-Quadratic mixing rule for cubic equations of state

q-Quadratic mixing rule for cubic equations of state

Leonardo S Souza an Ernesto P Borges b Fernando LP Pessoa a

a Escola de Quiacutemica Universidade Federal do Rio de Janeiro RJ Brazilb Instituto de Fiacutesica Universidade Federal da Bahia Salvador BA Brazil

H I G H L I G H T S

The q-Mixing rule is a new nonquadratic mixing rule Correlations between species evaluated by means of the q-product The proposal is a generalization of van der Waals mixing rules The q-Mixing rule has been tested by evaluating vaporndashliquid equilibrium

a r t i c l e i n f o

Article historyReceived 8 January 2015Received in revised form7 April 2015Accepted 11 April 2015Available online 28 April 2015

KeywordsMixing rulesEquation of stateVaporndashliquid equilibrium

a b s t r a c t

We introduce a new nonquadratic mixing rule that models correlations between species by means of theq-product of mole fractions instead of the ordinary product The use of the q-product is a procedureadopted within the nonextensive statistical mechanics formalism in the generalization of the centrallimit theorem for strongly correlated systems The proposal is a generalization of the ordinary van derWaals quadratic law that is recovered when the so-called nonquadraticity binary parameter takes thelimit qij-1 The proposed q-mixing rule has been tested by evaluating vaporndashliquid equilibrium atdifferent temperatures for systems containing alcohol hydrocarbons and CO2 species that yielddepartures from ideality Two distinct approaches the Akaike Information Criterion and the F-testwere used to compare the q-mixing rule and the van der Waals mixing rule

amp 2015 Elsevier Ltd All rights reserved

1 Introduction

There have been several proposals of equations of state overmore than a century some based on empirical knowledge andothers with a theoretical background (Poling et al 2001) The firstsuccessful approach was the van der Waals equation of state (vander Waals and Rowlinson 1988) This equation is based on a semi-empirical theory in which pressure is considered as the sum oftwo contributions a repulsion term and an attraction term Theequation of state may be represented by

P frac14RTvb

a

f ethvTHORN eth1THORN

where the parameters a is the attractive parameter and b is themolar co-volume They depend on thermodynamic properties ofpure substances and on the composition for mixtures Better

accuracy in the predictions of thermodynamic properties likevapor pressure liquid density equilibrium ratios is achieved bymodifications of the original van der Waals equation Probably themost useful are those of Soave (1972) and Peng and Robinson(1976)

The evaluation of the parameters a and b has significantimportance for the accurate representation of experimental dataThe expressions for the parameters a and b used with PengndashRobinson equation of state was proposed in Peng and Robinson(1976) as follows

aethTTHORN frac14 045724R2T2

cPc

1thornk 1TTc

05 ( )2

kfrac14 037464thorn15422ω026922ω2

bfrac14 007780RTc

Pc eth2THORN

where ω is the acentric factor Tc and Pc are the critical tempera-ture and pressure respectively and R is the universal gas constantMixing rules can be derived from statistical mechanics or by using

Contents lists available at ScienceDirect

journal homepage wwwelseviercomlocateces

Chemical Engineering Science

httpdxdoiorg101016jces2015040220009-2509amp 2015 Elsevier Ltd All rights reserved

n Corresponding authorE-mail addresses leosilvasouza2001gmailcom (LS Souza)

ernestoufbabr (EP Borges) pessoaufrjbr (FLP Pessoa)

Chemical Engineering Science 132 (2015) 150ndash158

a phenomenological point of view from classical thermodynamics(Hall et al 1993) The widely used van der Waals Mixing Rule(vdw-MR) (Vidal 1978) are

aM frac14Xc

i

Xc

j

xixjaij eth3THORN

bM frac14Xc

i

Xc

j

xixjbij eth4THORN

where xi is the molar fraction of species i c is the number ofdifferent species in the mixture aii and bii are the pure componentparameters (aii ai and bii bi) and aij and bij (ia j) are crossparameters (Wei and Sadus 2000 Peng and Robinson 1976Valderrama 2003) that are represented by the combining rules(Vidal 1978)

aij frac14 aiiajj $05 1kij

$ eth5THORN

bij frac14biithornbjj

21 lij $

eth6THORN

where kij frac14 kji kii frac14 0 lij frac14 lji lii frac14 0 and consequently aij frac14 aji andbij frac14 bji (ia j) kij and lij are binary interaction parameters asso-ciated with a and b (Poling et al 2001) that can be adjusted byexperimental data

The vdW-MR have a quadratic dependence on composition (bM isreduced to a linear dependence if lij frac14 0) This functional form is notable to describe all mixtures and several works have proposeddifferent nonquadratic rules eg Wong and Sandler (1992) Adachiand Sugie (1986) Won (1983) and Higashi et al (1994) Someproposed models take into account the excess Gibbs energy atinfinite pressure calculated from a liquid-phase activity coefficientmodel (Huron and Vidal 1979 Michelsen 1990)

The present work introduces a new nonquadratic mixing rulefor cubic equations of state that differs from vdW-MR once it mayconsider strong correlations following the lines of nonextensivestatistical mechanics (Tsallis 1988 Borges 2004 Moyano et al2006) through a nonquadraticity parameter qij

The work is organized as follows Section 2 introduces somebasic concepts of nonextensive statistical mechanics and intro-duces the semi-empirical model for the q-Mixing Rule (q-MR) Themethodology results and discussions appear in Section 3 andfinally Section 4 is dedicated to the conclusions

2 Nonextensive statistical mechanics

With the advent of the nonextensive statistical mechanics as ageneralization of the BoltzmannndashGibbs statistical mechanics it becameclear over the years that the new formalism was able to representcertain classes of systems that were not properly described byBoltzmannndashGibbs statistical mechanics Indeed since the original paper(Tsallis 1988) more than two decades ago many works have beendeveloped concerning nonlinear systems The generalized entropy is

Sq kB1

PWi frac14 1 p

qi

q1 eth7THORN

with pi being the probability of the microscopic state i W being thenumber of microscopic states kB being a positive constant (theBoltzmann constant) and q being the entropic index If q-1 Eq (7)recovers the celebrated BoltzmannndashGibbs entropy

S kBXW

i frac14 1

pi ln pi eth8THORN

Formal and logical arguments considering an analogy betweenEqs (8) and (7) have lead to the definition of generalized functions

(Tsallis 1994) particularly the q-logarithm function

lnqethxTHORN x1q11q

ethx40THORN eth9THORN

and its inverse the q-exponential function

expqethxTHORN exq frac121thorneth1qTHORNx)1=eth1qTHORN if frac121thorneth1qTHORNx)400 if frac121thorneth1qTHORNx)r0

(

eth10THORN

The usual logarithmic and exponential functions are recovered in thelimit q-1

The maximization of the Tsallis entropy Eq (7) with theconstraint of constant generalized mean energy leads to thegeneralized canonical ensemble distribution (Tsallis 1988 Tsalliset al 1998)

p Eieth THORN prop expq βqEi amp

eth11THORN

where βq is the Lagrange parameter that is related to the inversetemperature Ei is the energy of the ith state and p(Ei) is theprobability of the state with energy Ei be occupied One of thecentral differences between Tsallis and Boltzmann weights is thatthe former presents power law tails (long-lasting for q41 andabruptly vanishing for qo1) while the latter has exponential tails

The difference between Boltzmann and Tsallis weights may be alsounderstood within the following comparison It is known that thepressure P of a hypothetical static isothermal ideal gas atmospherewith gradient given by dP=dhfrac14 mgρ decreases with height accord-ing to an exponential law (Halleys law see for instance Feynmanet al 1963) PethhTHORN frac14 P0expethmgh=kBTTHORN where P0 is the pressure atheight hfrac140 g is the gravitational acceleration A similar expressionholds for the number density ρ (number of moleculesvolume) Thenumerator of the exponent is the potential energy of a molecule withmass m and the denominator is its thermal energy Thermal equili-brium is a very strong simplifying hypothesis that certainly is not validto our atmosphere If we consider that the temperature linearlydecreases with height ie with a constant temperature gradientinstead of a zero temperature gradient mdash the second simplesthypothesis possible mdash then we find that the pressure decays asPethhTHORN frac14 P0 expqethmgh=kBT0THORN with qfrac14 1kBγ=gm and γ frac14 dT=dh isthe negative of the atmospheric temperature gradient Once P frac14 ρkBTand TethhTHORN frac14 T0γh ρethhTHORN frac14 ρ0frac12expqethmgh=kBT0THORN)q In both cases wehave a ratio between the potential and the thermal energies but in theformer we have an exponential law (Boltzmann weight) while in thelatter we have a q-exponential law (Tsallis weight) The case γ frac14 0recovers Halleys law

The introduction of the generalized q-logarithm and q-expo-nential functions has allowed the development of a consistentgeneralized nondistributive algebra based on the q-algebraicoperators (Nivanen et al 2003 Borges 2004) q-addition xqyq-difference x⊖qy q-product x+qy and q-ratio x⊘qy representedas

xqyfrac14 xthornythorneth1qTHORNxy

x⊖qyfrac14xy

1thorneth1qTHORNyya

1q1

x+qyfrac14 x1qthorny1q1 (1=eth1qTHORN

thorn ethx y40THORN

x⊘qyfrac14 x1qy1qthorn1 (1=eth1qTHORN

thorn ethx y40THORN eth12THORN

The subscript thorn in the q-product and in the q-ratio is a short-cutthat means frac12A)thorn frac14maxethA0THORN ie whenever the basis is negative orzero (x1qthorny1q1r0 for the q-product and x1qy1qthorn1r0for the q-ratio) the q-operation (q-product or q-ratio) is defined aszero instead of being calculated by its corresponding mathematicalexpression This feature known as cut-off is essential for theconsistency of the formalism and it is also present in the definitionof the q-exponential see Eq (10) The q-product is commutative

LS Souza et al Chemical Engineering Science 132 (2015) 150ndash158 151

(x+qyfrac14 y+qx) but it is not distributive in relation to the ordinaryproduct aethx+qyTHORNa ethaxTHORN+qyax+qethayTHORN It is also not distributive inrelation to the q-addition ethaqbTHORN+qxa etha+qxTHORNqethb+qxTHORN 8qa1This q-algebra particularly the q-product has a central role in thegeneralization of the central limit theorem and in the general-ization of the Fourier transform (Umarov et al 2008 Jauregui andTsallis 2010) It will be clear in the following that the present workis based squarely in the q-product Some properties of q-functionsand q-algebra may be found in Yamano (2002) Naudts (2002)Cardoso et al (2008) Lobao et al (2009) and Tsallis (2009)

There have been proposed generalizations of equations of statefollowing the lines of nonextensive statistical mechanics The idealgas has been revisited in Silva Jr et al (1998) Plastino and Lima(1999) and the equation of state is written as PV frac14 IethqTHORNNRTnwhere

IethqTHORN frac141

1eth1qTHORNβn12mv2

) ethq=eth1qTHORNTHORNd3v

Z1eth1qTHORNβn1

2mv2

) 1=eth1qTHORN

dv

βn frac14 1=ethRTnTHORN and Tn is a function that recovers the temperature T inthe limit q-1 The van der Waals-EoS has also been revisitedalong these lines as Martiacutenez et al (2001)

Pthornaq1V

2

ethVbTHORN frac14NRT eth13THORN

where aq frac14 a βq=βeth1qTHORNh i

and βq=βfrac14 1thorneth1qTHORNβ (

aN2=ω with

βfrac14 1=ethRTTHORN There are other works dealing with nonextensiveclassical gas Negative specific heat and polytrope-type relationare analytically explored in Abe (1999) Equipartition and virialtheorems are shown to be valid 8q in Martiacutenez et al (2000)Internal energy and energy correlation are considered in Abe et al(2001) The kinetic nonextensive generalization of the Maxwellianideal gas is discussed in Lima and Silva (2005) The Joule JoulendashThomson and second virial coefficients are evaluated in Zheng andDu (2007) It was recently shown that the ideal gas in a finite heatreservoir requires the q-entropy (Biroacute 2013) Heat capacities ofsimple diatomic gases (N2 O2 CO) present nonextensive behavior atlow temperatures (Guo and Du 2009) A recent model based on theq-exponential for evaluating solubility of solids in supercriticalsolvent was advanced in Tabernero et al (2014)

The present work is confined to mixtures rather than purecomponents and we focus on the mixing rules that are applied to

the parameters of cubic equations of state We have chosen thePengndashRobinson equation of state (Peng and Robinson 1976) but theexpressions for the mixing rules presented below in Section 21 mayequally be applied to other equations of state eg Soave equation ofstate (Soave 1972)

21 q-mixing rules

Within the pairwise additivity approximation with shortndashrangeinteractions the parameter aM of the mixture is considered as anaverage of the interactions between binaries weighted by the jointprobability of finding that particular pair of molecules together ie aM frac14

Pijpijaij If independence between molecules is consid-

ered the joint probability of finding species i in the neighborhoodof species j is simply the product of finding i and j in the bulksolution that is represented by its molar fraction pij frac14 pipj xixjThus the quadratic nature of the mixing rule derives from thehypothesis of independence Deviation from this scenario isusually considered by the introduction of an additional parameterkij according to Eq (3) This structure preserves the quadraticnature ie the independence hypothesis The present modelfollows the procedure used in the generalization of the centrallimit theorem It considers the joint probability of stronglycorrelated systems (that may be a consequence of local inhomo-geneities long-range interactions or other features that leads tothe failure of the independence hypothesis) as the q-product of theindividual probabilities rather than the ordinary product (Tsallis2005 Tsallis et al 2005) The proposed q-Mixing Rule is

aqM frac14P

ijaijethxi+qij xjTHORNPijethxi+qij xjTHORN

eth14THORN

where qji frac14 qij qii frac14 1 and xi+qij xj is the q-product between thenon-negative numbers xi and xj The normalizing factorP

ijethxi+qij xjTHORN in the denominator of Eq (14) is necessary to avoidthe MichelsenndashKistenmacher syndrome (Michelsen andKistenmacher 1990) The parameter qij radically changes the usualmixing rule it exhibits a nonquadratic dependence on the com-position (that is obviously recovered as qij-1) as can be seenbelow in Figs 1 and 2 This break with the quadratic dependenceon the composition in the mixing rule has led us to call qij asnonquadraticity parameter within the present context

Fig 1 illustrates the dependence of aqM with the molar fractionfor different values of q12 in a hypothetical binary mixture withk12 frac14 0 aqM is limited by q12 frac14 0 with

a0M frac14x21a11thornx22a22

x21thornx22 eth15THORN

Fig 1 q-Mixing rule Eq (14) for a binary hypothetical mixture with arbitrarilychosen values a1 frac14 1 and a2 frac14 2 and k12 frac14 0 The choice was driven by the need togive a good visual representation of the influence of the parameter q12 on thecomposition Different values of the parameter q12 are indicated q12 frac14 1 (dashedline) q12 frac14 0 (dashedndashdotted line red online) q12 frac141 (dashedndashdashedndashdottedline blue online) A straight line that corresponds to q12 frac14 1 andk12 frac14 1etha11thorna22THORN=eth2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia11a22

pTHORN is indicated by a solid line for comparison (For

interpretation of the references to color in this figure caption the reader is referredto the web version of this paper)

Fig 2 Effect of the parameter k12 in the q-MR Eq (14) for a binary hypotheticalmixture with a1 frac14 1 and a2 frac14 2 q12 frac14 1 (dashed line) q12 frac14 01 (solid line) andk12 frac14 0 (black) k12 frac14 03 (red online) k12 frac14 03 (blue online) (For interpretationof the references to color in this figure caption the reader is referred to the webversion of this paper)

LS Souza et al Chemical Engineering Science 132 (2015) 150ndash158152

that does not depend on the cross parameter a12 and q12 frac141with

a1M frac14x21a11thornx22a22thorn2a12frac12x1Hethx2x1THORNthornx2Hethx1x2THORN)

x21thornx22thorn2frac12x1Hethx2x1THORNthornx2Hethx1x2THORN) eth16THORN

and H(u) is the Heaviside step function The parameter q12a1introduces two distinct regions with positive and negative concavities(the concavity of the usual vdW-MR is fixed given by a11thorna222a12)Proper fitting of the parameter k12 leads to a shift of the inflectionpoint as shown in Fig 2 and thus allowing better description ofexperimental data It is worth noting that this shift may be such thatthe inflection point can be located outside the physical region0rx1r1 so the model with q12a1 is able to represent datawithout the inflection point but in a nonquadratic form

Derivatives of Eq (14) are necessary for the calculation ofvarious thermodynamic properties of mixtures eg fugacities andthey are given by

partaqMpartNi

frac141P

ijethxi+qij xjTHORN

X

ij

partaijethxi+qij xjTHORNpartNi

aqM

X

ij

partethxi+qij xjTHORNpartNi

8lt

9=

eth17THORN

with

X

ij


frac142NT

X

i

xi+qij xj amp

xj xqiji xqijj 1δij $ amp( )

2NT

xj+qij xj ampqij

x1qijj

X

ij


frac142NT

X

i

aij xi+qij xj amp


2ajjNT

xj+qij xj ampqij

x1qijj eth18THORN

where NT is the total number of moles and δij is the Dirac delta Asit happens in this generalized formalism all equations are reducedto the ordinary ones in the limit qij-1

3 Parameter estimation results and discussions

In this section a comparative analysis between the performanceof q-MR and vdW-MR for the prediction of vaporndashliquid equilibrium(VLE) of systems at different temperatures is shown The PengndashRobinson equation of state (Peng and Robinson 1976) together withvdW-MR and q-MR were used for calculating the vaporndashliquidequilibrium for the following binary systems at different tempera-tures CO2thorn2-butanol (Elizalde-Solis and Galicia-Luna 2010) etha-nolthornglycerol (Shimoyama et al 2009) ethanethorndecane (Gardeleret al 2002) CO2thornethanol (Galicia-Luna et al 2000) CO2thornstyrene(Tenoacuterio Neto et al 2013) CO2thornacetic acid (Bamberger et al2000) CO2thornH2S (Chapoy et al 2013) and ethylenethorn1-decanol(Gardeler et al 2002) The fitting parameters of the model are k12and q12 (we remind the reader that the vdW-MR uses fixed q12 frac14 1so it is not a fitting parameter for this case but only for q-MR) The

Table 1Relative errors between experimental and calculated values of pressure (in bar) and vapor mole fraction of species 1

System Temperature (K) Classical mixing rule q-mixing rule

k12 ΔpethTHORN Δy k12 q12 ΔpethTHORN Δy

CO2thorn2-butanol 31321 01144 62 00005 01238 10506 24 0000533312 01052 108 0003 01220 10695 40 000136306 00917 99 0007 01069 10683 50 0008All temperatures 01012 92 0002 01162 10656 46 0003

Ethanolthornglycerol 49300 01047 51 0002 00487 11698 54 000354300 00789 72 002 00477 11662 37 00257300 00926 62 002 00502 11877 32 003All temperatures 00848 68 001 00487 11699 47 001

Ethanethorndecane 41095 00166 57 0005 00118 09754 21 000544425 00015 28 001 00064 09831 17 001All temperatures 00071 45 0008 00024 09817 30 0008

CO2thornethanol 31282 00931 12 0006 00928 09966 10 000634840 00889 63 0006 01004 10411 11 00137300 00852 83 001 01013 10620 18 001All temperatures 00881 61 0006 00963 10352 34 0008

CO2thornstyrene 30300 00550 53 ndash 00383 07982 26 ndash

31300 00781 49 ndash 00213 08151 23 ndash

32300 00735 52 ndash 00392 07890 25 ndash

All temperatures 00755 51 ndash 00328 07984 25 ndash

CO2thornacetic acid 31320 00492 30 00009 00512 10178 09 0000833320 00586 30 0002 00623 10284 07 000235320 00697 48 0005 00681 10300 17 0006All temperatures 00623 43 0003 00636 10320 35 0003

Ethylenethorn1-decanol 30815 00314 45 00003 00238 09859 38 0000431815 00323 42 0004 00260 09881 23 0004All temperatures 00319 43 0002 00251 09875 29 0002

CO2thornH2S 25841 00977 11 001 01044 08488 09 00127315 00980 12 001 00959 10884 02 00129347 01019 09 001 01053 08729 05 001All temperatures 01004 11 001 01031 09043 09 001


parameters were estimated using the boiling point calculationprogrammed in FORTRAN language and the particle swarm opti-mization algorithms from ESTIMA packages (Schwaab et al 2008Alberton et al 2013) was used to minimize the objective functiongiven by Eq (19)

Considering that the dependent variables yexp and Pexp areuncorrelated (yexp is the mole fraction of the vapor phase) and

assuming that the residuals are independent and normally dis-tributed with zero mean the parameters can be evaluated byminimizing the objective function

FobethθTHORN frac14XNexp

n

ycalcn ethx T θTHORNyexpn $2

σ2yexp n

thornPcalcn ethx T θTHORNPexp

n

amp2

σ2Pexp n

eth19THORN

where θ represents the set of fitting parameters Nexp is the numberof experiments and Pexp

n and Pcalcn are respectively the experimental

Fig 3 Pressurendashcomposition diagram for the CO2 (1)thorn2-butanol (2) systemExperimental data with error bars and calculated values with q-MR (solid line)and vdW-MR (dashed line) are displayed The main panel (graph at the left) showsthe mole fraction of species 1 (x1) of the liquid phase and the three thin panels atthe right show the mole fraction of species 1 (y1) of the vapor phase

Fig 4 Pressurendashcomposition diagram for the ethanol (1)thornglycerol (2) systemExperimental data with error bars and calculated values with q-MR (solid line) andvdW-MR (dashed line) are displayed The main panel (graph at the left) shows themole fraction of species 1 (x1) of the liquid phase and the three thin panels at theright show the mole fraction of species 1 (y1) of the vapor phase

Fig 5 Pressurendashcomposition diagram for the ethane (1)thorn1-decane (2) systemExperimental data with error bars and calculated values with q-MR (solid line) andvdW-MR (dashed line) are displayed The main panel (graph at the left) shows themole fraction of species 1 (x1) of the liquid phase and the two thin panels at theright show the mole fraction of species 1 (y1) of the vapor phase

Fig 6 Pressurendashcomposition diagram for the CO2 (1)thornethanol (2) system Experi-mental data with error bars and calculated values with q-MR (solid line) and vdW-MR (dashed line) are displayed The main panel (graph at the left) shows the molefraction of species 1 (x1) of the liquid phase and the three thin panels at the rightshow the mole fraction of species 1 (y1) of the vapor phase

Fig 7 Pressurendashcomposition diagram for the CO2 (1)thornstyrene (2) system at 303 K313 K and 323 K Experimental data with error bars and calculated values with q-MR (solid line) and vdW-MR (dashed line) are displayed The graph shows the molefraction of species 1 (x1) of the liquid phase and the mole fraction of species 1 (y1)of the vapor phase

Fig 8 Pressurendashcomposition diagram for the CO2 (1)thornacetic acid (2) systemExperimental data with error bars and calculated values with q-MR (solid line) andvdW-MR (dashed line) are displayed The main panel (graph at the left) shows themole fraction of species 1 (x1) of the liquid phase and the three thin panels at theright show the mole fraction of species 1 (y1) of the vapor phase


and calculated pressure of system for the nth data set and σ is theestimated standard deviation for each experimental point Uncer-tainty of experimental data for each considered systemwas providedby the corresponding references according to two cases they wereexplicitly given for the following systems CO2thornacetic acid CO2thornH2Sand CO2thornstyrene Estimated uncertainties were provided for thesystems CO2thorn2-butanol ethanolthornglycerol ethanethorn1-dodecaneCO2thornethanol and ethylenethorn1-decanol systems

Table 1 shows the fitted parameters k12 and q12 and deviationsbetween experimental data and predicted results according to

ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1

jyexpi ycalci j eth21THORN

It can be seen that the PengndashRobinson equation of state with q-MR presents a good prediction of the phase behavior for allsystems with smaller deviations in the pressure than those usingvdW-MR

Deviations in the vapor phase concentration using PengndashRobin-son equation of state were small for both mixing rules Ref TenoacuterioNeto et al (2013) does not bring data for the vapor phase of thesystem CO2thornstyrene At low pressures the results calculated withq-MR for the system CO2thorn2ndashbutanol are better than those

Fig 9 Pressurendashcomposition diagram for the ethylene (1)thorn1-decanol (2) system Experimental data with error bars and calculated values with q-MR (solid line) and vdW-MR (dashed line) are displayed Both graphs show the mole fraction of species 1 (x1) of the liquid phase and the mole fraction of species 1 (y1) of the vapor phase

Fig 10 Pressurendashcomposition diagram for the CO2 (1)thornH2S (2) system Experimental data with error bars and calculated values with q-MR (solid line) and vdW-MR(dashed line) are displayed The graph shows the mole fraction of species 1 (x1) of the liquid phase and the mole fraction of species 1 (y1) of the vapor phase

Fig 11 Influence of the parameters q12 and k12 on the pressure composition diagram The curve for the system CO2 (1)thornethanol (2) at 34840 K is taken as a baseline forcomparison (solid black line) as it is in Fig 6 Left panel displays different values of q12 for fixed k12 frac14 01004 Right panel displays different values of k12 for fixedq12 frac14 10411


calculated with vdW-MR however the predicted values by thesemodels for higher pressures are similar The results obtained for thesystems CO2thornacetic acid and CO2thornstyrene with q-MR are signifi-cantly better than with the conventional mixing rule by providingqualitatively correct predictions of pressure and vapor concentra-tion However both mixing rules show similar results for theCO2thornH2S system The q-MR is able to present a smooth inflectionfor the system CO2thornstyrene which is in agreement with the dataFigs 3ndash10 show pressurendashcomposition diagrams with comparisonbetween experimental data and calculated results for the consid-ered examples

Some pressurendashcomposition diagrams particularly Figs 3 4 and 7display an inflection point This change in the concavity is due to thestronger influence of interactions between molecules under highpressures and the ability of the q-MR model to represent thisbehavior is a consequence of the inflection point of the curves ofthe interaction parameter of the mixture aqM versus composition seeFig 1 and 2 Fig 11 shows the effect of the parameters q12 and k12 onthe inflection point It can be seen that as the parameter q12 departsfrom unit the difference in concavities becomes more pronouncedwhereas the parameter k12 is related to a shift of the curves

The q-MR model has two binary parameters while the originalvdW-MR has only one In order to make a unbiased comparison wehave implemented the Akaike Information Criterion (AIC) (Akaike1974) that is a suitable tool for these situations It is defined as

AICi frac14 2log ethmaximum likelihoodTHORNthorn2ki eth22THORN

If the standard deviations σi are known and the residuals are normallydistributed the AIC is written as

AICi frac14 Foptob ethθTHORNthorn2kithorn2kiethkithorn1THORNnki1

eth23THORN

where ki is the number of parameters of the model i Nexp is thenumber of experiments and θopt is the set of optimal parameters (inour case parameters estimated for all temperatures as showed inTable 1) According to the Akaike procedure the more appropriatemodel among those that are being compared is that one which resultsin the lower AIC value The results are shown in Table 2 It is alsouseful to compare the relative AIC values and this is shown in the lastcolumn of the Table Values less than 1 indicate that the q-MR is betterthan vdW-MR All considered systems present relative AIC indexsmaller than 1 indicating that the improvement achieved by the q-MR is not simply due to the existence of an additional parameter

We have also considered the F-test as an additional criteriumfor discrimination between models with different numbers ofparameters (Ludden et al 1994 Motulsky and Christopoulos2004) The F-ratio is defined as

Fratio frac14FvdWMRob ethθoptTHORNFqMR

ob ethθoptTHORN

FqMRob ethθoptTHORN

dfqMR

dfvdWMRdfqMR eth24THORN

where dfqMR and dfvdWMR are the degrees of freedom(df i frac14Nki) of the equation of state with q-MR and the equationof state with vdW-MR respectively The calculated F-ratios areshowed in Table 3 and they are compared to the critical F-ratiostaken from the F-Distribution Table (with the same number ofdegrees of freedom) for three different values of significance levelαfrac14 001 005 and 01

The greater the value of the F-ratio the greater the statisticalsignificance of the q-MR model compared to the vdW-MR modelOnly one system considered in the present work presented F-ratiosmaller than the critical F-ratio the calculated F-ratio for thesystem CO2thornacetic acid is slightly smaller than F-ratioα frac14 001

Table 2AIC values (all temperatures have been considered)

System Number of data points FqMRob ethθoptTHORN FvdWMR

ob ethθoptTHORN AICqMR AICvdWMR AICqMR

AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038

Ethanolthornglycerol 22 511312 841957 511712 842157 061

Ethanethorndecane 22 668610 10 31719 669010 10 31919 065

CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016

CO2thornacetic acid 20 17 86615 24 49074 17 87015 24 49274 073

Ethylenethorn1- decanol 20 13 12617 19 34319 13 13017 19 34519 068

CO2thornH2S 18 402409 64 53757 402809 64 53957 006

Table 3F-ratio (all temperatures considered) and significance levels of αfrac14 001 αfrac14 005 and αfrac14 01

System F-ratio F-ratioα frac14 001 F-ratioα frac14 005 F-ratioα frac14 01

CO2thorn2-butanol 431 77 42 29

Ethanolthornglycerol 129 81 43 30

Ethanethorndecane 109 81 43 30

CO2thornethanol 708 73 41 28

CO2thornstyrene 81983 85 45 30

CO2thornacetic acid 67 82 44 30

Ethylenethorn1-decanol 85 83 44 30

CO2thornH2S 2406 85 45 30


4 Conclusions

This work introduces a new nonquadratic mixing rule that is ageneralization of the usual quadratic mixing rule which is recoveredin a special limit qij-1 The q-mixing rule here introduced general-ized through the nonquadraticity parameter qij is free from theMichelsenndashKistenmacher syndrome It is based on the q-product thatis able to reflect correlations Departures from the quadratic nature ofthe mixing rule as described by the q-mixing rule may be so strongas to present an inflection point in the graph of the parameter of themixture aqM as a function of composition This is distinctive from theoriginal parabolic mixing rule which has constant concavity Thisfeature is also responsible for the change in concavity in thepressurendashcomposition diagram specially at high pressures wheninteraction between molecules becomes more significant It is acommon practise to use the binary parameter kij with quadraticmixing rules to increase the fitting power of the models though thisparameter preserves the quadratic nature of the rule We also use theparameter k12 in our proposal Indeed it is essential in the presentmodel once it allows proper adjustment of the inflection point of themixing rule

We have used the PengndashRobinson equation of state in thepresented examples but the procedure can readily be applied tovirtually any cubic equation of state that uses mixing rules agrave la vander Waals

We have exemplified the usefulness of the proposed q-mixingrule with vaporndashliquid equilibrium calculations for binary systemscontaining carbon dioxide (CO2thorn2-butanol CO2thornethanol CO2thorn-styrene CO2thornacetic acid CO2thornH2S) and also for the mixturesethanolthornglycerol ethanethorndecane and ethylenethorn1-decanol Wehave used two procedures to compare the proposed model withthe usual one the Akaike Information Criterion and the F-test andthe q-MR has succeeded in all instances considered

Acknowledgements

Leonardo S Souza and Fernando LP Pessoa thank ANP (Brazi-lian agent) for financial support Ernesto P Borges acknowledgesNational Institute of Science and Technology for Complex Systemsand FAPESB (Brazilian agencies)

References

Abe S Martiacutenez S Pennini F Plastino A 2001 Classical gas in nonextensiveoptimal Lagrange multipliers formalism Phys Lett A 278 (5) 249ndash254 ISSN0375-9601

Abe S 1999 Thermodynamic limit of a classical gas in nonextensive statisticalmechanics negative specific heat and polytropism Phys Lett A 263 (4ndash6)424ndash429 ISSN 0375-9601

Adachi Y Sugie H 1986 A new mixing rule-modified conventional mixing ruleFluid Phase Equilibria 28 (2) 103ndash118

Akaike H 1974 A new look at the statistical model identification IEEE TransAutom Control 19 (6) 716ndash723

Alberton KPF Alberton AL di Maggio JA Diaz MS Secchi AR 2013Accelerating the parameters identifiability procedure set by set selectionComput Chem Eng 55 181ndash197

Bamberger A Sieder G Maurer G 2000 High-pressure (vaporthorn liquid) equili-brium in binary mixtures of (carbon dioxidethornwater or acetic acid) at tem-peratures from 313 to 353 K J Supercrit Fluids 17 (2) 97ndash110

Biroacute TS 2013 Ideal gas provides q-entropy Physica A Stat Mech Appl 392 (15)3132ndash3139 ISSN 0378-4371

Borges EP Nonextensive local composition models in theories of solutions arxivpreprint arXiv12060501

Borges EP 2004 A possible deformed algebra and calculus inspired in nonexten-sive thermostatistics Physica A Stat Mech Appl 340 (1ndash3) 95ndash101 ISSN0378ndash4371

Cardoso PGS Borges EP Lobatildeo TCP Pinho STR 2008 Nondistributivealgebraic structures derived from nonextensive statistical mechanics J MathPhys 49 (9) 093509

Chapoy A Coquelet C Liu H Valtz A Tohidi B 2013 Vapourndashliquid equilibriumdata for the hydrogen sulphide (H2S)thorncarbon dioxide (CO2) system at tem-peratures from 258 to 313 K Fluid Phase Equilibria 356 223ndash228

Elizalde-Solis O Galicia-Luna LA 2010 Vaporndashliquid equilibria and phasedensities at saturation of carbon dioxidethorn1-butanol and carbon dioxidethorn2-butanol from 313 to 363 K Fluid Phase Equilibria 296 (1) 66ndash71

Feynman RP Leighton RB Sands ML 1963 The Feynman Lectures on Physicsvol 1 Addison-Wesley Publishing Co Reading Massachusetts

Galicia-Luna LA Ortega-Rodriguez A Richon D 2000 New apparatus for thefast determination of high-pressure vaporndashliquid equilibria of mixtures and ofaccurate critical pressures J Chem Eng Data 45 (2) 265ndash271

Gardeler H Fischer K Gmehling J 2002 Experimental determination of vaporndashliquid equilibrium data for asymmetric systems Ind Eng Chem Res 41 (5)1051ndash1056

Guo L Du J 2009 Heat capacity of the generalized two-atom and many-atom gasin nonextensive statistics Physica A Stat Mech Appl 388 (24) 4936ndash4942ISSN 0378-4371

Hall KR Iglesias-Silva GA Mansoori GA 1993 Quadratic mixing rules forequations of state origins and relationships to the virial expansion Fluid PhaseEquilibria 91 (1) 67ndash76

Higashi H Furuya T Ishidao T Iwai Y Arai Y Takeuchi K 1994 An exponent-type mixing rule for energy parameters J Chem Eng Jpn 27 (5) 677ndash679

Huron M-J Vidal J 1979 New mixing rules in simple equations of state forrepresenting vapourndashliquid equilibria of strongly non-ideal mixtures FluidPhase Equilibria 3 (4) 255ndash271 ISSN 0378-3812

Jauregui M Tsallis C 2010 New representations of π and Dirac delta using thenonextensive statistical-mechanics q-exponential function J Math Phys 51 (6)063304

Lima JAS Silva R 2005 The nonextensive gas a kinetic approach Phys Lett A338 (35) 272ndash276 ISSN 0375-9601

Lobao TCP Cardoso PGS Pinho STR Borges EP 2009 Some properties ofdeformed q-numbers Braz J Phys 39 (2A) 402ndash407

Ludden TM Beal SL Sheiner LB 1994 Comparison of the Akaike InformationCriterion the Schwarz criterion and the F test as guides to model selectionJ Pharmacokinet Biopharm 22 (5) 431ndash445

Martiacutenez S Pennini F Plastino A 2000 Equipartition and virial theorems in anonextensive optimal Lagrange multipliers scenario Phys Lett A 278 (12)47ndash52 ISSN 0375-9601

Martiacutenez S Pennini F Plastino A 2001 Van der Waals equation in a non-extensive scenario Phys Lett A 282 (4ndash5) 263ndash268 ISSN 0375-9601

Michelsen ML Kistenmacher H 1990 On composition-dependent interactioncoefficients Fluid Phase Equilibria 58 (12) 229ndash230 ISSN 0378-3812

Michelsen ML 1990 A modified HuronndashVidal mixing rule for cubic equations ofstate Fluid Phase Equilibria 60 (1) 213ndash219

Motulsky H Christopoulos A 2004 Fitting Models to Biological Data using Linearand Nonlinear Regression A Practical Guide to Curve FittingOxford UniversityPress

Moyano LG Tsallis C Gell-Mann M 2006 Numerical indications of a q-generalised central limit theorem Europhys Lett 73 (6) 813ndash819

Naudts J 2002 Deformed exponentials and logarithms in generalized thermo-statistics Physica A Stat Mech Appl 316 (1ndash4) 323ndash334 ISSN 0378-4371

Nivanen L le Mehaute A Wang QA 2003 Generalized algebra within anonextensive statistics Rep Math Phys 52 (3) 437ndash444

Peng DY Robinson DB 1976 A new two-constant equation of state Ind EngChem Fundam 15 (1) 59ndash64

Plastino AR Lima JAS 1999 Equipartition and virial theorems within generalthermostatistical formalisms Phys Lett A 260 (1) 46ndash54

Poling BE Prausnitz JM OConnell JP 2001 The Properties of Gases and Liquidsvol 5 McGraw-Hill New York

Schwaab M Biscaia Jr EC Monteiro JL Pinto JC 2008 Nonlinear parameterestimation through particle swarm optimization Chem Eng Sci 63 (6)1542ndash1552

Shimoyama Y Abeta T Zhao L Iwai Y 2009 Measurement and calculation ofvaporndashliquid equilibria for methanolthorn glycerol and ethanolthornglycerol systemsat 493ndash573 K Fluid Phase Equilibria 284 (1) 64ndash69

Silva Jr R Plastino AR Lima JAS 1998 A Maxwellian path to the nonextensivevelocity distribution function Phys Lett A 249 (5ndash6) 401ndash408 ISSN 0375-9601

Soave G 1972 Equilibrium constants from a modified RedlichndashKwong equation ofstate Chem Eng Sci 27 (6) 1197ndash1203

Tabernero A Vieira de Melo SAB Mammucari R Martiacuten EM del Valle NR2014 Foster Modelling solubility of solid active principle ingredients in sc-CO2

with and without cosolvents a comparative assessment of semiempiricalmodels based on Chrastils equation and its modifications J Supercrit Fluids93 91ndash102

Tenoacuterio Neto ET Kunita MH Rubira AF Leite BM Dariva C Santos AFFortuny M Franceschi E 2013 Phase Equilibria of the Systems CO2thornStyreneCO2thornSafrole and CO2thornStyrenethornSafrole J Chem Eng Data 58 (6) 1685ndash1691

Tsallis C Mendes RS Plastino AR 1998 The role of constraints within general-ized nonextensive statistics Physica A Stat Theor Phys 261 (3ndash4) 534ndash554ISSN 0378-4371

Tsallis C Gell-Mann M Sato Y 2005 Asymptotically scale-invariant occupancyof phase space makes the entropy Sq extensive Proc Natl Acad Sci USA 102(43) 15377ndash15382

Tsallis C 1988 Possible generalization of Boltzmann Gibbs statistics J Stat Phys52 (1ndash2) 479ndash487 ISSN 0022-4715 (Print) 1572ndash9613 (Online)


Tsallis C 1994 What are the numbers that experiments provide Quim Nova 17(6) 468ndash471

Tsallis C 2005 Nonextensive statistical mechanics anomalous diffusion andcentral limit theorems Milan J Math 73 (1) 145ndash176

Tsallis C 2009 Introduction to Nonextensive Statistical Mechanics Approaching aComplex WorldSpringer New York

Umarov S Tsallis C Steinberg S 2008 On a q-central limit theorem consistentwith nonextensive statistical mechanics Milan J Math 76 (1) 307ndash328 ISSN1424-9286 (Print) 1424-9294 (Online)

Valderrama JO 2003 The state of the cubic equations of state Ind Eng ChemRes 42 (8) 1603ndash1618

van der Waals JD 1988 On the Continuity of the Gaseuous and Liquid StatesEdited with an Introductory Essay by Rowlinson JS Studies in StatisticalMechanics North-Holland Amsterdam

Vidal J 1978 Mixing rules and excess properties in cubic equations of state ChemEng Sci 33 (6) 787ndash791

Wei YS Sadus RJ 2000 Equations of state for the calculation of fluid-phaseequilibria AIChE J 46 (1) 169ndash196

Won K 1983 Thermodynamic calculation of supercritical-fluid equilibria newmixing rules for equations of state Fluid Phase Equilibria 10 (2) 191ndash210

Wong DSH Sandler SI 1992 A theoretically correct mixing rule for cubicequations of state AIChE J 38 (5) 671ndash680

Yamano T 2002 Some properties of q-logarithm and q-exponential functions inTsallis statistics Physica A Stat Mech Appl 305 (3ndash4) 486ndash496 ISSN 0378-4371

Zheng Y Du J 2007 An application of nonextensive parameter the nonextensivegas and real gas Int J Mod Phys B 21 (06) 947ndash953


a phenomenological point of view from classical thermodynamics(Hall et al 1993) The widely used van der Waals Mixing Rule(vdw-MR) (Vidal 1978) are

aM frac14Xc

i

Xc

j

xixjaij eth3THORN

bM frac14Xc

i

Xc

j

xixjbij eth4THORN

where xi is the molar fraction of species i c is the number ofdifferent species in the mixture aii and bii are the pure componentparameters (aii ai and bii bi) and aij and bij (ia j) are crossparameters (Wei and Sadus 2000 Peng and Robinson 1976Valderrama 2003) that are represented by the combining rules(Vidal 1978)

aij frac14 aiiajj $05 1kij

$ eth5THORN

bij frac14biithornbjj

21 lij $

eth6THORN

where kij frac14 kji kii frac14 0 lij frac14 lji lii frac14 0 and consequently aij frac14 aji andbij frac14 bji (ia j) kij and lij are binary interaction parameters asso-ciated with a and b (Poling et al 2001) that can be adjusted byexperimental data

The vdW-MR have a quadratic dependence on composition (bM isreduced to a linear dependence if lij frac14 0) This functional form is notable to describe all mixtures and several works have proposeddifferent nonquadratic rules eg Wong and Sandler (1992) Adachiand Sugie (1986) Won (1983) and Higashi et al (1994) Someproposed models take into account the excess Gibbs energy atinfinite pressure calculated from a liquid-phase activity coefficientmodel (Huron and Vidal 1979 Michelsen 1990)

The present work introduces a new nonquadratic mixing rulefor cubic equations of state that differs from vdW-MR once it mayconsider strong correlations following the lines of nonextensivestatistical mechanics (Tsallis 1988 Borges 2004 Moyano et al2006) through a nonquadraticity parameter qij

The work is organized as follows Section 2 introduces somebasic concepts of nonextensive statistical mechanics and intro-duces the semi-empirical model for the q-Mixing Rule (q-MR) Themethodology results and discussions appear in Section 3 andfinally Section 4 is dedicated to the conclusions

2 Nonextensive statistical mechanics

With the advent of the nonextensive statistical mechanics as ageneralization of the BoltzmannndashGibbs statistical mechanics it becameclear over the years that the new formalism was able to representcertain classes of systems that were not properly described byBoltzmannndashGibbs statistical mechanics Indeed since the original paper(Tsallis 1988) more than two decades ago many works have beendeveloped concerning nonlinear systems The generalized entropy is

Sq kB1

PWi frac14 1 p

qi

q1 eth7THORN

with pi being the probability of the microscopic state i W being thenumber of microscopic states kB being a positive constant (theBoltzmann constant) and q being the entropic index If q-1 Eq (7)recovers the celebrated BoltzmannndashGibbs entropy

S kBXW

i frac14 1

pi ln pi eth8THORN

Formal and logical arguments considering an analogy betweenEqs (8) and (7) have lead to the definition of generalized functions

(Tsallis 1994) particularly the q-logarithm function

lnqethxTHORN x1q11q

ethx40THORN eth9THORN

and its inverse the q-exponential function

expqethxTHORN exq frac121thorneth1qTHORNx)1=eth1qTHORN if frac121thorneth1qTHORNx)400 if frac121thorneth1qTHORNx)r0

(

eth10THORN

The usual logarithmic and exponential functions are recovered in thelimit q-1

The maximization of the Tsallis entropy Eq (7) with theconstraint of constant generalized mean energy leads to thegeneralized canonical ensemble distribution (Tsallis 1988 Tsalliset al 1998)

p Eieth THORN prop expq βqEi amp

eth11THORN

where βq is the Lagrange parameter that is related to the inversetemperature Ei is the energy of the ith state and p(Ei) is theprobability of the state with energy Ei be occupied One of thecentral differences between Tsallis and Boltzmann weights is thatthe former presents power law tails (long-lasting for q41 andabruptly vanishing for qo1) while the latter has exponential tails

The difference between Boltzmann and Tsallis weights may be alsounderstood within the following comparison It is known that thepressure P of a hypothetical static isothermal ideal gas atmospherewith gradient given by dP=dhfrac14 mgρ decreases with height accord-ing to an exponential law (Halleys law see for instance Feynmanet al 1963) PethhTHORN frac14 P0expethmgh=kBTTHORN where P0 is the pressure atheight hfrac140 g is the gravitational acceleration A similar expressionholds for the number density ρ (number of moleculesvolume) Thenumerator of the exponent is the potential energy of a molecule withmass m and the denominator is its thermal energy Thermal equili-brium is a very strong simplifying hypothesis that certainly is not validto our atmosphere If we consider that the temperature linearlydecreases with height ie with a constant temperature gradientinstead of a zero temperature gradient mdash the second simplesthypothesis possible mdash then we find that the pressure decays asPethhTHORN frac14 P0 expqethmgh=kBT0THORN with qfrac14 1kBγ=gm and γ frac14 dT=dh isthe negative of the atmospheric temperature gradient Once P frac14 ρkBTand TethhTHORN frac14 T0γh ρethhTHORN frac14 ρ0frac12expqethmgh=kBT0THORN)q In both cases wehave a ratio between the potential and the thermal energies but in theformer we have an exponential law (Boltzmann weight) while in thelatter we have a q-exponential law (Tsallis weight) The case γ frac14 0recovers Halleys law

The introduction of the generalized q-logarithm and q-expo-nential functions has allowed the development of a consistentgeneralized nondistributive algebra based on the q-algebraicoperators (Nivanen et al 2003 Borges 2004) q-addition xqyq-difference x⊖qy q-product x+qy and q-ratio x⊘qy representedas

xqyfrac14 xthornythorneth1qTHORNxy

x⊖qyfrac14xy

1thorneth1qTHORNyya

1q1

x+qyfrac14 x1qthorny1q1 (1=eth1qTHORN

thorn ethx y40THORN

x⊘qyfrac14 x1qy1qthorn1 (1=eth1qTHORN

thorn ethx y40THORN eth12THORN

The subscript thorn in the q-product and in the q-ratio is a short-cutthat means frac12A)thorn frac14maxethA0THORN ie whenever the basis is negative orzero (x1qthorny1q1r0 for the q-product and x1qy1qthorn1r0for the q-ratio) the q-operation (q-product or q-ratio) is defined aszero instead of being calculated by its corresponding mathematicalexpression This feature known as cut-off is essential for theconsistency of the formalism and it is also present in the definitionof the q-exponential see Eq (10) The q-product is commutative




IethqTHORN frac141

1eth1qTHORNβn12mv2


Z1eth1qTHORNβn1

2mv2

) 1=eth1qTHORN

dv


Pthornaq1V

2




aN2=ω with




21 q-mixing rules




aqM frac14P


eth14THORN

















partaqMpartNi

frac141P

ijethxi+qij xjTHORN

X

ij


aqM

X

ij


8lt

9=

eth17THORN

with

X

ij


frac142NT

X

i

xi+qij xj amp


2NT

xj+qij xj ampqij

x1qijj

X

ij


frac142NT

X

i

aij xi+qij xj amp


2ajjNT

xj+qij xj ampqij

x1qijj eth18THORN












31300 00781 49 ndash 00213 08151 23 ndash

32300 00735 52 ndash 00392 07890 25 ndash










n


σ2yexp n


n

amp2

σ2Pexp n

eth19THORN












ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1














eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References





























































IethqTHORN frac141

1eth1qTHORNβn12mv2


Z1eth1qTHORNβn1

2mv2

) 1=eth1qTHORN

dv


Pthornaq1V

2




aN2=ω with




21 q-mixing rules




aqM frac14P


eth14THORN

















partaqMpartNi

frac141P

ijethxi+qij xjTHORN

X

ij


aqM

X

ij


8lt

9=

eth17THORN

with

X

ij


frac142NT

X

i

xi+qij xj amp


2NT

xj+qij xj ampqij

x1qijj

X

ij


frac142NT

X

i

aij xi+qij xj amp


2ajjNT

xj+qij xj ampqij

x1qijj eth18THORN












31300 00781 49 ndash 00213 08151 23 ndash

32300 00735 52 ndash 00392 07890 25 ndash










n


σ2yexp n


n

amp2

σ2Pexp n

eth19THORN












ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1














eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References
































































partaqMpartNi

frac141P

ijethxi+qij xjTHORN

X

ij


aqM

X

ij


8lt

9=

eth17THORN

with

X

ij


frac142NT

X

i

xi+qij xj amp


2NT

xj+qij xj ampqij

x1qijj

X

ij


frac142NT

X

i

aij xi+qij xj amp


2ajjNT

xj+qij xj ampqij

x1qijj eth18THORN












31300 00781 49 ndash 00213 08151 23 ndash

32300 00735 52 ndash 00392 07890 25 ndash










n


σ2yexp n


n

amp2

σ2Pexp n

eth19THORN












ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1














eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References































































n


σ2yexp n


n

amp2

σ2Pexp n

eth19THORN












ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1














eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References





























































ΔP frac14100Nexp

XNexp

i frac14 1

jPexpi Pcalc

i jPexpi

eth20THORN

and

Δyfrac141

Nexp

XNexp

i frac14 1














eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References

































































eth23THORN




ob ethθoptTHORN


dfqMR







AICvdWMR

CO2thorn2-butanol 29 80 33928 208 48345 80 34328 208 48545 038



CO2thornethanol 43 687052 18 73905 687452 18 74105 034

CO2thornstyrene 18 215463 13 18747 215863 13 18947 016



CO2thornH2S 18 402409 64 53757 402809 64 53957 006










CO2thornH2S 2406 85 45 30


4 Conclusions




Acknowledgements


References



























































4 Conclusions




Acknowledgements


References



























































-Quadratic mixing rule for cubic equations of state

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Transcript of -Quadratic mixing rule for cubic equations of state