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$\%$ C$1$ 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 gas chromatograph, which yielded a first tie line $($TL $1)$ as shown in Table $1.$

Table $1.$ Overall composition of Mixture $1,$ and resulting tie line $($TL $1).$

$\backslash$table$[[,$Overall composition,Vapor phase composition,Liquid phase composition$],[$C$1,0\mathrm{.}85,0\mathrm{.}95,0\mathrm{.}40],[$C$4,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[$C$8,0\mathrm{.}15,0\mathrm{.}05,0\mathrm{.}60]]$

a$.$ Analytically calculate the mole fraction of the vapor phase by using material balance and the data 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$).$

$1$

b$.$ Clearly indicate Mixture $1$ on the ternary diagram.

c$.$ Using the data table, clearly indicate TL $1$ on the ternary diagram.

Then, she added a small amount of $C4,$ and obtained a $2^{nd}$ tie line $($TL $2)$ at the same temperature and pressure $(140$ bars and $400K)$ as shown in Table $2.$ She repeated this process of adding a certain amount of pure $C4$ and measuring equilibrium phase compositions. She obtained TLs $3,4,5,$ and $6.$ The tables below show the compositions $($in mole percentage$)$ of equilibrated liquid and vapor phases in her measurements.

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

e$.$ 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.

Parts b c d e