Please answer parts A through E below, including the diagram for part B$.$ Thank you!

$(50$ points$)$ Let us calculate the compositions and quantities of the gas and liquid when $1$ mole of the following mixture is brought to equilibrium at $160F$ and $245$ psia. Raoult's law is assumed to be valid in this question.

a$.(10$ points$)$ The vapor pressure is $385\mathrm{.}7$ psia for $C_3,121\mathrm{.}5a$ for $n{C}_4,$ and $2\mathrm{.}5$ psia for $n{C}_8$ at $160F.$ See below for the vapor pressure curves for the three components using an equation of state. Calculate the $K$ values for the three components at $160F$ and $245$ psia, assuming Raoult's law.

b$.(10$ points$)$ Plot the Rachford$-$Rice $($RR$)$ function. In your figure, the y axis should be the RR function, and the $x$ axis be $"v"$ the vapor$-$phase mole fraction. Indicate the values for $"v"$ at poles of the RR function, if any. What is the mole fraction for the vapor phase $($i$.$e$.,{}_v$ that satisfies the RR equation$)?$

c$.(10$ points$)$ Calculate the compositions for the vapor and liquid phases using the root from part b$.$

d$.(10$ points$)$ On the basis of Raoult's law, estimate the bubble$-$point and dew$-$point pressures at $160F$ for this $3-$component mixture.

e$.(10$ points$)$ You mixed $x$ mol$\%$ of pure propane with $(100-x)$ mol $\%$ of the ternary mixture $($given in the table$).$ This new mixture was brought to equilibrium at $160F$ and $300a,$ and you found that this is a bubble point. Find the value for $x.$ You are assuming Rault's law for this question; you do not use an equation of state here.