Construct three x$-$y diagrams following three different approaches listed below for n$-$hexane $($component $2)/$n$-$heptane $($component $1)$ binary system at $101$ kPa. We know that the boiling points of hexane and heptane at $101$ kPa are $68\mathrm{.}7\backslash$deg C and $98\mathrm{.}4\backslash$deg C$,$ respectively.

$($a$)$ use the heuristic equation for calculating relative volatility to construct the x$-$y diagram. To do so$,$ you need to find out all the parameters needed to apply the heuristic equation online and explicitly list them down in your answer. Also, write down the assumptions you need to make in order to use the heuristic equation.

$($b$)$ Second, let$$s use the Antoine equations given for both components:

logP$2($sat$)=4\mathrm{.}00266-(1171\mathrm{.}53/($T$-48\mathrm{.}784))$ and logP$1($sat$)=4\mathrm{.}02832-(1268\mathrm{.}636/($T$-56\mathrm{.}199))$

where the unit of Pi$($sat$)$ is in bar, and T is in K$.$ What assumptions do you need to make in order to use Antoine equations to construct x$-$y diagram? List them in your answer.