= Activity equality for coexisting liquids
{title2=$a_i^\alpha=a_i^\beta$}
For exchanging neutral components at a common <temperature> and <pressure>, \b[activity equality for coexisting liquids] expresses <liquid-liquid equilibrium> as $a_i^\alpha=a_i^\beta$, provided both <thermodynamic activities> use the same <standard state>. The basic equilibrium condition is equality of <chemical potentials>; equality of numerical activities follows only after their references have been matched.
Write the definition of <thermodynamic activity> as
$$
\mu_i^r=\mu_i^{\circ,r}+RT\ln a_i^r,\qquad r\in\{\alpha,\beta\},
$$
where $R$ is the <molar gas constant> and $T$ is <temperature>. Subtraction at equilibrium gives
$$
\frac{a_i^\beta}{a_i^\alpha}
=\exp\!\left(\frac{\mu_i^{\circ,\alpha}-\mu_i^{\circ,\beta}}{RT}\right).
$$
The ratio is one when the two standard chemical potentials are the same. IUPAC defines activity relative to the specified standard chemical potential and distinguishes pure-phase and solute reference conventions.<Quantities, Units and Symbols in Physical Chemistry by IUPAC>[§2.11, pp.57,59; §2.11.1(iv), pp.61–62.]{ref} Mixing conventions can therefore produce unequal activities for physically coexisting phases. For instance, if one reference exceeds the other by $RT\ln2$, its corresponding activity is half as large at the same <chemical potential>. <Standard-state activity conversion> changes the numerical coordinate, not the physical driving force.
Equal <thermodynamic activities> also need not mean equal <mole fractions>. With a common pure-liquid reference, $a_i=x_i\gamma_i$, where $\gamma_i$ is an <activity coefficient>. An original symmetric <regular solution> example makes this quantitative. Set the dimensionless interaction parameter to $\chi=2\ln3$. The model has coexisting component-A mole fractions $x_A^\alpha=1/4$ and $x_A^\beta=3/4$, with
$$
\ln\gamma_A=\chi(1-x_A)^2,\qquad
\ln\gamma_B=\chi x_A^2.
$$
Thus $\gamma_A^\alpha=3^{9/8}\approx3.44161$ and $\gamma_A^\beta=3^{1/8}\approx1.14720$, while
$$
a_A^\alpha=\tfrac14 3^{9/8}
=\tfrac34 3^{1/8}=a_A^\beta\approx0.860402.
$$
Component B satisfies the analogous equality by symmetry. The different compositions are compensated by different nonideal environments. These are chosen model values, not experimental compositions. As in <liquid-liquid equilibrium>, satisfying equalities must be supplemented by stability against competing phase arrangements.
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