6. Coping with Non-Ideality SVNA 10.3

CHEE 311 J.S. Parent 1

6. Coping with Non-Ideality SVNA 10.3

Up until now, we have considered only ideal mixtures that (unfortunately) do not represent many cases experienced in practice.

Non-ideality can take two forms: Deviations in pure component behaviour e.g. pure gases at

high pressure Deviations in mixture behaviour e.g. V xi Vi

So far, our treatment of non-ideality has involved: the development of a method for describing non-ideal, single-

component, gas behaviour the extension of this treatment to the description of pure

liquids

What remains is to revise our treatment of perfect gas mixtures and ideal solutions to account for non-ideal mixing effects.


Partial Properties: Thought Experiment

Suppose we add a drop of waterto pure acetone.

What change in volume would result?

If the resulting water/acetone mixture is ideal (recall definition of ideal) the volume increase is simply that of the volume of the water droplet.

If the mixture behaves non-ideally, the volume increase will not equal the volume of the water droplet. The effect may, in fact, be quite different.

Non-ideality in mixtures results from complex intermolecular interactions that we cannot predict.

We still have to solve engineering problems (separations, property calculations, …) using these non-ideal systems.

na moles Acetone

nw moles H2O


Partial Properties: Thought Experiment

If the volume change of the acetone-water mixture does not equal the volume of the water droplet, then the properties of pure water are irrelevant.

We like to assign values or “contributions” to each component in non-ideal mixtures to account for the variation of a property with respect to composition.

This leads us to define partial molar properties, which in our thought experiment gives us the partial molar volume for water in an acetone-water solution.

This quantity represents change in solution volume as the number of moles of water is varied at a given P,T, and nacetone.

acetone2

2

n,T,POHOH n

)nV(V


Partial Molar Quantities

We prefer to think of mixtures in terms of their components: Overall property has a contribution from each component in

the mixture. In non-ideal systems, the properties of the pure components

have little meaning, forcing us to find an alternate way of defining molar quantities.

If nM represents the total thermodynamic property of interest:

(10.7)

where is a partial molar property, also a function of (T,P, nj)

A partial molar property is specific to the P,T and composition from which it is derived by equation 10.7

It is difficult to predict, but can be measured experimentally.

nj,T,Pii n

)nM(M

iM


Total Properties of Non-Ideal Mixtures

Ideal mixtures result from a lack of molecular interactions (ideal gas) or equivalent molecular interactions (ideal solution). In these cases, a total thermodynamic property (nM) for a mixture is:

nM = ni Mi where Mi represents the pure

component property of i.

Non-ideal systems do not obey this simple formula, as cross-component molecular interactions differ from pure component interactions.

nM = ni where represents the partial molar

property of component i.

In terms of mole fractions (dividing by n):

M = xi (10.11)

If we know the partial properties of the components of the mixture (from experimental data) we can derive its total property.

This is summability relation, which is opposite to 10.7 that defines a partial property

iM iM

iM


The Gibbs-Duhem Equation

An important question we need to answer is: how do the partial molar properties of a mixture relate?

Start with the definition of total Gibbs Energy at T,P:nG = ni Gi = ni i

If we change the composition of the system at constant T,P, the Gibbs energy responds accordingly:

dnG = d(ni i) = ni di + i dni

But we know that the total change in Gibbs energy is defined by:

d(nG) = nV dP - nS dT + i dni (10.3)

which at constant P,T (dP=0, dT=0), is:

d(nG) = i dni


The Gibbs-Duhem Equation

d(nG) provided by these two relations must be equal. Therefore,d(nG) = ni di + i dni

= i dni For this to be true,

ni di = 0 (10.14)

This is the Gibbs-Duhem equation applied to chemical potential

Why is it useful? It states that partial molar properties cannot change

independently, if one partial property increases, others must decrease

Estimates of partial molar properties (from experimental data, correlations…) can be checked for consistency


Calculating Partial Molar Props. - Binary Systems

Partial molar properties (Mi) can always be derived from an equation for the solution property (M) as a function of composition:

(10.7)

Comparatively simple relations exist for binary systems:(10.11)

therefore

The Gibbs-Duhem equation tells us that:(10.14)

Leaving us with:

nj,T,Pii n

)nM(M

22221111

2211

dxMMdxdxMMdxdM

MxMxM

0MdxMdx 2211

2211 dxMdxMdM


Calculating Partial Molar Props. - Binary Systems

We left off with:

Since x1+x2=1, dx1 = - dx2 and

which we can write as:

Substituting the total solution property:

(10.15, 10.16)

What we need to calculate a partial molar property is an expression for the total molar property (M) as a function of composition.

2211 dxMdxMdM

1211 dxMdxMdM

211

MMdxdM

2211 MxMxM

112

121 dx

dMxMMand

dxdM

xMM


Notation for the Course

Our superscripts and subscripts are propagating rapidly, so let’s revisit our definitions:

S = entropy of one mole of the mixture

Si = entropy of one mole of pure i

Si = entropy of one mole of pure i in the mixture= partial molar entropy

Suggested ReadingsExamples 10.1, 10.2, 10.3, 10.4 - Partial Molar Properties


6. Non-Ideal Mixtures

In our attempt to describe the Gibbs energy of real gas and liquid mixtures, we examine two “sources” of non-ideal behaviour:

Pure component non-ideality»concept of fugacity

Non-ideality in mixtures»partial molar properties»mixture fugacity and residual properties

We will begin our treatment of non-ideality in mixtures by considering gas behaviour.

Start with the perfect gas mixture model derived earlier. Modify this expression for cases where pure component non-

ideality is observed. Further modify this expression for cases in which non-ideal

mixing effects occur.


Perfect Gas Mixtures

We examined perfect gas mixtures in lecture 9. The assumptions made in developing an expression for the chemical potential of species i in a perfect gas mixture were:

all molecules have negligible volume interactions between molecules of any type are negligible.

Based on this model, the chemical potential of any component in a perfect gas mixture is:

where the reference state, Giig(T,P) is the pure component Gibbs

energy at the given P,T. We can choose a more convenient reference pressure that is

standard for all fluids, that is P=unit pressure (1 bar,1 psi,etc) In this case the pure component Gibbs energy becomes:

iigi

igi ylnRT)P,T(G)P,T(

PlnRT)T()P,T(G iigi


Perfect Gas Mixtures

Substituting for our new reference state yields:

(10.28)which is the chemical potential of component i in a perfect gas mixture at T,P.

The total Gibbs energy of the perfect gas mixture is provided by the summability relation:

(10.11)

(10.29)

PylnRT)T(

ylnRT)PlnRT)T((

ylnRT)P,T(G)P,T(

ii

ii

iigi

igi

PylnyRT)T(y

)P,T(y

)P,T(Gy)P,T(G

iiii

igii

igi

ig

i


Ideal Mixtures of Real Gases

One source of mixture non-ideality resides within the pure components. Consider an ideal solution that is composed of real gases.

In this case, we acknowledge that molecules have finite volume and interact, but assume these interactions are equivalent between components

The appropriate model is that of an ideal solution:

where Gi(T,P) is the Gibbs energy of the real pure gas:

(10.30)

Our ideal solution model applied to real gases is therefore:

iiidi ylnRT)P,T(G)P,T(

iii flnRT)T()P,T(G

iiiidi fylnRT)T()P,T(


Non-Ideal Mixtures of Real Gases

In cases where molecular interactions differ between the components (polar/non-polar mixtures) the ideal solution model does not apply

Our knowledge of pure component fugacity is of little use in predicting the mixture properties

We require experimental data or correlations pertaining to the specific mixture of interest

To cope with highly non-ideal gas mixtures, we define a solution fugacity:

(10.42)

where fi is the fugacity of species i in solution, which replaces the product yiP in the perfect gas model, and yifi of the ideal solution model.

vi

vii

f̂lnRT)T()P,T(


Non-Ideal Mixtures of Real Gases

To describe non-ideal gas mixtures, we define the solution fugacity:

and the fugacity coefficient for species i in solution:

(10.47)

In terms of the solution fugacity coefficient:

Notation:fi, i - fugacity and fugacity coefficient for pure species i

fi, i - fugacity and fugacity coefficient for species i in solution

vii

v f̂lnRT)T()P,T(i

Py

f̂ˆi

viv

i

PyˆlnRT)T()P,T( ivii

vi


Calculating iv from Compressibility Data

Consider a two-component vapour of known composition at a given pressure and temperature

If we wish to know the chemical potential of each component, we must calculate their respective fugacity coefficients

In the laboratory, we could prepare mixtures of various composition and perform PVT experiments on each.

For each mixture, the compressibility (Z) of the gas can be measured from zero pressure to the given pressure.

For each mixture, an overall fugacity coefficient can be derived at the given P,T:

How do we use this overall fugacity coefficient to derive the fugacity coefficients of each component in the mixture?

P

0

dPP

)1Z(ln


Calculating iv from Compressibility Data

It can be shown that mixture fugacity coefficients are partial molar properties of the residual Gibbs energy, and hence partial molar properties of the overall fugacity coefficient:

In terms of our measured compressiblity:

P

0 nj,P,Ti

vi P

dPn

)nnZ(ˆln

j

j

n,P,Ti

n,P,Ti

Rvi

n)lnn(

n)RT/Gn(ˆln


Calculating iv from the Virial EOS

We have used the virial equation of state to calculate the fugacity and fugacity coefficient of pure, non-polar gases at moderate pressures.

Under these conditions, it represents non-ideal PVT behaviour of pure gases quite accurately

The virial equation can be generalized to describe the calculation of mixture properties.

The truncated virial equation is the simplest alternative:

(3.31)

where B is a function of temperature and composition according to:

(10.65)Bij characterizes binary interactions between i and j; Bij=Bji

RTBP

1Z

ij

n

1

n

1ji ByyB

6. Coping with Non-Ideality SVNA 10.3

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