A binary mixture containing acetone and ethylacetate, is vaporized in a flash unit at atmospheric pressure to determine the equilibrium temperature and compositions. The liquid and condensed vapor samples are taken for $6$ times and analyzed by the gas chromatography. The gas chromatography results are given in Table $2.$ The calibration curves for acetone and ethyl acetate are defined by the relation between the peak area and injected volume as $A=5{10}^{16}L$ and $A=4{10}^{16}L,$ respectively, where $A$ is peak area in area unit and $v$ is injected volume in $m^3.$

a$)$ Calculate the mole fractions of the components both in liquid and vapor phases. $[10$ points$]$

b$)$ Plot the $"y$ vs $x"$ and $"T$ vs $x-$y$"$ diagrams. $[5$ points$]$

c$)$ Calculate the activity coefficients. $[5$ points$]$

Data: $,M{W}_{\text{a}\text{c}\text{e}\text{t}\text{o}\text{n}\text{e}}=58\mathrm{.}08k\frac{g}{k}$mol

$M{W}_{\text{e}\text{t}\text{h}\text{y}\text{l}\text{a}\text{c}\text{e}\text{t}\text{a}\text{t}\text{e}}=88\mathrm{.}1k\frac{g}{k}$mol

${}_{\text{e}\text{t}\text{h}\text{y}\text{l}\text{a}\text{c}\text{e}\text{t}\text{a}\text{t}\text{e}}=901k\frac{g}{m^3}$

$T_{bp,\text{e}\text{t}\text{h}\text{y}\text{l}\text{a}\text{c}\text{e}\text{t}\text{a}\text{t}\text{e}}=77C$

${}_{\text{a}\text{c}\text{c}\text{e}\text{t}\text{o}\text{n}\text{e}}=792k\frac{g}{m^3}$

$T_{bp,\text{a}\text{c}\text{e}\text{t}\text{o}\text{n}\text{e}}=56\mathrm{.}5C$

$ln{F}_i^0=A_i-\frac{B_i}{C_i+T}$ where $P_i^0$ is in $mmHg$ and $T$ in $K.$

Table $1.$ Antoine constants for acetone and ethyl acetate

$\backslash$table$[[$Compounds$,$A$,$B$,$C$],[$Acetone$,16\mathrm{.}6513,2940\mathrm{.}46,-35\mathrm{.}93],[$Ethylacetate$,16\mathrm{.}1516,2790\mathrm{.}5,-57\mathrm{.}15]]$

Table $2.$ Gas chromatography data for the acetone$-$ethyl acetate mixture

$\backslash$table$[[\backslash$table$[[$Sample$],[$No$.]],\backslash$table$[[$Temperature$],[(C)]],$Liquid phase, $x,$Vapor phase, y$],[\backslash$table$[[$Ret$.$ Time$],[($min$)]],$Area,$\backslash$table$[[$Ret$.$ Time$],[($min$)]],$Area$],[1,61,2\mathrm{.}185,5814983,2\mathrm{.}139,8437152],[3\mathrm{.}986,500030,3\mathrm{.}048,453660],[2,64,2\mathrm{.}176,5434683,2\mathrm{.}138,8125713],[3\mathrm{.}978,2581258,3\mathrm{.}048,1705494],[3,67,2\mathrm{.}187,5226133,2\mathrm{.}141,6587629],[3\mathrm{.}990,3875996,3\mathrm{.}047,2259091],[4,70,2\mathrm{.}191,3386553,2\mathrm{.}135,3463522],[3\mathrm{.}995,4784510,3\mathrm{.}042,3459012],[5,74,2\mathrm{.}105,2765190,2\mathrm{.}126,3922934],[3\mathrm{.}809,5338498,3\mathrm{.}033,5176801],[6,76,2\mathrm{.}113,1001065,2\mathrm{.}123,1888574],[3\mathrm{.}814,5973082,3\mathrm{.}025,5176871]]$

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