 Activity equality for coexisting liquids

For exchanging neutral components at a common temperature and pressure, 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 aswhere $R$ is the molar gas constant and $T$ is temperature. Subtraction at equilibrium givesThe 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.^[ref] Mixing conventions can therefore produce unequal activities for physically coexisting phases. For instance, if one reference exceeds the other by $RT\ln 2$, 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 $\chi =2\ln 3$. The model has coexisting component-A mole fractions $x_A^{\alpha }=1/4$ and $x_A^{\beta }=3/4$, withThus ${\gamma }_A^{\alpha }=3^{9/8}\approx 3.44161$ and ${\gamma }_A^{\beta }=3^{1/8}\approx 1.14720$, whileComponent 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.