The Gibbs energy can be used to predict phase separation. We derived the Gibbs energy of a mixture as g$=$ x$\_($i$)$g$\_($i$)+$RT$$ x$\_($i$)$lnx$\_($i$)+$g$^($E$).$ Use a reference state of the pure species at system T and P so the first term is zero. This is then equivalent to $\backslash$Delta g$\_($mix $)(\backslash$Delta g$\_($mix $)=$RT$$ x$\_($i$)$lnx$\_($i$)+$g$^($E$)).$ For the following mixtures, assume the three$-$suffix Margules equation accurately models g$^($E$).$ a$)$ Given a mixture of species a and b$,\backslash$gamma $\_($a$)^(\backslash$infty $)=6$ and $\backslash$gamma $\_($b$)^(\backslash$infty $)=32.$ Plot $\backslash$Delta $($g$\_($mix $))/($R$)$T vs x$\_($a$)$ and determine the number of phases, the composition of each phase, and overall $\backslash$Delta $($g$\_($mix$))/($R$)$T for the solution at $($i$)$ x$\_($a$)=0\mathrm{.}1,($ii$)$ x$\_($a$)=0\mathrm{.}5,$ and $($iii$)$ x$\_($a$)=0\mathrm{.}8.$ b$)$ Is a miscible, partially miscible, or immiscible in b $?$ c$)$ Given a mixture of species a and c$,\backslash$gamma $\_($a$)^(\backslash$infty $)=0\mathrm{.}8$ and $\backslash$gamma $\_($c$)^(\backslash$infty $)=0\mathrm{.}2.$ Plot $\backslash$Delta $($g$\_($mix$))/($R$)$T vs x$\_($a$)$ and determine the number of phases, the composition of each phase, and overall $\backslash$Delta $($g$\_($mix$))/($R$)$T for the solution at $($i$)$ x$\_($a$)=0\mathrm{.}1,($ii$)$ x$\_($a$)=0\mathrm{.}5,$ and $($iii$)$ x$\_($a$)=0\mathrm{.}8.$ d$)$ Is a miscible, partially miscible, or immiscible in c $?$