Perturbation theory of dipolar hard spheres: The vapour-liquid equilibrium of strongly polar mixtures

Fluid Phase Equilibria, 1 I (1983) 207-224 Elsevier Science Publishers B.V., Amsterdam

PERTURBATION VAPOUR-LIQUID MIXTURES

207 -

Printed in The Netherlands

THEORY OF DIPOLAR HARD SPHERES: EQUILIBRIUM OF STRONGLY POLAR

THE

J. WINKELMANN Department of Chemistry, Karl- Marx University Leieig,

PDR -7010 Leipzig (G.D.R.)

(Received August 2, 1982; accepted in final form OGobg

28, 1982) 3

ABSTRACT Winkelmann, J., 1983. Perturbation theory of dipolar hard spheres: the vapour-liquid equilibrium of strongly polar mixtures. Fluid Phase Equihbria 11: 207-224. Thermodynamic perturbation theory has been applied to calculate vapour-liquid equilibria (VLE) for binary mixtures of polarizable, dipolar hard-sphere fluids. Effective potential parameters determined previously from the vapour pressures of pure compounds are used to represent the intermolecular interactions in strongly polar and associating fluids. The one-fluid conformal-solution approach is adopted. With a binary correction factor k,, the theory is found able to predict VLE over wide ranges of temperature and pressure. A similar procedure yields a calculated azeotropic line. In the case of hydrogen-bond formation, a temperature-dependent k,, reproduces both GE and HE at various temperatures and results in a rather exact azeotropic line.

INTRODUCTION

In the design of separation operations, chemical engineers often need to perform phase equilibrium calculations and thus face the problem of how to predict phase equilibria in fluid mixtures. Besides empirical (e.g., Soave, 1979) and semi-theoretical equations of state (Vera and Prausnitz, 1971; Oellrich et al., 1978; Deiters, 1979; Grenzheuser and Gmehling, 1981; methods such as Whiting and Prausnitz, 1982) and group-contribution UNIFAC, the thermodynamic perturbation theory of polar fluids and their mixtures has proved outstandingly successful in predicting the complete range of phase behaviour, including VLE, LLE, azeotropy and critical behaviour (Gubbins and Twu, 1978; Twu and Gubbins, 1978; Twu et al., 1976). The first polar perturbation theory was proposed by Stell et al. (1972, 1974) who introduced the Pad& approximant, and by Rushbrooke et al. ( 1973) McDonald ( 1974) and, in the case of mixtures, by Chambers and 0378-3812/83/$03.00

0 1983 Elsevier Science Publishers

B.V.

McDonald (1975). Melnyk and Smith (1974) adopted a quasi-conformal approach to extend the theory of Rushbrooke et al. (1973) to a mixture of dipolar and non-polar hard spheres of equal diameter. Using effective diameters and dipole moments they calculated phase diagrams and excess mixing functions. A perturbation scheme for multipolar Lennard-Jones systems has been worked out (Ananth et al., 1974; Twu et al., 1975; Flytzani-Stephanopoulos et al., 1975; Gray et al., 1978) which includes inductive, dispersion and charge-overlap forces. This theory has been found able to predict VLE and LLE for binary and ternary, dipolar and quadrupolar systems. Sevcik and Boublik (1979) applied it to calculate excess functions for mixtures composed of polar, polarizable Lennard-Jones molecules. Gibbs (1978) predicted VLE and the critical-locus lines for the system methanol-ethane. In a series of papers (Calado et al., 1978; Lobo et al., 1980, 1981; Machado et al., 1980), Staveley, Gubbins and co-workers tested the mixture theory against experimental results for polar-non-polar mixtures. We began a systematic study by considering a model of polarizable, dipolar hard spheres. Previous papers (Winkelmann, 1979, 1981) have dealt with VLE calculations for pure fluids. The effect of real vapour-phase behaviour was demonstrated. Using effective potential parameters it was possible to extrapolate the vapour pressure curve up to almost the critical temperature. In the present paper VLE calculations are reported for binary mixtures of dipolar hard spheres. For strongly polar components, such as chloroform, acetone, diethyl ether, dimethyl formamide, methanol and water, effective parameters obtained in the previous paper (Winkelmann, 1981) are used. The procedure for solving the equations of phase equilibrium is extended to binary mixtures, and a similar procedure is used to determine azeotropic lines. PERTURBATION

THEORY

The intermolecular pair potential uij between two axially symmetric molecules of components i and j can be written as the sum of an isotropic part u’(r) and several anisotropic contributions, of which only the dipolar and inductive terms will be considered:

(1) where r ‘is the intermolecular separation vector from the centre of molecule 1 to that of molecule 2, and wi = 8,(p, is the angular orientation of molecule i. The various contributions to the anisotropic potential are approximated by the first terms in a spherical-harmonics expansion (Twu et al., 1975;

209

Flytzani-Stephanopoulos et al., 1975; Calado et al., 1978). As previously (Winkelmann, 1981) consideration is restricted to the potential according to eqn. (l), i.e., 24;; = /Q/kjril)(s;sjcjj - 2c,c,) and u”‘= IJ

--y-6 [“&(3C; ‘J

+ 1) + a#;(3cj

+ 1)]/2

where p is the dipole moment and cr the scalar polarizability, ci = cos 8,, si = sin Bi, and cii = cos cp, where 0, and cprepresent the angular orientations of the axially symmetric molecules. In thermodynamic perturbation theory the properties of a real system, where the molecules interact according to u(r, ol, w2), are expanded about the values for a reference system with a potential u”. Series expansion of the Helmholtz free energy in powers of the anisotropic potential about the value FHS for the reference system of hard spheres yields f=fHs+f~lr+f~CL+0(~8)+fiay.+f20LP+O((y11.6) where f = F/N.

(2)

For a binary mixture, the perturbation terms are

and

where p = N/V and /3= (kT)Ki?P” =

J

‘. The integral KjykDDis given by

#Z(_Y, ST t)W~~~(y,~,tphhi~

with y = r,2/~jj, s = r2s/ujjkand t = r13/uik. The Padt approximant is used to sum the dipolar and inductive terms separately: f=fHS+fy(l

-fy/fy)-‘+fpq1

-f;g/fpp)-’

(7)

In solving the integrals (eqns. (3)-(6)), we follow Twu et al. (1975) and adopt the one-fluid conformal approach of Mansoori and Leland ( 1972) to the mixture RDF: gEs( r/Uij, pu:) z: gHS(r/uii> P”_J)

(8)

210

with rJ3 X = ~X;X/J,j

(9)

Introducing this approximation tegrals at p,* = puJ:

into eqns. (3)-(6)

gives the pure-fluid

in-

where r* = r/u. In the same way, K,‘jkDDreduces to the pure-fluid integral. For these integrals Larsen’s approximate formulae (Larsen et al., 1977) are used, i.e.,

I,““( p*) =

c

Jn,ip*i

(10)

i=O

which are in excellent agreement with Monte Carlo results (Patey and Valleau, 1976). Once the free energy is known, the compressibility, internal energy and chemical potentials can be obtained from standard thermodynamic relations. For the reference mixture of hard spheres, the Carnahan-Starling equation of state (Mansoori et al., 1971) and the chemical potential given by Masuoka et al. (1977) and by Uno et al. (1976) are applied. THERMODYNAMICS

OF VLE

For a binary mixture, surfaces in the P--T-x-y given by the isothermal equilibrium conditions &( PL, Xl, T) = II?(L+?Y~, d(PL>

T)

Xl? T) = CL:(P~>Y~> T)

P”( PL, Xl, T)=PG(&y,>

diagram for the VLE are

(11)

T)

which can be solved for the densities and compositions of the coexisting phases of the mixture at equilibrium; y, is the mole fraction of component 1 in the vapour phase (G) and x, its mole fraction in the liquid (L). For a fixed T and xi, eqn. (11) is solved and the pressure calculated from the equation of state. This procedure is superior to the method of using activity coefficients and different equations of state for the different phases, except for the computational simplicity of the latter. After having found the densities in each step, the excess functions GE, HE and lJE at the actual equilibrium pressure are calculated. Since it is no longer necessary to remain within the zero-pressure approximation, the computa-

211

tion of the excess functions can be extended into the high-pressure region up to the critical-locus line. In the same way, the azeotropic line for a binary mixture is obtained by solving

&( PL,q, q = g( PG,xy, q PL( pL, xf-, r) = P”( p, xy, T) for pL, pG and the azeotropic composition then obtained from the equation of state. COMPUTATIONAL

(12) x7’. The azeotropic

pressure is

PROCEDURE

The computational procedure for establishing the phase diagrams and finding azeotropic states is not a new one. In the past it has been applied mostly to cubic equations of state, and critical behaviour has been predicted (Teja and Rowlinson, 1973; Hicks and Young, 1975; Gubbins and Twu, 1978). This method is adopted here using the minimization procedure of Powell (1965). Implementation of the algorithm leads to questions of the physical significance of solutions, and the serious problem of trivial solutions is encountered (Huron et al., 1977, 1978; Asselineau et al., 1979). By adding a suitable penalty function to the objective function, i.e., SQP = SQ + c [PNL,/(C,

- CLB;) + PNL;/(CUB;

- Ci)]

i

it is possible to extend the phase diagram computation up to the critical-locus line without encountering trivial roots. In the case of phase diagram construction, the solution of eqn. (11) for a particular concentration is used as the initial guess for the next step. For the azeotropic locus line, computations are started at a low temperature where the azeotropic state is known as a solution of eqn. (11). Again, the solution of eqn. (12) at a particular T is made the initial guess for the next step. RESULTS AND DISCUSSION

The dipolar hard-sphere potential oversimplifies intermolecular interactions because the dipole moment has to account for all the different types of interaction. Therefore effective parameters were determined, adopting a procedure used earlier (Winkelmann, 1981). These parameters were fitted to experimental vapour pressures up to 0.1 MPa; they are listed in Table 1. The pure components considered may be arranged into two classes,

212 TABLE 1 Effective

potential

parameters

Component

u( X 10”) (m)

cl(D)”

a( X 1030) (m3)

Chloroform Acetone Diethyl ether Dimethyl formamide Water Methanol

4.053 3.808 3.982 3.481 2.304 2.153

4.187 3.831 3.919 3.872 2.051 1.767

1.001 0.028 0.114 1.537 0.381 0.475

a 1 D (Debye) = 3.337

x

10e3’ Cm.

according to their interactions: ordinary polar, and hydrogen-bond-forming compounds. The two classes differ in their respective parameters u and p, and this gap leads to an apparent miscibility gap for binary mixtures containing an associated and a non-associated compound (e.g., water-acetone). Thus in calculating mixing properties, consideration must be restricted to components belonging to the same class. For binary mixtures the reference system is composed of additive hard spheres, uii = 0.5( ui + a,), whereas a priori no mixing rules are needed for TABLE 2 Excess functions

for equimolar

mixtures at 298.15 K

GE (J mol-‘)

HE (J mol-‘)

VE (cm3 mol-‘)

Chloroform - acetone Exp. a k,, = 1.00 k,, = 1.0440

-610 35 -614

- 1914 90 - 902

- 0.094 0.03 - 0.97

Chloroform -- diethyl ether Exp. b k,, = 1.00 k,, = 1.0565

- 775 4 - 785

- 2645 -8 - 1235

- 1.39 0.13 - 1.59

314 64 316

- 786 172 537

Methanol - water Exp. ’ k,, = 1.00 k,, = 0.9865 * From Handa and Nikolaev, b From Handa ’ From Kooner

- 1.00 0.024 0.063

and Benson, 1979; Handa and Fenby, 1975; Apelblat et al., 1980; Rabinovich 1960; R&k and Schroder, 1957. and Benson, 1979;‘Beath and Williamson, 1969; Becker et al., 1974. et al., 1980; Lama and Lu, 1965; Westmeier, 1976.

213

h12=1.0U0

22. Ql

a5

0.3

Fig. 1. Comparison chloroform-acetone.

0.7

Xl

0.9

of calculated (solid line) with experimental

(points, dashed line) VLE for

either dipole moment or polarizability. However, in binary mixtures containing chloroform with acetone or with diethyl ether, hydrogen bonds are formed which are not present in the pure components and which the potential parameters for the pure compounds cannot account for. Formation

Dlethylethedli

-Chloroformi

50..

,

, OJ

,

, 0.3

,

, 0.5

.,

Fig. 2. VLE for diethyl ether-chloroform 1974) (points).

0.7

, Xl

, 0.9

I

from theory (line) and experiment

(Becker et al.,

214

0.1

0.3

0.5

Fig. 3. VLE for methanol-water (points).

0.7

x,

0.9

from theory

(line) and experiment

(Kooner

et al., 1980)

of these complexes enhances the actual dipole moment p,,, and hence parameter k,, is introduced to take this enhancement into account:

a

(13) This pLf, replaces pipj in eqns. (3) and (6). The value of k,, is close to unity. It was fitted to G& at a single temperature and then used in all calculations for a particular system. Table 2 shows the influence of k,, on the excess functions, and this effect is even more striking when the computed P-x-y diagrams are compared with experimental data (Figs. l-3). Both of the chloroform systems exhibit negative azeotropes, the compositions and pres-

Fig. 4. VLE for acetone-dimethyl al., 1966) (points).

formamide

from theory (line) and experiment

(Quitzsch

et

215

sures of which are well reproduced by the theory. The ll~n-jdeal~ty of these systems results almost entirely from the binary parameter k,, which represents the hydrogen bonds formed in the mixture. Therefore the excess functions are highly sensitive to k12, but they are nearly s~I~rn~tr~cin shape. A correction factor to uii also affects the magu~tude of the excess functions, but leaves the shape almost unch~ged. it does not resuit in the S-shaped VE function found e~pe~mental~y for chloroform-acetone (Wanda and Benson, 1979).

k&,5440 CntOr5fOrm(l)-Acetone(2)

b

01

0‘3

05

07

0.9

Y$

Fig. 5. Predicted VLE for c~orofo~m-acetone up to the ~~!i~3~-~~~s line ~~a~~~~~ and ~xper~rnenta~ data (points) at 328 K from ~abi~o~c~ and Nikofaev (196Oj and at 423 K from Campbell and MusbalI~ (1970).

216

r

I

Dlethylether(ll 6.0..

I I I

W”

I I I I I I ( I I

co

3.0..

-ChloroformI

b298.1 SK .-*-

I ~_._._. \

I

._.-.r==fL-‘-

._._.-r;;’ .A

*’

0.l

03,.

d5’

. 0.7

xt”

0.9

’

Fig. 6. Predicted VLE for diethyl ether-chloroform using k ,2= 1.0565: the dashed line is the azeotropic locus; the experimental points are from Becker et al. (1974).

Figure 3 shows a comparison of theory with experiment for the system methanol-water. Here the VLE is less non-ideal, although the components are both hydrogen-bond-forming. The P-x-y diagram agrees well with the data of Kooner et al. (1980). For the mixture acetone-dimethyl formamide (Fig. 4), the theoretical predictions are in excellent agreement with results measured by Quitzsch et al. (1966). The system is almost ideal, with a large difference in the pure-component vapour pressures, and this is confirmed by our computations. To test the predictive ability of the perturbation approach, VLE for the three systems in Table 2 were computed over wide ranges of temperature and pressure. Figure 5 shows the prediction up to the critical line for chloroform-acetone. Only a few high-pressure VLE data have been reported for this system (Campbell and Musbally, 1970), which at 423 K are in close agreement with the computations of the equilibrium ratio K, = y/x, but which exhibit rather large deviations in pressure. The negative azeotrope is not bound above. In Fig. 6 the predicted VLE for diethyl ether-chloroform

217

250-

200-

150-

.--_*_.-*~:.7 ._.-

,,/L-----

1 ,.’

L

.A*-

, 01

,

.

_.---•-

,

,

03

OS

07

09

Xl

Fig. 7. VLE for methanol-water; comparison of theory (line) with experimental data from Kooner et al. (1980) at 298 K, Vohland (1977) at 323 and 343 K, and Hirata et al. (1975) at 373 K.

is presented, and Fig. 7 shows that for methanol-water, together with experimental data of Kooner et al. (1980), Vohland (1977) and Hirata et al. (1975). Figure 7 demonstrates that the theory with a binary parameter k,, adjusted at 298 K is able to predict the VLE up to 373 K with reasonable accuracy. For both of the chloroform systems the entire azeotropic lines from 273 K to the critical line have been calculated. The results for chloroform-acetone are in good agreement with experimental data (see Apelblat et al., 1980). In Fig. 8 the computed P-T projection is shown. On the basis of Malesinski’s model, Tamir (1980, 198 1) derived correlating equations for both Paz and P as a function of temperature: log Paz = A, + B,/T+ X

-=A,+B,T+C2T=

C,T+

D,T3

I

I

493

5;3

553=

5i3 TIKI

Fig. 8. P-T projection for chloroform-acetone calculated using k,, = 1.0440: solid lines represent the vapour pressure curves and the critical-locus line; the triangles are calculated azeotropic points.

The parameters in eqn. (14) were obtained by minimizing a relative objective function; they are given in Table 3. Despite the fact that the present theory is not restricted to an ideal vapour phase and a regular-solution approach, the azeotropic line is described quite well by Malesinski’s model. This may TABLE 3 Parameters

in eqn. (14) A

B

Diethyl ether- chloroform Paz 9.68291 Xar

0.19681

Chloroform -acetone Pa= 9.606 10

xaz

D( x 109)

0 @)

2.88

0.015

K, k,, = 1.0565) - 3.678

1.23E-04

K, k, z = 1.0440) - 1922.71 - 3.366 6.312E-04 6.48E-05

0.0034

(T = 288-547

0.38564

1.35214

(T = 273-533 - 1900.57

- 3.556E-04

Chloroform-acetone (T= Paz 10.08615 X*=

C(X103)

288-490

K, k,, =1.0832-1.315X - 2008.J7 - 4.060 - 5.48E-03 9.6E-03

2.45 10m4 T) 2.94

0.177 0.014 0.113 0.866

219 TABLE 4 Excess functions for equimolar chloroform-acetone mixture: comparison results using k ,2 = 1.044 and k,, = 1.0832- 1.315 x 10P4 T T (K)

298

308 313 318 323 328

GE (J mol-‘)

of experiment

HE (J mol-‘)

Exp.

k I2

k,, U-1

Exp.

k I2

k,,

(T)

-610 - 589 -580

-614 - 603 - 599 -594 -589 -584

-614 - 584 - 570 -555 -541 - 527

- 1914 - 1900 - 1855

-

-

1916 1881 1864 1846 1829 1812

- 532 -516

with

- 1740

903 899 898 897 897 897

be due to a GE which does not vary significantly with temperature. From the results in Table 2 it is seen that the correction factor k,, leads to a good description of GE but still fails to yield HE. In assigning k,, to hydrogen-bond association it is necessary to take into account its temperature dependence. From spectroscopic investigations it is evident that a monomer-dimer equilibrium constant varies linearly with T. Thus, the bond strength of an H-bond is assumed and k,2 is taken to decrease linearly with temperature: k12(T)

= k, - k,T

(15)

Substitution of this k,,(T) into eqn. (13) yields an additional contribution to the internal energy and finally to HE. Test calculations were performed for the system chloroform-acetone to see how the use of eqn. (15) affects the excess functions as well as the azeotropic line. In Table 4, experimental G’ and HE values for equimolar mixtures are compared with values computed using either a constant k,, or eqn. (15). Use of the temperature-dependent k,,(T) yields a considerable improvement in the values of HE, and also gives the correct temperature dependence of GI’. Formerly, using a constant k,*, unlimited azeotropy was obtained, which was supported by the calculations of Deiters and Schneider (1976) who used a Redlich-Kwong equation of state. Equation (15) results in a change of this behaviour. Now an upper limit to the azeotropic line is found at T = 490 K (Fig. 9). Rowlinson (1959) expected the system to be no longer azeotropic at the critical point; the experimental data of Campbell and Musbally (1970) in the range 4233453 K tend to an upper limit, the temperature of which is in the close vicinity of the endpoint predicted here. Again, eqns. (14) were fitted to Paz and xaz for this azeotropic line; the parameters are included in Table 3.

220

Chloroform(l)-Acetone\21

P 1MPa

3,O

2,5

1

280

1,5

I

‘A

I

Fig. 9. Predicted VLE for chloroform-acetone 0, experimental VLE, and A, experimental Musbally (1970).

using k ,2 (If): A, calculated azeotropic loci; 0, azeotropic points, are from Campbell and

CONCLUSIONS

It has been shown that thermodynamic perturbation theory gives an accurate account of the thermodynamic properties of highly non-ideal polar mixtures. Although the potential model of dipolar hard spheres is rather simple, it gives a good qualitative prediction of binary VLE over wide ranges of temperature and pressure, and for systems where hydrogen bonds are formed. In the latter case, the use of a temperature-dependent binary parameter is expected to yield a quantitative prediction of VLE and excess functions. Generally, however, deviations in I/E and HE show that the

221

potential is not sufficiently tions in polar mixtures.

complex to account

for all anisotropic

LIST OF SYMBOLS

c,

CLB; CUB,

f ;z” Gi% HE k k P” PNLi r

SQ, SQP T ‘i

V+ W( 123) XE x, Yi ; I-1, l-l* P u

parameter i lower bound to parameter i upper bound to parameter i free energy per particle (F/N) hard-sphere pair correlation function excess Gibbs free energy excess Gibbs free energy for equimolar mixture excess enthalpy Boltzmann constant binary mixture parameter pressure penalty coefficient molecular distance objective functions temperature intermolecular pair potential excess volume triplet interaction potential X,( P, T) - Z,x,X,(P, T), where X denotes G, H. V liquid mole fraction of component i vapour mole fraction of component i polarizability 1,‘kT dipole moment chemical potential number density hard-sphere diameter

Subscripts i, i, k m

components mixture

Superscripts

:s L G

azeotropic hard-sphere liquid vapour

interac-

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