Chapter 15, Problem 15.29P

Consider a ternary system comprising solute species 1 and a mixed solvent (species 2 and 3). Assume that

$\frac{G^E}{RT}=A_{12}{x}_1{x}_2+A_{13}{x}_1{x}_3+A_{23}{x}_2{x}_3$

Show that Henry’s constant for species 1 in the mixed solvent is related to Henry’s constants ${\mathcal{H}}_{1,2}$ and ${\mathcal{H}}_{1,3}$ for species 1 in the pure solvents by:

$In{\mathcal{H}}_1=x_2^{\text{'}}In{\mathcal{H}}_{1,2}+x_3^{\text{'}}In{\mathcal{H}}_{1,3}-A_{23}{x}_2^{\text{'}}{x}_3^{\text{'}}$

Here $x_2^{\text{'}}$ and $x_3^{\text{'}}$ are solute-free mole fractions:

$x_2^{\text{'}}\equiv \frac{x_2}{x_2+x_3}$ , $x_3^{\text{'}}\equiv \frac{x_3}{x_2+x_3}$