Question:

For the vapor liquid equilibrium $($VLE$),$ the following information is available for a binary system containing species $1$ and $2$:

$\frac{G^E}{RT}=3**x_1**x_2**e^{\frac{-280}{T}}$

$ln(P_1^{\text{p}\text{u}\text{t}})=16\mathrm{.}6-\frac{3600}{T-33}$

$ln(P_2^{\text{n}\text{a}\text{t}})=16\mathrm{.}3-\frac{3800}{T-47}$

in the above equations, $G^E$ is the excess Gibbs energy, $P_{\text{I}}^{\text{s}\text{u}\text{t}}$ is the vapor pressure of species $i$ in

kPa,$T$ is the absolute temperature in $K$ and $x_i$ is the mole fraction of species $i$ in the liquid phase.

For the above system, the modified Raoult's law:

$y_{i}P=x_i{}_i{P}_i^{\text{i}\text{a}\text{t}},(i=1,2)$

is adequate for the description of the VLE, where $y_i,x_i$ are the mole fractions of species $i$ in the

vapor and liquid phase, respectively, $P$ is the total pressure, and $$ is the activity coefficient of

species i defined as:

$ln({}_1)=\frac{G^E}{RT}+x_2\frac{\text{d}\text{e}\text{l}\frac{{}^E}{H_T}}{del{x}_1},$ and $,ln({}_2)=\frac{G^E}{RT}-x_1\frac{\text{d}\text{e}\text{l}\frac{de{l}^E}{RT}}{del{x}_1}$

Note that according to Gibbs phase rule, the above binary system has two degrees of freedom.

Also, $y_1+y_2=1,x_1+x_2=1$ and $\frac{del{x}_1}{del{x}_2}=-1.$

One can show that the following two equations are applicable for the above system:

$y_1**P=x_1{e}^{(3**(e^{\frac{-280}{T}})**(1-x_1{)}^2+(16\mathrm{.}6-\frac{3600}{T-33}))}$

$(1-y_1)**P=(1-x_1)e^{(3**(e^{\frac{-280}{T}})**(x_1{)}^2+(16\mathrm{.}3-\frac{3800}{T-47}))}$

where $y_1$ and $T$ are the unknowns and $x_1$ and $P$ are input variables.

Perform the following tasks:

$($a$)$ Derive the Jacobian matrix for the above system $($Hint: put the equations in residual

form first!$).$

$($b$)$ Using the residuals and Jacobian above, solve this problem using MATLAB, Use $P=100$

kPa and $x_1=0\mathrm{.}3.$

$($c$)$ Solve this problem for $P=100$kPa and different values of $,$

$...1\mathrm{.}0).$ Plot $T$ and $y_1$ as functions of $x_1.$