Chapter 12.7, Problem 28P

(A)

Interpretation Introduction

Interpretation:

Plot using the Raoult’s law.

Concept Introduction:

Calculate the Antoine equation for the component (1).

${\log }_{10}\left(P_i{}^{sat}\right)=A-\frac{B}{T+C}$

Write the bubble point pressure from Raoult’s law.

$P=x_1{P}_1{}^{sat}+x_2{P}_2{}^{sat}$

Write the vapor phase mole fraction.

$y_1=\frac{x_1{P}_i{}^{sat}}{P_{\text{model}}}$

(B)

Interpretation Introduction

Interpretation:

Plot using Ideal solution.

Concept Introduction:

Write the Antoine equation for the component.

${\log }_{10}\left(P_1{}^{sat}\right)=A-\frac{B}{T+C}$

Write the expression to calculate the parameter $a$ for component.

$a=\frac{27R^2{T}_c{}^2}{64P_c}$

Here, gas constant is $R$, critical temperature and critical pressure for component are $T_c$ and $P_c$ respectively.

Write the expression to calculate the parameter $b$ for component.

$b=\frac{R{T}_c}{8P_c}$

Write the expression for the van Laar parameters.

$L_{12}=\frac{b_1}{RT}{\left(\frac{\sqrt{a_1}}{b_1}-\frac{\sqrt{a_2}}{b_2}\right)}^2$

$L_{21}=\frac{b_2}{RT}{\left(\frac{\sqrt{a_1}}{b_1}-\frac{\sqrt{a_2}}{b_2}\right)}^2$

Here, temperature is T, parameters of the van Laar equation using the VDW definitions is $L_{12},L_{21}$, and equation of state parameter $a_1,b_1,a_2,\text{and}{b}_2$.

Write the activity coefficients from the van Laar model for components 1 and 2.

$\ln \left[{\gamma }_1\right]=L_{12}{\left(1+\frac{L_{12}{x}_1}{L_{21}{x}_2}\right)}^{-2}$

$\ln \left[{\gamma }_2\right]=L_{21}{\left(1+\frac{L_{21}{x}_2}{L_{12}{x}_1}\right)}^{-2}$

Here, liquid phase mole fraction for component 1 and 2 is $x_1\text{and}{x}_2$ respectively.

Write the expression to calculate the value of ${B^{\prime }}_1$.

$\begin{array}{c}{B^{\prime }}_1=\frac{\left(B^0+{\omega }_1{B}^1\right)R{T}_{c,1}}{P_{c,1}}\\ =\frac{\left[\left(0.083-\frac{0.422}{T_{r,1}^{1.6}}\right)+{\omega }_1\left(0.139-\frac{0.172}{T_{r,1}^{4.2}}\right)\right]R{T}_{c,1}}{P_{c,1}}\end{array}$

Here, reduced temperature of component 1 is $T_{r,1}$, acentric factor of component 1 is ${\omega }_1$, and second virial coefficients are $B^0\text{and}{B}^1$.

Write the expression to calculate the value of ${B^{\prime }}_2$.

$\begin{array}{c}{B^{\prime }}_2=\frac{\left(B^0+{\omega }_2{B}^1\right)R{T}_{c,2}}{P_{c,2}}\\ =\frac{\left[\left(0.083-\frac{0.422}{T_{r,2}^{1.6}}\right)+{\omega }_2\left(0.139-\frac{0.172}{T_{r,2}^{4.2}}\right)\right]R{T}_{c,2}}{P_{c,2}}\end{array}$

Here, reduced temperature of component 1 is $T_{r,1}$, acentric factor of component 1 is ${\omega }_1$, and second virial coefficients are $B^0\text{and}{B}^1$.

Write the expression to calculate the fugacity coefficient of component 1$\left({\widehat{\phi }}_{1,sat}\right)$.

$\ln \left({\widehat{\phi }}_{1,sat}\right)=\frac{P_1^{sat}{B^{\prime }}_1}{RT}$

Write the expression to calculate the fugacity coefficient of component 2$\left({\widehat{\phi }}_{2,sat}\right)$.

$\ln \left({\widehat{\phi }}_{2,sat}\right)=\frac{P_2^{sat}{B^{\prime }}_2}{RT}$

Write the expression for Raoult’s law equation.

$P=x_1{P}_1^{sat}+x_2{P}_2^{sat}$

$y_1=\frac{x_1{P}_1^{sat}}{P}$

Write the expression for the vapor definitions of the fugacity coefficients for component 1 and 2.

$\ln \left({\widehat{\phi }}_1\right)=\frac{P}{RT}\left[{B^{\prime }}_{11}+y_2^2\left(2{B^{\prime }}_{12}-{B^{\prime }}_{11}-{B^{\prime }}_{22}\right)\right]$

$\ln \left({\widehat{\phi }}_2\right)=\frac{P}{RT}\left[{B^{\prime }}_{22}+y_1^2\left(2{B^{\prime }}_{12}-{B^{\prime }}_{11}-{B^{\prime }}_{22}\right)\right]$

Here, system pressure is P, vapor phase mole fraction for components 1 and 2 is $y_1\text{and}{y}_2$, second-virial coefficient of component 1 interacting with component 2 is ${B^{\prime }}_{12}$,

second-virial coefficient of pure component 1 and 2 is ${B^{\prime }}_{11}\text{and}{B^{\prime }}_{22}$ respectively.