= Activity conversion of an equilibrium constant
<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.<Quantities, Units and Symbols in Physical Chemistry by IUPAC>[§2.11, printed pp. 57–58: relative activity, reaction quotient, standard reaction Gibbs energy and standard equilibrium constant; notes 12, 13 and 18]{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 $\widetilde a_i=c_i a_i$, where each $c_i$ is positive and independent of composition at the chosen conditions. From the product defining the <reaction quotient>,
$$
Q=\prod_i a_i^{\nu_i},\qquad
\widetilde Q=Q\prod_i c_i^{\nu_i}.
$$
At <chemical equilibrium>, $Q=K$ and $\widetilde Q=\widetilde K$, so
$$
\boxed{\widetilde K=K\prod_i c_i^{\nu_i}.}
$$
A 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 $\widetilde\mu_i^\circ=\mu_i^\circ-RT\ln c_i$, where $R$ is the <molar gas constant>. Consequently
$$
\Delta_r\widetilde G^\circ=\Delta_rG^\circ-RT\sum_i\nu_i\ln c_i,
\qquad
\Delta_r\widetilde G^\circ+RT\ln\widetilde Q
=\Delta_rG^\circ+RT\ln Q.
$$
The 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 $\widetilde a_A=10a_A$, $\widetilde a_B=2a_B$. Then $\widetilde K=25(2/10^2)=0.5$. A composition with $Q=5$ has $\widetilde Q=0.1$; both conventions give $Q/K=0.2$ and $\Delta_rG=RT\ln0.2$. The original <standard Gibbs energy of reaction> is $-RT\ln25$, whereas the converted value is $RT\ln2$. 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.
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