Please answer parts A through $F$ below, including a ternary diagram that includes all $of$ its necessary components and complete the tables below.

$(30$ points$)$ A student in UT PGE department studied a simple ternary system consisting of methane $($C$1),$ n$-$butane $($C$4),$ and n$-$octane $($C$8).$ She planned an experiment to measure tie lines for this ternary system using a PVT cell. First, she made a mixture of $85$ mol$\%Cl$ and $15$ mol$\%$ C$8($Mixture $1)$ in a PVT cell at $140$ bars and $400K,$ which resulted in vapor$-$liquid equilibrium. The compositions of equilibrium gas and liquid phases were measured by using Eas chromatograph, which yielded a first tie line $($TL$.1)$ as

shown in T$$ble $1.$

a$.(5$ points$)$ Analytically calculate the mole fraction af the vapor phase by using material tralarice and the cata given in Table $1.$ Give three digits after the decimal point. Do not use a graphical method $($i$.$e$.,$ no need to use a ruler$).$

b$.(5$ points$)$ Clearly indicate Mixture $1$ on the ternary diagram.

c$.(5$ points$)$ Usinf, the data table, clearly indicate TL $1$ on the ternary diagram. Then, she added a small amount of $C4,$ and obtained a $2^{nst}$ tie line $($TL $2)$ at the same temperature and pressure bars and $400kj$ as shown in Table $2.$ She repeated this process of adding a certain amount of pure $C4$ and measuring equilibrium phase compositions. She abtained TLS $3,$

$4,5,$ and $6.$ The tables below show the compositions $($in mole percentage$)$ of equil brated liquid

and vapor phases in her measurements.

d$.(5$ points$)$ Draw TLs $2,3,4,5,$ and $6$ on the diagram.

e$.(5$ points$)$ Make a two$-$phase envelope. There must be a critical point on your diagram. Clearly indicate the critical point on the diagram. Clearly indicate the bubble$-$point curve, and the dew$-$point curve.

f$.(5$ points$)$ The student recorded the overall composition used to measure each tie line, but forgot to show them in Tables $2,3,4,5,$ and $6.$ Please fill the blanks for these tables $(2$ through $6),$ basecl on your diagram. Hint: Oraw the mixing line between Mixture $1$ and

$100\%Cl,$ on which all the mixtures must lie. Table $3.$ Overall composition of Mixture $3,$ and resultine, tie line $($TL $3).$ Table $4.$ Overall composition of Mixture $4,$ and resulting tie line $($TL$4).$

Table $5.$ Overall composition of Mixture $5,$ and resultine tie line $($TL $5).$ Table $6.$ Overall composition of Mixture $6,$ and resulting tie line $($TL $6).$