilibrium; changing the reference changes the numerical activity quotient because two reactant molecules form one product molecule.

In transition state theory, $K^{‡}$ is the dimensionless equilibrium-like constant for forming the activated complex from the reactants under the chosen standard-state convention. It is not the equilibrium constant for forming the stable product.

${\Delta }_{\mathrm{r}}{G}^{\circ ,‡}$ is the standard activation free energy, ${\Delta }_{\mathrm{r}}{H}^{\circ ,‡}$ the standard activation enthalpy, and ${\Delta }_{\mathrm{r}}{S}^{\circ ,‡}$ the standard activation entropy. Each is the activated-complex quantity minus the corresponding sum for the reactants, with the unstable reaction-coordinate motion treated by the transition-state flux construction.

$[{]}^{\circ }=c^{\circ }$ is the standard concentration, conventionally $1\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ for the rate units used here. Its inverse converts the attempt frequency $k_{\mathrm{B}}T/h$ into the units of a bimolecular rate constant. The stated Eyring equation assumes unit transmission coefficient.

The activation enthalpy is obtained from the slope across the measured $1/T$ interval. The activation entropy comes from the intercept at $1/T=0$, far outside that interval, so a small slope uncertainty can produce a much larger uncertainty in the extrapolated intercept. Slope and intercept errors are consequently strongly correlated.

The entropy is also sensitive to the assumed prefactor and standard concentration, whereas a constant multiplicative calibration error shifts the intercept without changing the slope. This explains why its inferred precision is often poorer. Since enthalpy and entropy have different units, a claim about which absolute numerical uncertainty is larger requires a stated comparison convention.

Use decimal logarithms and write $x={\log }_{10}{a}_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4}$ and $y={\log }_{10}(a_{{\mathrm{Mg}}^{2+}}/a_{{\mathrm{H}}^{+}}^2)$. The question's concentration notation is interpreted as dimensionless thermodynamic activities, or concentrations divided by a fixed standard concentration with activity coefficients set to one.

For the chlorite–gibbsite reaction, unit solid activities giveHence $y=10.53-0.60x$. It is a descending straight mineral activity diagram boundary, passing through $(0,10.53)$ and $(-5,13.53)$. Its continuation alone does not establish which portions are stable against kaolinite.

For an elementary reversible reaction with concentration-based rates obeying the law of mass action, writewhere stoichiometric coefficients ${\nu }_i$ are positive for products. At detailed balance, $r_f=r_r$, soThis $K_c$ carries a power of concentration when $\Delta \nu ={\sum }_i{\nu }_i\ne 0$.

For an ideal solution with activities $a_i=c_i/c^{\circ }$, the dimensionless thermodynamic equilibrium constant is $K={\prod }_i(c_{i,\mathrm{eq}}/c^{\circ }{)}^{{\nu }_i}$. ConsequentlyKeeping the standard concentration factor is what makes the units of both rate coefficients consistent. For $A+B\rightleftharpoons C$, $\Delta \nu =-1$ and $k_r=k_f{c}^{\circ }/K$: a second-order forward coefficient becomes a first-order reverse coefficient.

Thermodynamics fixes this ratio only under the stated elementary, reversible rate-law model. It does not determine either coefficient separately or replace a multistep reaction mechanism by one elementary step.