Compu e simula ion o apo -liquid equilib ia o linea dipola luids:
Depa u es om he p inciple o co esponding s a es
Beni o Ga zo
´n, San iago Lago,a) and Ca los Vega
Depa amen o de Quı
´mica Fı
´sica, Facul ad de Ciencias Quı
´micas, Uni e sidad Complu ense de Mad id.
28040 Mad id, Spain
Luis F. Rull
Depa amen o de Fı
´sica A o
´mica Molecula y Nuclea , Uni e sidad de Se illa, Ap do 1065, Se illa 41080,
Spain
~Recei ed 14 No embe 1994; accep ed 25 Janua y 1995!
Liquid- apo equilib ium o linea dipola luids has been de e mined by using he Gibbs ensemble
simula ion echnique. Se e al elonga ions and alues o he dipole momen we e conside ed. Dipole
momen inc eases he c i ical empe a u e and a ec s sligh ly he c i ical densi y and p essu e.
Comp essibili y ac o a he c i ical poin dec eases as he dipole momen o he molecule inc eases.
Dipole momen p o okes de ia ions om he p inciple o co esponding s a es. I is shown ha he
empe a u e-densi y coexis ence cu e is b oadened and ha he slope o he apo p essu e cu e
inc eases wi h inc easing dipole momen . We p opose a new way o educing he dipole momen so
ha he inc ease o he c i ical empe a u e becomes almos independen on he molecula
elonga ion. We ha e also ob ained he apo -liquid equilib ium o models ha ing bo h a dipole and
a quad upole momen . The ob ained da a we e used o desc ibe he beha io o some ela i ely
complex luids, namely, 1,1,1- i luo oe hane and 2,2,2- i luo oe hanol. Good ag eemen o
coexis ence densi ies and p essu es was ob ained. The esul s p esen ed in his wo k o linea
dipola luids along wi h p e ious wo k on linea quad upola luids p o ide a e y comp ehensi e
iew o he e ec o pola o ces on he apo -liquid equilib ium o linea luids. © 1995 Ame ican
Ins i u e o Physics.
I. INTRODUCTION
The p inciple o co esponding s a es is one o he mos
use ul concep s in liquid s a e heo y. This p inciple i s
enuncia ed by an de Waals in 1873 s a es ha he equa ion
o s a e ~EOS!o a luid when educed by he c i ical p op-
e ies is he same o all subs ances. This p inciple is qui e
success ul o desc ibing he beha io o sphe ical o quasi-
sphe ical molecules. A molecula de i a ion o his p inciple
was ca ied ou by Pi ze 1and Guggenheim.2Howe e , i
was soon clea ha his p inciple is only app oxima e and
does no hold o all kind o subs ances. De ia ions om he
p inciple o co esponding s a es3we e clea ly isible in sub-
s ances ha ing sho - ange epulsi e aniso opic o ces ~non-
sphe ical shape!and in luids p esen ing long- ange a ac-
i e o ces p o oked by mul ipole momen s.
F om an empi ical poin o iew, de ia ions om he
p inciple o co esponding s a es a e usually desc ibed by he
acen ic ac o , in oduced by Pi ze e al.4Howe e , i is
clea ha a molecula unde s anding o he o igin o he de-
ia ions om he p inciple o co esponding s a es should be
p e e able.
The ole o he molecula shape on apo -liquid equilib-
ium ~VLE!o linea and simple nonlinea luids is now well
unde s ood. Pe u ba ion heo ies o nonpola linea sys ems
ha e been de eloped du ing he las decade o he wo-
cen e Lenna d-Jones model5–7 and o he Kiha a model.8–10
These heo ies we e able o desc ibe he e ec o molecula
aniso opy on VLE. Recen Gibbs ensemble Mon e Ca lo
~GEMC!simula ions o he VLE o Gay-Be ne,11,12 Kiha a,13
and wo-cen e Lenna d-Jones14 luids ha e comple ed he
pic u e o how he molecula shape p o okes depa u es
om he p inciple o co esponding s a es.
The nex s ep is o y o unde s and he ole o pola
o ces on VLE o molecula luids. Wi h ha pu pose we
ha e ecen ly pe o med Gibbs ensemble simula ions o de-
e mine he VLE o linea quad upola luids.15 F om his
s udy we lea ned he e ec o a quad upole momen on he
VLE o a linea luid. The nex na u al s ep is o analyze he
e ec o a dipole momen on he coexis ence p ope ies o a
linea luid. This is he pu pose o he p esen wo k. P e ious
wo k conce ning he e ec o a dipole momen on VLE
should be men ioned. The e ec o a dipole momen on he
coexis ence p ope ies o a sphe ical model has been s udied
by GEMC.16–19 Howe e , sphe ical molecules wi h a pe ma-
nen dipole momen a e no commonly ound in na u e. Typi-
cally, molecules ha ing a pe manen dipole momen p esen
also a nonsphe ical shape. Lupkowsky and Monson ha e de-
eloped a pe u ba ion heo y o he wo-cen e Lenna d-
Jones model wi h an embedded dipole momen .20 Mo eo e ,
Dubey e al. ha e s udied by compu e simula ion he VLE
o his model and ound good ag eemen wi h he heo e ical
p edic ions.21 In his wo k we ollow his line o wo k and
we s udy he coexis ence p ope ies o a linea luid wi h a
pe manen dipole momen . We choose he Kiha a po en ial o
desc ibe he molecula shape and a dipole–dipole in e ac ion
e m is added. The choice o he Kiha a po en ial p esen s
some ad an ages, in pa icula , ha he VLE o bo h he non-
pola and quad upola Kiha a luids has been p e iously ob-
ained om simula ions.13,15 In his way we a e able o dis-
a!Au ho o whom co espondence should be add essed.
7204 J. Chem. Phys. 102 (18), 8 May 1995 0021-9606/95/102(18)/7204/12/$6.00 © 1995 Ame ican Ins i u e o Physics
cuss he e ec o he dipole momen on he coexis ence
p ope ies which is common wi h he wo k o Dubey e al.21
Mo eo e , we can also discuss di e ences and simila i ies
be ween he e ec o a dipole momen on VLE wi h espec
o he e ec o he quad upole momen o wi h espec o he
ole o he molecula shape.
The ob ained esul s may be use ul no only in p o iding
an unde s anding o he ole o pola o ces on coexis ence
p ope ies bu desc ibing he apo -liquid equilib ium o eal
luids as well. In an a emp o asses he abili y o he dipola
Kiha a model, we ha e applied ou simula ions o desc ibe
he VLE o a luid o echnical in e es like he e ige an
1,1,1- i luo oe hane.
Finally we shall conside he apo -liquid equilib ium o
linea models ha ing bo h a dipole and a quad upole mo-
men . The beha io o hese sys ems will be compa ed wi h
ha o a pu ely dipola o a pu ely quad upola model. Tha
allows o s udy he addi i i y o di e en mul ipole momen s
on he apo -liquid coexis ence p ope ies o a gi en luid.
Simula ion da a o a dipola model wi h quad upole we e
used o desc ibe he apo -liquid coexis ence p ope ies o a
complex luid like 2,2,2- i luo oe hanol.
The scheme o he pape is as ollows. In Sec. II he
molecula model and simula ion me hod a e desc ibed. In
Sec. III he esul s o he simula ions, and he ob ained e-
sul s o models a e desc ibed. The in luence o he dipole
momen upon he coexis ence p ope ies, c i ical pa ame e s
and depa u es om he p inciple o co esponding s a es is
analyzed and a compa ison wi h expe imen al esul s o
1,1,1- i luo oe hane is also gi en. Sec ion IV p esen s he
esul s o models ha ing bo h a dipole and a quad upole
momen , and hei applica ion o he desc ip ion o VLE o
2,2,2- i luo oe hanol. Conclusions o his wo k a e p esen ed
in Sec. V.
II. SIMULATION METHOD
Le us conside a dipola linea luid consis ing o ods
o leng h Lwi h an embedded poin dipole
m
, in e ac ing
h ough a po en ial gi en by
u~ ,
1,
2!5uK~ ,
1,
2!1u
mm
~ ,
1,
2!,~1!
whe e is he dis ance be ween he cen e s o mass o he
molecules and
i[
$
u
i,
i
%
s ands o he pola angles o
molecule iwi h espec o a e e ence ame ha ing i s pola
axis aligned along he cen e o mass sepa a ion ec o , .uK
is he Kiha a po en ial,22 gi en by
uK~ ,
1,
2!54
e
F
S
s
~ ,
1,
2!
D
12
2
S
s
~ ,
1,
2!
D
6
G
~2!
and u
mm
is he dipole–dipole po en ial,23
u
mm
~ ,
1,
2!5
m
1•
m
2
323~
m
1• !~
m
2• !
5.~3!
In Eq. ~2!,
~ ,
1,
2!is he sho es dis ance be ween
he molecula co es ~see Fig. 1!,
e
is an ene ge ic pa ame e
and
s
a size pa ame e . In Eq. ~3!,
m
iis he dipole ec o
loca ed in he cen e o molecule i, aligned wi h he molecu-
la axis. When L50 and
m
50, he po en ial unc ion gi en
by Eqs. ~1!–~3! educes o he well known Lenna d-Jones
po en ial. When L50 and
m
Þ0 he luid unde conside a ion
is he S ockmaye luid.23 Finally, i LÞ0 and
m
50, we ha e
a Kiha a luid.
The Kiha a po en ial is a eliable model o desc ibing
he modynamic beha io o luids. I has been used o de-
sc ibe he gas,24 liquid,25–28 and solid phases29 o eal sub-
s ances. Howe e , we should ecognize a his poin ha he
Kiha a dipola model is somewha a i icial in one espec .
Linea dipola luids a e usually made up by he e onuclea
dia omic molecules ~ o ins ance, HCl!. The use o he Ki-
ha a po en ial gi en by Eq. ~1!implies a sphe ocylinde -like
co e ha is adequa e only when he wo a oms o g oups
o ming he molecule ha e simila sizes. Howe e , he use o
his model p esen s an impo an ad an age. Since he mo-
lecula co e used is he same as in p e ious wo k on
nonpola 13 and quad upola 15 models any di e ence in he
beha io o he dipola model will be exclusi ely a ibu ed
o he dipole momen .
E alua ion o he Kiha a po en ial equi es he calcula-
ion o he sho es dis ance be ween wo linea ods. This
seems a e y ime consuming ask, bu e y e icien algo-
i hms o i s de e mina ion a e a ailable,28,30–32 so ha he
compu e ime expended in he e alua ion o he Kiha a po-
en ial be ween wo linea ods is simila o he ime equi ed
o e alua e he wo-cen e Lenna d-Jones in e ac ion be ween
wo molecules.
To simula e a dipola luid, one has o deal wi h he
long- ange dipola in e ac ions. Two me hods ha e been de-
signed o deal wi h long- ange e ec s in o simula ions o
pola luids: he Ewald summa ions ~EW!me hod33 and he
eac ion ield ~RF!app oach.34 In he EW me hod, he cen al
simula ion box is su ounded by an in ini e numbe o epli-
cas. To conside he long- ange dipola in e ac ions, la ice
ec o sums a e aken o e sphe ical shells o an in ini e
sphe ical la ice su ounded by a con inuum. The RF ap-
FIG. 1. Sho es dis ance
be ween wo linea ods o leng h L.
7205Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
p oach eplaces pa icles beyond a cu o dis ance by a di-
elec ic con inuum. The e ec o his con inuum is aken in o
accoun by including a new e m in o he dipola pai po en-
ial. In p e ious wo k he VLE o dipola luids has been
de e mined wi h he GEMC echnique in combina ion wi h
he EW me hod. Howe e , he RF echnique can be imple-
men ed in GEMC simula ions yielding compa ible esul s.
We ha e ecen ly pe o med19 GEMC o he S ockmaye 23
model ~Lenna d-Jones1dipole!by using he RF echnique.
Fo his po en ial model GEMC simula ions using he EW
me hod a e also a ailable.16–18 Coexis ence densi ies and
p essu es and c i ical magni udes ob ained om bo h me h-
ods we e undis inguishable.19 Tha p o es ha bo h RF and
EW can be implemen ed in GEMC simula ions yielding
iden ical esul s. In his wo k we shall use he RF echnique
o accoun o long- ange o ces since i is simple and less
compu a ionally demanding han he EW echnique. This is
an impo an ac o since he simula ions e en in he absence
o dipola o ces a e al eady qui e demanding om a com-
pu a ional poin o iew.
Wi hin he RF geome y, he dipola pai in e ac ion po-
en ial is35
uRF
mm
~ ,
1,
2!5
5
m
1•
m
2
323~
m
1• !~
m
2• !
522~
e
RF21!
2
e
RF11
3
m
1•
m
2
c
3, , c
0, > c~4!
whe e cis he cu o dis ance and
e
RF he dielec ic cons an
o he con inuum.
To de e mine he VLE o dipola linea Kiha a mol-
ecules we use he Gibbs ensemble Mon e Ca lo simula ion
echnique. This me hod, de eloped by Panagio opoulos,36 al-
lows he di ec de e mina ion o he coexis ence cu e, simu-
la ing simul aneously bo h phases. A mo e de ailed desc ip-
ion o his me hod can be ound in he o iginal pape s.36,37
We ha e ob ained VLE o linea dipola Kiha a luids o
educed leng h L*5L/
s
50.3, 0.6 and 0.8 using he Gibbs
ensemble echnique. F om a p e ious wo k,13 we know he
coexis ence cu e o hese sys ems o
m
50. GEMC simula-
ions o 512 molecules we e pe o med. A empe a u es
close o he c i ical poin , he ini ial con igu a ion was aken
om an
a
-N2la ice, wi h 256 molecules in each box. A
lowe empe a u es, inal con igu a ions om p e ious uns
we e used. The Kiha a in e ac ion was unca ed a
53
s
and long- ange co ec ions we e applied by ha he luid was
uni o m beyond he cu o .38 The long- ange dipola in e ac-
ion was conside ed wi hin he RF geome y by including a
RF e m in o he pai po en ial @see Eq. ~4!#, plus he addi ion
o a RF sel - e m as long- ail co ec ion.39 The dipola in e -
ac ion was unca ed a c53
s
1L, and he RF dielec ic
cons an was se equal in bo h phases:
e
RF~liquid!5
e
RF~ apo !5`. I has been p o ed ha his ap-
p oach does no a ec o he coexis ence p ope ies.19 To
ob ain a poin o he coexis ence cu e, we pe o med 3000–
6000 s eps o equilib a ion plus 4000–8000 s eps o a e -
ages. A s ep consis s o one a emp o mo ing each pa icle
in bo h phases, ollowed by one a emp o changing he
olume and Nex a emp s o exchanging pa icles be ween
he simula ion boxes. Accep ance a ios o pa icle mo es
and olume exchanging we e kep in he ange 30–60 % and
Nex was chosen o ge an exchange a io o 1–3 %. We ob-
ained VLE o T>0.75Tc. To ob ain a poin o he coex-
is ence cu e, we need abou 6ho CPU ime on a DEC
3000/600 wo ks a ion.
The c i ical empe a u e, Tc
*, densi y, nc
*, and p essu e,
Pc
*we e es ima ed by i ing he simula ion da a o he ex-
p essions
n1
*1ng
*
25a1bT*,~5!
n1
*2ng
*5c
S
12T*
Tc
*
D
b
,~6!
ln P*5d1e
T*,~7!
whe e n1
*and ng
*a e he liquid and apo educed densi ies
~n*5n
s
3, wi h nbeing he numbe densi y!,T*5kT/
e
is
he educed empe a u e, and P*5P
s
3/
e
is he educed a-
po p essu e. Equa ion ~5!is he ec ilinea diame e s law.2In
Eq. ~6!, we assumed a c i ical exponen
b
51/3, close o he
uni e sal alue gi en by he eno maliza ion g oup heo y.40
Equa ion ~7!is he Clausius–Clapey on equa ion o he a-
po p essu e.41 We ha e also es ima ed he acen ic ac o ,
i s de ined by Pi ze e al.,4
52log
S
P
Pc
D
T50.7Tc
21~8!
which is a measu e widely used in Chemical Enginee ing o
he depa u es om he p inciple o co esponding s a es.
III. RESULTS AND DISCUSSION
We ha e ob ained he coexis ence cu e o linea Kiha a
luids o he men ioned elonga ions and o wo di e en
alues o he educed dipole o each elonga ion. The e-
duced dipole,
m
*2, is de ined as
m
*25
m
2
es
3.~9!
We ha e s udied he ollowing sys ems: L*50.3 and
m
*251.5, 3; L*50.6 and
m
*252, 4; and L*50.8 and
m
*252.3, 4.6.
The esul s o simula ions a e p esen ed in Tables I–III,
including apo and liquid densi ies and p essu es on he
coexis ence cu e o di e en empe a u es. The es ima ed
e o s we e ob ained om he s anda d de ia ions o e
blocks o 100 s eps. Fo some empe a u es, se e al addi-
ional uns we e also pe o med.
In Figs. 2, 3, and 4 we compa e he esul s o his wo k
o L*50.3, 0.6, and 0.8, espec i ely, wi h he p e ious da a
ob ained o he nonpola Kiha a model.13 Table IV shows
he c i ical p ope ies ~ empe a u e, densi y, p essu e, pack-
ing ac ion, and comp essibili y ac o !as es ima ed om
he simula ion esul s making use o Eqs. ~5!–~7!, and an
7206 Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
es ima e o acen ic ac o s. Resul s o he Lenna d-Jones
c i ical p ope ies om Re . 42 a e also shown.
These esul s show ha he c i ical empe a u e inc eases
signi ican ly as he educed dipole
m
*2inc eases. The e o e,
he boiling empe a u e o a luid inc eases wi h he dipole
momen , a ac widely quo ed in gene al chemis y ex
books.43,44 Dipole momen does no a ec s ongly he c i i-
cal densi y. Fo mode a e dipole momen s he c i ical densi y
is almos iden ical wi h he co esponding nonpola model o
he same elonga ion. Fo la ge dipole momen s he e is a
sligh dec ease o he c i ical densi y. These indings ag ee
wi h p edic ions o a an de Waals like heo y o pola luids
ecen ly p oposed.45 Mo eo e , he packing ac ion a he
c i ical poin
h
c, de ined as
h
c5ncVm,~10!
emains almos cons an and independen on bo h he dipole
momen and he molecula elonga ion. App oxima ely he
16% o he a ailable olume is occupied by molecules a he
c i ical poin . In Eq. ~10!,Vmis he molecula olume,
Vm5
p
6
s
3~113
2L*!.~11!
Quad upole and dipole momen s seem o a ec c i ical
densi ies in a di e en way. Quad upole momen inc eases
c i ical densi y15 whe eas he dipole momen p o okes a
sligh dec ease in he c i ical densi y. C i ical p essu e shows
a simila beha io . Fo mode a e dipole momen s he c i ical
p essu e is almos iden ical wi h ha o he nonpola model.
Fo la ge dipole momen s he e is a sligh dec ease o he
c i ical p essu e. In Table IV we p esen alues o he com-
TABLE I. Resul s o phase coexis ence p ope ies o linea dipola Kiha a
luids o L*5L/
s
50.3 and
m
*25
m
2/~
es
3!51.5 and 3. All he modynamic
p ope ies a e gi en in educed uni s. The numbe s in pa en heses indica e
he unce ain y in uni s o he las decimal digi i.e., 0.474~11!means 0.474
60.011.
T*ng
*Pg
*nl
*Pl
*
m
*251.5
1.18 0.1102~76!0.0657~35!0.351~23!0.067~29!
1.16 0.0903~41!0.0571~25!0.362~19!0.048~19!
1.14 0.0674~54!0.0490~27!0.352~23!0.041~20!
1.10 0.0627~32!0.0427~14!0.4096~92!0.046~21!
1.075 0.0548~37!0.0384~18!0.425~12!0.034~17!
1.05 0.0434~32!0.0315~18!0.4333~89!0.016~20!
1.025 0.0359~13!0.026 98~86!0.4535~85!0.020~22!
1 0.0333~16!0.024 35~56!0.4690~95!0.016~21!
0.975 0.0273~17!0.0203~12!0.474~11!0.013~30!
0.95 0.021 51~96!0.016 31~61!0.4891~83!0.007~20!
0.925 0.015 83~71!0.012 56~53!0.4978~59!0.004~18!
0.9 0.012 67~74!0.010 30~63!0.5084~66!20.008~27!
m
*253
1.28 0.0653~15!0.0475~14!0.377~22!0.043~42!
1.26 0.0533~25!0.0410~16!0.386~19!0.021~35!
1.25 0.0447~12!0.036 64~98!0.373~30!0.005~50!
1.225 0.0425~20!0.0337~12!0.412~19!0.024~46!
1.2 0.0358~14!0.029 79~82!0.419~19!20.001~46!
1.175 0.0315~13!0.0261~11!0.444~11!0.016~29!
1.175 0.0327~16!0.026 70~93!0.4358~85!0.003~33!
1.15 0.0288~17!0.0235~13!0.4485~76!20.022~41!
1.125 0.022 57~94!0.019 11~79!0.4681~61!20.002~34!
1.1 0.018 82~87!0.016 18~83!0.4767~80!20.008~29!
1.04 0.011 57~50!0.010 87~54!0.4940~62!20.034~48!
0.975 0.008 65~61!0.007 24~87!0.5255~47!20.029~34!
0.95 0.006 05~22!0.005 47~31!0.5326~52!20.051~33!
0.925 0.004 62~15!0.004 56~36!0.5409~59!20.011~61!
TABLE II. Resul s o phase coexis ence p ope ies o linea dipola Kiha a
luids o L*5L/
s
50.6 and
m
*25
m
2/~
es
3!52 and 4.
T*ng
*Pg
*nl
*Pl
*
m
*252
1.035 0.0630~24!0.0356~19!0.286~17!0.035~24!
1.025 0.0588~23!0.0343~10!0.299~13!0.041~29!
1.015 0.0531~19!0.0324~14!0.298~12!0.031~19!
1 0.0517~28!0.0306~14!0.311~14!0.032~30!
0.975 0.0383~22!0.0249~12!0.3209~87!0.024~22!
0.95 0.0281~17!0.019 33~82!0.332~10!0.012~23!
0.925 0.022 73~41!0.015 84~41!0.3476~64!0.012~23!
0.9 0.016 62~58!0.012 28~36!0.350~12!20.015~47!
0.875 0.016 33~80!0.011 41~37!0.3671~64!0.006~20!
0.825 0.009 57~30!0.007 10~34!0.3853~38!20.009~15!
0.8 0.007 03~15!0.005 80~30!0.3958~66!20.011~26!
m
*254
1.21 0.0978~72!0.0446~41!0.240~22!0.041~15!
1.2 0.0904~71!0.0421~40!0.246~37!0.035~25!
1.175 0.0602~83!0.0356~35!0.2705~96!0.024~10!
1.15 0.0466~35!0.300~11!0.292~15!0.025~21!
1.125 0.0382~12!0.025 89~86!0.3120~95!0.014~26!
1.1 0.0337~15!0.2297~71!0.3242~98!0.008~20!
1.075 0.023 11~91!0.017 21~53!0.3297~99!20.001~25!
1.05 0.020 3~11!0.015 16~69!0.3404~70!20.010~23!
1.025 0.018 6~10!0.013 58~60!0.3562~53!20.014~18!
1 0.014 61~55!0.010 94~38!0.3640~53!20.011~23!
0.975 0.010 31~48!0.008 09~31!0.3675~55!20.022~22!
0.95 0.009 25~61!0.007 21~43!0.3835~49!20.016~24!
TABLE III. Resul s o phase coexis ence p ope ies o linea dipola Kiha a
luids o L*5L/
s
50.8 and
m
*25
m
2/~
es
3!52.3 and 4.6.
T*ng
*Pg
*nl
*Pl
*
m
*252.3
1 0.0575~18!0.0314~11!0.214~28!0.022~20!
0.99 0.0495~42!0.0289~14!0.210~30!0.028~15!
0.985 0.0489~38!0.0279~14!0.222~23!0.024~17!
0.975 0.0436~32!0.02608~93!0.244~11!0.027~16!
0.95 0.0422~22!0.02396~84!0.2706~72!0.026~20!
0.925 0.0308~25!0.0193~11!0.2825~75!0.021~18!
0.9 0.0221~18!0.01461~96!0.2909~92!0.015~23!
0.875 0.0180~11!0.012 04~67!0.3035~59!0.016~17!
0.85 0.016 43~66!0.010 87~47!0.3145~47!0.009~14!
0.825 0.011 36~48!0.007 89~28!0.3227~52!0.002~16!
0.8 0.008 91~89!0.006 27~48!0.3295~35!0.002~15!
0.775 0.007 15~22!0.005 10~19!0.3376~43!20.002~22!
m
*254.6
1.145 0.0545~25!0.0321~22!0.221~13!0.034~24!
1.135 0.0484~26!0.0298~13!0.233~15!0.042~30!
1.125 0.0548~30!0.0304~19!0.234~14!0.033~18!
1.1 0.0446~22!0.0259~13!0.256~17!0.035~29!
1.075 0.0325~20!0.020 78~84!0.264~11!0.018~12!
1.05 0.300~37!0.0192~19!0.2828~63!0.010~13!
1.025 0.208~24!0.0145~14!0.2872~64!0.008~16!
1 0.0158~11!0.011 38~64!0.2961~59!20.004~15!
0.95 0.011 57~41!0.008 59~41!0.3192~55!0.002~19!
0.925 0.009 10~98!0.007 05~60!0.3243~51!20.006~18!
0.9 0.007 24~58!0.006 09~39!0.3315~63!20.008~20!
7207Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
p essibili y ac o a he c i ical poin , Zc. Dipole momen
educes signi ican ly he alue o Zc. This ag ees wi h ex-
pe imen al da a3and esul s o Table IV p o ide an illus a-
ion o his ac . Dipole momen does no s ongly a ec
ei he c i ical densi y o p essu e. Howe e , i p o okes an
impo an inc ease o he c i ical empe a u e. The e o e, he
dec ease o Zcwi h he dipole is mos ly due o he inc ease
o he c i ical empe a u e. All he esul s p esen ed he e o
dipola linea luids ag ee wi h p e ious indings o sphe i-
cal dipola models.16–19
The e ec o dipole momen on he apo p essu e is
shown in Fig. 5 o L*50.3. Dipole momen dec eases he
apo p essu e a a gi en empe a u e. The same e ec is
obse ed o he dipola Lenna d-Jones luid.16–19 Acco ding
o he Clausius equa ion, which becomes accu a e a low
empe a u es, he slope o a ln P* s 1/T*plo is ela ed wi h
he apo iza ion en halpy by
dln P*
d1/T*52 DH
N
e
52DH
*.~12!
In Fig. 5 he loga i hm o he educed apo p essu e is
ep esen ed s in e se educed empe a u e a di e en e-
duced dipole momen s. We can conclude ha since he slope
o he lines is almos cons an ha implies, om Eq. ~12!,
ha a low empe a u es DH
*is ela i ely cons an . We see
om Fig. 5 ha DH
*inc eases as he dipole is inc eased.
This con i ms he idea ~widely quo ed on ex books o
chemis y! ha apo iza ion en halpy inc eases when mol-
ecules ha e a dipole momen .43,44
In Fig. 6~a! he ela i e a ia ion o he c i ical empe a-
u e, DTc/Tc
0 o se e al elonga ions and dipole momen s is
shown. The magni ude DTcis de ined as
DTc5Tc~
m
*!2Tc
0~13!
and Tc
0is he c i ical empe a u e o a nonpola model o he
same elonga ion.
Acco ding o he esul s p esen ed in Fig. 6~a! he same
educed dipole momen p o okes la ge changes in c i ical
empe a u e as he molecule becomes mo e sphe ical. In ou
p e ious wo k on quad upola Kiha a luids,15 we al eady
poin ed ou ha molecules wi h di e en elonga ions should
be compa ed when hey p esen he same densi y o mul i-
pole. In his way we we e able o show ha he inc ease o
he c i ical empe a u e as a unc ion o he densi y o quad-
upole p esen s uni e sal beha io o , in o he wo ds, is in-
FIG. 2. Vapo -liquid coexis ence densi ies o dipola Kiha a luids o
L*5L/
s
50.3. The esul s a e gi en in educed uni s T*5kT/
e
and
n*5n
s
3. Da a co esponding o a educed dipole
m
*25
m
2/~
es
3!53 a e
plo ed wi h iangles. Squa es co espond o
m
*251.5. Ci cles ep esen
da a o he nonpola sys em o Re . 13. Lines a e i ings o simula ion da a
o Eqs. ~5!and ~6!o main ex .
FIG. 3. Same as in Fig. 2, bu o L*5L/
s
50.6. Da a co esponding o a
educed dipole
m
*25
m
2/~
es
3!54 a e plo ed wi h iangles. Squa es co e-
spond o
m
*252. Ci cles ep esen da a o nonpola sys em o Re . 13.
FIG. 4. Same as in Fig. 2, bu o L*5L/
s
50.8. Da a co esponding o a
educed dipole
m
*25
m
2/~
es
3!54.6 a e plo ed wi h iangles. Squa es co -
espond o
m
*252.3. Ci cles ep esen da a o non-pola sys em o Re . 13.
7208 Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995

dependen o he molecula elonga ion. He e we shall ollow
he same app oach by de ining he educed densi y o dipole,
m,as~see Re . 15 o de ails!
m25
m
2
e
Vm
3/3 .~14!
Figu e 6~b!shows ha he magni ude o DTc/Tc
0 ollows
now a uni e sal cu e as a unc ion o m2, ega dless o he
alue o L*. The uni e sali y o he cu e in Fig. 6~b!sup-
po s ou choice o m2 o compa ing luids wi h di e en
L*. Consequences o he uni e sali y shown in Fig. 6~b!a e
qui e in e es ing. Fo ins ance, i he c i ical empe a u e o
a linea Kiha a luid wi h elonga ion L*and
m
*50 is known
(Tc
0), Tcmay be p edic ed o any alue o
m
*2~m2!,by
eading DTc/Tc
0in Fig. 6~b!and sol ing o Tc. The si ua-
ion conce ning he e ec o pola o ces on he c i ical em-
pe a u e o linea luids can be summa ized as ollows. Fo
quad upola luids15 when DTc~ educed by
e
/k!is plo ed as
a unc ion o he educed densi y o quad upole esul s co -
esponding o molecules wi h di e en elonga ions also all
on a single line. Fo dipola luids when DTc/Tc
0is plo ed as
FIG. 5. Loga i hm o he educed apo p essu es, P*5P
s
3/
e
s he in-
e se o educed empe a u e, 1/T*51/(kT/
e
), o L*50.3 and
m
*250
~solid ci cles!,
m
*251.5 ~open iangles!, and
m
*253~solid iangles!.
FIG. 6. ~a!Rela i e a ia ion o he educed c i ical empe a u e @see Eq.
~13!in he ex #as a unc ion o he educed dipole o L*50~ci cles!
~ esul s ob ained om Re s. 16–19!,L*50.3 ~squa es!,L*50.6 ~open i-
angles!, and L*50.8 ~open diamonds!.~b!Rela i e a ia ion o he educed
c i ical empe a u e as a unc ion o he educed densi y o dipole, m, de-
ined in Eq. ~14!. Symbols a e as in ~a!. Lines a e plo ed as a guide o he
eye.
TABLE IV. C i ical p ope ies o di e en dipola linea luids.
h
cis he c i ical packing ac ion, de ined in Eq.
~10!,Zcis he comp essibili y ac o a he c i ical poin and
is he acen ic ac o , de ined in Eq. ~8!.
L*
m
*2m2Tc
*nc
*Pc
*
h
cZc
0 0 0 1.310a0.314a0.126a0.164a0.306a20.03a
0.3 0 0 1.114~12!0.219 ~6!0.073~10!0.166 ~5!0.30~4!0.00~12!
1.5 2 1.210~40!0.220~17!0.075~17!0.167~15!0.28~9!0.04~13!
3 4 1.365~48!0.206~13!0.065~15!0.156 ~9!0.23~5!0.06~13!
0.6 0 0 1.000~12!0.161 ~5!0.051~10!0.160 ~5!0.32~6!0.15~16!
2 2 1.090~29!0.167~10!0.050~15!0.166~10!0.28~10!0.15~19!
4 4 1.225~20!0.161~16!0.047~13!0.160~16!0.24~7!0.23~16!
0.8 0 0 0.952~11!0.140 ~3!0.038 ~8!0.161 ~4!0.29~6!0.11~12!
2.3 2 1.026~24!0.143~15!0.037 ~7!0.165~17!0.25~6!0.15~11!
4.6 4 1.179~37!0.137~13!0.038~13!0.158~15!0.24~8!0.15~20!
aResul s om Re . 42.
7209Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
a unc ion o he educed densi y o dipole esul s co e-
sponding o molecules wi h di e en elonga ions all on a
single line. The sea ch o an explana ion o hese uni e sali-
ies should be a challenge o heo ies o molecula pola
luids so a p esen ed. In ac we ha e ecen ly p oposed a
simple heo y45 explaining some o hese indings bu u he
wo k is s ill needed.
In Fig. 7 we show he coexis ence cu e, when he em-
pe a u e and densi y a e educed by hei co esponding c i i-
cal pa ame e s, o wo alues o L*. Al hough he esul s
shown in Fig. 7 a e qui e sensi i e o e o s in he de e mi-
na ion o c i ical p ope ies, so ha cau ion is needed, we
obse e ha end is a b oadening o he VLE coexis ence
cu e due o he dipole momen . The e o e, dipola o ces
p o oke a b oadening o he coexis ence cu e and his is in
common wi h quad upola o ces.15
Depa u es om he p inciple o co esponding s a es in
he apo p essu e due o he dipole a e illus a ed by plo s o
ln (P/Pc) sT
c
/T. This is ep esen ed in Fig. 8 o L*50.3
and 0.8. We ha e also ep esen ed he apo p essu e o a
Lenna d-Jones luid, aken om Re . 42. Thus, i can be
obse ed he e ec ha bo h shape and dipole exe upon
de ia ions om co esponding s a es o apo p essu e. The
slope ~in absolu e alue!o he ln(P/Pc) cu e inc eases
wi h he molecula aniso opy ~i.e., compa e L*50,
m
*250
wi h L*50.3,
m
*250!also wi h he dipole momen ~i.e.,
compa e L*50.3,
m
*250 wi h L*50.3,
m
*253!. Al hough
hese esul s should be aken wi h ca e due o di icul ies in
ob aining accu a ely c i ical magni udes, we belie e his
FIG. 7. Reduced coexis ence densi ies, n/nc, as a unc ion o he educed
empe a u e, T/Tc o se e al dipola sys ems. ~a!L*50.3 and
m
*250~solid
line!,
m
*251.5 ~long-dashed line!and
m
*253~sho -dashed line!.~b!
L*50.8 and
m
*250~solid line!,
m
*252.3 ~long-dashed line!and
m
*254.6
~sho -dashed line!.
FIG. 8. Reduced apo p essu es as a unc ion o he in e se o educed
empe a u e o L*50, 0.3 and 0.8. ~a!Resul s o ~ om op o bo om!
L*50 and
m
*250, and L*50.3 wi h
m
*250, 1.5, and 3. ~b!Resul s o
~ om he op o he bo om!L*50 and
m
*250, and L*50.8 wi h
m
*250,
2.3 and 4.6.
7210 Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
end o be co ec . We conclude ha he e ec o a dipole
momen on co esponding s a es plo s is simila o ha ound
p e iously o quad upola models. An impo an p ope y
widely used in Chemical Enginee ing is he acen ic ac o
. In he las column o Table IV compu ed alues o
a e
p esen ed. Al hough a p ecise de e mina ion o
om
GEMC is a qui e di icul ask ~see he la ge e o ba s o
!
some endencies a e clea . Aniso opy and dipole momen
inc ease he magni ude o he acen ic ac o .
In p e ious wo k15 we ha e shown ha he Kiha a qua-
d upola model is a good e ec i e po en ial o desc ibing
apo -liquid equilib ium o eal luids. In ac he compu ed
coexis ence p ope ies o he model we e in excellen ag ee-
men wi h expe imen al esul s o ca bon dioxide. He e we
shall illus a e how his is also he case o he Kiha a dipola
model. As an example we ake 1,1,1- i luo oe hane
~CH3–CF3!. This choice is mo i a ed by he ac ha he size
o F is simila o ha o H ~Re . 46!and ha makes ou
choice o he molecula shape ~a sphe ocylinde ! easonable.
Mo eo e , his molecule p esen s a dipole momen aligned
wi h he C–C bond which is in common wi h ou model.
C i ical empe a u e and densi y ob ained om simula ions
o L*50.6 ~a simila alue o L*was used by Fische e al.
o desc ibe he he modynamic p ope ies o e hane7!and
m
*254 we e i ed o he expe imen al c i ical empe a u e
and densi y o CH3–CF3. In his way we ob ain he pa am-
e e s
s
53.73 Å and
e
/k5282.54 K.
Expe imen al47 and simula ion coexis ence cu e and a-
po p essu es o he e ige an 1,1,1- i luo oe hane a e
shown in Fig. 9. The dipole momen ob ained wi h he
GEMC da a,
m
GEMC52.8310218 esu cm, in easonable
ag eemen wi h he expe imen al alue,48
m
52.32310218
esu cm. We ha e ob ained an excellen desc ip ion o coex-
is ence p ope ies o his luid by using a simple model o i s
in e ac ion ene gy. The e o e, we can conclude ha he in e -
ac ion po en ial o Eqs. ~1!–~3!is a good e ec i e pai po-
en ial o he he modynamic desc ip ion o a ela i ely
complex dipola luid, as he e ige an 1,1,1-
i luo oe hane.
We ha e hus a p esen ed esul s o models p esen ing
a dipole model. In ou p e ious wo k we ob ained esul s o
models p esen ing only a quad upole momen . Howe e , i is
o en ound in na u e ha molecules p esen ing dipole mo-
men ha e also a signi ican quad upole momen ~ o in-
s ance, wa e !.49 In Sec. IV we p esen esul s o models
ha ing simul aneously bo h a dipole and a quad upole mo-
men .
IV. DIPOLAR MODELS WITH A QUADRUPOLE
In his sec ion we p esen GEMC esul s o a Kiha a
model wi h L*50.8 and
m
*252.3 and Q*251.5. The pai
po en ial is gi en by
u~ ,
1,
2!5uK~ ,
1,
2!1u
mm
~ ,
1,
2!1uQQ1u
m
Q.
~15!
Exp essions o uKand u
mm
a e gi en by Eqs. ~1!and
~2!. Exp ession o uQQ was aken om Re . 49. The u
m
Q
e m is gi en by49
u
m
Q53
m
Q
2 4@~c12c2!~115c1c222e1•e2!#.~16!
In Eq. ~16!,ci5cos
u
i~see Fig. 1!and eiis a uni ec o
in he di ec ion o
m
i. In Table V he VLE o his model is
p esen ed. Es ima ed c i ical pa ame e s a e Tc
*
51.153(30), nc
*50.139(15), Pc
*50.037(8), and
Zc50.23~5!. The acen ic ac o is
50.20~12!. Resul s o a
pu ely dipola model, a pu ely quad upola and o a nonpo-
la model o he same elonga ion a e also p esen ed in Fig.
10. As expec ed he c i ical empe a u e o he dipole
1quad upole model is highe han ha o he pu ely dipola
o ha o he pu ely quad upola model. The alue o
DTc
*, de ined as [Tc(
m
*,Q*)2Tc
0]/(
e
/k), o he dipole
1quad upole model is DTc
*50.201. I he alues o
DTc
* o he pu ely dipola o he pu ely quad upola model
a e added hen one ob ains DTc
*50.136. The e o e he
inc ease o he c i ical empe a u e o he dipola model wi h
quad upole is la ge han he summa ion o he inc ease un-
de gone by he pu ely dipola and he pu ely quad upola
FIG. 9. Coexis ence p ope ies o 1,1,1- i luo oe hane. ~a!Coexis ence den-
si ies. ~b!Vapo p essu es. Symbols ep esen he expe imen al da a ~ aken
om Re . 47!. Lines a e i ings o he GEMC da a o L*50.6 and
m
*254,
wi h he pa ame e s ob ained as desc ibed in he ex .
7211Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995
luid. The explana ion o ha is as ollows. The dipola
model wi h quad upole as desc ibed by Eq. ~15!p esen s no
only he uQQ and u
mm
e ms bu also he addi ional con ibu-
ion a ising om he in e ac ion be ween he dipole and he
quad upole gi en by he u
m
Q e m. As a gene al ule e ms in
he Hamil onian o a sys em dec easing he ee ene gy a a
gi en densi y and empe a u e aise he c i ical empe a u e.
Con e sely, e ms in he Hamil onian inc easing he ee en-
e gy o he sys em dec ease he c i ical empe a u e. The
e ms uQQ,u
mm
, and u
m
Qdec ease he ee ene gy o he
sys em and, he e o e, all o hem aise he c i ical empe a-
u e. The dipola model wi h quad upole is mo e han he
summa ion o con ibu ions due o he dipole and con ibu-
ions due o he quad upole and his is so because o he
p esence o he u
m
Q e m. Ou esul s ag ee wi h heo e ical
p edic ions made by Bena ides e al.50
In Fig. 11 co esponding s a es plo s o he VLE o he
dipola model wi h quad upole a e p esen ed. As an icipa ed,
he dipola model wi h quad upole p esen s la ge de ia ions
om he p inciple o co esponding s a es han he pu ely
quad upola o he pu ely dipola model. In Fig. 11~a!i can
also be obse ed ha he b oadening o he coexis ence cu e
is la ge o he dipola model wi h quad upole han o he
pu e dipola o quad upola model. Fig. 11~b!shows ha he
slope o he ln P/Pc s Tc/Tplo is la ge o he dipola
model wi h quad upole luid.
Ou conclusions a e simila o he conclusions exposed
by Dubey and O’Shea o Lenna d-Jones dipola plus qua-
d upola luids.51
As i has been al eady said, dipola molecules in na u e
o en also p esen a quad upole momen .49 To desc ibe he
liquid- apo coexis ence p ope ies o a eal dipola luid
wi h quad upole like he 2,2,2- i luo oe hanol ~TFE!,
b oadly used as sol en o p o eins,52 we ha e es ima ed he
molecula pa ame e s o a linea Kiha a dipola model wi h
quad upole ha could gi e an accu a e desc ip ion o VLE o
TFE. We ha e assumed he same shape o TFE han o
1,1,1- i luo oe hane, ha is, L*50.6, and he same
e
and
s
Kiha a pa ame e s. Then, we ha e es ima ed he dipole and
quad upole momen s o eal TFE om he expe imen al di-
pole momen s o 1,1,1- i luo oe hane and e hanol. By as-
suming a disc e e cha ge model we ha e es ima ed ha he
mul ipole momen s o eal TFE a e
m
TFE51.8310218 esu cm
and QTFE55.2310226 esu cm2. The ial educed mul ipole
momen s ob ained by ha way o TFE a e
m
*25
m
TFE
2/~
es
3!>1.5 and Q*25QTFE
2/~
es
5!>1. We ha e
pe o med GEMC simula ions o ha model ~see Table V!.
The c i ical pa ame e s ob ained o his model ~L*50.6,
m
*251.5, and Q*251!a e Tc
*51.143(14), nc
*
50.160(6) and Pc
*50.050(8). C i ical empe a u e and
densi y we e i ed o he expe imen al c i ical empe a u e
and densi y o eal TFE, ob aining he ollowing Kiha a
pa ame e s:
e
/k5436.66 K and
s
53.83 Å.
Expe imen al53–56 and simula ion coexis ence cu e and
apo p essu es o TFE a e shown in Fig. 12. The dipole and
quad upole momen s ob ained wi h he simula ion da a a e
m
GEMC52.26310218 esu cm and QGEMC57.06310226
esu cm2, in good ag eemen wi h he es ima ed expe imen al
mul ipoles. The desc ip ion made by simula ion o he dipo-
la Kiha a model wi h quad upole o liquid- apo coexis -
ence o TFE is easonably good.
A ema kable ac obse ed in all ou simula ion desc ip-
FIG. 10. Liquid- apo coexis ence cu es o pola Kiha a luids wi h
L*50.8. Resul s o ~ om he op o he bo om!a dipola model wi h
quad upole ~
m
*252.3, Q*251,5! luid, a pu e dipola ~
m
*252.3!, a pu e
quad upola ~Q*251.5!, and he nonpola luid. Lines a e i ings o he
GEMC da a. Da a o he pu e quad upola luid ob ained om Re . 15. Da a
o he nonpola Kiha a luid ob ained om Re . 13.
TABLE V. Coexis ence p ope ies o dipola luids wi h quad upole. See he
main ex o an es ima e o c i ical p ope ies.
T*ng
*Pg
*nl
*Pl
*
L*50.8,
m
*252.3, Q*251.5
1.1 0.0406~22!0.0278~14!0.234~20!0.024~22!
1.075 0.0322~17!0.0233~11!0.267~13!0.017~24!
1.05 0.0279~14!0.0202~10!0.2808~95!0.017~31!
1.025 0.021 70~83!0.016 52~70!0.289~11!0.009~27!
1 0.016 35~66!0.012 68~47!0.294~15!0.008~54!
0.975 0.015 24~42!0.011 45~42!0.3173~78!0.005~37!
0.975 0.015 63~32!0.011 80~33!0.3147~12!0.000~48!
0.95 0.012 65~47!0.009 59~40!0.3263~93!0.011~33!
0.925 0.010 02~54!0.007 63~40!0.3378~70!20.001~33!
0.9 0.008 36~27!0.006 29~24!0.3457~66!0.001~33!
0.85 0.004 97~15!0.003 69~12!0.3569~59!20.020~36!
L*50.6,
m
*251.5, Q*251
1.07 0.0466~17!0.0324~15!0.296~37!0.034~40!
1.06 0.0436~27!0.0302~14!0.299~25!0.021~40!
1.05 0.0391~11!0.0281~13!0.312~15!0.019~33!
1.025 0.0371~16!0.0256~10!0.333~12!0.017~25!
1 0.0257~13!0.019 27~94!0.338~14!0.016~31!
0.975 0.0220~10!0.016 43~73!0.352~13!0.008~38!
0.95 0.0207~10!0.015 18~61!0.3653~79!0.002~29!
0.95 0.0191~19!0.0143~12!0.3677~84!0.006~32!
0.925 0.0157~12!0.011 65~71!0.3759~64!0.002~31!
0.9 0.011 78~60!0.008 91~41!0.3872~74!20.011~33!
0.875 0.009 46~53!0.007 13~35!0.3946~62!20.006~31!
0.875 0.011 39~26!0.008 34~22!0.3951~65!20.006~36!
0.85 0.008 08~59!0.005 95~43!0.4060~59!20.003~29!
0.8 0.004 43~38!0.003 21~27!0.4220~50!0.000~29!
7212 Ga zo
´n
e al.
: Vapo -liquid equilib ia in dipola luids
J. Chem. Phys., Vol. 102, No. 18, 8 May 1995