Binary phase transition boiling rate

(Callen 9.7-2)


The phase diagram of a solution of A in B, at a pressure of 1 atm, has a coexistence region in the $(X,T)$ plane (e.g., $X\_A$ is a liquid molar fraction with $N=N\_A+N\_B$ and T temperature and $Y\_A$ is a gas molar fraction). The upper bounding curve of the two phase region can be represented by


$T=T\_0-(T\_0-T\_1)Y\_A^2$


and the lower bounding curve of the two-phase region is represented by


$T=T\_0-(T\_0-T\_1)X\_A(2-X\_A)$


Show that if a small fraction of the liquid $−dN/N$ is boiled off, the change in the fraction of the remaining liquid is,


$dX\_A =-\left[(2X\_A-X\_A^2)^{\frac{1}{2}}-X\_A \right]\frac{-dN}{N}$.


All I knew was that $dX\_A$ could be obtained from $dN = dN\_A+dN\_B$


and from there


$dN\_A+dN\_B = N dX\_A + X\_A dN + N dX\_B + X\_B dN$... I previously found that, when temperature is such that $X\_A=X\_B=1/2$, $Y\_A=0.8660$...


