Critical scaling laws and an excess Gibbs energy model

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Fluid Phase Equilibria 150–151 Ž1998. 429–438

Critical scaling laws and an excess Gibbs energy model Thomas A. Edison, Mikhail A. Anisimov, Jan V. Sengers

)

Department of Chemical Engineering and Institute for Physical Science and Technology, UniÕersity of Maryland, College Park, MD 20742, USA

Abstract Excess Gibbs energy models which are widely used in modelling thermodynamic properties of multicomponent liquid phases are based on the assumption that long-range density or concentration fluctuations can be neglected. This assumption is no longer valid near a system’s critical point, where large density or concentration fluctuations effectively mask the identity of the system and produce universal phenomena which have been well studied in simple liquid–vapor and liquid–liquid systems. Based on the Landau–Ginzburg–Wilson theory of fluctuations, a crossover procedure has been developed to incorporate the effects of critical fluctuations into a classical excess Gibbs energy model. As an example, we have applied our crossover procedure to the Non-Random Two-Liquid ŽNRTL. excess Gibbs energy model. This crossover procedure involves the use of transformed variables for temperature and concentration and the addition of a fluctuation term to the classical excess Gibbs energy. The resulting transformed Gibbs energy has the universal scaling behavior near the consolute critical point and has a smooth crossover to classical behavior far away from the consolute critical point. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Critical phenomena; Gibbs free energy; Isobaric binary mixtures; Liquid–liquid equilibrium; NRTL model

1. Introduction Liquid–liquid equilibria play an important role in the chemical industry. In an attempt to describe liquid–liquid equilibria in fluids, various phenomenological and semi-theoretical models have been proposed in the chemical engineering literature. Some commonly used models are the regular solution model, the Wilson model, the NRTL model, and UNIQUAC w1x. All these models are classical Žmean-field. in nature, and they fail to account for the effects of long-range concentration fluctuations in the vicinity of a consolute critical point. Classical models predict power-law behavior with classical critical exponents, and these drawbacks are now well understood w2x. )

Corresponding author

0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 2 9 0 - 8

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In an attempt to describe both the critical region and the region remote from the critical point, various modifications of classical equations have been proposed. de Pablo and Prausnitz w3x have proposed a phenomenological correction to classical equations and have obtained a good representation of liquid–liquid equilibria in some binary and ternary systems. To account for non-classical behavior, de Pablo and Prausnitz w4,5x have also have applied a transformation proposed by Fox w6x, and extended it to binary and ternary systems. Since Fox’s method uses field variables explicitly, the methodology is less transparent than the present method which uses the concentration as a density-like variable which is readily experimentally accessible. In addition the transformation proposed by Fox fails to reproduce correct universal amplitude ratios asymptotically close to the critical point w7x. A systematic way of incorporating long-range fluctuations into a classical equation can be developed on the basis of renormalization-group theory w8x. Specifically, Sengers et al. have shown how long-range fluctuations can be incorporated in a Landau-type expansion w9–11x. The approach has been extended to the Carnahan–Starling–DeSantes equation of state w12,13x. In this paper, a general transformation of the temperature and the mole fraction variable in Gibbs energy models for liquid–liquid equilibria is proposed. This transformation can be applied to any classical Gibbs energy model. The transformed Gibbs energy model has the correct singular behavior in the vicinity of the consolute critical point and has a smooth crossover to classical behavior far away from the consolute critical point.

2. Thermodynamic potential For a binary liquid mixture containing components 1 and 2, the molar Gibbs energy g satisfies a differential relation of the form d g s ysdT q Õd p q m 1d x 1 q m 2 d x 2 ,

Ž1.

where s the molar entropy, T the temperature, Õ the molar volume, p the pressure, m i the chemical potential of component i, and x i the mole fraction of component i. For a system at constant pressure, Eq. Ž1. can be written as d g s ysdT q m 21d x 2 ,

Ž2.

where m 21 s m 2 y m 1. Here the thermodynamic potential, the molar Gibbs free energy g Ž T, x 2 ., is a function of a field variable T and a density-like variable x 2 . Binary liquid–liquid mixtures near a consolute point and pure fluids near a vapor–liquid critical point belong to the same universality class as the 3D Ising model w2,14x. Close to the critical point, large density andror concentration fluctuations effectively mask the microscopic identity of the system and produce universal phenomena. Near a consolute point of a binary mixture, the difference in the concentration of either component in the two coexisting liquid phases at constant pressure asymptotically behaves as w15x: xXi y x i s 2 B

T y Tc Tc

b

,

Ž3.

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

431

where the prime refers to the mole fraction in the more concentrated phase, Tc is the consolute temperature, b is a universal critical exponent, and B is a system-dependent coefficient. Another property that is readily accessible to experiments is the osmotic susceptibility, which is defined as w15x:

E x1

ž / Em 1

sy p ,T

E x2

y1

E 2g

ž / ½ ž / 5 s x2

Em 1

p ,T

E x 22

,

Ž4.

p ,T

and which diverges at x s x c asymptotically as x2

E x1

ž / Em 1

sG p ,T

T y Tc

yg

,

Tc

Ž5.

where the critical temperature is approached in the one-phase region. Here g is another universal critical exponent, x c is the critical concentration and G is a system-dependent coefficient. The heat capacity at constant pressure has a weak divergence near the consolute point, when measured along a path of constant critical composition x s x c w15x: C p, x s A

T y Tc Tc

ya

,

Ž6.

where A is a system dependent coefficient and where the critical exponent a is related to b and g by a s 2 y 2 b y g .

3. Excess Gibbs function In this work, the reduced molar Gibbs energy of mixing D grRT is used as a generating function: Dg RT

s x 1 ln x 1 q x 2 ln x 2 q

gE RT

,

Ž7.

where R is the universal gas constant and g E is the excess Gibbs energy. One g E model which is widely used to describe partially miscible systems Ž at fixed p . is the NRTL equation w16x: gE RT

s x1 x 2

t 21G 21 x 1 q x 2 G 21

q

t 12 G 12 x 2 q x 1G 12

Ž8.

where G 21 s exp Ž ya 12t 21 . , G 12 s exp Ž ya 12t 12 . u 21 y u11 u12 y u 22 t 21 s , t 12 s . RT RT

Ž9. Ž 10 .

Here u i j is a temperature-dependent energy parameter characteristic of the i–j interaction. The parameter a 12 is related to the nonrandomness of the molecular distribution in the mixture. When a 12 is zero, the molecular distribution is completely random, and Eq. Ž8. reduces to the regular solution

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

432

model. The NRTL equation has three parameters t 21, t 12 , and a 12 , but a 12 is generally fixed at a value between 0.2 and 0.470 w17x.

4. Classical molar Gibbs energy of mixing For a complete specification of the thermodynamic properties, the reduced molar Gibbs energy of mixing as a function of mole fraction and temperature is considered here. All variables are made dimensionless: T˜s y

Tc T

,

x˜ s

x xc

,

g˜ s

g RT

,

m ˜ 21 s

m 21 RT

,

DT˜s T˜q 1, D x˜ s x˜ y 1.

Ž 11 .

Here the mole fraction x refers to the mole fraction x 2 of component 2. In terms of these reduced variables, the NRTL parameters in Eq. Ž9. can be rewritten as:

˜ ˜., t 21 s a˜ Ž 1 y bDT

Ž 12.

˜ ˜., t 12 s c˜ Ž 1 y dDT

where we have introduced the customary approximation that u i j can be represented by linear functions of temperature. By applying the conditions of incipient immiscibility at the consolute point

E 2D g

ž / E x2

E 3D g

s xsx c

ž / E x3

s 0,

Ž 13.

xsx c

the number of parameters can be reduced by two. For the NRTL equation, a˜ s 1.3474 and c˜ s 0.94366. The classical molar Gibbs energy of mixing is not valid near the consolute point, where the thermodynamic properties calculated from a classical equation exhibit power-laws, Eqs. Ž 3. and Ž 5., with classical exponent values of b s 0.5 and g s 1.0. Experiments and theory have shown that b f 0.325 and g f 1.24 w18x. In order to reproduce the correct singular behavior, the classical Gibbs ˜ x˜ . is separated into a regular and a critical part: energy of mixing D g˜ clŽ T, D g˜ cl Ž T˜ , x˜ . s D g˜ reg Ž T˜ , x˜ . q D g˜ cr Ž T˜ , x˜ . .

Ž 14.

˜ The The regular part D g˜ reg is the Gibbs energy of mixing at x s x c and is an analytic function of T. ˜ Ž . critical part D g˜ cr T, x˜ will become a singular function after the effects of fluctuations are incorporated. This separation of the Gibbs free energy of mixing can be accomplished by expanding D g˜ around the consolute point: D g˜ cl Ž T˜ , x˜ . s D g˜ Ž 1,T˜ . q

E E x˜

Ž D g˜ .

Ž x˜ y 1 . q xs1 ˜

1 E2 2! E x˜

2

2 Ž x˜ y 1 . q PPP .

Ž D g˜ . xs1 ˜

Ž 15.

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In this expansion, the terms of second and higher order in x˜ are combined to form D g˜ cr : D g˜ cr Ž T˜ , x˜ . s D g˜ cl Ž T˜ , x˜ . y D g˜ cl Ž T˜ ,1 . y

E E x˜

Ž D g˜cl .

Ž x˜ y 1 . .

Ž 16.

xs1 ˜

The regular part of the NRTL equation is: D g˜ reg s x ln

xc 1 y xc

t 21G 21

q ln Ž 1 y x c . q Ž x c2 q x y 2 xx c .

t 21G 21Ž 1 y G 21 .

y Ž x y x c . x c Ž1 y x c .

x c q Ž 1 y x c . G 21

x c q Ž 1 y x c . G 21 2 q

t 12 G 12

q

Ž 1 y x c . q x c G12

t 12 G 12 Ž G 12 y 1 .

Ž 1 y x c . q x c G12

2

,

Ž 17.

and the critical part of the NRTL equation is: D g˜ crs x ln

x 1yx

q ln

1yx 1 y xc

t 21G 21

=

y Ž x y x c . ln

t 21G 21

=

Ž 1 y x . q xG12

Ž 1 y x c . q x c G12

t 21G 21Ž 1 y G 21 . x c q Ž 1 y x c . G 21

2

q

q x Ž1 y x .

y Ž x y 2 xx c q x c2 .

t 12 G 12

q

x c q Ž 1 y x c . G 21 =

1 y xc

t 12 G 12

q

x q Ž 1 y x . G 21

xc

q Ž x y x c . x c Ž1 y x c .

t 12 G 12 Ž G 12 y 1 .

Ž 1 y x c . q x c G12

2

.

Ž 18 .

The variables in Eqs. Ž 17. and Ž18. are DT˜ and x s x c Ž1 q D x˜ . but in the sequel we shall consider D g˜ cr as a function of DT˜ and D x. ˜ In addition we consider an inverse correlation length k which is a measure of the spatial extent of the critical fluctuation. In the classical theory it is defined as:

k

2 cl s

E 2D g˜ cr E D x˜ 2

.

Ž 19 .

˜ x˜ . as a function of DT˜ and D x. ˜ x˜ . we shall consider k cl2 Ž DT,D Just as D g˜ cr Ž DT,D ˜

5. Renormalized molar Gibbs energy of mixing The critical part of the classical molar Gibbs energy of mixing D g˜ cr , defined by Eq. Ž 16. , has to be transformed close to the critical point to take into account the effects of critical fluctuations. One such transformation has been developed by Chen et al. w9x for the Helmholtz free-energy density of a one-component fluid near the vapor–liquid critical point, where the order parameter can be identified

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

434

with the overall density. In incompressible liquid mixtures, the order parameter can be identified with the concentration x. The renormalized Gibbs energy of mixing D g˜ x is given by: D g˜ x s D g˜ cr DT˜x ,D x˜ x y

ž

/

1 2

DT˜ 2 K ,

Ž 20.

with

˜T U 1r2 , DT˜x s DTT

D x˜ x s D x˜ D 1r2 U 1r4 .

Ž 21.

The term y1r2 DT˜ K is an additive kernel term, which includes fluctuation-induced contributions U and K in these transformations to the Gibbs energy of mixing w11x. The rescaling functions T , D,U are defined as: 2

TsY

2 n y1

1

nv

v

,

UsY , a n Ks Yy n v y 1 / , ž a uL

h

D s Yy v ,

Ž 22.

where a , h , n , and v are universal critical exponents. The exponents n and h are related to the exponents a , b and g by g s Ž 2 y h . n and 3n s 2 y a w2,18x. The exponent v is an additional exponent related to corrections to the asymptotic power laws given by Eqs. Ž 3. , Ž 5. and Ž 6. . The function Y in Eq. Ž 22. is a crossover function, which is defined implicitly through w9,11x:

L

1 y Ž1 y u . Y s u 1 q

ž / k

2

1 2

1

v

Y .

Ž 23.

˜ x˜ . through The variable k 2 is obtained from k cl2 Ž DT,D 1 2

k sY

2v

k cl2 DT˜x ,D x˜ x .

ž

/

Ž 24.

k 2 serves as a measure of the distance from the critical point. As k 2 ™ 0, Y ™ 0 and one recovers from Eq. Ž20. the universal scaled asymptotic critical behavior w13x. As k 2 ™ `, Y ™ 1, and D g˜ x reduces to the critical part of the classical free energy of mixing D g˜ cr . The crossover behavior of D g˜ x is governed by two system-dependent constants u and L. The inverse ‘length’ L reflects the discrete structure of matter, and u is an effective coupling constant.

6. Results and discussion We now demonstrate the applicability of the renormalized NRTL equation to a simple binary liquid–liquid system with molecules of each component approximately the same size. One such system that has been studied close to the consolute point is n-heptaneq acetic anhydride. Nagarajan et al. w19x have measured the coexistence curve of the above system by observing the transition temperature of 76 samples over a wide range of compositions. The coexistence data when plotted as mole fraction vs. temperature seem to be more symmetric than when plotted as volume fraction vs. temperature. The critical parameters Ž at 0.1 MPa. as measured by Nagarajan et al. w19x are

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

435

Tc s 341.658 K Žwhen converted into ITS-90. and x c Ž n-heptane. s 0.4707. Even though the authors have taken into account the effects of gravity in the experimental data, Vnuk w20x has found some inconsistencies in the data close to the critical point, which he attributes to gravitational effects. In the present analysis of the experimental data, similar inconsistencies were noted. In fitting the renormalized NRTL model we have kept the values Tc s 341.658 K, x c Ž n-heptane. s 0.4707, Ž 25. as determined by Nagarajan et al., but we omitted data points between Tc and 341.258 K. The optimized NRTL parameters and crossover parameters, along with critical exponents that are used in our analysis are listed in Table 1. The results of our calculation were insensitive to the value of a 12 , which was fixed at 0.2. Fig. 1 shows a comparison of the phase boundary calculated from the renormalized NRTL equation with the experimental data. In this figure, the calculated phase boundary of the classical NRTL equation is also shown. The parameters of the classical NRTL equation have been determined from the experimental data far from the consolute point. For a 12 s 0.2, the parameters of the classical NRTL equation are found to depend on temperature as: u12 y u 22 s 2.092T q 4.8, Ž 26. u 21 y u11 s y6.200T q 3136.0 t , with T in K and m i j in Jmoly1. The calculated classical critical parameters are T0 s 363.658 K and x 0 s 0.4845. Asymptotically close to the critical point, by fixing b at the theoretical value Ž b s 0.325., we obtained the system Table 1 Parameters and universal constants in the renormalized NRTL equation Universal critical exponents a s 0.110 n s 0.630 Universal NRTL constants as1.347424 ˜

h s 0.033 v s 0.810

System-I (n-heptaneq acetic anhydride) Critical parameters Tc s 341.658 K x c Ž n-heptane. s 0.4707 Crossover parameters us 0.555 L s1.0302 NRTL parameters ˜ 3.9827 ˜ bs ds1.9736 System-II (nitroethaneq cyclohexane) Critical parameters Tc s 296.956 K x c Žnitroethane. s 0.453 Crossover parameters us 0.566 L s1.000 Žfixed. NRTL parameters ˜ 2.2479 ˜ 0.6504 bs ds Caloric background parameters m m ˜ 2 sy6.275 ˜ 3 sy10.655

c˜ s 0.944626

a 12 s 0.2 Žfixed.

a 12 s 0.2 Žfixed. m ˜ 4 s 31.936

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

436

Fig. 1. Coexistence curve for the system n-heptaneqacetic anhydride at atmospheric pressure. The circles ` indicate experimental phase-boundary data, the solid curve represents the phase boundary calculated from the renormalized NRTL equation and the dashed curve represents the phase boundary calculated from the original classical NRTL equation. v indicates the classical critical point ŽT0 , x 0 . and e indicates the actual critical point ŽTc , x c ..

specific amplitude B s 1.81. This value of B is comparable to the value B s 1.85 found by Nagarajan et al. w19x from an asymptotic scaling analysis. The renormalized NRTL equation can also be used to represent caloric properties like the molar heat capacity at constant pressure C p, x and the molar excess enthalpy H E. To calculate caloric properties from the renormalized NRTL equation, we need to add a caloric background function m ˜ 0Ž DT˜ . to the regular part of the Gibbs energy of mixing D g˜ reg to account for a background contribution to the heat capacity. In practice we represent this function by a Taylor expansion with adjustable coefficients: 4

m ˜ 0 Ž DT˜ . s

Ý m˜ j Ž DT˜ .

j

.

Ž 27 .

js0

The coefficients m ˜ 0 and m˜ 1 are related to the zero point energy and entropy and are arbitrary. The system-dependent coefficients m ˜ 2 , m˜ 3 and m˜ 4 are found by fitting to caloric property data far away from the consolute point. For the system n-heptaneq acetic anhydride there are hardly any caloric data in the literature, but caloric data are available for the nitroethaneq cyclohexane mixture. Thoen et al. w21x have measured C p, x of the nitroethaneq cyclohexane mixture along x s x c as a function of temperature. Marsh et al. w22x and Marsh w23x have measured both H E and G E of the nitroethaneq cyclohexane mixture along different isotherms in the one-phase region. The classical NRTL model

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

437

Fig. 2. The molar heat capacity C p, x of nitroethaneqcyclohexane at x s x c as a function of temperature. ` and e represent two-phase and one-phase values of C p, x from the experiments of Thoen et al. w21x and the solid curve represents values calculated from the renormalized NRTL equation.

predicts a jump in C p, x at the critical point. Our renormalized NRTL implies a weak divergence in C p, x at the critical point in accordance with Eq. Ž 6.. The optimized NRTL parameters and crossover parameters for the nitroethaneq cyclohexane mixture are included in Table 1. The values for C p, x calculated from the renormalized NRTL equation are in good agreement with the experimental data obtained by Thoen et al. w21x, as shown in Fig. 2.

7. Conclusions A procedure for transforming Gibbs energy models to reproduce the universal scaling behavior in the vicinity of the critical point has been presented. Specifically, we have shown how the crossover procedure can be applied to an isobaric incompressible liquid–liquid system which belongs to the universality class of the 3D Ising model. Even though we have applied the procedure to a mixture with molecular components of approximately the same size, it can be extended to mixtures with molecular components of different sizes. This can be done by mixing of the field variables DT˜ and ˜ x˜ . y m˜ 21Ž T,1 ˜ .. For systems with molecular components of very different sizes, the Dm ˜ 21 s m˜ 21Ž T, volume fraction may be a better choice for the order parameter than the mole fraction. Hence, it is advantageous to develop classical Gibbs energy models also in terms of the volume fraction. Simultaneous prediction of both the phase boundary and the calorific properties is also possible with a

438

T.A. Edison et al.r Fluid Phase Equilibria 150–151 (1998) 429–438

small modification of our transformed NRTL model w13x. Further research towards extending this method to three-component systems at constant T and p will be considered in the future.

Acknowledgements We are indebted to J.M. Prausnitz for some valuable comments. The research is supported by the Division of Chemical Sciences of the Office of Basic Energy Sciences of the Department of Energy under Grant DE-FG02-95ER14509.

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