7. liquid phase properties from vle data (11.1)

CHEE 311 J.S. Parent 1

7. Liquid Phase Properties from VLE Data (11.1)

The fugacity of non-ideal liquid solutions is defined as:

(10.42)from which we derive the concept of an activity coefficient:

(10.89)

that is a measure of the departure of the component behaviour from an ideal solution.

Using the activity coefficient, equation 10.42 becomes:

How do we calculate/measure these properties?

lii

li f̂lnRT)T()P,T(

lii

li

i fx

f̂

liiii

li fxlnRT)T()P,T(


Liquid Phase Properties from VLE Data

Suppose we conduct VLE experiments on our system of interest. At a given temperature, we vary the system pressure by

changing the cell volume. Wait until equilibrium is established (usually hours) Measure the compositions of the liquid and vapour


Liquid Solution Fugacity from VLE Data

Our understanding of molecular dynamics does not permit us to predict non-ideal solution fugacity, fi

l . We must measure them by experiment, often by studies of vapour-liquid equilibria.

Suppose we need liquid solution fugacity data for a binary mixture of A+B at P,T. At equilibrium,

The vapour mixture fugacity for component i is given by,(10.47)

If we conduct VLE experiments at low pressure, but at the required temperature, we can use the perfect gas mixture model,

by assuming that iv = 1.

Pyˆf̂ ivi

vi

Pyf̂ ivi

vi

li f̂f̂


Liquid Solution Fugacity from Low P VLE Data

Since our experimental measurements are taken at equilibrium,

according to the perfect gas mixture model

What we need is VLE data at various pressures (all relatively low)

Py

f̂f̂

i

vi

li


Activity Coefficients from Low P VLE Data

With a knowledge of the liquid solution fugacity, we can derive activity coefficients. Actual fugacity

Ideal solution fugacityOur low pressure vapour fugacity simplifies fi

l to:

and if P is close to Pisat:

leaving us with

sati

sati

lsati

sati

li

P

RT

)PP(VexpPf

i

lii

li

i fx

f̂

lii

ii fx

Py

satii

ii Px

Py



Our low pressure VLE data can now be processed to yield experimental activity coefficient data:

satii

ii Px

Py


7. Correlation of Liquid Phase Data

The complexity of molecular interactions in non-ideal systems makes prediction of liquid phase properties very difficult.

Experimentation on the system of interest at the conditions (P,T,composition) of interest is needed.

Previously, we discussed the use of low-pressure VLE data for the calculation of liquid phase activity coefficients.

As practicing engineers, you will rarely have the time to conduct your own experiments.

You must rely on correlations of data developed by other researchers.

These correlations are empirical models (with limited fundamental basis) that reduce experimental data to a mathematical equation.

In CHEE 311, we examine BOTH the development of empirical models (thermodynamicists) and their applications (engineering practice).


Correlation of Liquid Phase Data

Recall our development of activity coefficients on the basis of the partial excess Gibbs energy :

where the partial molar Gibbs energy of the non-ideal model is provided by equation 10.42:

and the ideal solution chemical potential is:

Leaving us with the partial excess Gibbs energy:

(10.90)

idii

Ei GGG

li

li ii

f̂lnRT)T(G

liii

idi

idi fxlnRT)T(G

i

lii

li

lii

li

Ei

lnRT

fx

f̂lnRT

fxlnRTf̂lnRTG



The partial excess Gibbs energy is defined by:

In terms of the activity coefficient,

(10.94)

Therefore, if as practicing engineers we have GE as a function of P,T, xn (usually in the form of a model equation) we can derive i.

Conversely, if thermodynamicists measure i, they can calculate GE using the summability relationship for partial properties.

(10.97)

With this information, they can generate model equations that practicing engineers apply routinely.

nj,P,Ti

E

i n)RT/nG(

ln

i

ii

E

lnxRTG

nj,P,Ti

EEi

n)nG(

G



We can now process this our MEK/toluene data one step further to give the excess Gibbs energy,

GE/RT = x1ln1 + x2ln2



Note that GE/(RTx1x2) is reasonably represented by a linear function of x1 for this system. This is the foundation for correlating experimental activity coefficient data

sat11

11 Px

Pylnln

sat22

22 Px

Pylnln

2211E lnxlnxRT/G



The chloroform/1,4-dioxane system exhibits a negative deviation from Raoult’s Law.

This low pressure VLE data can be processed in the same manner as the MEK/toluene system to yield both activity coefficients and the excess Gibbs energy of the overall system.



Note that in this example, the activity coefficients are less than one, and the excess Gibbs energy is negative.

In spite of the obvious difference from the MEK/toluene system behaviour, the plot of GE/x1x2RT is well approximated by a line.


8.4 Models for the Excess Gibbs Energy

Models that represent the excess Gibbs energy have several purposes:

they reduce experimental data down to a few parameters they facilitate computerized calculation of liquid phase

properties by providing equations from tabulated data In some cases, we can use binary data (A-B, A-C, B-C) to

calculate the properties of multi-component mixtures (A,B,C)

A series of GE equations is derived from the Redlich/Kister expansion:

Equations of this form “fit” excess Gibbs energy data quite well. However, they are empirical and cannot be generalized for multi-component (3+) mixtures or temperature.

)Tttancons()xx(D)xx(CBxRTx

G 22121

21

E


Symmetric Equation for Binary Mixtures

The simplest Redlich/Kister expansion results from C=D=…=0

To calculate activity coefficients, we express GE in terms of moles: n1 and n2.

And through differentiation,

we find:

BxRTx

G

21

E

221

21E

)nn(

nnBRTnG

2n,P,T1

E

1 n)RT/nG(

ln

212

221 BxlnandBxln


7. Excess Gibbs Energy Models

Practicing engineers find most of the liquid-phase information needed for equilibrium calculations in the form of excess Gibbs Energy models. These models:

reduce vast quantities of experimental data into a few empirical parameters,

provide information an equation format that can be used in thermodynamic simulation packages (Provision)

“Simple” empirical models Symmetric, Margule’s, vanLaar No fundamental basis but easy to use Parameters apply to a given temperature, and the models

usually cannot be extended beyond binary systems.

Local composition models Wilsons, NRTL, Uniquac Some fundamental basis Parameters are temperature dependent, and multi-

component behaviour can be predicted from binary data.


Excess Gibbs Energy Models

Our objectives are to learn how to fit Excess Gibbs Energy models to experimental data, and to learn how to use these models to calculate activity coefficients.

sat11

11 Px

Pylnln

sat22

22 Px

Pylnln

2211E lnxlnxRT/G


Margule’s Equations

While the simplest Redlich/Kister-type expansion is the Symmetric Equation, a more accurate model is the Margule’s expression:

(11.7a)

Note that as x1 goes to zero,

and from L’hopital’s rule we know:

therefore,

and similarly

21212121

E

xAxAxRTx

G

1

210xln

xRTxG

lim

E

1

12

0x21

E

AxRTx

G

1

112 lnA 221 lnA


Margule’s Equations

If you have Margule’s parameters, the activity coefficients are easily derived from the excess Gibbs energy expression:

(11.7a)

to yield:

(11.8ab)

These empirical equations are widely used to describe binary solutions. A knowledge of A12 and A21 at the given T is all we require to calculate activity coefficients for a given solution composition.

21212121

E

xAxAxRTx

G

]x)AA(2A[xln 1122112221

]x)AA(2A[xln 2211221212


van Laar Equations

Another two-parameter excess Gibbs energy model is developed from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end results are:

(11.13)for the excess Gibbs energy and:

(11.14)

(11.15)

for the activity coefficients.

Note that: as x10, ln1 A’12

and as x2 0, ln2 A’21

2/121

/21

/21

/12

21

E

xAxA

AAxRTx

G

2

2/21

1/12/

121xA

xA1Aln

2

1/12

2/21/

212xA

xA1Aln


Local Composition Models

Unfortunately, the previous approach cannot be extended to systems of 3 or more components. For these cases, local composition models are used to represent multi-component systems.

Wilson’s Theory Non-Random-Two-Liquid Theory (NRTL) Universal Quasichemical Theory (Uniquac)

While more complex, these models have two advantages: the model parameters are temperature dependent the activity coefficients of species in multi-component liquids

can be calculated from binary data.

A,B,C A,B A,C B,C

tertiary mixture binary binary binary


Wilson’s Equations for Binary Solution Activity

A versatile and reasonably accurate model of excess Gibbs Energy was developed by Wilson in 1964. For a binary system, GE is provided by:

(11.16)

where(11.24)

Vi is the molar volume at T of the pure component i.aij is determined from experimental data.

The notation varies greatly between publications. This includes, a12 = (12 - 11), a12 = (21 - 22) that you will encounter in

Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62, 21-21.

)xxln(x)xxln(xRTG

2112212211

E

RTa

expVV

RTa

expVV 21

2

121

12

1

212


Wilson’s Equations for Binary Solution Activity

Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property:

When applied to equation 11.16, we obtain:

(11.17)

(11.18)

2112

21

1221

12212211 xxxx

x)xxln(ln

2112

21

1221

12121122 xxxx

x)xxln(ln

jn,P,Ti

EEii n

nGGlnRT


Wilson’s Equations for Multi-Component Mixtures

The strength of Wilson’s approach resides in its ability to describe multi-component (3+) mixtures using binary data.

Experimental data of the mixture of interest (ie. acetone, ethanol, benzene) is not required

We only need data (or parameters) for acetone-ethanol, acetone-benzene and ethanol-benzene mixtures

The excess Gibbs energy is written:

(11.22)

and the activity coefficients become:

(11.23)

where ij = 1 for i=j. Summations are over all species.

i j

ijji

E

xlnxRTG

k

jkjj

kik

iijji x

xxln1ln


Wilson’s Equations for 3-Component Mixtures

For three component systems, activity coefficients can be calculated from the following relationship:

Model coefficients are defined as (ij = 1 for i=j):

3322311

i33

2332211

i22

1331221

i113i32i21i1i

xxx

x

xxx

x

xxx

x)xxxln(1ln

RT

aexp

V

V ij

i

jij


Comparison of Liquid Solution Models

Activity coefficients of 2-methyl-2-butene + n-methylpyrollidone.

Comparison of experimental values with those obtained from several equations whose parameters are found from the infinite-dilution activity coefficients. (1) Experimental data. (2) Margules equation. (3) van Laar equation. (4) Scatchard-Hamer equation. (5) Wilson equation.

7. liquid phase properties from vle data (11.1)

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