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Discussion around the paradigm of ideal mixtures with emphasison the denition of the property changes on mixingRomain Privat, Jean-No el Jaubert n

Universite de Lorraine, E cole Nationale Supe rieure des Industries Chimiques, Laboratoire Re actions et Ge nie des Proce desUPR CNRS 3349, 1 rue Grandville, BP 20451,Nancy cedex 9, France

H I G H L I G H T S

c

Analysis and comparison of the various concepts of ideal mixtures.c The mixing enthalpy can be different from the excess enthalpy.c Volume change can be observed on the formation of an ideal mixture.c Correct expressions of the ideal-mixture property change on mixing.c Usefulness of the double-tangent construction when the gamma-phi approach is used.

a r t i c l e i n f o

Article history:Received 20 January 2012Received in revised form15 July 2012Accepted 21 July 2012Available online 2 August 2012

Keywords:Ideal mixtureIdeal solutionIdeal gaseous mixtureIdeal-mixture property changes on mixingDouble-tangent construction of coexistingphasesRaoults law

a b s t r a c t

In this paper, the various concepts of ideal mixtures found in the open literature are analyzed andcompared. It is in particular shown that expressions conventionally admitted for the ideal-mixtureproperty changes on mixing are incorrect when the ideal mixture is obtained from pure componentswhich are not in the same aggregation state as the mixture. Among these properties, a special attentionis devoted to the ideal-mixture Gibbs energy change on mixing since it is used to determine theequilibrium conditions. After explaining how to properly estimate this quantity, it is shown how to

correctly perform the double-tangent construction of coexisting phases for binary systems modeledwith the gamma-phi approach.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of ideal mixture is widely accepted in chemicalengineering thermodynamics. The three following proposalshowever show that some points still need to be claried.

Statement 1. About a century ago, the French chemist andphysicist Franc - ois Marie Raoult (18301901) published a simplerelation for vaporliquid equilibrium (VLE):

P U yi P sat i T U xi 1

generally known as Raoults law. P is the pressure, xi and yi arerespectively the mole fractions of component i in the liquid and

gas phases and P sat i T is the vapor pressure of pure i at tempera-ture T . It was experimentally found that this relation could beapplied under moderate pressure for very similar molecules, e.g.for the systems toluene ethyl benzene, propane n-butane,ethanol propan-1-ol. Raoults law is thus the VLE conditionwhen the gas phase can be considered as a perfect-gas mixtureand when the liquid phase can be regarded as an ideal solution.

Statement 2. Gibbs (1873) demonstrated that by plotting atconstant temperature and pressure the molar Gibbs energy ( g )of a binary system vs. the molar composition, the existence of amultiple tangent (typically a double tangent) was a necessary andsufcient condition so that a stable multiphase (typically a two-phase) equilibrium exists at this temperature and pressure.As highlighted in the second part of this paper, the sameconclusion can be drawn by plotting the Gibbs energy changeon mixing ( g m ). This theorem, usually called the double-tangent

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/ces

Chemical Engineering Science

0009-2509/$- see front matter & 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ces.2012.07.030

n Corresponding author. Tel.: 33 3 83 17 50 81; fax: 33 3 83 17 51 52.E-mail address: [email protected] (J.-N. Jaubert) .

Chemical Engineering Science 82 (2012) 319333
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construction of coexisting phases , established by Gibbs and used byVan der Waals in his thesis can obviously not be questioned.

Statement 3. All well-known textbooks of thermodynamics statethat the Gibbs energy change on mixing for an ideal liquid binarymixture ( idl) and for a perfect-gas binary mixture ( pgm) arerespectively given by:

g m, idl T , x RT X

2

i 1 xiUln xi 2

g m , pgm T , y RT X2

i 1 yiUln yi 3

Eq. (2) is even frequently used to dene an ideal solution.

Both g m functions in Eqs. (2) and (3) have exactly the samemathematical form. As a consequence, by plotting g m ,idl /( RT ) vs. x1and g m , pgm /( RT ) vs. y1 , one obtains a unique curve which corre-sponds to the superimposition of these two plots. This feature isillustrated in Fig. 1.

It is clearly impossible to draw a double tangent on the

corresponding curve. Following Gibbss theorem (see Statement 2 ),it thus can be argued that VLE cannot arise when an ideal liquidphase is in equilibrium with a perfect-gas mixture what is instraight contradiction with Raoults law previously mentioned, theexperimental truthfulness of which cannot be questioned.

We can thus conclude that the three previous proposals lead toapparently-contradictory results. The goal of this article is thus toclarify this apparent paradox.

2. Discussion around the paradigm of ideal mixtures

2.1. The ideal-mixture concept

Even though most real gases depart appreciably in theirbehavior from the perfect-gas law, that equation is very valuableboth as an approximate representation of the properties of realgases at low pressure and as a leading term in more complexequations of state. In a somewhat similar manner, an ideal-mixture equation should be useful both as an approximation tothe properties of many real mixtures and as a leading term formore complex equations for mixture properties.

2.2. The ideal-mixture denition

A statistical reasoning allows demonstrating that the chemicalpotential of a perfect gas i in a perfect-gas mixture ( pgm) is given by:

g pgmi T , P , y g pg i pure T , P RT ln yi 4

where g pg i pure T , P is the chemical potential of the pure perfect gas atthe same temperature T and pressure P as the pgm.

The ideal mixture, which may be either a gaseous or liquidmixture, may be built in a similar manner. Following Lewis andRandall (1961) , we dene an ideal system as a multicomponentsystem in which the chemical potentials of all the componentshave the following form:

g idi T , P , z g

same agg : stateas the mixturei pure T , P RT ln z i 5

for all temperatures, pressures and compositions. The superscript

id stands for ideal mixture and g same agg : stateas the mixturei pure T , P is the chemical

potential of pure i in the same state of aggregation as the consideredmixture .

Notes :

(1) by changing the overall composition of a binary system atconstant temperature and pressure, the mixture can go fromthe liquid state to the gas state after crossing a vaporliquidequilibrium (VLE) region. Following Prigogine and Defay(1954) , the mixture will be considered as ideal if (i) in theliquid region the liquid phase is ideal, (ii) if in the gas domainthe gas phase is ideal and (iii) if in the two-phase region eachof the phases is ideal.

(2) it is not always simple to decide in which aggregation state amixture is. This is particularly true at high temperature andpressure where the adjective uid is more appropriate thanliquid or gaseous . If at a given temperature and pressure,

regardless of the composition, the binary system is always ina one-phase region, the chemical potential of pure i in Eq. (5)is the chemical potential in its actual (stable) state at theconsidered temperature and pressure.

For a component i in an ideal liquid mixture ( idl), we thus canwrite:

g idli T , P , x g liquidi pure T , P RT ln xi 6

where g liquidi pure T , P represents the chemical potential of pure compo-nent i in the liquid state. It is however very important to be awarethat this reference state may sometimes be a hypothetical, i.e.unstable, state. It is indeed not scarce that the actual (stable) statefor the pure component at T and P is the gas state and not the liquid

state. This situation will be discussed in detail in a next section.For a component i in an ideal gaseous mixture ( idg ), one obtains:

g idg i T , P , y g gasi pure T , P RT ln yi 7

where g gasi pure T , P represents the chemical potential of pure compo-nent i in the gaseous state. Once again, the reference state g gasi pure T , P may be a hypothetical state, i.e. a non-physically realiz-able state. Indeed, the actual state for the pure component may beliquid. By comparing Eqs. (4) and (7) , it becomes obvious that aperfect-gas mixture is a particular case of an ideal gaseous mixture.

Note: all binary mixtures behave as a perfect-gas mixture i.e. asan ideal gaseous mixture when the pressure tends to zero. However,all binary mixtures are obviously not ideal mixtures. Indeed, to bequalied of ideal, each component of a multicomponent system

must obey Eq. (5) at all temperatures, pressures and compositions.

0.0 0.5 1.0

-0.5

0.0

x1 & y1

&gm,idl gm,pgm

RTRT

Fig. 1. Reduced Gibbs energy change on mixing for an ideal liquid binary mixture

(idl) and for a perfect-gas binary mixture ( pgm) as dened by Eqs. (2) and (3) .

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Eq. (5) may also be written using the concept of fugacity. Anideal mixture is thus a mixture in which we can write the fugacityof each component in the form:

^

f idi T , P , z z iU f

same agg : stateas the mixturei pure T , P 8

where f same agg : stateas the mixturei pure T , P is the fugacity of pure i at T and P . The

pure liquid is selected for an ideal liquid mixture and the pure gasis selected for an ideal gaseous mixture even if these states arehypothetical.

Introducing the fugacity coefcients, Eq. (8) writes:

^

j idi T , P , z jsame agg : stateas the mixturei pure T , P 9

so that in an ideal mixture the fugacity coefcient of a componenti,

^

j idi T , P , z, is the same as the fugacity coefcient of the purecomponent taken at the same T and P and in the same aggregationstate as the ideal mixture. Following Michelsen and Mollerup(2007) , we could also state that in an ideal mixture, at a given T and P , the fugacity coefcients

^

j idi are composition independent.In classical thermodynamics, the activity coefcient of a

component i in a homogeneous phase is dened by:

giT , P , z ^

j iT , P , z

jsame agg : stateas the mixturei pure T , P

10

From Eq. (9) , we can thus dene an ideal mixture as a mixturein which the activity coefcients of all the components are unity:

gidi T , P , z 1 11

As a corollary, we can state that in an ideal mixture, theactivity a i of each component, classically dened by:

a iT , P , z

^

f iT , P , z

f

same agg : state

as the mixturei pure T , P

z iUgiT , P , z 12

is identical to the mole fraction z i of component i. We thus canwrite:

a idi T , P , z z i 13

2.3. Contribution of statistical thermodynamics to the denitionof an ideal mixture

The question we now need to address is: do ideal mixturesactually exist? The answer is given by the statistical thermodynamicswhich allows demonstrating that Eqs. (5), (8), (9), (11) and (13) canbe obtained by assuming that molecules A and B are sufciently alikefrom the point of view of the molecular interactions which they exerton one another, and from the point of view of their shapes and sizes.We thus can say that in ideal mixtures there are interactions but theaverage energy of AB interactions in the mixture is the same as theaverage energy of AA and BB interactions in the pure compounds.On the other hand, let us recall that a perfect-gas mixture is an idealgaseous mixture in which not only are the interactions the same(as in any ideal mixture), but they are also zero. The laws of idealmixtures are thus limiting laws which are obeyed more closely, thegreater the similarity with regard to size, shape and energy, betweenthe components. Ideal mixtures are thus useful both as an approx-imation to the properties of many real mixtures and as a leading termfor more complex equations for mixture properties which was the

basis to dene them.

2.4. Raoults law

Let us consider an ideal liquid mixture in equilibrium with aperfect-gas mixture; then:

g idli T , P , x g pgmi T , P , y 14

From Eqs. (4) and (6) , we thus obtain:

RT lnyi xi g liquidi pure T , P g pg i pure T , P RT ln f

liquidi pure T , P

f pg i pure T , P " # 15 From basic thermodynamics, we know that:

f liquidi pure T , P P sat i T Uj i pure T , P

sat i T Uexp 1RT R P P sat i T v liquidi pure dP h i f pg i pure T , P P 8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

17

we can thus conclude that the partial molar volume, enthalpy,internal energy, and constant pressure heat capacity of a componenti in an ideal mixture is equal to the molar property of purecomponent i taken at the same temperature at the same pressureand in the same aggregation state as the ideal mixture. Like thefugacity coefcients

^

j idi and the activity coefcients gidi , at a givenT and P , such partial molar properties ( v idi , h

idi , u

idi , c

idP , i) are thus

independent of the ideal-mixture composition.

R. Privat, J.-N. Jaubert / Chemical Engineering Science 82 (2012) 319333 321
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The molar properties m id of an ideal mixture containing pcomponents thus verify:

g id T , P , z X p

i 1 z iU g

same agg : stateas the mixturei pure T , P RT X

p

i 1 z iUln z i

v id T , P , z X p

i 1 z iUv

same agg : stateas the mixturei pure T , P

sid T , P , z X p

i 1 z iUs

same agg : stateas the mixturei pure T , P RX

p

i 1 z iUln z i

h id T , P , z X p

i 1 z iUh


u id T , P , z X p

i 1 z iUu


a id T , P , z X p

i 1 z iUa


p

i 1 z iUln z i

c idP T , P , z

X

p

i 1 z iUc

same agg : stateas the mixtureP , i pure T , P

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

18

Equivalently, one can write the following equations which willbe discussed in Section 2.6 .

v id T , P , zX p

i 1 z iUv


h id T , P , zX p

i 1 z iUh


u id T , P , zX p

i 1 z iUu


c idP T , P , z

X

p

i 1 z iUc

same agg : stateas the mixtureP , i pure T , P 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:and

sid T , P , zX p

i 1 z iUs

same agg : stateas the mixturei pure T , P RX

p

i 1 z iUln z i

a id T , P , zX p

i 1 z iUa


p

i 1 z iUln z i

g id T , P , zX p

i 1 z iU g


p

i 1 z iUln z i

8>>>>>>>>>>>>>>>>>>>>>>>>>:

19

2.6. Ideal-mixture property changes on mixing M m,id

Confusion has often appeared in the mathematical expressionof these properties. They are indeed often mixed up with thequantities dened in the left-hand side ( lhs) of Eq. (19) . It is thusdecided to make the point on their denition and usefulness.

2.6.1. DenitionThe property change on mixing M m , of any extensive property

M , is by denition the difference between the property M of theactual mixture and the sum of the properties of the purecomponents which make it up, all at the same temperature and pressure as the mixture . We here want to emphasize that such aquantity is called change on mixing because it is the change thatwould be measured if the mixing took place in a mixing device of a laboratory. As an example ( Sandler, 2006 ), the enthalpy change

on mixing can be measured using a calorimeter; two streams: one

of pure uid 1 and the second of pure uid 2, both at atemperature T and at pressure P , enter a mixing device, and asingle mixed stream also at T and P leaves. Heat is added orremoved to maintain the temperature of the outlet stream. Thisquantity of heat is by denition the enthalpy change on mixing.This well-known example was selected to emphasize that all thestates are physically realizable: the obtained mixture (liquid,gaseous or in the two-phase area) and the two pure uids (either

liquid or gaseous) are in their stable state at T and P .We can thus dene the molar property change on mixing m m

by the following equation:

m m mT , P , z

|fflfflfflfflffl{zfflfflfflffl property of the stableactual mixture X

p

i 1 z iUm stablei pure T , P 20

In the common case where the obtained mixture is in the one-phase area (liquid or gaseous), we thus have:

m m X p

i 1 z iU m i m stablei pure T , P h i 21

where m i and z i respectively stand for the partial molar property

and the mole fraction of i in the stable obtained mixture.In the particular case where the obtained mixture is in thetwo-phase area, i.e. a phase a in equilibrium with a phase b [e.g.VLE or liquidliquid equilibrium (LLE)], the following equationapplies:

m m t a Um phase a T , P , x a 1 s a Um phase b T , P , x b

|fflfflfflfflfflfflfflfflfflfflmT , P , z X

p

i 1 z iUm stablei pure T , P

t a UX p

i 1 xia U m i , phase a m

stablei pure T , P h i

|fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflffl ffl} mm , phase a 1 t a U

X p

i 1

xib U m i, phase b mstablei pure T , P

h i |fflfflfflfflfflfflfflfflfflmm , phase b22

where t a is the molar relative amount of phase a . xia and xib standfor the mole fractions of i respectively in the a and b phases.

It is important to be aware that there is no link between theaggregation state of the obtained mixture and the state of aggregation of the pure uids. As shown in Fig. 2, by mixingtwo pure components: two liquids or two gases or one liquid andone gas, it is possible to obtain a liquid solution, a gas mixture or atwo-phase system (in vaporliquid or in liquidliquid equili-brium). All the possible combinations are in fact observable.

Remark. None of the four binary systems shown in Fig. 2 canbe considered as ideal. The ideal-mixture properties are how-

ever introduced into the thermodynamic description of realmixtures as convenient standards of normal behavior. It is indeedalways possible to divide a property of a real mixture ( m rm ) in twoparts: the ideal-mixture contribution ( m id ) and the excess con-tribution ( mE ):

m rm m id m E 23

In Eq. (23) , the real mixture and the associated ideal mixturehave the same temperature, the same pressure, the same compo-sition and are in the same aggregation state.

Such an equation also applies to the property changes onmixing and we always can write:

mm

mm , id

mE

24

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we thus must be able to properly estimate the ideal-mixtureproperty changes on mixing ( mm ,id ) even if the considered systemshows deviations from ideality.

Following the denition of a molar property change on mixinggiven by Eq. (20) , we dene the ideal-mixture property changeson mixing by:

m m , id m id T , P , z

X p

i 1

z iUm stablei pure T , P 25

we however know that the molar properties ( m id ) in idealmixtures (gaseous or liquid) obey Eq. (18) .

We thus obtain for the ideal-mixture property changes onmixing:

g m , id T , P , z X p

i 1 z iU g

same agg : stateas the mixturei pure T , P g

stablei pure T , P 24 35

RT X p

i 1 z iUln z i

vm , id T , P , z X p

i 1 z iU v

same agg : stateas the mixturei pure T , P v


sm , id T , P , z X p

i 1 z iU s

same agg : stateas the mixturei pure T , P s


RX p

i 1 z iUln z i

hm , id T , P , z X p

i 1 z iU h

same agg : stateas the mixturei pure T , P h


um , id T , P , z X p

i 1 z iU u

same agg : stateas the mixturei pure T , P u


a m , id T , P , z X p

i 1 z iU a

same agg : stateas the mixturei pure T , P a


RT X p

i 1 z iUln z i

c m , idP T , P , z X p

i 1 z iU c

same agg : stateas the mixtureP , i pure T , P c

stableP , i pure T , P 24 35

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 26

Eq. (26) highlights that it can be wrong to state that:

J the ideal-mixture Gibbs energy change on mixing obeyssystematically Eqs. (2) and (3) .

J there are no volume or enthalpy or internal energy or heatcapacity changes on the formation of an ideal mixture from itspure components at the same temperature and pressure.

Although these statements are written in most of the chemicalengineering thermodynamics textbooks, they are only true whenthe considered ideal mixture and the p pure components are all inthe same aggregation state (liquid or gas). However, an idealmixture can be obtained by mixing pure components in differentaggregation states [as stated above, the ideal-mixture concept canalso be used to model the properties of real mixtures throughEq. (23) and in this case, the aggregation state of the hypotheticalideal solution is often different from the aggregation state of thepure components at the same temperature and pressure (seeFig. 2)].

Let us take as an example, the mixing at t 20 1C and P 2 kPaof 1 mol of toluene (1) and 9 mol of ethylbenzene (2), leading to aliquid mixture the composition of which is x1 0.1 and x2 0.9(see point A in Fig. 3a). This liquid mixture is a good example of anideal solution since the two molecules are more or less identicalfrom the point of view of the molecular interactions they exert onone another, and from the point of view of their shapes and sizes.To be convinced of this assertion, the molar excess enthalpy hE ,the ideal-mixture molar enthalpy change on mixing hm ,id and themolar enthalpy change on mixing hm have been calculated att 20 1C and P 2 kPa using the PengRobinson equation of state

(Peng and Robinson, 1976 ) and temperature-dependent kij

( Jaubert and Mutelet, 2004 ; Jaubert et al., 2010 ). As shown inFigs. 3bd whatever the composition, hE is extremely close to zeroand as a consequence hm ,id cannot be distinguished from hm byeye. The studied system is thus doubtless with an excellentapproximation an ideal mixture. However in the consideredconditions, pure toluene is in the gaseous state P sat 1 20 1C %3 kPa and pure ethylbenzene is in the liquid state P sat 2 20 1C %1 kPa . As a consequence, the molar Gibbs energy change on

mixing is completely different from RT x1 ln x1 x2 ln x2 (see point Ain Fig. 5) and the molar enthalpy change on mixing is far fromzero (see point A in Fig. 3d); it is negative and close to x1 UDvapH 1(20 1C)E 4 kJ mol

1 . On the contrary, the molar excess enthalpyis only 0.0017 kJ mol 1 (see point A in Fig. 3b).

Similarly by mixing at t 20 1C and P 2 kPa, 9 mol of toluene(1) and 1 mol of ethylbenzene (2), we obtain a gas mixture (seepoint B in Fig. 3a) which can be considered as a perfect-gasmixture that is a gaseous ideal mixture the composition of whichis y1 0.9 and y2 0.1. Once again: g m T , P , za RT y1 ln y1 y2 ln y2 (see point B in Fig. 5) and hm (T ,P ,z) a 0 (see point B in Fig. 3d).In this case, the molar enthalpy change on mixing is positive andclose to y2 UDvap H 2 (20 1C)E 4 kJ mol

1 whereas the molarexcess enthalpy is as weak as 0.000014 kJ mol 1 (see point B inFig. 3b). More information on how Fig. 3 was calculated can befound in Appendix A .

2.6.2. Calculation of the ideal-mixture Gibbs energy changeon mixing

The knowledge of the ideal-mixture Gibbs energy change onmixing allows calculating all the other ideal-mixture propertychanges on mixing and allows applying the double-tangentconstruction of coexisting phases. Moreover, ideal-mixture prop-erties are the starting point to estimate the properties of realmixtures. The correct calculation of g m ,id (T ,P ,z) is thus of thehighest importance. Starting from Eq. (26) , we can write:

g m,

id T , P , z RT X p

i 1 z iUln

z iU f


f stablei pure T , P 26664 3777527

Following Eq. (27) , the mathematical expressions to be used tocalculate the ideal-mixture Gibbs energy change on mixing for abinary liquid or gas phase are given here below [for clarity,component 1 is the most volatile and component 2 is the heaviestone so that P sat 1 T 4 P

sat 2 T ].

Remark. At low to moderate pressure f liquidi pure T , P P sat i T and

f gasi pure T , P P so that Eq. (27) can generally be simplied asshown below.

Case 1. The pressure is higher than the vapor pressure of component 1 at the working temperature P 4 P sat 1 T 4 P

sat 2 T .

In these conditions the stable state for the 2 pure components isthe liquid state and from Eq. (27) :

g m , idliquid T , P , z RT z 1 ln z 1 z 2 ln z 2

g m , id gas T , P , z RT z 1 lnz 1 U f

gas1 pure T , P

f liquid1 pure T , P ! z 2 ln z 2 U f gas2 pure T , P f liquid2 pure T , P ! !8>>>:28 ; 29

At low to moderate pressure, one has:

g m , id gas T , P , z RT z 1 lnz 1 UP

P sat 1 T " # z 2 ln z 2 UP P sat 2 T " #" # 30 Case 2. The pressure is between the two vapor pressures

P sat 1 T 4 P 4 P

sat 2 T . In these conditions, the stable state for

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component 1 is the gas state but the stable state for component 2is the liquid state. From Eq. (27) :

g m , idliquid T , P , z RT z 1 lnz 1 U f

liquid1 pure T , P

f gas1 pure T , P ! z 2 ln z 2 ! g m , id gas T , P , z RT z 1 ln z 1 z 2 ln

z 2 U f gas2 pure T , P

f liquid2 pure T , P ! !8>>>>>:

31 ; 32


g m , idliquid T , P , z RT z 1 lnz 1 UP sat 1 T

P h i z 2 ln z 2h i g m , id gas T , P , z RT z 1 ln z 1 z 2 ln

z 2 UP P sat 2 T h ih i8>:

33 ; 34

Case 3. The pressure is lower than the vapor pressure of compo-nent 2 at the working temperature P o P sat 2 T o P

sat 1 T . In these

conditions the stable state for the 2 pure components is the gasstate. From Eq. (27) :

g m , idliquid T , P , z RT z 1 lnz 1 U f

liquid1 pure T , P

f gas1 pure T , P ! z 2 ln z 2 U f liquid2 pure T , P f gas2 pure T , P ! ! g m , id

gasT , P , z RT z 1 ln z 1 z 2 ln z 2

8>:

35 ; 36


g m , idliquid T , P , z RT z 1 lnz 1 UP sat 1 T

P ! z 2 ln z 2 UP sat 2 T P ! ! 37 Eqs. (28) and (36) are similar to Eqs. (2) and (3) and can be

found in any textbook of thermodynamics ( Ahlstrom et al., 2010 ).They are even often used to dene an ideal mixture. We howeverbelieve that such a denition should be avoided because thesetwo equations only constitute a particular case of the mathema-tical expression of g m ,id (T ,P ,z). However, to our knowledge, nothermodynamics textbook gives Eqs. (30), (33) , (34) and (37) .

In conclusion, this section clearly demonstrates that the ideal-mixture Gibbs energy change on mixing obeys Eqs. (2) and (3)(and consequently hm ,id vm ,id 0) only when the two purecomponents and the mixture are in the same aggregation state.Such a situation never occurs when at a given temperature thepressure is between the vapor pressures of the two pure compo-nents. This region wherein VLE arises and wherein Raoults lawapplies is however extremely important since the composition of the two phases in equilibrium can be determined from theknowledge of g m ,id (T ,P ,z).

The mathematical expressions of the other ideal-mixture

property changes on mixing are given in Appendix B .

0.0 0.5 1.0

10.0

11.0

12.0

x1,y1

P/kPa

L + L = V

V + V = V

L + L = L

L + V = V

L+ L = VLE

a

0.0 0.5 1.016.0

20.0

24.0

x1,y1

P/kPa

V + V = L

L+

V=

L

V + V = VLE

L + V = VLE

b

c

0. 0.4 0.8

4.00

8.00

12.0

x1,y1

P/kPaL + L = LLE

L + V = LLE

3-phase line

d

x1,y1

0.0 0.5 1.04.0

8.0

12.0P/kPa

V + V= LLE

3-phase line

Fig. 2. Highlight of the various reachable aggregation states by mixing two pure components. (a) System cyclohexane (1) benzene (2) at 20 1C. (b) System acetone(1) chloroform (2) at 20 1C. (c) System nitromethane (1) 1,2-ethanediol (2) at 40 1C. (d) Binary system giving simultaneously rise to a negative azeotrope and a liquidliquidequilibrium (such systems are so infrequent that it was preferred to show on a ctitious example that by mixingtwo pure gases a liquidliquid equilibrium could be reached).



7/15

2.7. Pengs denition of an ideal mixture

Twenty years ago, in two interesting articles ( Peng, 1989 ,1990 ), Peng also noticed that Raoults law was not compatiblewith the unique frequently used denition of the ideal-mixtureGibbs energy change on mixing recalled hereafter:

g m , id T , z RT z 1 ln z 1 z 2 ln z 2 38

In this article, we decided to give up Eq. (38) and proof wasgiven that this equation could be incorrect (see Tables B1 and B2 )since it is only a particular case of the mathematical expression of g m ,id . As shown previously, our approach remains totally compa-tible with Raoults law when an ideal liquid phase is in equili-brium with a perfect-gas mixture. Peng made the opposite anddecided to dene an ideal mixture as a mixture for which Eq. (38)is always true. In his approach, the chemical potentials of all the

components of an ideal mixture have thus the following form:

g idi T , P , z g stablei pure T , P RT ln z i 39

where g stablei pure T , P is the chemical potential of pure i in itsstable (actual) state at T and P [note that to switch from Pengsdenition to the denition we adopted here for ideal mixtures,

the term g stablei pure T , P has to be replaced by g same agg : stateas the mixture

T , P

i pure ]. Eq. (39)may also be written using the concept of fugacity. According toPeng, an ideal mixture is thus a mixture in which we can write thefugacity of each component in the form:

^

f

id

i T ,

P ,

z z iU f

stable

i pure T ,

P 40

where f stablei pure T , P is the fugacity of pure i in its stable state at T and P .From Eq. (39) , we immediately always have:

vm , id T , P , z 0sm , id T , P , z R z 1 ln z 1 z 2 ln z 2 hm , id T , P , z 0

8>>>:41

Using Pengs denition, there are thus never volume orenthalpy changes on the formation of an ideal mixture from itspure components at the same temperature and pressure.

Note : Eq. (41) is often written in many textbooks of thermody-namics in which the ideal mixture is yet conventionally dened byEq. (5) and Appendix B was devoted to give proof that it couldbe wrong.

It is noticeable that Pengs denition is equivalent to the conven-tional denition of an ideal mixture when the pure components andthe mixture are all in the same aggregation state. However, in ouropinion, various arguments conict with Pengs approach:

J rstly, an ideal liquid phase (as dened by Peng) can never bein equilibrium with an ideal gaseous phase (as dened byPeng). Indeed, if in each phase, the chemical potentials followEq. (39) , the classical vaporliquid equilibrium condition:

g liqi T , P , x g gasi T , P , y 42

reduces to the trivial following equation:

xi yi 43

We should thus conclude that the VLE relation between an ideal

liquid phase and an ideal gas phase (Eq. (43) ) is no more Raoults

0.0 0.5 1.0

1.0

2.0

3.0

x1,y1

P/kPa

A B

C

D

L V

t = 20 C

0.0 0.5 1.0

-10.0

0.0

10.0

z1

hm,id (kJ . mol 1)

A

B

L

V

Ideal liquid solution for which

h 0m,id liquid

Ideal gaseousmixture for which

h 0m,id gas

Ideal liquid solution inequilibrium with an ideal

gaseous mixture

t = 20 C and P = 2 kPa

0.0 0.5 1.00.000

0.002

0.004

z1

h E (kJ . mol 1)L

Vt = 20 C and P = 2 kPa

A

B

0.0 0.5 1.0

-10.0

0.0

10.0

z1

hm (kJ . mol 1)

A

B

L

Vt = 20 C and P = 2 kPa

Fig. 3. System toluene (1) ethylbenzene (2) which may be regarded as an ideal binary mixture because the two molecules are of closely similar chemical structure.(a) Isothermal phase diagram at t 20 1C. (b) hE vs. z 1 at t 20 1C and P 2 kPa. (c) hm ,id vs. z 1 at t 20 1C and P 2 kPa. (d) hm vs. z 1 at t 20 1C and P 2 kPa.

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law (Eq. (1) ) but is instead a particular case of this law in whichP P sat 1 P

sat 2 . As a consequence, the two-phase region at a given

temperature would be represented by a horizontal segmentcovering the whole composition range. Such an isothermal(P , x1 , y1 ) phase diagram would stem from the attening of aclassical Raoults law phase diagram (as in Fig. 3a) in which thetwo pure components would have the same vapor pressure (theisothermal linear bubble-point curve would become horizontal

and would merge with the dew-point curve). Such phasediagrams cannot exist for real binary systems. Indeed, thecondition P sat 1 P

sat 2 at all temperatures implies that the two

molecules are so close each other that they are actually identical.This ideal-mixture concept would not be anymore an approx-imation to the properties of many real mixtures and we believethat it would be really a pity for practicing engineers.

J Secondly, Peng loses the structural denition of an idealmixture issued from the statistical thermodynamics. Indeed amixture containing alike compounds from the point of view of the molecular interactions which they exert on one another,and from the point of view of their shapes and sizes could notbe qualied as ideal. Only mixtures containing rigorouslyidentical molecules could be called ideal. In other words anideal mixture would degenerate in a kind of pure compound. Itthus would be a poorly appropriate model to describe mixtures.

J Thirdly, by changing the classical denition of an ideal solution,it would be necessary to change the values of the activitycoefcients collected in specialized books since many years.Indeed in Pengs approach, the activity coefcients are denedby:

gi , Peng T , P , z ^

j iT , P , zj stablei pure T , P

44

whereas in the conventional approach, they are dened byEq. (10) . We thus can write:

gi , Peng T , P , z gi, conventional T , P , zU f


f stablei pure T , P 45

As an example, Table 1 gives for a conventionally dened idealmixture (obeying Eq. (5) ) in which the conventional activitycoefcients are by denition unity, the values of the activitycoefcients obtained following Pengs approach. For convenience,it is assumed that the pressure of the mixture is low to moderate.

This table shows once again that the two approaches becomeidentical when the pure components and the mixture are in thesame aggregation state [ gliquidi , Peng 1 when P 4 P

sat 1 T 4 P

sat 2 T and

g gasi, Peng 1 when P o P sat 2 T o P

sat 1 T ]. More annoying, this table

puts in evidence that a binary mixture can never be considered asideal (employed in Pengs sense) for all temperatures, pressures

and compositions. Indeed by inspecting Table 1 , which gives theactivity coefcient values for a mixture obeying Raoults law, it canbe concluded that in Pengs approach, such a mixture is only ideal:

at high pressure P 4 P sat 1 T when the two pure componentsand the mixture are in the liquid state andat low pressure P o P sat 2 T when the two pure componentsand the mixture are in the gaseous state but becomes non-ideal when VLE may exist P sat 1 T 4 P 4 P

sat 2 T .

2.8. Epilogue

Our opinion is that Pengs papers have the great advantage to

highlight a serious weakness in the ideal-mixture concept even if

we do not subscribe to the solution he proposes. Peng spent manyyears to develop the famous PengRobinson equation of state andhe can thus be qualied of expert in modeling uids withequations of state. For him, no doubt that the property changeson mixing must be dened as proposed in this paper by Eq. (20) .This is indeed the unique denition which guarantees that suchproperties can be calculated from an equation of state (all theconsidered states are physically realizable). Since in Pengs approach,

the ideal-mixture property changes on mixing obey Eqs. (38) and(41) , the excess properties ( mE mm mm ,id) can thus be easilyestimated from an equation of state. Such an approach is howeverincompatible with Raoults law and with a statistical reasoning.Following Peng, Fig. 1 would correspond to the correct situationwhere an ideal-liquid mixture is in equilibrium with a perfect-gasmixture and in accordance with his approach, Peng would concludethat no vaporliquid equilibrium could exist for such systems. Wethus can say that Pengs approach is consistent.

This paper and Pengs theory both highlight that it is notpossible to nd an ideal-solution denition which would besimultaneously compatible with:

1. a statistical reasoning,2. Raoults law and3. very simple expressions of the ideal-mixture property changes

on mixing.

Peng decided to give up the two rst points whereas we decidedto only give up the third one and to give the correct expressions of mm ,id . We indeed saw no reason to change the ideal-mixturedenition used worldwide and introduced by Lewis and Randall. Itis however important to be aware that since the classical denitionof an ideal solution (Eq. (5)) may refer for the pure components to anon-stable state, it is not always possible to calculate with anequation of state the ideal-mixture property changes on mixing andthus the excess properties (the property changes on mixing are inreturn always calculable). Such a situation could arise if the equation

of state did not have a volume-root corresponding, for the purecomponent, to this non-stable state.

Remark. The derivation of mixing rules for cubic EoS by equating ata reference pressure, the reduced excess Gibbs energy calculated fromthe EoS noted g E EoS with the reduced excess Gibbs energyexpressed from an activity-coefcient model, g E g , is a well-knownapplication requiring the calculation of molar excess Gibbs energy of the liquid phase from an EoS g E liquid , EoS (Michelsen and Mollerup,2007 ). When using the HuronVidal approach, an innite referencepressure is taken. Under these conditions, any uid (mixture and purecomponents) is a homogeneous liquid and it is thus always possibleto calculate g E liquid , EoS (a liquid root the covolume always exists atP N ). However, when considering the MHV approach (ModiedHuronVidal), the reference pressure is zero and it is not alwayspossible to calculate g E liquid , EoS since it is not always possible to nd aliquid root for the pure components and the mixture at a giventemperature and pressure. This problem has clearly been alreadyidentied as a shortcoming of the MHV approach and is a practicalillustration of the impossibility for the EoS to calculate excessproperties in some cases.

We here want to state that in many textbooks, the authors in factswitch (and they do not seem aware to do it) from Pengs denition of an ideal mixture (when they dene the ideal-mixture propertychanges on mixing) to the classical denition used in this paperwhen Raoults law is introduced. To remain consistent, such authorsin fact use the ideal-mixture concept in a restrictive way by

considering that an ideal mixture can always be obtained by the



9/15

mixing of pure components which are in the same aggregation stateas the mixture. With this restriction, the ideal-mixture enthalpychange on mixing is obviously always equal to zero. In return, thispaper thus extends the ideal-mixture concept to mixtures obtainedby the mixing of pure components which are not necessarily in thesame aggregation state as the resulting mixture. This extension leadsto the conclusion that the ideal-mixture enthalpy (or volume) changeon mixing can be different from zero.

To conclude this section, we want to emphasize on an examplethat the use of the traditional ideal-mixture concept [in whichhm ,id (T ,P ,z) 0] can lead to inconsistent results. Let us consider thesystem toluene ethylbenzene at t 20 1C and P 2 kPa. We heremix very similar molecules and such a mixture can thus beconsidered, with a very good approximation, as ideal.

By computing the excess enthalpy with the following equa-tion:

hE T , P , z hT , P , zh id T , P , z

hT , P , z z 1 Uhsame agg : stateas the mixture1 pure T , P z 2 Uh

same agg : stateas the mixture2 pure T , P 24 35

46 results identical to those shown in Fig. 3b are going to be

obtained. Thus hE (T ,P ,z) values will be extremely small what isconsistent for an ideal mixture.

By computing the excess enthalpy with the following equa-tion:

hE T , P , z hm T , P , zhm , id T , P , z 47

we should obviously obtain the same result. This is effectivelytrue if hm ,id (T ,P ,z) is computed properly using Eq. (26) recalledhereafter (see also Fig. 3c):

hm , id T , P , z z 1 U hsame agg : stateas the mixture1 pure T , P h

stable1 pure T , P 24 35

z 2 U hsame agg : stateas the mixture2 pure T , P h

stable2 pure T , P 24 35

48

however by setting arbitrarily that hm ,id (T ,P ,z) 0, we are going toobtain for hE (T ,P ,z), the values shown in Fig. 3d. Such values arefar from zero and the considered mixture must be declared asnon-ideal. In conclusion by ignoring the fact that hm ,id (T ,P ,z) can

be different from zero, the values of hE (T ,P ,z) estimated from

Eq. (47) become identical to those of hm (T ,P ,z) but are differentfrom those calculated by Eq. (46) . Such a big confusion leadseveral research groups (see e.g. Oscarson et al., 1996 ) tointroduce the concept of excess enthalpies of mixing when theypublished heats of mixing measured in regions where VLE arises.They indeed do not know if their measurements must be calledexcess or mixing enthalpies and we perfectly understand theirquestioning. We hope that this article will convince them to call

their next measurements mixing enthalpies.

3. Necessary and sufcient conditions to apply the Gibbsdouble-tangent construction of coexisting phases

The Gibbs double-tangent construction of coexisting phases isof the highest interest to determine the compositions of twophases in equilibrium. This section is devoted to dene theapplication range of this graphical approach which is oftendescribed ambiguously.

From basic thermodynamics, it is well known that by plottingat constant temperature and pressure, any molar property m of abinary system vs. x1 , and by adding the tangent line for acomposition z 1 , the tangent intercepts at x1 1 and x1 0 directlygive the values of the two molar partial properties: m 1 T , P , z 1 andm 2 T , P , z 1 .

This graphical technique can be used to determine the composi-tion of the coexisting phases by plotting g (the molar Gibbs energy)as a function of x1 at constant T and P . Indeed, for a binary system,the equilibrium condition between two phases I and II writes:

g 1 T , P , z 1 , I g 1 T , P , z 1 , II g 2 T , P , z 1 , I g 2 T , P , z 1 , II( 49

As a consequence, the presence of a double tangent allowsdening two phases, the composition of which are z 1,I and z 1,IIwhich satisfy Eq. (49) . Let us however recall that Eq. (49) is anecessary but not a sufcient condition to reach a stable equili-brium. It means that the graphically determined compositions donot necessarily correspond to a stable equilibrium. Strictly speaking:

if a unique double tangent can be drawn, it always highlights astable two-phase system. More generally, the existence of aunique multiple tangent always highlights a stable multiphasesystem.if several multiple tangents (typically double tangents) existon a g vs. composition plot at a given temperature andpressure, some can characterize non-stable multiphase sys-tems but at least one of them denes a stable multiphasesystem. In other words, the existence of a multiple tangentassociated to a non-stable two-phase system systematicallyinvolves the existence of a multiple tangent associated to astable multiphase system. This instance is illustrated in Fig. 4wherein the isothermal and isobaric reduced Gibbs energy of agiven binary system is represented. It appears that threedouble tangents: (AB), (CD) and (EF) can be drawn. In thepresent case, the double tangent (AB) which lowers the Gibbsenergy characterizes a stable equilibrium whereas equilibria(CD) and (EF) are non-stable. Fig. 4 thus highlights that themere existence of a non-stable equilibrium, e.g. (CD), involvesthe existence of the stable equilibrium (AB) explaining why as written in Statement 2 (see the introduction) the exis-tence of a multiple tangent is a necessary and sufcient condition so that a stable equilibrium between several phasesof different compositions exists.

This technique can be applied to graphically determine any

kind of phase equilibrium. In the case of a LLE, the molar Gibbs

Table 1Activity coefcients values issued from Pengs approach ( gi,Peng ) for a convention-ally ideal mixture obeying Raoults law ( gi,conventional 1).

Pressure range gi,Peng

P 4 P sat 1 T 4 P sat 2 T

gliquid1 , Peng 1 gliquid2 , Peng 1

Component 1 is liquidComponent 2 is liquidThe mixture ideal in the conventional sense is

in the liquid state

P sat 1 T 4 P 4 P sat 2 T gliquid1 , Peng

P sat 1P g

liquid2 , Peng 1

g gas1 , Peng 1 g gas2 , Peng

P P sat 2

Component 1 is gaseousComponent 2 is liquidThe mixture ideal in the conventional sense is

either in the liquid state or in the gaseous stateor in VLE

P o P sat 2 T o P sat 1 T

g gas1 , Peng 1 g gas2 , Peng 1

Component 1 is gaseousComponent 2 is gaseousThe mixture ideal in the conventional sense is

in the gaseous state

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energy of the liquid phase assumed to be a single phase that is:

g liquid T , P , x X2

i 1 xiU g i , liquid T , P , x 50

has to be plotted.In the case of a VLE, the molar Gibbs energy of the one-phase

liquid postulated system ( g liquid ) and the molar Gibbs energy of the one-phase gas postulated system:

g gas T , P , y X2

i 1 yiU g i, gas T , P , y 51

are calculated. For a given value of the overall composition, onlythe smallest value between g liquid and g gas which identies themost stable homogeneous system needs to be plotted. The molarGibbs energy of the system is thus dened by:

g min g liquid , g gasn o 52 When the two phases in equilibrium are not in the same aggrega-

tion state (e.g. a liquid phase in equilibrium with a gaseous phase), inorder a double tangent indicates the composition of the phases inequilibrium, it is compulsory that the pure-component reference stateused to render the chemical potential of a component i is the same inthe two phases. Indeed, by doing so, one obtains for a VLE:

g i, liquid T , P , z g ref :statei pure T , P RT ln

^

f i, liquid T , P , z f ref :statei pure T , P !

g i, gas

T , P , z g ref :statei pure

T , P RT ln^

f i, gasT , P , z

f ref :statei pure T , P

!8>>>>>:

53

so that:

g liquid T , P , zX2

i 1 z iU g

ref :statei pure T , P RT X

2

i 1 z iUln

^

f i , liquid T , P , z

f ref :statei pure T , P 24 35

g gasT , P , zX2

i 1 z iU g

ref :statei pure T , P RT X

2

i 1 z iUln

^

f i, gas T , P , z

f ref :statei pure T , P 24 35

8>>>>>>>>>>>>>: 54 The lhs of Eqs. (54) are thus the molar Gibbs energy of each

phase (liquid or gas) translated by the same quantity. Such atranslation does not alter in any way the double-tangent construc-

tion of coexisting phases. The pure-component reference state can be

chosen freely and can for instance be the pure liquid, the pure gas,the pure perfect gas or the stable (actual) state. Although any valuewould be correct, it is convenient to arbitrarily assign a value of zeroto g ref :statei pure T , P so that g can be easily dened by:

g T , P , zRT X

2

i 1 z iUln

^

f iT , P , z f ref :statei pure T , P 24 35

55

The non-inuence of the pure component reference state on thedouble-tangent construction of coexisting phases is illustrated inFig. 5 for the system toluene (1) ethylbenzene (2) at t 20 1C andP 2 kPa. In such a gure, two different reference states wereselected:

if the chemical potentials of the pure perfect gases g pg i pure atsystem temperature and at unit pressure, are taken as refer-ence and equal to zero, the reduced Gibbs energy of ahomogeneous system containing p components of mole frac-tions z i is according to Eq. (55) :

g T , P , zRT

X p

i 1 z iUln

^

f iT , P , z 56

Assuming that the considered liquid phase is an ideal solution^

f i, liquid T , P , x P sat i U xi and that the gas phase is a perfect-gas

mixture ^

f i, gas T , P , y P U yi , one has:

g liquid T , P , x RT x1 Uln x1 UP

sat 1 T x2 Uln x2 UP sat 2 T g gas T , P , y

RT y1 Uln y1 UP y2 Uln y2 UP 8


11/15

With the considered reference state, we thus obtain: g /RT g m /RT .Such a reference state is often chosen by researchers working withequations of state (see for example the book by Michelsen andMollerup, 2007 or the paper by Baker et al., 1982) . Both quantities^

f iT , P , z and f stablei pure T , P can indeed be calculated without efforts and g

is always zero when the composition is zero or one. For the studiedsystem which can be regarded as ideal and by remembering that inthe considered conditions, pure toluene is in the gaseous state andpure ethylbenzene is in the liquid state, one obtains:

g liquid T , P , x RT x1 ln

x1 UP sat 1 T P h i x2 ln x2 g gasT , P , y

RT y1 ln y1 y2 lny2 UP

P sat 2 T h i8>:

59

Expressions in Eq. (59) are obviously similar to the g m , id

formulations given by Eqs. (33) and (34) and Fig. 5 highlightson the selected system that this second reference state leads asexpected to the same liquid and gas phase compositions as therst one.

This second reference state leads us to conclude that Gibbsstheorem asserts that by plotting at constant temperature andpressure the Gibbs energy change on mixing ( g m ) of a binary

system vs. the molar composition, the existence of a doubletangent is a necessary and sufcient condition so that an equili-brium between two phases of different compositions exists at thistemperature and pressure which was the second statement of ourintroduction.

Remark:

(1) In Fig. 5, the curves g vs. composition are concave upward(i.e. convex) for all composition and there is thus no change inconcavity. Let us however recall that a change in concavity isa prerequisite to represent a two-phase equilibrium only if the two phases are in the same aggregation state (e.g. liquidliquid equilibrium) and if the same model (e.g. the sameequation of state or the same activity coefcient model) isapplied to both phases.

(2) By coming back to Fig. 1, we should conclude that an ideal liquidphase cannot be in VLE with an ideal gaseous mixture. Thisassertion is obviously wrong. The reason does not come from anincorrect use of the double-tangent construction of coexistingphases but from an erroneous expression for g m ,id . Eqs. (2) and(3) never describe the ideal-solution Gibbs energy change onmixing at a temperature and pressure where a VLE exists. In thislatter case, the correct mathematical expressions to be used aregiven by Eqs. (33) and (34) . We can also state that Eq. (2) is acorrect mathematical expression for the molar Gibbs energy of an ideal liquid phase when the pure liquid is chosen as the pure-component reference state to render the chemical potentials.Symmetrically Eq. (3) is a correct mathematical expression forthe molar Gibbs energy of an ideal gaseous mixture when thepure gas is chosen as the pure-component reference state torender the chemical potentials. As previously explained, sincethe pure-component reference state is different for the twophases, the double-tangent construction cannot be applied withsuch expressions.

(3) As previously explained, all well-known textbooks of thermo-dynamics, by dening the ideal-mixture Gibbs energy change onmixing by Eqs. (2) and (3) can obviously never apply the Gibbsdouble-tangent construction of coexisting phases to vaporliquid equilibrium. Unfortunately, the deep reason of why itdoes not work is from our knowledge never explained thuscausing confusion for students and practicing engineers. How-ever, in the same books, the graphical construction is exten-

sively used to determine with success LLE what reinforces the

confusion. The reason of this success is simple: to graphicallyhighlight a LLE, we only need to plot g mliquid =RT vs. x1 . FollowingEq. (2) , all textbooks thus plot the following function:

G mliquidRT

x1 Uln x1 Ug1 x2 Uln x2 Ug2 60

When the liquidliquid equilibrium is obtained by mixing twopure liquids, then G mliquid g mliquid (the stable state is the pure

liquid state for the two components) and we nd back theclassical Gibbs criterion. A double tangent will thus give thecomposition of the two liquid phases in equilibrium.When a liquidliquid equilibrium is obtained by mixing a pureliquid and a pure gas (or possibly two pure gases), Eq. (2) is notthe correct equation to calculate g m ,idl (T , x) and Eq. (60) is notthe correct equation to express g mliquid but a double tangenton the curve G mliquid =RT vs. x1 (Eq. (60) ) will allow to determinethe composition of the two liquid phases in equilibrium.Indeed, in Eq. (60) , the same pure-component reference state here the pure liquid state is used to render the chemicalpotential of a component i in the two liquid phases. In otherwords, by improperly stating that G mliquid (Eq. (60) ) is the molarGibbs energy change on mixing of the liquid phase, the

graphical construction works.

4. Conclusion

The concept of ideal mixture is illustrated in thousands of booksand scientic publications and is widely used in chemical engineer-ing thermodynamics. The aim of this paper was however to showthat some points on the properties of ideal mixtures needed to beclaried. From a practical point of view, we are convinced that anideal-mixture equation must be useful both as an approximation tothe properties of many real mixtures and as a leading term for morecomplex equations for mixture properties. This is the reason why,the structural denition of an ideal mixture is in our opinion themost convenient: we will state that a binary mixture can beconsidered as ideal if the two molecules A and B are sufcientlyalike from the point of view of the molecular interactions whichthey exert on one another, and from the point of view of theirshapes and sizes. Such ideal mixtures have many remarkableproperties. Among them, the simple mathematical expression of the chemical potential of a component i in an ideal mixture givenby Eqs. (6) and (7) is certainly the most important.

In many textbooks of thermodynamics, an important miscon-ception however arises during the denition of the propertychange on mixing and the goal of this article was to give thecorrect expressions of these quantities. The key point to under-stand this mistake is to be aware that the pure-component stateswhich appear in Eqs. (6) and (7) can be non-physically realizablestates. Such a situation always occurs when at a given tempera-ture, the pressure of the ideal mixture is between the vaporpressures of the two pure components. In this pressure range(where VLE necessarily arises), it is always observed volume andenthalpy change on the formation of an ideal mixture from its purecomponents at the same temperature and pressure . For the systemtoluene (1) ethylbenzene (2) at t 20 1C and P 2 kPa, the heatof mixing may be as high as 17.7 kJ mol 1 but the molar excessenthalpy never exceeds 0.005 kJ mol 1 . Ideal mixtures have thusin fact less remarkable properties than expected.

The knowledge of the correct mathematical expressions tocalculate ideal-mixture property changes on mixing as those givenin this article are extremely important for at least two reasons:

Ideal-mixture properties are the starting point to estimate the

properties of real mixtures. A non-correct estimation of the

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ideal-mixture property changes on mixing, will inevitably leadto a wrong estimation of the property changes on mixing of real mixtures through Eq. (24) .The molar Gibbs energy change on mixing is a key property onwhich the double-tangent construction of coexisting phases canbe applied. An incorrect estimation of g m ,id(T ,P ,z), can thus lead tounreliable and doubtful results like to state that an ideal liquidmixture cannot be in VLE with a perfect-gas mixture.

To conclude, we can assert that by using the ideal-solutionconcept, the chemical potential of a component i in a real (non-ideal) liquid phase ( rlp ), always writes:

61

By denition of the partial molar Gibbs energy change onmixing g mi , the chemical potential of a component i in a real(non-ideal) liquid phase ( rlp ), also always writes:

62

we however always can nd situations for which: g liquidi pure T , P a g stablei pure T , P , and by consequence: g

m , idi a RT ln xi and g

mi a RT ln a i

which in return implies: vm , idi a 0 and hm , idi a 0 that is v

m a vE andhm a hE .

Nomenclature

Latin letters

a molar Helmholtz energya i activity of component i

c P molar heat capacity at constant pressureEoS equation of state f i fugacity of pure component i

^

f i fugacity of component i in a mixture g molar Gibbs energy g =RT reduced Gibbs energyh molar enthalpykij binary interaction parameter involved in the classical

mixing rules of the PengRobinson EoSlhs left hand sideP pressureP sat i T vapor pressure of pure component i at temperature T R gas constant (8.314472 J mol 1 K 1 )rhs right hand side

s molar entropyT absolute temperatureu molar internal energyv molar volume xi mole fraction of component i in the liquid phase yi mole fraction of component i in the gas phase z i overall mole fraction of component iz overall mole fraction vector

Greek letters

gi activity coefcient of component i in a mixturej i fugacity coefcient of pure component i

^

j i fugacity coefcient of component i in a mixture

t a molar proportion of phase a

Subscripts and superscripts

E excess propertyid ideal mixtureidg ideal gas mixtureidl ideal liquid mixturem property change on mixing pgm perfect-gas mixturerlp real liquid phaserm real mixture

Acknowledgements

The authors would like to thank Professors Michael L. Michel-sen and Georgios M. Kontogeorgis from the Technical Universityof Denmark warmly for the exciting discussions we had whiledoing this research.

Appendix A. h E (T , P ,z), h m (T , P , z ) and h m , id (T , P ,z) calculationsfrom an equation of state

We believe it is important to explain how Figs. 3 bd werecalculated with the PengRobinson equation of state. Fig. 3ahighlights a vaporliquid equilibrium at t 20 1C and P 2 kPa:the liquid phase composition is x1 0.47 and the gas phasecomposition is y1 0.72.

(1) hE (T ,P ,z) calculation:For z 1 o 0.47, the mixture is thus liquid and the excessenthalpy was calculated as:

hE , liquid T , P , z h liquid T , P , z

|fflfflfflfflfflfflfflfflmolar enthalpy of the real liquid mixture h id , liquid T , P , z

|fflfflfflfflfflfflsee Eq : 18 h liquid T , P , z z 1 Uh

liquid1 pure T , P z 2 Uh

liquid2 pure T , P h i

A1

The stable state for component 1 is however gaseous andh liquid1 pure T , P was thus calculated with the smallest volume(non-stable liquid root) when the equation of state wassolved for pure component 1.For z 1 4 0.72, the mixture is gaseous and the excess enthalpywas calculated as:

hE , gas T , P , z h gas T , P , z

|fflfflfflfflfflfflffl{zmolar enthalpy of the real gas mixture h id , gas T , P , z

|fflfflfflfflfflfflsee Eq : 18 h gas T , P , z z 1 Uh

gas1 pure T , P z 2 Uh

gas2 pure T , P A2

The stable state for component 2 is however liquid andh gas2 pure T , P was thus calculated with the largest volume(non-stable gas root) when the equation of state was solvedfor pure component 2. A straight line was added in the two-phase region (0.47 r z 1 r 0.72).

(2) hm (T ,P ,z) calculation:Whatever the composition ( z 1 o 0.47 and z 1 4 0.72), theenthalpy change on mixing (see Eq. (20) ) was calculated as:

hm T , P , z hT , P , z

|fflfflfflfflffl{zfflfflfflmolar enthalpyof the mixture z 1 Uh

stable1 pure T , P z 2 Uh

stable2 pure T , P h i

hT , P , z

|fflfflfflfflffl{zfflfflfflmolar enthalpyof the mixture z 1 Uh

gas1 pure T , P z 2 Uh

liquid2 pure T , P h iA3

and a straight line was added in the two-phase region.

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(3) hm ,id (T ,P ,z) calculation:The ideal-mixture enthalpy change on mixing can obviouslybe simply determined as: hm ,id hm hE . A direct calculationcan also be performed following Eq. (26) . At t 20 1C andP 2 kPa, the stable states for component 1 and 2 are respec-tively gaseous and liquid. For z 1 o 0.47, the mixture is liquidand hm ,id (Eq. (26) ) is thus given by:

hm , id T , P , z z 1 U h liquid1 pure T , P hstable1 pure T , P h i z 2 U h liquid2 pure T , P hstable 2 pure T , P h i

z 1 U hliquid1 pure T , P h

gaseous1 pure T , P h i% z 1 UDvap H 1 20 1C A4

For z 1 4 0.72, the mixture is gaseous and hm ,id (Eq. (26) ) isgiven by:

hm , id T , P , z z 1 U h gaseous1 pure T , P h

stable1 pure T , P h i

z 2 U h gaseous2 pure T , P h

stable 2 pure T , P

h i z 2 U h gaseous2 pure T , P h

liquid2 pure T , P h i% z 2 UDvap H 2 20 1C A5

By the end, a straight line is added in the two-phase region(0.47 r z 1 r 0.72).

Appendix B. Calculation of the ideal-mixture volume,entropy, enthalpy, internal energy and constant pressure heatcapacity change on mixing.

The knowledge of the ideal-mixture Gibbs energy change onmixing allows calculating the other ideal-mixture propertychanges on mixing. Indeed, from basic thermodynamics, one has:

vm, id @

g m , id@P T , zsm , id @ g m , id@T P , z

hm , id g m , id T Usm , id

um , id hm , id P Uvm , id

c m , idP @hm , id

@T P , z

8>>>>>>>>>>>>>>>>>>>>>:

B1

For the three cases studied in Section 2.6.2 and for which theideal-mixture Gibbs energy change on mixing is respectivelygiven by Eqs. (28), (30), (33) , (34), (36) and (37) , the mathematicalexpressions for the other property changes on mixing are sum-marized in Tables B1 and B2 . The derivative of the vapor pressurewith respect to temperature may be needed to calculate sm ,id . Ithowever can be estimated from the ClausiusClapeyron equation:dP sat i =dT Dvap S i=Dvap V i. At low to moderate pressure, the volumeof vaporization veries:

Dvap V i %RT P sat i

B2

Table B1Mathematical expressions of the property changes on mixing for a binary liquid ideal mixture.

Case 1 g m , idliquid T , P , x RT x1 ln x1 x2 ln x2 Eq: 28

P 4 P sat 1 T 4 P sat 2 T v

m , idliquid T , P , x 0

sm , idliquid T , P , x g m , idliquid

T R x1 ln x1 x2 ln x2

hm , idliquid T , P , x 0

um , idliquid T , P , x 0

c m , idP , liquid T , P , x 0

Component 1 is liquidComponent 2 is liquid

The ideal mixture is liquid

Case 2 g m , idliquid T , P , x RT x1 lnx1 UP sat 1 T

P h i x2 ln x2h iEq: 33 P sat 1 T 4 P 4 P sat 2 T vm , idliquid T , P , x x1 RT P sm , idliquid T , P , x

g m , idliquidT

x1 UDvap S 1

R x1 lnx1 UP sat 1 T

P ! Dvap S 1R ! x2 ln x2 !hm , idliquid T , P , x x1 UDvap H 1 T

um, idliquid T , P , x x1 U RT Dvap H 1 c m , idP , liquid T , P , x 0

Component 1 is gaseousComponent 2 is liquid


Case 3 g m , idliquid T , P , x RT x1 ln

x1 UP sat 1 T P h i x2 ln x2 UP

sat 2 T P h ih iEq: 37 P o P sat 2 T o P sat 1 T Component 1 is gaseous vm , idliquid T , P , x x1 RT P x2 RT P RT P

sm , idliquid T , P , x g m , idliquid

T x1 UDvap S 1 x2 UDvap S 2

R x1 lnx1 UP sat 1 T

P ! Dvap S 1R ! x2 ln x2 UP sat 2 T P ! Dvap S 2R ! !

hm , idliquid T , P , x x1 UDvap H 1 T x2 UDvap H 2 T

um , idliquid T , P , x x1 U RT Dvap H 1 x2 U RT Dvap H 2 c m , idP , liquid T , P , x 0

Component 2 is gaseous


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Sandler, S.I., 2006. Chemical, biochemical, and engineering thermodynamics, 4thedition John Wiley and Sons, Inc., New York.


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