2. Thermodynamics of the separationThermodynamic properties play an important role in separation operations with respect to energy requirements, phase equilibria, biological activity, and equipment sizing [1]. In distillation, since liquid activity coefficient and excess properties of equilibrium state can give predictions for K-value or property changes of mixing, analyzing phase equilibrium state can be significant [2]. There are several models for the prediction of vapor-liquid equilibrium (VLE). Here, we will deal with the underlying principles of VLE and the models for the excess Gibbs energy, and show them as diagrams of ethanol-water system comparing with experimental data.

2.1 Underlying principles

The definition of the activity coefficient is , where is the activity coefficient of

species i, is the fugacity of species i in solution, is the fugacity of pure species i , and

is liquid mole fraction of i in solution.

By Lewis/Randall rule, this equation converts into where is the

fugacity of species i in ideal solution [2].

At VLE, the Gibbs free energy of the two phases should be equal, which means the fugacity

of the two phases also should be the same.

By the definition of fugacity : . VLE measurements are very often made at

pressure low enough that the vapor phase may be assumed an ideal gas. In this case, =1,

leads the equation to [2].

With these principles and assumptions, the activity coefficient is re-defined as

, and this is modified Raoult’s law allowing calculation of activity coefficients

from experimental low-pressure VLE data [2].

Activity coefficient is also partial molar property of solution, which gives

(for binary system). By summability relation :

. With this equation, the models for the excess Gibbs energy suggest

different relation between liquid composition and activity coefficient, which gives the derived

liquid activity coefficient of the system. Vapor composition, pressure, and temperature also

can be derived from the activity coefficient and modified Raoult’s law [2].

Furthermore, the definition of K-value for modified Raoult’s law, , can give

predictions for K-value from the derived activity coefficient [1].

For the analyzing of ethanol/water equilibrium state, the experimental data is from

KDB(Korea Thermophysical Properties Data Bank) of CHERIC(Chemical Engineering

Research Information Center). The information is isobaric T-x-y data at 760 mmHg.

T, deg.C P, mmHg x y100.0 760 0 099.3 760 0.0028 0.03296.9 760 0.0118 0.11396.0 760 0.0137 0.15796.0 760 0.0144 0.13595.6 760 0.0176 0.15694.8 760 0.0222 0.18693.8 760 0.0246 0.21293.5 760 0.0302 0.23192.9 760 0.0331 0.24890.5 760 0.0519 0.31890.5 760 0.0530 0.31489.4 760 0.0625 0.33988.4 760 0.0673 0.37088.6 760 0.0715 0.36287.2 760 0.0871 0.40685.4 760 0.126 0.46884.5 760 0.143 0.48784.0 760 0.172 0.50583.4 760 0.206 0.53083.0 760 0.210 0.52782.3 760 0.255 0.55282.0 760 0.284 0.56781.4 760 0.321 0.58681.5 760 0.324 0.58381.2 760 0.345 0.59180.9 760 0.405 0.61480.5 760 0.430 0.62880.2 760 0.449 0.63380.0 760 0.506 0.661

79.5 760 0.545 0.67378.8 760 0.663 0.73378.5 760 0.735 0.77678.4 760 0.804 0.81578.3 760 0.917 0.90678.3 760 1.000 1.000

For the vapor pressure of the components, the equation,

[3], is used.

For ethanol, C1=73.304, C2=-7122.3, C3=-7.1424, C4=2.8853*10^(-6), C5=2.For water C1=73.649, C2=-7258.2, C3=-7.3037, C4=4.1653*10^(-6), C5=2 [3].

2.2 Models for the excess Gibbs energy

2.2.1 Margules equation

In Margules equation, it assumes is a linear function of .

Activity coefficient is derived by the differentiation of the equation with respect to .

[2].

2.2.2 Van Laar equation

This equation results when the reciprocal expression is expressed as a linear

function of .

New parameters are defined by the equations, , , and from

these definitions,

[2]

2.2.3 Local composition Models

The prediction of the molecular thermodynamics of liquid-solution behavior is often based

on the concept of local composition. In this concept, within a liquid solution, local

compositions, different from the overall mixture composition, are presumed to account for

the short-range order and nonrandom molecular orientations that result from different in

molecular size and intermolecular forces [2].

2.2.3.1 Wilson equation

The Wilson equation contains two parameters like the Margules and van Laar equations.

Here, ,

and are the molar volumes at temperature T of pure liquid.

and are constants independent of composition and temperature.

Activity coefficient is also derived by differentiation of .

[2]

For ethanol/water binary system, =339.1160(J/mol), and =4450.75932(J/mol) [4].

2.2.3.2 NRTL(Non-Random-Two-Liquid) equation

The NRTL equation has three independent parameters for binary system.

, , and are constants that are dependent on particular pair of species, and independent

of composition and temperature.

[2]

For ethanol/water binary system, =0.0417, =9868.83912(J/mol), and

=15937.56528(J/mol) [4].

2.2.3.3 UNIFAC(Universal Quasi-Chemical Functional-group Activity Coefficients)

method

In UNIFAC method, activity coefficient is expressed as the summation of two parts. The

first part is combinatorial part which contains pure-species parameters only. The second part

is residual part to account for molecular interactions. This method considers interactions

between the functional groups(subgroups) of the molecules in solution [2].

Here,

, , , , , ,

, ,

: species

: dummy index

: subgroup

: dummy index

: the number of subgroup k that belongs to species i

, , and are constants that can be found in references [2].

For ethanol/water binary system, (A : Ethanol, B : Water)

Main group Subgroup

1 "CH2"CH3 1 0.9011 0.848 1 0

CH2 2 0.6744 0.540 1 0

5 "OH" OH 15 1.0000 1.200 1 0

7 "H20" H20 17 0.9200 1.400 0 1

= = = = = = 0 K

=986.5 K, =986.5 K, =1318.00 K, =1318.00 K

=156.40 K, =156.40 K, =300.00 K, =300.00 K

=353.50 K, =-229.10 K [2].

, , ,

, , ,

, , ,

, , ,

,

,

, , ,

, , ,

,

,

,

,

,

,

,

,

, , ,

[1] Seader J.D., Henley E.J., and Roper D.K., “Separation Process Principles”. John Wiley &

Sons, 3rd Edition, 2011.

[2] Smith J.M., Van Ness H.C., and Abbott M.M., “Introduction to Chemical Engineering

Thermodynamics”. McGrawHill, 7th edition, 2005.

[3] Perry R.H., and Green D.W., “Perry's Chemical Engineer's Handbook”. McGrawHill, 8th

edition, 2008

[4] Gmehling J., Onken U., and Arlt W. “Vapor-Liquid Equilibrium Data Collection”.

Chemistry Data Series, Volume I, parts 1-8, 1974-1990

Appendix – Matlab Coding

- Margules equation

clear all; clc; % Margulest=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4 88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2 80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHGT=t+273.15;P=101.33; % kPax1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033 0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210 0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735 0.804 0.917 0.999];y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318 0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567 0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906 0.999];x2=1-x1;y2=1-y1; C1=73.304;C2=-7122.3;C3=-7.1424;C4=2.8853*10^(-6);C5=2;P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000; C1=73.649;C2=-7258.2;C3=-7.3037;C4=4.1653*10^(-6);C5=2;P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;R=8.314; % ---- exp ---- gamma1_e=y1.*P./(x1.*P1_sat);gamma2_e=y2.*P./(x2.*P2_sat);GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e)); % ---- derived ---- for i=1:36 if i>4 && i<36 func(i-4)=GE_e(i)./(R.*T(i).*x1(i).*x2(i)); x_a(i-4)=x1(i); end endx=0:0.01:1;c=polyfit(x_a,func,1);y=c(2)+c(1).*x; A12=y(1);A21=y(101); gamma1_d=exp((1-x1).^2.*(A12+2*(A21-A12).*x1));gamma2_d=exp((x1.^2.*(A21+2*(A12-A21).*(1-x1))));y_d=x1.*gamma1_d.*P1_sat./P;K_e=y1./x1;K_d=y_d./x1;

y=x; figure(1)plot(x1,y1,'r^');hold onplot(x1,y_d, 'b');hold onplot(x, y, 'k')hold onlegend('Exp. Data', 'Margules equation', '45¢ª line', 0);title('x-y diagram for ethanol/water VLE');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('Vapor-phase mole fraction of ethanol(y1)');grid on figure(2)plot(x1, gamma1_e, 'r^');hold onplot(x1, gamma2_e, 'bo');hold onplot(x1, gamma1_d, 'r');hold onplot(x1, gamma2_d, 'b');hold onlegend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol by Margules equation', '¥ã2-water by Margules equation', 0);title('Liquid-phase activity coefficients for ethanol/water system at 1 atm');xlabel('Liquid phase mole fraction of ethanol(x1)');ylabel('Liquid-phase activity coefficient');axis([0 1 0 7]);grid on figure(3)plot(x1, K_e, 'ro');hold onplot(x1, K_d, 'bo');hold onlegend('K-value by exp. data', 'K-value by Margules equation', 0);title('K-value of ethanol/water system at 1 atm');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('K-value');axis([0 1 0 17]);grid on

- van Laar equationclear all; clc; % Van Laart=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4 88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2 80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHGT=t+273.15;P=101.33; % kPax1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033 0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210 0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735 0.804 0.917 0.999];y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318 0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567 0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906

0.999];x2=1-x1;y2=1-y1; C1=73.304;C2=-7122.3;C3=-7.1424;C4=2.8853*10^(-6);C5=2;P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000; C1=73.649;C2=-7258.2;C3=-7.3037;C4=4.1653*10^(-6);C5=2;P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;R=8.314; % ---- exp ---- gamma1_e=y1.*P./(x1.*P1_sat);gamma2_e=y2.*P./(x2.*P2_sat);GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e)); % ---- derived ---- x=0:0.01:1; for i=1:36 if i>1 && i<36 func(i-1)=x1(i).*x2(i).*R.*T(i)./GE_e(i); x_a(i-1)=x1(i); endend c=polyfit(x_a,func,1);y=c(2)+c(1).*x;A12=1/y(1);A21=1/y(101); GE_d=R*T.*x1.*(1-x1).*(A12*A21./(A12.*x1+A21.*(1-x1)));gamma1_d=exp(A12.*(1+(A12.*x1)./(A21.*(1-x1))).^(-2));gamma2_d=exp(A21.*(1+(A21.*(1-x1))./(A12.*x1)).^(-2));y_d=x1.*gamma1_d.*P1_sat./P;K_e=y1./x1;K_d=y_d./x1;y=x; figure(1)plot(x1,y1,'r^');hold onplot(x1,y_d, 'b');hold onplot(x, y, 'k')hold onlegend('Exp. Data', 'Van Laar equation', '45¢ª line', 0);title('x-y diagram for ethanol/water VLE');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('Vapor-phase mole fraction of ethanol(y1)');

grid on figure(2)plot(x1, gamma1_e, 'r^');hold onplot(x1, gamma2_e, 'bo');hold onplot(x1, gamma1_d, 'r');hold onplot(x1, gamma2_d, 'b');hold onlegend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol by Van Laar equation', '¥ã2-water by Van Laar equation', 0);title('Liquid-phase activity coefficients for ethanol/water system at 1 atm');xlabel('Liquid phase mole fraction of ethanol(x1)');ylabel('Liquid-phase activity coefficient');axis([0 1 0 7]);grid on figure(3)plot(x1, K_e, 'ro');hold onplot(x1, K_d, 'bo');hold onlegend('K-value by exp. data', 'K-value by Van Laar equation', 0);title('K-value of ethanol/water system at 1 atm');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('K-value');axis([0 1 0 17]);grid on

- Wilson equationclear all; clc; % Wilsont=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4 88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2 80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHGT=t+273.15;P=101.33; % kPax1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033 0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210 0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735 0.804 0.917 0.999];y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318 0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567 0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906 0.999];x2=1-x1;y2=1-y1; C1=73.304;C2=-7122.3;C3=-7.1424;C4=2.8853*10^(-6);C5=2;P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000; C1=73.649;C2=-7258.2;

C3=-7.3037;C4=4.1653*10^(-6);C5=2;P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;R=8.314; % ---- exp ---- gamma1_e=y1.*P./(x1.*P1_sat);gamma2_e=y2.*P./(x2.*P2_sat);GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e)); % ---- derived ---- x=0:0.01:1;a12=80.7419*4.2;a21=1059.7046*4.2;V1=58.68;V2=18.0;lambda12=V2./V1.*exp(-a12./(R.*T));lambda21=V1./V2.*exp(-a21./(R.*T)); GE_d=R.*T.*(-x1.*log(x1+(1-x1).*lambda12)-(1-x1).*log((1-x1)+x1.*lambda21));gamma1_d=exp(-log(x1+(1-x1).*lambda12)+(1-x1).*(lambda12./(x1+(1-x1).*lambda12)-lambda21./((1-x1)+x1.*lambda21)));gamma2_d=exp(-log((1-x1)+x1.*lambda21)-x1.*(lambda12./(x1+(1-x1).*lambda12)-lambda21./((1-x1)+x1.*lambda21)));y_d=x1.*gamma1_d.*P1_sat./P;K_e=y1./x1;K_d=y_d./x1;y=x; figure(1)plot(x1,y1,'r^');hold onplot(x1,y_d, 'b');hold onplot(x, y, 'k')hold onlegend('Exp. Data', 'Wilson equation', '45¢ª line', 0);title('x-y diagram for ethanol/water VLE');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('Vapor-phase mole fraction of ethanol(y1)');grid on figure(2)plot(x1, gamma1_e, 'r^');hold onplot(x1, gamma2_e, 'bo');hold onplot(x1, gamma1_d, 'r');hold onplot(x1, gamma2_d, 'b');hold onlegend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol by Wilson equation', '¥ã2-water by Wilson equation', 0);title('Liquid-phase activity coefficients for ethanol/water system at 1 atm');xlabel('Liquid phase mole fraction of ethanol(x1)');

ylabel('Liquid-phase activity coefficient');axis([0 1 0 7]);grid on figure(3)plot(x1, K_e, 'ro');hold onplot(x1, K_d, 'bo');hold onlegend('K-value by exp. data', 'K-value by Wilson equation', 0);title('K-value of ethanol/water system at 1 atm');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('K-value');axis([0 1 0 17]);grid on

- NRTL equationclear all; clc; % NRTLt=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4 88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2 80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHGT=t+273.15;P=101.33; % kPax1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033 0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210 0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735 0.804 0.917 0.999];y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318 0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567 0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906 0.999];x2=1-x1;y2=1-y1; C1=73.304;C2=-7122.3;C3=-7.1424;C4=2.8853*10^(-6);C5=2;P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000; C1=73.649;C2=-7258.2;C3=-7.3037;C4=4.1653*10^(-6);C5=2;P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;R=8.314; % ---- exp ---- gamma1_e=y1.*P./(x1.*P1_sat);gamma2_e=y2.*P./(x2.*P2_sat);GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e)); % ---- derived ---- x=0:0.01:1;

b12=-2349.7236*4.2;b21=3794.6584*4.2;alpha=0.0417;tau12=b12./(R.*T);tau21=b21./(R.*T);G12=exp(-alpha.*tau12);G21=exp(-alpha.*tau21); GE_d=R.*T.*x1.*(1-x1).*((G21.*tau21)./(x1+(1-x1).*G21)+((G12.*tau12)./((1-x1)+x1.*G12)));gamma1_d=exp((1-x1).^(2).*(tau21.*(G21./(x1+(1-x1).*G21)).^(2)+G12.*tau12./(((1-x1)+x1.*G12).^(2))));gamma2_d=exp(x1.^(2).*(tau12.*(G12./((1-x1)+x1.*G12)).^(2)+G21.*tau21./((x1+(1-x1).*G21).^(2))));y_d=x1.*gamma1_d.*P1_sat./P;K_e=y1./x1;K_d=y_d./x1;y=x; figure(1)plot(x1,y1,'r^');hold onplot(x1,y_d, 'b');hold onplot(x, y, 'k')hold onlegend('Exp. Data', 'NRTL equation', '45¢ª line', 0);title('x-y diagram for ethanol/water VLE');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('Vapor-phase mole fraction of ethanol(y1)');grid on figure(2)plot(x1, gamma1_e, 'r^');hold onplot(x1, gamma2_e, 'bo');hold onplot(x1, gamma1_d, 'r');hold onplot(x1, gamma2_d, 'b');hold onlegend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol by NRTL equation', '¥ã2-water by NRTL equation', 0);title('Liquid-phase activity coefficients for ethanol/water system at 1 atm');xlabel('Liquid phase mole fraction of ethanol(x1)');ylabel('Liquid-phase activity coefficient');axis([0 1 0 7]);grid on figure(3)plot(x1, K_e, 'ro');hold onplot(x1, K_d, 'bo');hold onlegend('K-value by exp. data', 'K-value by NRTL equation', 0);title('K-value of ethanol/water system at 1 atm');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('K-value');axis([0 1 0 17]);grid on

- UNIFAC methodclear all; clc; % UNIFACt=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4 88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2 80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHGT=t+273.15;P=101.33; % kPax1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033 0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210 0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735 0.804 0.917 0.999];y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318 0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567 0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906 0.999];x2=1-x1;y2=1-y1; C1=73.304;C2=-7122.3;C3=-7.1424;C4=2.8853*10^(-6);C5=2;P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000; C1=73.649;C2=-7258.2;C3=-7.3037;C4=4.1653*10^(-6);C5=2;P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;R=8.314; % ---- exp ---- gamma1_e=y1.*P./(x1.*P1_sat);gamma2_e=y2.*P./(x2.*P2_sat);GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e)); % ---- derived ---- % species A = ethanol, species B = water % subgroup 1 = CH3, subgroup 2 = CH2, subgroup 15 = OH, subgroup 17 = H20 x=0:0.01:1; R1=0.9011;R2=0.6744;R15=1.0000;R17=0.9200; Q1=0.848;Q2=0.540;Q15=1.200;Q17=1.400; V1A=1;

V2A=1;V15A=1;V17A=0; V1B=0;V2B=0;V15B=0;V17B=1; a_1_2=0;a_2_1=0;a_1_1=0;a_2_2=0;a_15_15=0;a_17_17=0;a_1_15=986.5;a_2_15=986.5;a_1_17=1318.00;a_2_17=1318.00;a_15_1=156.40;a_15_2=156.40;a_17_1=300.00;a_17_2=300.00;a_15_17=353.50;a_17_15=-229.10; t_1_2=1;t_2_1=1;t_1_1=1;t_2_2=1;t_15_15=1;t_17_17=1;t_1_15=exp(-a_1_15./T);t_2_15=exp(-a_2_15./T);t_1_17=exp(-a_1_17./T);t_2_17=exp(-a_2_17./T);t_15_1=exp(-a_15_1./T);t_15_2=exp(-a_15_2./T);t_17_1=exp(-a_17_1./T);t_17_2=exp(-a_17_2./T);t_15_17=exp(-a_15_17./T);t_17_15=exp(-a_17_15./T); rA=V1A*R1+V2A*R2+V15A*R15+V17A*R17;rB=V1B*R1+V2B*R2+V15B*R15+V17B*R17; qA=V1A*Q1+V2A*Q2+V15A*Q15+V17A*Q17;qB=V1B*Q1+V2B*Q2+V15B*Q15+V17B*Q17; e1A=V1A*Q1/qA;e2A=V2A*Q2/qA;e15A=V15A*Q15/qA;e17A=V17A*Q17/qA;e1B=V1B*Q1/qB;e2B=V2B*Q2/qB;e15B=V15B*Q15/qB;e17B=V17B*Q17/qB; bA1=e1A.*t_1_1 + e2A.*t_2_1 + e15A.*t_15_1 + e17A.*t_17_1;bA2=e1A.*t_1_2 + e2A.*t_2_2 + e15A.*t_15_2 + e17A.*t_17_2;bA15=e1A.*t_1_15 + e2A.*t_2_15 + e15A.*t_15_15 + e17A.*t_17_15;

bA17=e1A.*t_1_17 + e2A.*t_2_17 + e15A.*t_15_17 + e17A.*t_17_17;bB1=e1B.*t_1_1 + e2B.*t_2_1 + e15B.*t_15_1 + e17B.*t_17_1;bB2=e1B.*t_1_2 + e2B.*t_2_2 + e15B.*t_15_2 + e17B.*t_17_2;bB15=e1B.*t_1_15 + e2B.*t_2_15 + e15B.*t_15_15 + e17B.*t_17_15;bB17=e1B.*t_1_17 + e2B.*t_2_17 + e15B.*t_15_17 + e17B.*t_17_17; c1=(x1.*qA.*e1A + (1-x1).*qB.*e1B)./(x1.*qA + (1-x1).*qB);c2=(x1.*qA.*e2A + (1-x1).*qB.*e2B)./(x1.*qA + (1-x1).*qB);c15=(x1.*qA.*e15A + (1-x1).*qB.*e15B)./(x1.*qA + (1-x1).*qB);c17=(x1.*qA.*e17A + (1-x1).*qB.*e17B)./(x1.*qA + (1-x1).*qB); s1=c1.*t_1_1 + c2.*t_2_1 + c15.*t_15_1 + c17.*t_17_1;s2=c1.*t_1_2 + c2.*t_2_2 + c15.*t_15_2 + c17.*t_17_2;s15=c1.*t_1_15 + c2.*t_2_15 + c15.*t_15_15 + c17.*t_17_15;s17=c1.*t_1_17 + c2.*t_2_17 + c15.*t_15_17 + c17.*t_17_17; JA=rA./(rA.*x1+rB.*(1-x1));JB=rB./(rA.*x1+rB.*(1-x1)); LA=qA./(qA.*x1+qB.*(1-x1));LB=qB./(qA.*x1+qB.*(1-x1)); gammaA_C=exp(1-JA+log(JA)-5.*qA.*(1-JA./LA+log(JA./LA)));gammaB_C=exp(1-JB+log(JB)-5.*qB.*(1-JB./LB+log(JB./LB))); gammaA_R=exp(qA.*(1-(c1.*bA1./s1 + c2.*bA2./s2 + c15.*bA15./s15 + c17.*bA17./s17 - e1A.*log(bA1./s1) - e2A.*log(bA2./s2) - e15A.*log(bA15./s15) - e17A.*log(bA17./s17))));gammaB_R=exp(qB.*(1-(c1.*bB1./s1 + c2.*bB2./s2 + c15.*bB15./s15 + c17.*bB17./s17 - e1B.*log(bB1./s1) - e2B.*log(bB2./s2) - e15B.*log(bB15./s15) - e17B.*log(bB17./s17)))); gamma1_d=gammaA_C.*gammaA_R;gamma2_d=gammaB_C.*gammaB_R; GE_d=R.*T.*(x1.*log(gamma1_d)+(1-x1).*log(gamma2_d));y_d=x1.*gamma1_d.*P1_sat./P;K_e=y1./x1;K_d=y_d./x1;y=x; figure(1)plot(x1,y1,'r^');hold onplot(x1,y_d, 'b');hold onplot(x, y, 'k')hold onlegend('Exp. Data', 'UNIFAC method equation', '45¢ª line', 0);title('x-y diagram for ethanol/water VLE');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('Vapor-phase mole fraction of ethanol(y1)');grid on figure(2)plot(x1, gamma1_e, 'r^');hold onplot(x1, gamma2_e, 'bo');hold onplot(x1, gamma1_d, 'r');hold on

plot(x1, gamma2_d, 'b');hold onlegend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol by UNIFAC method', '¥ã2-water by UNIFAC method', 0);title('Liquid-phase activity coefficients for ethanol/water system at 1 atm');xlabel('Liquid phase mole fraction of ethanol(x1)');ylabel('Liquid-phase activity coefficient');axis([0 1 0 7]);grid on figure(3)plot(x1, K_e, 'ro');hold onplot(x1, K_d, 'bo');hold onlegend('K-value by exp. data', 'K-value by UNIFAC method', 0);title('K-value of ethanol/water system at 1 atm');xlabel('Liquid-phase mole fraction of ethanol(x1)');ylabel('K-value');axis([0 1 0 17]);grid on