= Standard-state activity conversion
\b[Standard-state activity conversion] changes the numerical <thermodynamic activity> assigned to a fixed physical state when its reference <chemical potential> changes. The observable equilibrium and physical <chemical potential> stay the same. The activity and its standard chemical potential must therefore be converted together.
At fixed <temperature> $T$, write
$$
\mu_i=\mu_i^\circ+RT\ln a_i
=\widetilde\mu_i^\circ+RT\ln\widetilde a_i.
$$
If $\delta_i=\widetilde\mu_i^\circ-\mu_i^\circ$, subtraction gives
$$
\boxed{\widetilde a_i=a_i\exp[-\delta_i/(RT)].}
$$
Here $R$ is the <molar gas constant>. This follows from the IUPAC definition of relative activity and its specified <standard state>.<Quantities, Units and Symbols in Physical Chemistry by IUPAC>[§2.11, pp.57,59, and §2.11.1(iv), pp.61–62: activity and reference conventions.]{ref} Changing just the activity in the first equation would instead change the calculated physical chemical potential.
For an invented conversion $\widetilde a_i=10a_i$, the compensating reference shift is $\delta_i=-RT\ln10$. An old activity $0.03$ becomes $0.30$; the two expressions for $\mu_i$ are identical. An activity of one consequently identifies a reference condition under the chosen convention, not an invariant amount of material.
An <activity conversion of an equilibrium constant> follows the same logic. For the reaction A $\rightleftharpoons$ B, take $K=a_B/a_A$ at equilibrium and convert by $\widetilde a_A=10a_A$, $\widetilde a_B=2a_B$. Then $\widetilde K=K/5$. If $K=5$, the converted value is $1$, while
$$
\Delta_r\widetilde G^\circ
=\Delta_rG^\circ-RT\ln(2/10)=0
$$
because originally $\Delta_rG^\circ=-RT\ln5$. For any physical composition, the corresponding reaction quotient also changes by $1/5$, so $\Delta_rG^\circ+RT\ln Q$ remains unchanged. These are reference changes, not altered reaction chemistry.
The same requirement governs <activity equality for coexisting liquids>: compare activities only after identifying their <standard states>, composition scales and reference conventions. Equal numerical values expressed under incompatible conventions do not, by themselves, establish <liquid-liquid equilibrium>.
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