A nonrandom lattice fluid hydrogen bonding theory for phase equilibria of associating systems

Fluid Phase Equilibria 158–160 Ž1999. 143–149

A nonrandom lattice fluid hydrogen bonding theory for phase equilibria of associating systems Min Sun Yeom a , Ki-Pung Yoo a , Byung Heung Park b , Chul Soo Lee a

b, )

Department of Chemical Engineering, Sogang UniÕersity, C.P.O. Box 1142, Seoul, South Korea b Department of Chemical Engineering, Korea UniÕersity, Seoul 136-701, South Korea Received 2 April 1998; accepted 30 September 1998

Abstract The Veytsman statistics for hydrogen bonding ŽHB. contribution is combined with the nonrandom lattice fluid ŽNLF. model developed recently by the present authors. The physical contribution is characterized by two temperature dependent molecular parameters representing molecular size and interaction energy for a pure fluid, and a binary interaction parameter for a binary mixture. The chemical part is represented by internal energy and entropy of HB. The resulting NLF-HB equation of state was applied to describe thermodynamic properties of pure fluids and phase equilibrium of mixtures. The results for alkane–alcohol mixtures showed significant improvements over those of the NLF EOS Žequation of state. and HM Žhole quasichemical group-contribution. model. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Vapor liquid equilibria; Mixture Žalkane–alcohol.

1. Introduction Systems of molecules interacting with strong specific intermolecular forces, such as hydrogen bonding ŽHB., deviate remarkably from ideal solution behavior. Prausnitz et al. w1x discussed classical approaches. Most models for associated solutions in the literature are for liquids. However, there is a growing interest in equation of state Ž EOS. thermodynamics w2x. Ikonomou and Donohue w3x proposed the APACT EOS by considering chemical equilibria between the various molecular species present in a mixture. Chapman et al. w4x and Huang and Radosz w5x derived SAFT. The lattice-fluid model offers an alternative basic framework for HB-EOS approaches. Smirnova and Victorov w6x proposed the hole quasichemical group-contribution model Ž HM. . Deak ´ et al. w7x presented high pressure VLE data of alkanol–alkane mixtures and compared APACT EOS, PR EOS )

Corresponding author. Tel.: q82-2-3290-3290; fax: q82-2-926-6102; e-mail: [email protected]

0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 1 1 3 - 2

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

144

with the Wong–Sandler mixing rules ŽWSrPR., and HM model. Sanchez and Panayiotou w8x recently reviewed EOS thermodynamics for polymer solutions with HB effects. In their review, HB effects were modeled by invoking association complexes w9x or by counting the number of arrangements of hydrogen bonds w10x. The latter approach yields less complicated solutions for complex systems. Veytsman w11x proposed a convenient method for the counting association complexes. This HB model is judged compatible with lattice fluids. In this study, we present a new approach combining the Veytsman statistics for HB contribution and the explicit-form nonrandom lattice fluid Ž NLF. model w12,13x for physical description of fluids by present authors. The new model was applied to VLE of alkane–alkanol mixtures and discussed.

2. Derivation We begin with the lattice configurational partition function for HB systems approximated as a product of physical and chemical contributions. The physical term is written in the Guggenheim–Miller approximation. Veytsman statistics w11x allows us to write the partition function for the HB . between donor group of type k and acceptor contribution in terms of the number of HB pair Ž NkHB l group of type l. We assumed the two contributions are independent and found the maximum terms at constant volume and temperature for a C component mixture. The volume change on association was neglected. The physical term is taken from the NLF model of You et al. w12,13x and the HB contribution is given by the condition L

ž

k NkHB l Nr s Nd y

Ý

K

NkHB m

ms 1

/ž

Nal y

Ý NnlHB ns1

/

exp Ž yb AHB kl .

Ž k s 1,2, . . . , K , l s 1,2, . . . , L . Ž1.

HB HB where AHB k l s Uk l y TS k l . k and l are indices of donor types and acceptor types. The number of k donor type k, Nd , and that of acceptor type l, Nal, are defined in terms of the number of donor type k in species i, d ki , and that of acceptor type l in species j, a lj, as

C

Ndk s

C

Ý d ki Ni

Nal s

is1

Ý a lj Nj

Ž2.

js1

The Helmholtz free energy is given by a sum of a physical and a chemical contribution. The former is given by You et al. w12,13x. The HB contribution is derived to give K

K

L

HB HB l l l b AcHB s NHB ln Nr y Ý Ž Ndk ln Ndk y Ndk . q Ý Ž NkHB 0 ln Nk 0 y Nk 0 . y Ý Ž Na ln Na y Na .

k

k

L

qÝ Ž l

K

N0HB l ln

HB N0HB l y N0 l

l

L

HB HB HB HB . q Ý Ý Ž b NkHB l A k l q Nk l ln Nk l y Nk l .

k

l

Ž3.

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

145

where K

L

NHB s Ý

L

Ý NkHB l ,

k

K

k HB NkHB 0 s Nd y Ý Nk j ,

l

l HB N0HB l s Na y Ý Ni l

j

Ž4.

i

the volume is given by V s VH Nr , and the EOS is obtained by the standard method. Psy

E Ac

ž / EV

1 s T

z

b VH

qM

½ ž 2 / ln 1 q ž r

5

/

y 1 r y ln Ž 1 y r . y n HB r y

M

z

´M

ž 2 /u ž V / 2

Ž5.

H

where V H is the unit lattice volume and

´ M s Ž 1ru 2 . Ý Ý u i u j ´ i j q Ž br2 u 2 . Ý Ý Ý Ý u i u j u k u l ´ i j Ž ´ i j q ´ k l y ´ i k y ´ jk . K

L

C

ks1 ls1

c

Ý x i ri ,

qM s

is1

Ý x i qi

Ž7.

is1

C

Ý ri

ž ž

r i s Ni rir N0 q

is1 C

us

rM s

is1

C

rs

c

Ý Ý NkHB l r Ý Ni r i ,

V HB s

Ž6.

Ý ui

u i s Ni qir N0 q

is1

Ý Ni ri is1 C

Ý Ni qi is1

/ /

Ž8. Ž9.

Other thermodynamic functions may be obtained from the Helmholtz free energy.

3. Results and discussions We set the coordination number z at 10 and V H at 9.75 cm3rmol. We use the hydrogen bond HB w x parameters for 1-alkanols, Ui HB j s y25.1 kJrmol of Renon and Prausnitz 14 and S i j s y2.65 = y2 10 kJrŽmol K. as given by Ref. w15x. Two molecular parameters, r 1 and ´ 11, are fitted to density Table 1 Coefficients of molecular parameters for Eqs. Ž10. and Ž11. Chemicals

ea

eb

ec

ra

rb

rc

Range ŽK.

n-Pentane n-Hexane n-Heptane n-Octane Methanol Ethanol 1-Propanol 1-Pentanol 1-Hexanol 1-Octanol

94.484 97.278 99.068 100.590 110.005 105.014 106.242 108.297 108.358 109.741

0.0369 0.0313 0.0352 0.0356 0.0504 0.0333 0.0165 0.0011 0.0348 0.0101

0.0189 y0.0245 y0.0187 y0.0243 0.1309 0.0820 0.0009 y0.0911 y0.0048 y0.0739

9.924 11.460 13.035 14.594 3.680 5.353 6.924 10.086 11.556 15.144

y0.0021 y0.0015 y0.0019 y0.0019 0.0001 y0.0012 y0.0017 y0.0009 0.0005 0.0006

0.0012 0.0061 0.0060 0.0075 y0.0036 y0.0050 y0.0059 0.0048 0.0091 0.0092

303–443 273–473 273–513 273–533 273–503 273–516 273–537 293–453 313–433 325–633

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

146

and vapor pressure data for liquids and pressure–volume data for gases at constant temperatures. They are correlated using the form w16x, ´ 11rk s ea q e b Ž T y T0 . q e c T ln Ž T0rT . q T y T0 Ž 10. r 1 s ra q r b Ž T y T0 . q rc T ln Ž T0rT . q T y T0 Ž 11. where the reference temperature, T0 , is 298.15 K. Coefficients for these equations are summarized in Table 1 for selected pure alkanes and 1-alkanols. For mixtures, a binary interaction energy parameter, l12 , is introduced, which is defined by the relation ´ 12 s Ž ´ 11 ´ 22 . 1r2 Ž1 y l12 .. For fluids or fluid mixtures without chemical effects the present model reduces to NLF EOS by You et al. w12,13x. This model provides an approximate and explicit solution to the Guggenheim Table 2 Comparison of results with experimental data System

T ŽK.

P ŽkPa.

a l12

a n-Pentaneq methanol n-Pentaneq 1-pentanol n-Hexaneq ethanol n-Hexaneq ethanol n-Hexaneq propanol n-Hexaneq 1-hexanol Methanolq n-heptane n-Heptaneq 1-octanol n-Hexaneq 1-pentanol n-Hexaneq 1-hexanol n-Hepaneq 1-hexanol Ethanolq n-butane 1-Propanolq n-butane 1-Butanolq n-butane a

AADP c

AADX d

AADY e

References

b

100.0

0.040

–

0.27

–

–

0.029

w19x

303.15

–

0.009

–

–

–

0.016

0.002

w20x

293.15–333.15

–

0.0

9.967

–

4.68

–

0.020

w21,22x

–

–

101.3

0.035

–

0.17

–

–

0.016

w23x

–

101.3

0.021

–

0.48

–

–

0.023

w24x

0.011

–

–

1.18

–

–

w25x

0.040

–

0.40

–

–

–

w26x

293.15–313.15

0.0

1.466

–

–

0.011

–

w27,28x

298.15–323.15

0.0

3.339

–

–

0.005

0.005

w20,29x

313.15–353.15 –

101.3

–

101.3

0.006

–

–

–

0.008

0.033

w30x

–

101.3

0.026

–

–

–

0.022

0.027

w30x

14.615

–

0.16

–

–

w7x

423.15–463.15

0.004

333.15–473.15

0.0

6.052

–

1.03

–

–

w7x

333.15–473.15

0.002

4.595

–

1.71

–

–

w7x

l12 s aq br T. AADT s Ž100rN .Ý iN < Tical yTiexp
AADT b

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

147

Fig. 1. Comparison of experimental data with calculated values by NLF-HB EOS and NLF EOS for 1-pentanol– n-pentane system at 303.15 K.

combinatory and was found to give better results than the random lattice fluid Ž RLF. model of Lacombe and Sanchez w17x. Various lattice fluid models for physical interactions were recently reviewed w18x. Although Eqs. Ž10. and Ž 11. are slightly different from those reported by You et al. w12x, the fitting error in vapor pressure and density is essentially the same. For HB systems Panayiotou and Sanchez w10x maximized the partition function based on slightly different assumptions. The present NLF-HB model conveniently yields the Helmholtz free energy, whereas Panayiotou and Sanchez have the Gibbs free energy. Generally speaking, pure physical models have difficulties with associating mixtures. These difficulties are most pronounced for alkane–alkanol mixtures with similar carbon numbers. Results

Fig. 2. Comparison of experimental data with calculated values by NLF-HB EOS and HM for 1-propanol– n-butane system at various temperatures. Ž`. Critical points.

148

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

for selected systems in Table 2 shows that the present NLF-HB EOS is very accurate. The results for 1-pentanol–n-pentane mixture are shown in Fig. 1 where they are compared with those for NLF model. As expected NLF-HB model gives much better results than NLF model for associating systems. The calculated results for VLE of 1-propanol–n-butane mixture are compared in Fig. 2 with Deak ´ et al.’s w7x data and results of HM model. For high pressure VLE of alkanol–alkane mixtures they concluded that HM and APACT EOS give better predictions than WSrPR. The present NLF-HB model is seen better than HM model especially for VLE at elevated pressures. However, both models are not satisfactory for critical loci.

4. Conclusion The Veytsman statistics for HB contribution is combined with the NLF model developed recently by the present authors. The volume change on HB was neglected and the canonical partition function was derived. The resulting NLF-HB EOS was tested for VLE of alkanol–alcohol mixtures at low and elevated pressures. NLF-HB EOS gave better agreements with experimental data than NLF EOS and HM over a wide range of pressure and temperature.

5. List of symbols a, d Nr q r z Greeks letters ´ l n HB u r Superscripts HB i, j, k, l Subscripts 0 a, d i, j, k, l ij, kl HB

number of acceptor groups or donor group, respectively defined by the relation Nr s N0 q Ý Ni ri surface area parameter segment number coordination number interaction energy parameter binary interaction parameter defined by Eq. Ž6. surface area fraction defined by Eq. Ž9. reduced density defined by Eq. Ž8. chemical contribution by HB index for component, donor group or acceptor group related to holes acceptor or donor, respectively index for component, donor group or acceptor group interaction pair chemical contribution by HB

M.S. Yeom et al.r Fluid Phase Equilibria 158–160 (1999) 143–149

149

Acknowledgements The authors are grateful to KOSEF and KMOTIE for financial support.

References w1x J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 2nd edn., Prentice-Hall, Englewood Cliffs, NJ, 1986. w2x K.P. Johnston, in: J.M.L. Penninger ŽEd.., Supercritical fluid science and technology, ACS Symp. Ser. 406, ACS, Washington, DC, 1989. w3x G.D. Ikonomou, M.D. Donohue, AIChE J. 32 Ž1986. 1716–1725. w4x W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Ind. Eng. Chem. Res. 29 Ž1990. 1709–1721. w5x S.H. Huang, M. Radosz, Ind. Eng. Chem. Res. 30 Ž1991. 1994–2005. w6x N.A. Smirnova, A.I. Victorov, Fluid Phase Equilib. 66 Ž1993. 333–344. w7x A. Deak, ´ A.I. Victorov, Th.W. de Loss, Fluid Phase Equilib. 107 Ž1995. 277–301. w8x I.C. Sanchez, C.G. Panayiotou, in: S.I. Sander ŽEd.., Models for Thermodynamic and Phase Equilibria Calculations, Marcel Dekker, New York, 1993. w9x C.G. Panayiotou, I.C. Sanchez, Macromolecules 24 Ž1991. 6231–6237. w10x C.G. Panayiotou, I.C. Sanchez, J. Phys. Chem. 95 Ž1991. 10090–10097. w11x B.A. Veytsman, J. Phys. Chem. 94 Ž1990. 8499–8500. w12x S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilib. 93 Ž1994. 193–213. w13x S.S. You, K.-P. Yoo, C.S. Lee, Fluid Phase Equilib. 93 Ž1994. 215–232. w14x H. Renon, J.M. Prausnitz, Chem. Eng. Sci. 22 Ž1967. 299–307. w15x C. Panayiotou, J. Phys. Chem. 92 Ž1988. 2960–2970. w16x H.V. Kehiaian, J.P.E. Grolier, G.C. Bebson, J. Chem. Phys. 75 Ž1978. 1031–1048. w17x R.H. Lacombe, I.C. Sanchez, J. Phys. Chem. 80 Ž1976. 2568–2580. w18x C.S. Lee, K.-P. Yoo, Fluid Phase Equilib. 144 Ž1998. 13–22. w19x F.G. Tenn, R.W. Missen, Can. J. Chem. Eng. 41 Ž1963. 12. w20x M. Ronc, G.R. Ratcliff, Can. J. Chem. Eng. 54 Ž1976. 326. w21x N. Ishii, J. Soc. Chem. Ind. Jpn. 38 Ž1935. 659. w22x G.W. Lindberg, D. Tassios, J. Chem. Eng. Data 16 Ž1971. 52. w23x J.E. Sinor, J.H. Weber, J. Chem. Eng. Data 5 Ž1960. 243. w24x P.S. Prabhu, M. van Winkle, J. Chem. Eng. Data 8 Ž1963. 210. w25x H. Wolff, A. Shadiakhy, Fluid Phase Equilib. 7 Ž1981. 309. w26x M. Benedict, C.A. Johnson, E. Solomon, L.C. Rubin, Trans. Am. Inst. Chem. Eng. 41 Ž1945. 371. w27x G. Geiseler, K. Quitzsch, H.G. Vogel, D. Pilz, H. Sachse, J. Phys. Chem. Neue. Folge 56 Ž1967. 288. w28x A.M. Kinyukhin, T.M. Lesteva, N.P. Markuzin, L.S. Budantseva, Prom-st. Sint. Kauch. 7 Ž1982. 4. w29x S.G. Sage, G.A. Ratcliff, J. Chem. Eng. Data 21 Ž1976. 71. w30x P.R. Rao, C. Chiranjivi, C.J. Dasarao, J. Appl. Chem. ŽLondon. 18 Ž1968. 166.