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Activity conversion of an equilibrium constant changes the numerical equilibrium constant when a different standard state convention rescales the thermodynamic activities used to define it. The equilibrium composition is unchanged. The essential requirement is to convert the reaction quotient, the equilibrium constant and the standard Gibbs energy of reaction together.^[ref]

At fixed temperature $T$, write a chemical reaction as ${\sum }_i{\nu }_i{B}_i=0$, with positive stoichiometric coefficients for products and negative ones for reactants. Let a standard-state activity conversion give ${\tilde{a}}_i=c_i{a}_i$, where each $c_i$ is positive and independent of composition at the chosen conditions. From the product defining the reaction quotient,At chemical equilibrium, $Q=K$ and $\tilde{Q}=\tilde{K}$, soA scale change for a reactant enters with a negative exponent; multiplying every activity by the same number need not leave $K$ unchanged when the sum of stoichiometric coefficients is nonzero.

The reference chemical potentials must compensate. Since ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$, the converted standard values are ${\tilde{\mu }}_i^{\circ }={\mu }_i^{\circ }-RT\ln c_i$, where $R$ is the molar gas constant. ConsequentlyThe physical Gibbs energy of reaction and the ratio $Q/K$ are invariant. A change in the numerical equilibrium constant alone therefore does not show that the reaction has become more favorable.

For an original example, consider $2\mathrm{A}\rightleftharpoons \mathrm{B}$, with $K=25$, and choose ${\tilde{a}}_A=10a_A$, ${\tilde{a}}_B=2a_B$. Then $\tilde{K}=25(2/10^2)=0.5$. A composition with $Q=5$ has $\tilde{Q}=0.1$; both conventions give $Q/K=0.2$ and ${\Delta }_{r}G=RT\ln 0.2$. The original standard Gibbs energy of reaction is $-RT\ln 25$, whereas the converted value is $RT\ln 2$. Their opposite signs identify different reference conventions, while the actual reaction driving Gibbs energy at the chosen composition agrees exactly. This calculation assumes a reference rescaling, rather than a composition-dependent alteration of the physical activity model.

Chemical affinity is the thermodynamic driving tendency of a specified chemical reaction, defined at fixed temperature and pressure aswhere $G$ is Gibbs energy, $\xi$ the extent of reaction, ${\nu }_i$ signed stoichiometric coefficients and ${\mu }_i$ chemical potentials.^[ref]

For the chosen forward direction, $dG=-\mathcal{A}d\xi$: positive chemical affinity favours a small positive advance. At an unconstrained equilibrium, $\mathcal{A}=0$. For a reaction quotient $Q$ and equilibrium constant $K$, $\mathcal{A}=RT\log (K/Q)$ under the usual activity conventions. This is a thermodynamic condition, not a rate prediction; a favourable reaction can remain slow because of an Gibbs energy of activation.

Let a chemical reaction have signed stoichiometric coefficients ${\nu }_i$, positive for products. Changes of amount of substance satisfy $d{n}_i={\nu }_{i}d\xi$, with extent of reaction $\xi$. At fixed temperature and pressure, the Gibbs free energy differential isThus the Gibbs energy of reaction is ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$, using molar chemical potentials when $\xi$ is measured in moles.

An interior chemical equilibrium allows either sign of a small $d\xi$, so it requires ${\Delta }_{r}G=0$. A positive forward change lowers $G$ when ${\Delta }_{r}G<0$; a negative change lowers it when ${\Delta }_{r}G>0$. Stability additionally requires a constrained minimum, rather than stationarity alone.

At a boundary of allowed compositions, only one direction may be feasible. Then the necessary minimum condition is ${\Delta }_{r}Gd\xi \ge 0$ for every allowed small change. The zero-derivative equality must not be imposed on an infeasible reverse change.

For complete neutralization, the required masses of anhydrous potassium carbonate, anhydrous sodium carbonate and magnesium oxide are in the ratio $138:106:40=69:53:20$. Here “carbonate of potash” means potassium carbonate, “carbonate of soda” means sodium carbonate, and “caustic magnesia” means magnesium oxide, rather than magnesium hydroxide or a magnesium carbonate preparation. The hydration state of the sodium carbonate matters considerably.

Each formula unit of either carbonate consumes two protons; each formula unit of magnesium oxide also consumes two protons. Thus one mole of each base neutralizes one mole of sulfuric acid when the final salts are the normal sulfates:The carbon dioxide evolved in the first two chemical reactions escapes, whereas magnesium oxide produces no gas. The stoichiometric coefficients, rather than the visibly different behaviour of the bases, determine the required amounts.

Let $m$ be the mass of pure ${\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4$ in the given chemical solution. Using the rounded relative atomic masses $\mathrm{H}=1$, $\mathrm{C}=12$, $\mathrm{O}=16$, $\mathrm{Na}=23$, $\mathrm{Mg}=24$ and $\mathrm{K}=39$ givesThus 98 mass units of pure sulfuric acid require respectively 138, 106 and 40 mass units. Present atomic weights slightly change these rounded values, without changing the one-to-one mole ratios. If $m_s$ denotes the mass of the supplied acid and $w$ its mass fraction of ${\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4$, substitute $m=w{m}_s$. A volume alone is insufficient without its concentration or an equivalent density and composition.

If “carbonate of soda” is instead the crystalline decahydrate, sodium carbonate decahydrate, its rounded molar mass is $286\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, so the corresponding factor is $286/98$, rather than $106/98$. Likewise an impure base requires correction for its assayed content. The unspecified given quantity can therefore be handled exactly by proportional reasoning; no missing numerical measurement is needed.

At the same temperature and pressure, one volume of HCl corresponds to $\frac{1}{2}$ volume of hydrogen and $\frac{1}{2}$ volume of chlorine; one of steam to one of hydrogen and $\frac{1}{2}$ of oxygen; one of ${\mathrm{N}}_2\mathrm{O}$ to one of nitrogen and $\frac{1}{2}$ of oxygen; one of NO to $\frac{1}{2}$ of each element; and one of ammonia to $\frac{1}{2}$ of nitrogen and $\frac{3}{2}$ of hydrogen. All elemental volumes here refer to their ordinary diatomic gases.

The simplest way to avoid an erroneous factor of two is to balance each chemical reaction using ${\mathrm{H}}_2$, $\mathrm{C}{\mathrm{l}}_2$, ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$. Under Avogadro's law, their stoichiometric coefficients are also gas volume ratios, provided all gases are compared at the same temperature and pressure and ideal gas behaviour is a good approximation.

If the common chosen volume of each compound is $V$, the corresponding ordered elemental volumes are therefore $(V/2,V/2)$, $(V,V/2)$, $(V,V/2)$, $(V/2,V/2)$ and $(V/2,3V/2)$. The ratios concern molecular counts and hypothetical free elemental gases, not separate compartments of elemental gas still present inside a compound.

“Hydrochloric acid” must here mean gaseous hydrogen chloride, rather than its aqueous chemical solution; water must remain as steam. If water condenses or Ammonia dissolves in the collecting liquid, measured volumes no longer show these simple ratios. For an experimental comparison, dry-gas and hydrostatic pressure corrections are those discussed in 3.

Subtract reaction (I) from reaction (II), cancelling the common aqueous ions and corundum, and divide by two. Choose the albite-to-nepheline direction:This is albite plus water converting to nepheline and dissolved silica. Standard Gibbs energies of reaction add with their stoichiometric coefficients, so logarithms of equilibrium constants do too. For this single-formula normalization,The doubled reaction has ${\log }_{10}K=-22.06$, while reversing either reaction changes the sign. Specifying the direction and normalization removes that apparent ambiguity.

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.