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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.

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.