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Local composition models

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Local composition models

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Content LOCAL COMPOSITION MODELS by Wei-Hsiang Sun A Thesis Presented to the FACULTY OF THE SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN CHEMICAL ENGINEERING July 1984 UMI Number: EP41819 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete m anuscript and there are missing pages, th e se will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI* Dissertation Publishing UMI EP41819 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest* ProQ uest LLC. 789 E ast Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 C f ) ' W J<) 57 3£33P/-J$& This thesis, written by Wel-Hsiang Sun under the guidance of Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of Master of Science in Chemical Engineering Date.... Chairman Faculty Com m ittee ACKNOWLEDGMENTS I am deeply grateful to Professor K. Shing for her delightful and patient guide in this work. Moreover, I'm very indebted to Professor K. Shing for providing a lot of help through this work. I also wish to thank Professor R. Salovey and Professor Y. C. Yortsos for serving as members of my guidence committee and providing helpful suggestions. I sincerely thank Department of Chemical Engineering, USC for providing support, good training, and good environment for my graduate study and research. I am very grateful to my friends, Mr. J. M. Liao, Mr. S. N. Wang, and Mr. C. L. Sun, for their encourgement and spirit during the writing of this work. Finally, I would like to express my deepest gratitude to my father, my mother, and my family. Without their encourgement and inspiration this work would not be possible. TABLE OF CONTENTS ACKNOWLEDGMENTS Page ii LIST OF FIGURES V LIST OF TABLES vii ABSTRACT X CHAPTER 1 INTRODUCTION 1 2 THEORIES OF LIQUIDS AND LIQUID MIXTURES 3 2.1 Hard-Sphere (HS) Fluid 3 2.2 Lennard-Jones (LJ) Fluid 4 2.3 Leonard-Barker-Henderson (LBH) Perturbation Theory 8 2.4 The van der Waals 1-fluid (vdW1) Theory 13 3 LOCAL COMPOSITION MODELS 16 3.1 Radial Distribution Function (rdf) and Local 17 Composition (LC) 3.2 The Wilson Equation 2 1 3.3 The Nonrandom, Two-Liquid (NRTL) Equation 23 4 RESULTS AND DISCUSSIONS 28 4.1 Model 28 4.2 Helmholtz Free Energy and Excess Free Energy 28 - LBH Results i ii Page 4.3 Local Composition 35 4.4 Testing the Wilson Equation 60 4.5 Testing the NRTL Equation 79 4.6 Helmholtz Free Energy and Excess Free Energy 95 - vdW1 Results 5 CONCLUSIONS 101 i v Figure LIST OF FIGURES Page 2.1 A typical Lennard-Jones potential U(r) plotted versus the interraolecular separation r 6 2.2 The separations of LJ potential in LBH perturbation theory 11 3.1 Two types of cells according to Scott's two-liquid theory of binary mixtures r 26 4.1 Local compositions of X11 and X22 as a function of r for A-1 model at X1 = 0.10, T* = 0.7, 1.0 41 4.2 As in Fig. 4.1, but at X1 = 0.25 42 4.3 As in Fig. 4.1, but at X1 = 0.50 43 4.4 As in Fig. 4.1, but at X1 = 0.75 44 4.5 Local compositions of X11 and X22 as a function of r for B-1 model at X1 = 0.10, T* = 0.7, 0.9 45 4.6 As in Fig. 4.5, but at X1 = 0.25 46 4.7 As in Fig. 4.5, but at X1 = 0.50 47 4.8 As in Fig. 4.5, but at X1 =0.75 48 4.9 Local compositions of X11 and X22 as a function of r for C-1 model at X1 = 0.10 and T* = 0.7 49 4.10 As in Fig. 4.9, but at X1 = 0.25 50 4.11 As in Fig. 4.9, but at X1 = 0.50 51 4.12 As in Fig. 4.9, but at X1 = 0.75 52 4.13 Local compositions of X11 and X22 as a function of r for D-1 model at X1 = 0.10, T* = 0.7, 0.9 53 V Figure Page 4.14 As in Fig. 4.13, but at X1 = 0.50 54 4.15 As in Fig. 4.13, but at X1 = 0.75 55 4.16 Local compositions of X11 and X22 as for D-1 model at X1 = 0.10, T* = 0.7, a function 0.9, *1.1 of r 56 4.17 As in Fig. 4.16, but at X1 = 0.25 57 4.18 As in Fig. 4.16, but at X1 = 0.50 58 4.19 As in Fig. 4.16, but at X1 = 0.75 59 4.20 Excess Gibbs free energy of A-1 model Wilson equation fitted to 65 4.21 As in Fig. 4.20, but of B-1 model 66 4.22 As in Fig. 4.20, but of C-1 model 67 4.23 As in Fig. 4.20, but of D-1 model 68 4.24 As in Fig. 4.20, but of E-1 model 69 4.25 As in Fig. 4.20, but of D-2 model 70 4.26 As in Fig. 4.20, but of E-2 model 71 4.27 Excess Gibbs free energy of B-1 model fitted to NRTL equation, the fitting is carried out at T* = 0.7, 0.9 simultaneously 84 4.28 As in Fig. 4.27, but the fitting is carried out each temperature separately at 85 4.29 As in Fig. 4.27, but of D-1 model 86 4.30 As in Fig. 4.29, but the fitting is carried out each temperature separately at 87 4.31 As in Fig. 4.27, but of E-1 model 88 4.32 As in Fig. 4.31, but the fitting is carried out each temperature separately at 89 vi Table LIST OF TABLES Page .4.1 Model mixtures used, states studied, and the assignments of potential parameters 29 4.2 The Helmholtz free energy and excess Gibbs free energy of A-1 model as determined by LBH theory 30 4:3 As in Table 4.2, but of B-1 model 30 4.4 As in Table 4.2, but of C-1 model 31 4.5 As in Table 4.2, but of D-1 model 31 4.6 As in Table 4.2, but of D-2 model 32 4.7 As in Table 4.2, but of E-1 model 33 4.8 As in Table 4.2, but of E-2 model 34 4.9 Local compositions of nearest neighbor particles around a central particle for B-1 model at X1 = 0.5, T* = 0.7, 0.9 37 4.10 As in Table 4.9, but for D-2 model at X1 = 0.5, T* = 0.7, 0.9, 1.1 38 4.10.1 Local compositions for D-1 model at equimolar ratio: comparison of MC results with LBH results 39 4.10.2 Local compositions for D-2 model at equimolar ratio: comparison of MC results with LBH results 40 4.11 Parameters of Wilson Eq. fitted to the excess Gibbs free energy of model mixtures 72 4.12 Comparison of "experimental" local compositions with Wilson Eq. for A-1 model at X1 = 0.10 72 4.13 As in Table 4.12, but at X1 = 0.25 73 4.14 As in Table 4.12, but at X1 = 0.50 73 Vi i Table page 4.15 As in Table 4.12, but at X1 = 0.75 73 4.16 Comparison of "experimental” local compositions with Wilson Eq. for B-1 model at X1 = 0.10 - 74 4.17 As in Table 4.16, but at X1 = 0.25 74 4.18 As in Table 4.16, but at X1 = 0.50 74 4.19 As in Table 4.16, but at X1 = 0.75 75 4.20 Comparison of "experimental” local compositions with Wilson Eq. for C-1 model at X1 = 0.10, 0.25, 0*75, and T#’= 0.7 75 t 4.21 Comparison of "experimental" local compositions with Wilson Eq. for D-2 model at X1 = 0.10 76 4.22 As in Table 4.21, but at X1 = 0.25 76 4.23 As in Table 4.21, but at X1 = 0.50 77 4.24 As in Table 4.21, but at X1 = 0.75 77 4.25 Comparison of "experimental" local compositions with Wilson Eq. for E-2 model 78 4.26 Parameters of NRTL Eq. fitted to excess Gibbs free energy of model mixtures. The fitting is carried out for each model at T* = 0.7, 0.9 simultaneously. 90 4,27 As in Table 4.26, but the fitting is carried out for each model at each temperature separately. 90 4.28 Comparison of "experimental" local compositions with NRTL Eq. for B-1 model at X1 = 0.10 91 4.29 As in Table 4.28, but at X1 = 0.25 91 4.30 As in Table 4.28, but at X1 = 0.50 91 4.31 As in Table 4.28, but at X1 = 0.75 92 4.32 Comparison of "experimental" local compositions with NRTL Eq. for D-1 model at X1 = 0.10 92 Vi'i i Table Page 4.33 As in Table 4.32, but at X1 = 0.20, 0.25 92 4.34 As in Table 4.32, but at X1 = 0.50 93 4.35 As in Table 4.32, but at X1 = 0.75 93 4.36 Comparison of "experimental” local compositions 94 with NRTL Eq. for E-1 model 4.37 The Helmholtz free energy and excess Gibbs free 96 energy of A-1 model as determined by vdW1 theory 4.38 As in Table 4.2, but of B-1 model 96 4.39 As in Table 4.2, but of C-1 model 97 4.40 As in Table 4.2, but of D-1 model 97 4.41 As in Table 4.2, but of D-2 model 98 4.42 As in Table 4.2, but of E-1 model 99 4.43 As in Table 4.2, but of E-2 model 100 ABSTRACT In this work computer simulation and thermodynamic perturbation theory are used to calculate excess free energy and local composition (LC) for various types of Lennard-Jones (LJ) model fluids. Those results are obtained for a variety of LJ mixtures over a range of temperatures and compositions. We use such data to test existing mixture theories. Particular attention is paid to theories based on local composition concept of Wilson. The Wilson and Nonrandom, Two-Liquid (NRTL) equations are tested in this work. Our testings show that the Wilson equation can give an excellent representation of excess free energies for nearly ideal solutions, moderately good representation for associated solutions, but poor representation for phase separating solutions. Our comparisons also show that the Wilson parameters are temperature dependent. Furthermore, in some cases, the Wilson equation can give reasonable, though not accurate, estimates of the local compositions Xij. On the other hand, the NRTL equation can represent the excess free energies well for both nearly ideal solutions and phase separating solutions, and its two adjustable parameters are also temperature dependent. Moreover, in some cases, the NRTL equation can give not only reasonable, but quantitatively satisfactory estimates of the x local compositions Xij. Therefore, it is concluded that the success of the NRTL model in phase equilibria studies is directly attributable to its ability to better describe the microscopic environment of the molecules. CHAPTER 1 INTRODUCTION Computer simulation and thermodynamic perturbation theories are very powerful techniques to study the properties of liquids based on the molecular theory. It can provide detailed data on the structure and the thermodynamic properties of well defined model fluids. Such data cannot be obtained by experiment and are very useful to test existing mixture theories. The main aim of this work is to obtain such data for a variety of Lennard-Jones (LJ) mixtures over a range of temperatures and compositions and to test mixture theories. Particular attention is paid to those theories based on the local composition (LC) concept of Wilson [1] and those commonly used in industry, e.g., the Wilson and NRTL equations [2], The data on structure and thermodynamic properties of interest are local compositions and Helmholtz free energies, which are necessary in order to test existing LC models as well as to develop improved models. Those data can be calculated directly from computer simulation studies or from molecular theory. Monte Carlo (MC) 1 simulation and the Leonard-Barker-Henderson (LBH) perturbation theory [3] are used in this work. The results are obtained for various types of LJ mixtures over a range of temperatures and compositions. Then the results are used to test local composition models, in particular, the Wilson and NRTL equations. The outline of the thesis is as follows. We first introduce LBH perturbation theory in Chapter 2. We describe how the theory is applied to derive the expressions for the Helmholtz free energy of LJ model fluids. , Using this theory and Monte Carlo simulation, radial distribution functions and excess free energy data are obtained. Since the model fluids studied are conformal, we also introduce the most successful conformal solution theory, namely the van der Waals 1-fluid (vdW1) theory. In Chapter 3> we introduce the expressions for LC in terms of radial distribution function (rdf) from molecular point of view. Then we introduce the Wilson and NRTL equations and the expressions of LC proposed by each model. Finally, in Chapter 4, we present the LBH and MC simulation results for LC and Helmholtz free energy of various types of LJ mixtures studied in this work. Then we present comparisons with the Wilson and NRTL equations. CHAPTER 2 THEORIES OF LIQUIDS AND LIQUID MIXTURES In this chapter the Leonard-Barker-Henderson (LBH) perturbation theory [3] is introduced to give the theoretical expressions of Helmholtz free energy of Lennard-Jones binary mixtures. These expressions form the starting point of this work and the results obtained for model fluids are tested against local composition models, in particular, the Wilson and NRTL equations [1, 2]. The model fluids involved in this study are hard-sphere (HS) and Lennard-Jones (LJ) fluids. So the HS and LJ fluids are first introduced in Secs. 2.1 and 2.2 respectively, then the LBH theory is described in Sec. 2.3. Finally, in Sec. 2.4, the van der Waals 1-fluid (vdwl) theory is introduced, which can give the configurational free energy of conformal fluids directly. 2.1 Hard-Sphere (HS) Fluid The hard-sphere potential has the form Co for r > d U(r) = ( 2.1) o o for r < d where d is the HS diameter. This potential has no attractive part, but does simulate the steep repulsive part of realistic potentials by taking into account the non zero size of molecules. It considers molecules to be like billard balls; for HS molecules there are no forces between the molecules when their centers are separated by a distance larger than d, the HS diameter, but the force of repulsion becomes infinitely large when they touch, at a separation equal to d. Since Eq. (2.1) requires only one characteristic constant, the HS model is a one-parameter model. A system of particles with this potential is called a hard-sphere fluid. This is the simplest potential used. The thermodynamic properties and behavior of this fluid are already well developed in statistical thermodynamics (e.g., radial distribution function [4, 5], Helmholtz free energy and equation of state [6]). Therefore it is the potential often used by people to try to gain qualitative understanding. 2.2 Lennard-Jones (LJ) Fluid The form of Lennard-Jones 6-12 potential is U(r) = 4 [ (<7Vr)**12 - (CT/r)**6 ] (2.2) 4 where r is the intermoleeular distance between the centers of two molecules, £ is the depth of the energy well (minimum potential energy) and < T is the collision diameter, i.e., the separation where U = 0. Eq. (2.2) relates the potential energy of two molecules to their distance of separation in terms of two parameters: an energy paramater £ which, when multiplied by minus one, gives the minimum energy corresponding to the equilibrium separation; and a distance parameter (T which is equal to the separation when the potential energy is zero. An illustration of Eq. (2.2) is given in Fig. 2.1. This is the best known two-parameter potential for small, non-polar spherical molecules. In Lennard-Jones potential the repulsive wall is not vertical but has a finite slope; this implies that if two molecules have very high kinetic energy, they may be able to interpenetrate to separations smaller than the collision diameter o . Potential functions with this property are sometimes called "soft-sphere" potentials. LJ potential applies to isolated pairs of nonpolar, spherically symmetric molecules. The best values of LJ parameters € and C can be determined from experimental second virial coefficient data [7]. The model can also be used to study liquid properties using effective parameters fitted to experimental liquid properties. A system of particles with Lennard-Jones potential is called a Lennard-Jones fluid. Eq. (2.2) is the LJ potential for pure fluid; for the mixtures the LJ potential has the form 0 -€ Figure 2.1 A typical Lennard-Jones potential U(r) plotted versus the intermolecular separation r. 6 Uij(r) = 4£iJ [ ( Olj/r)*#12 - (crij/r)**6 ] (2.3) where i and j refer to each component, Uij(r) is the pair potential for a pair of ij molecules, £ij and <71 j are potential parameters for LJ mixtures. In view of the physical significance of the parameters (Eii and <Tii, it is possible to make reasonable predictions of what these parameters are for the interaction between different molecules. Thus, to a first approximation, it is suggested that for the interaction of two unlike molecules having nearly the same size and ionization potential, and, on the basis of hard-sphere model for molecular interaction, Eqs. (2.4) and (2.5) are known as the Lorentz-Berthelot combining rules and do, in fact, provide a basis for obtaining good results for the properties of a variety of mixtures. The combining rule for unlike size parameter <71 j in Eq. (2.5) generally works well for the study of the properties of LJ mixtures. But for the unlike energy parameter £ij, different combining rules are used to simulate the effects of association or solvation. A system of LJ binary mixtures €ij = ( €ii €jj )**(1/2) (2.4) <71 j = ( <71 i + <Tjj )/2 (2.5) 7 can be completely defined in terms of the following parameters. El 1 = 1 E12 = £12/£11 E22 = £22/£11 (2.6) S11 = 1 S12 = 0*12/C7"I1 S22 = £7:22/ £T11 (2.7) where £11* £22 are energy parameters of species 1 and 2 respectively, and £ 7 " 12 is obtained from Eq. (2.5). The cross interaction energy parameter is defined as £12 = C ( £11 £22 )**( 1/2) (2.8) where C is a constant, we call deviation factor. The state condition for LJ model fluid is specified by a reduced temperature T* = kT/£l 1 (2.9) and a reduced density p * = (N/V) £T11 3 where k is Boltzmann constant, T is the temperature of the system, N is the number of molecules in a volume V. 2.3 Leonard-Barker-Henderson (LBH) Perturbation Theory The perturbation theory of Barker and Henderson [8, 9] has been successfully applied to pure fluids and to mixtures of hard spheres. The Leonard-Barker-Henderson (LBH) theory is a direct extension of Barker-Henderson theory to binary mixtures in which attractive forces are present. In this theory the free energy of a binary mixture of LJ molecules is expanded (using a double expansion procedure) about that of hard spheres. The results obtained by LBH theory (Helmholtz free energy and local composition) are presented latter in Chapter 4. In the double expansion the free energy of LJ mixtures is first expanded about a reference fluid, then that of the reference fluid is expanded about second reference fluid. Two cases are considered by LBH in the second expansion, either a pure HS fluid or a mixture of hard spheres may be used as the reference fluid. We use the latter, since it gives better results without a great increase in complexity. The LJ pair potential Uij(r) is first separated into a reference part Uijrep and a perturbing part UijP. Where ! Uij r < C Tij 0 r > (7lj ! 0 r < . (7lj Uij r > crij 9 (2.10) ( 2. 11) I * 6 f Then the reference part Uij is separated into a HS part U° and P another perturbing part Uij . The splittings of potential are shown in Fig. 2.2. Thus the free energy of a mixture of LJ molecules is expanded about that of the reference fluid with ref potential Uij , and the free energy of which is again expanded about that of a hard sphere mixture. So we have AU = Aref ♦ AP (2.12) Aref - ' + M (2.13) where A0 is the free energy of a HS mixture. In the expansion of ref A in Eq. (2.13) the HS diameters are chosen by setting the first order term 4A to zero. Thus the Helmholtz free energy of a LJ mixture can be obtained using LBH perturbation theory, and is given to first order, by [1] A - A ------ = _ 4 7 r pX1 X2 d12 g12°(d12) [ d12 - dl2 ] NkT + Z 7 T P 0 Y Xi Xj f Uij(r) gij°(r) r2 dr (2.14) ij Jcrij where 0 - 1/kT, g12°(d12) is the contact value of g12°, A° and gij° are the free energy and radial distribution function of the reference HS mixture, with 10 .LJ REF AU REF Figure 2,2 The double expansion of LJ mixtures in L B H perturbation theory. 11 dij = ( dii + djj )/2 (2.15) dii = dii (2.16) faj dij = [ 1 - exp { -£Uij(r) } ] dr (2.17) Jo Uij(r) = 4£ij [ (<rij/r)«*12 - ( crij/r)**6 ] (2.18) In general, dij ^ dij for i ^ j. In order to calculate Helmholtz free energy from Eq. (2.1*1), we have to know the contact value g12°(d12), the free energy of HS mixture A0, and the HS radial distribution function gij°(r) for cri j < r < ©o. The A0 is taken from Mansoori-Carnahan-Starling-Leland equation of state [6] and the gij°(r) are the Percus-Yevic (PY) solutions generated using Perram's algorithm [4] based on Baxter's method [5]. In our calculations, about 300 intervals are used to generate each gij°(r), the cut-off point is greater than 4 0rij, and the tail corrections are made by setting g(r) = 1 for r/o~> rc, where re is the cut-off distance. Some error may arise due to the expansion of Ure^ about U° to first order only; but it is expected to be very small [10]. The LBH theory can calculate Helmholtz free energy A directly from Eq. (2.14). To calculate the excess properties we use the definition 12 AE = A - ( X1 A1 + X2 A2 ) (2.19) where A1 and A2 are the free energy of pure species 1 and 2, and A is the free energy of the mixture. All the properties are obtained at same temperature and pressure (p = 0). Since LBH is specific in density and not in pressure, the zero pressure point is located by iteration. From classical thermodynamics we have GE = AE + PVE , (2.20) Thus the excess Helmholtz free energy obtained at zero pressure is | equal to excess Gibbs free energy. I I | 2.4 The van der Waals 1-Fluid (vdW1) Theory ! In the so-called one-fluid theory of mixtures it is assumed that the configurational free energy of the mixture Aemix can be equated to that for a conformal pure fluid Axe, i.e. we can write i I Acmix = NkTJ Xi In Xi + Axe (2.21) i Here Axe is the configurational free energy for the pure fluid at the same temperature and density as the mixture. It is also usually assumed that all the constituents are conformal and have central potentials of the form 13 Uij(r) = €ij f(r/<Tij) (2.22) where f is a universal function. The essential problem in such an approach is to find mixing rules for the potential parameters €x and CTx of the pure reference substance that give the best results for Eq. (2.21). There are several versions of the theory of this [13-16], here we restrict our discussion to the van der Waals 1-fluid theory, since this seems to be the best of this family of theories [14, 15]. In vdW1 theory the mixing rules are €x Ox3 = X Xi Xj €ij <rij3 (2.23) ij CTx3 = X Xi Xj CTij3 (2.24) ij The configurational free energy of the pure reference fluid is conveniently replaced by the residual free energy Axr, using Axe = Axr + Acidgas = Axr + NkT 1 n p - NkT (2.25) and from corresponding states we can write Axr = N €x F(Tx*,/>x») (2.26) where Tx* = kT/<0c» f 3 * * = P . From Eqs. (2.21), (2.25), and (2.26) we have 14 Acmix = NkTX Xi In Xi + N €x F(Tx*,/>x«) + NkT In/? - NkT (2.27) In this expression F is a universal function of reduced temperature and density, to be obtained from experiment or computer simulation. For Lennard-Jones fluids it can be approximated by the equation of state of Nicolas et al. [17]. 15 CHAPTER 3 LOCAL COMPOSITION MODELS Local composition concept was first developed by Wilson in 196*1 [3] for* the study of vapor-liquid equilibrium of mixtures. Then, Renon in 1968 developed the NRTL Eq. [2], based on Scott's two-liquid theory [11] and on an assumption of nonrandomness. Both equations are derived through the use of local compositions for representation of excess Gibbs free energies of liquid mixtures. The derivations of these equations are presented in Secs. 3*2 and 3*3 respectively. The LC in existing models (such as Wilson and NRTL equations) is not clearly defined, so in Sec. 3*1 the LC based on molecular theory is introduced first. The LC is defined in terms of radial distribution function (rdf) from statistical mechanics and can be calculated directly. This will allow direct comparison of statistical and nonstatistical formulations of local composition model using data from computer simulation or molecular theory for model fluids. 16 3-1 Radial Distribution Function (rdf) and Local Composition The local composition in empirical and semiempirical local composition equations is not clearly defined, such as in Wilson and NRTL equations. But this can be done using statistical mechanical concept of radial distribution functions. First, we must define radial distribution function. We consider a pure liquid which consists of spherical molecules and we use a radially symmetric coordinate system whose origin is at the center of one of the molecules. Frpm this origin we proceed in a straight line to some distance r away and imagine a sphere of radius r drawn around the center of the central molecule at r = 0. On the surface of this sphere we imagine a thin spherical shell of thickness dr and we now define the radial distribution function g(r) I by the statement No. of molecular centers located in the spherical shell of thickness = 47Tr ^g(r)dr (3*0 dr on the surface of a sphere of radius r i where p = N/V, V is the volume of the liquid containing N molecules. From Eq. (3*1) we can see p g (r )d r is the probability of observing a second molecule in dr given that there is a molecule at r = 0. Note that this probability is not normalized to unity, but 17 we have instead R p g ( r ) 47Tr2 dr = N - 1 fZr N (3.2) 10 where R is the radius of spherical volume V containing N molecules. In fact, Eq. (3-2) shows that yj>g(r)47Tr2 dr is really the number of molecules between r and r + dr about a central molecule. The function g(r) can also be thought of as the factor that multiplies f the bulk density p to give a local density p ( r ) - p g ( r ) about some fixed molecule. The radial distribution function g(r) depends on temperature and density and has the following properties: When r is less than d, where d is the hard-core diameter of one molecule, the left side of Eq. (3-1) must be zero, and therefore g(r) = 0 , for r < < d (3-3) When r is very much larger than d, the positions of the molecules in the spherical shell are no longer affected by the presence of the central molecule and the left side of Eq. (3>1) must be equal to the volume of shell divided by the liquid volume per molecule. Therefore, g(r) = 1 , for r > > d (3*^) 18 The local composition is used to describe the local environment of molecules and is different from bulk composition due to imbalance of interaction strength among molecules. Useful information obtained from molecular theory computation (e.g., Monte Carlo simulation [18], LBH theory) gives detailed knowledge concerning the arrangement of neighboring molecules around a central molecule. The LC of specie i around a central molecule of specie j is defined as the ratio of number of molecules of specie i around a central molecule j to the number of molecules of all species around a central molecule j. For a binary mixture, we have Xij = Nij(Lij) / [ Nij(Lij) + Njj(Ljj) ] where i * j (3-5) Xii = 1 - Xji where Nij is the number of particle i around a central particle j, and Lij is the distance from the central particle. The coordination number Nij(Lij) of i molecules surrounding a central j molecule, within a sphere of radius Lij, can be expressed in terms of rdf as (Lij 2 Nij(Lij) = 4 (Ni/V) I 7T r gij(r) dr (3-6) JO where Ni is the total number of particle i. Physically, gij(r) gives the probability that a molecule of type i will be found 19 surrounding a molecule of type j at a radial distance r. Of course, the definition of LC depends on the radii Lij chosen for the calculation as we can see from Eq. (3*6). Let ij = 11, substitution of Eq. (3*6) into Eq. (3*5) yields the local composition X11 as X11 = N1KL11) / [ N11 (L11) + N2KL21) ] (3.7) fL2i rL11 2 = X1 / { X1 + X2 [ 4TTr g21 (r)dr/ I 47Trz g11(r)dr ] } J Q JO Similarly, X22 = N22(L22) / [ N22(L22) + N12(L12) ] (3-8) fL 1 2 2 fL 2 2 2 = X2 / { X2 + X1 [1 477" r g12(r)dr/ \ 47TiT g22(r)dr ] } Jo J0 where X1 and X2 are, respectively, overall mole fractions of species 1 and 2. Similar relations can be given for X21 and X12. The local compositions are related by X11 + X21 = 1 (3.9) X22 + X12 = 1 (3.10) Therefore, we can use molecular theory to obtain gij(r) first, then use Eqs. (3«7)-(3*10) to calculate local compositions directly. 20 3-2 The Wilson Equation Wilson is the first researcher using local compositions for representation of excess Gibbs energies of liquid mixtures [3]. The Wilson Eq. was proposed for the system where the molecules differ not only in molecular size but also in their intermolecular force. It was just a modification of Flory-Huggins Eq. [12] which was derived for mixtures of molecules which are chemically similar but which differ only in size. To take into account nonrandomness in liquid mixtures, Wilson uses LC concept to propose a relation between local mole fraction X11 of molecules 1 and local mole fraction X21 of molecules 2 which are in the immediate neighborhood of molecule 1: X21 X2 exp(-f12/RT) X2 = -------------- s ___ exp(-(f 12 - f11)/RT ) (3.11) XI1 X1 exp(-f11/RT) X1 and similarly X12 X1 exp(-f12/RT) X1 = ------------- = --- exp(-(f 12 - f22)/RT ) (3-12) X22 X2 exp(-f22/RT) X2 The local mole fractions are related by X11 + X21 = 1 (3-13) X22 + X12 = 1 (3.1*1) where the energy parameters f11, f12, f22 are not precisely defined, but are presumably related to the potential energies, respectively, of 1-1, 1-2, and 2-2 pairs of molecules. 21 Then Wilson defines local volume fractions using Eqs. (3*11) and (3.12). The local volume fractions of component 1 and 2 are respectively defined by e1 = v1 X11 /( v1 X11 + v2 X21 ) (3-15) e2 = v2 X22 /( v2 X22 + v1 X12 ) (3-16) where v1 and v2 are the molar volumes of components 1 and 2. Substitution Eq. (3*11) to (3-15) gives: e1 = ( X11 / X21 )/( X11 / X21 + v2 / v1 ) ( X1 / X2 ) exp(-(f11 - f12)/RT) = ------------------------------- ( 3.17) ( X1 / X2 ) exp(-(f11 - f12)/RT) + v2 / v1 i.e. X1 e1 = --------------------------------- (3* 18) X1 + X2 ( v2/v1 ) exp(-(f12 - f11)/RT) and similarly we have X2 e2 ------------------------------------- (3-19) X2 + X1 (v1/v2) exp(-(f12 - f22)/RT) Wilson proposes an expression for the excess Gibbs energy by analogy with the Flory-Huggins expression for a binary mixture, where he replaces overall volume fractions by local volume fractions: G E /RT = X1 ln(e1 / X1) + X2 ln(e2 / X2) (3-20) Eqs. (3.18) and (3*19)» substituted into Eq. (3*20), yield GE /RT = X1 ln{ 1/[ X1 + X2 (v2/v1) exp(-(f12 - f11)/RT) 3 } + X2 ln{ 1/C X2 + X1 (v1/v2) exp(-(f12 - f22)/RT) ] } (3.21) It is convenient to define two new parameters. These definitions are: A12 = (v2/v1) exp[-(f12 - f11)/RT] (3-22) A21 = (v1/v2) exp[-(f12 - f22)/RT] (3-23) i j | Wilson's modification of the Flory-Huggins equation then becomes: i i GE /RT = - X1 ln( X1 + A12 X2 ) - X2 ln( X2 + A21 X1 ) (3.24) Empirically, the Wilson Eq. has been shown to work well for mixtures of highly polar substances (e.g., water, acetone, and methanol) and nonpolar substances (e.g., benzene, carbon tetrachloride). But it fails to account for partial miscibility, ! which is the drawback of this model. 3.3 The Nonrandom, Two-Liquid (NRTL) Equation The NRTL Eq. was developed by Renon [2], based on Scott's two liquid model [11] and on an assumption of nonrandomness similar to 23 that used by Wilson. This equation is applicable to both partially and completely miscible systems by introducing a nonrandomness parameter ot , and gives an excellent representation of many types of liquid mixtures while other local composition equations appear to be limited to specific types. To take into account of nonrandomness of mixing, Renon assumes that the relation between the local mole fractions X21 and X11 is given by a modification of Eq. (3*11) X21 X2 exp(-ag21/RT) = - (3*25) X11 X1 exp(-ag11/RT) where a is a constant characteristic of the nonrandomness of the mixture (independent of temperature), and g11, g21 are parameters characteristic of 1-1 and 1-2 interactions (g12 = g21), respectively. Similarly we also have the relation between X12 and X22 X12 X1 exp(- ctg12/RT) — = - 0 .26) X22 X2 exp(-ag22/RT) The local mole fractions are again related by X11 + X21 = 1 X22 + X12 = 1 2b (3*27) (3*28) From Eqs. (3.25) and (3*27) we have X2 exp(-a(g21 - g 11)/RT) X21 (3-29) X1 + X2 exp(-a(g21 - g 11)/RT) and similarly from Eqs. (3.26) and (3-28) XI exp(-a(g12 - g22)/RT) X12 = (3-30) X2 + X1 exp(-a(g12 - g22)/RT) We now introduce Eqs. (3-29) and (3*30) into the two-liquid theory of Scott which assumes that there are two kinds of cells in a binary mixture : one for molecules 1 and one for molecules 2, as shown in Fig. 3*1. Let G1 and G2 stand, respectively, for the residual Gibbs energy (that is, compared with the ideal gas at the same temperature, pressure, and composition) of one mole of cells of type 1 and one mole of cells of type 2. These Gibbs energies are assumed to be related to the local mole fractions according to If we consider pure liquid 1, XI1 = 1, X21 = 0. In this case, the residual Gibbs energy for a cell containing a molecule 1 at its center, Glpure, is G1 = X11 g11 + X21 g21 (3-3D G2 = X12 g12 + X22 g22 (3-32) Glpure = g11 Similarly, for pure liquid 2, (3-33) 25 M O L E C U L E 1 A T C E N T E R M O L E C U L E 2 A T C EN TE R Figure 3.1 Two types of cells according to Scott's two-liquid theory of binary mixtures. 26 G2pure = g22 (3-34) The molar excess Gibbs energy for a binary solution is the sum of two changes in residual Gibbs energy: first, that of transfering X1 molecules from a cell of the pure liquid 1 into a cell 1 of the solution, X1(G1 - Glpure), and second, that of transfering X2 molecules from a cell of the pure liquid 2 into a cell 2 of the solution, X2(G2 - G2pure). Therefore, i £ j G = X1 (G1 - Glpure) + X2 (G2 - G2pure) (3*35) I Substituting Eqs. (3-27), (3-28), (3-31), (3-32), (3-33) and (3-34) into Eq. (3*35), we obtain ! GE = X1 X21 (g21 - g11) + X2 X12 (g12 - g22) (3-36) i i I where X21 and X12 are given by Eqs. (3*29) and (3*30). Eq. (3*36) coupled with Eqs. (3*29) and (3*30) are called the NRTL (nonrandom, two-liquid) equation. It is the simplest of the local composition equations and has perhaps the best semitheoretical basis. The adjustable parameters are (g12 - g22) and (g21 - g11). We may consider a as a third parameter, set it at a predetermined value. When justified by the data, NRTL Eq. can also be used as three-parameter equation for highly nonideal systems. 27 CHAPTER 4 RESULTS AND DISCUSSIONS ? 4.1 Model The model mixtures studied in the present calculation are binary mixtures containing Lennard-Jones 12-6 particles. Table 4.1 shows the assignments of potential parameters and states studied for the model mixtures, where Sij and Eij are ratio of the size and energy potential parameters respectively, and C is the deviation factor of E12 from the geometric mean of E11 and E22 (see Sec. 2.2). Seven models are studied at four bulk compositions, that is, X1 = 10%, 25%, 50%, and 75%, and at different temperatures. 4.2 Helmholtz Free Energy and Excess Free Energy - LBH Results In this section we present the results for Helmholtz free energy and excess Gibbs free energy of seven model mixtures studied in this work. The Helmholtz free energy is calculated using perturbation theory, and the calculation of excess Gibbs free energy 28 Table 4.1 Model mixtures used, states studied, and the assignments of potential parameters. Model A-1 B-1 C-1 D-1 D-2 - E-1 E-2 S11 1 1 1 1 1 1 1 S12 0.8 0.8 0.8 0.8 0.8 1 1 S22 0.6 0.6 0.6 0.6 0.6 1 1 El 1 1 1 1 1 1 1 1 E12 1 0.8944 0.7746 0.5 2 0.5 2 E22 1 0.8 0.6 1 1 1 1 C 1 1 1 0.5 2 0.5 2 T* 0.7 1 0.7 0.9 0.7 0.7 0.9 0.7 0.9 1.1 0.7 0.9 0.7 0.9 1.1 29 Table 4.2 The Helmholtz free energy and excess Gibbs free energy of A-1 model as determined by LBH theory at zero pressure, T# = 0.7, 1.0 X1 T* = 0.7 T* = 1.0 A / €11 g e / €11 A / €11 G E / €11 0 -3.1178 0 -2.2160 0 0.10 -3-2258 -0.0301 -2.4640 -0.0926 0.25 -3.4333 -0.0458 -2.7616 -0.1571 0.50 -3.7031 -0.0458 -3.1574 -0.1644 0.75 -3-9560 -0.0290 -3.4846 -0.1031 1 -4.1967 0 -3.7700 0 Table 4.3 As in Table 4.2, but of B-1 model at T* = 0.7, 0.9 X1 T* = 0.7 T* = 0.9 A / €11 ge / €11 A / €11 ge / €11 0 -1.9254 0 -1.5037 0 0.10 -2.2885 -0.1360 -1.8884 -0.1485 0.25 -2.7151 -0.2219 -2.3481 -0.2537 0.50 -3.2811 -0.2201 -2.9461 -0.2601 0.75 -3.7619 -0.1330 -3.4360 -0.1603 1 -4.1967 0 -3-8665 0 30 Table 4.4 As in Table 4.2, but of C-1 model at T* = 0.7 X1 T* A / €11 = 0.7 g e / €11 0 -1.1130 0 0.10 -1.5587 -0.1374 0.25 -2.1301 -0.2462 0.50 -2.9128 -0.2579 0.75 -3-5841 -0.1583 1 -4.1967 0 Table 4.5 As in Table 4.3> but of D-1 model. X1 T* = 0.7 T* = 0.9 A /01 ge / €11 A / €11 G E / €11 0 -3.178 0 -2.4705 0 0.10 -2.4560 0.7697 -1.9986 0.6115 0.20 -1.9032 0.8465 0.25 -2.2038 1.1837 0.50 -2.5627 1.0946 -2.3635 0.8050 0.75 -3-2918 0.6352 -3.0394 0.4780 1 -4.1967 0 -3.8664 0 Table 4.6 As in Table 1.1 4.2, but of D-2 model at T* = 0.7, 0.9, X1 T* = 0. .7 T# = 0.9 A / €11 g e / €11 A / €11 ge / €11 0 -3-1178 0 -2.4705 0 0.10 -5.0267 -1.8010 -4.3047 -1.6946 0.25 -6.2664 -2.8789 -5.5490 -2.7295 0.50 -6.3601 -2.7029 -5.7213 -2.5528 0.75 -5.4486 -1.5216 -4.9358 -1.4138 1 -4.1967 0 -3.8664 0 T* = 1.1 X1 A / €11 ge / €11 0 -2.0084 0 0.10 -3.7124 -1.5328 0.25 -4.9404 -2.5039 0.50 -5.1999 -2.3354 0.75 -4.5663 -1.2737 1 -3.7206 0 32 Table *1*7 As in Table 4*3 > but of E-1 model. T* = 0.7 T* - 0.9 X1 A / €11 G e / €11 A / €11 CE / €11 0 -4.1930 0 -3.8660 0 0.10 -3-7240 0.4722 -3.4626 0.4034 0.25 -3.2306 0.9657 -3.0586 0.8074 0.50 -2.9275 1.2688 -2.8324 1.0336 0.75 -3.2306 0.9657 -3.0586 0.8074 1 -4.1963 0 -3.8666 0 I i I i ! 33 Table 4.8 As in Table 4.6, but of E-2 model. X1 T* = 0.7 T* = 0.9 A / €11 G e / €11 A / €11 ge / €11 0 -4.1963 0 -3.8660 0 0.10 -5.2334 -1.0372 -4.7903 -0.9243 0.25 -6.3957 -2.1994" -5.8508 -1.9848 0.50 -7.1564 -2.9601 -6.5531 -2.6871 0.75 -6.3957 -2.1994 -5.8508 -1.9848 1 -4.1963 0 -3.8660 0 1 1 T* = 1.1 X1 A / €11 G E / €11 0 -3.7204 0 0.10 -4.5144 -0.7940 0.25 -5.4689 -1.7485 0.50 -6.1130 -2.3926 0.75 -5.4689 -1.7485 1 -3.7204 0 is described in Sec. 2.3. Tables 4.2-4.8 give these results for the model mixtures over a range of temperatures and compositions. All the results listed are obtained at zero pressure, where X1 is the mole fraction of componet E 1, A and G are, respectively, the Helmholtz free energy and excess Gibbs energy of the mixture. Among all the models, only D-1 and E-1 exhibit positive deviation from the the ideal solution while the others exhibit negative deviation. This is due to the weak interaction between unlike molecules ( C = 0.5 for D-1,and E-1). The excess free energy is symmetrical with respect to X1 for mdels E-1, E-2, moderately assymetrical for models A-1, B-1, C-1, and very assmmetrical for model D-1, D-2. It is because for E-1 and E-2, the size and energy parameters are the same for the two pure components (S22 = 1, E22 = 1); for D-1 and D-2, the size parameter is different (S22 = 0.6) and the cross interaction deviates from unity too much (E12 = 0.5, 2.0, respectively for D-1, D-2); for A-1, B-1, and C-1, the cross interaction does not deviate from unity too much. At higher temperature, A-1 and B-1 solutions show more nonideality, while D-1, D-2, E-1 and E-2 solutions show more ideality. As expected, when E11 $ E22 or S11 $ S22 the excess free energies are skewed towards component 1, the strongly interacting component. 4.3 Local Composition Local compositions of model mixtures are calculated directly 35 using LBH perturbation theory or Monte Carlo (MC) simulation. These results are obtained as a function of r for five model mixtures, A-1, B-1, C-1, D-1, and D-2 over a range of temperatures and compositions using LBH. Selected LBH results are shown in Tables 4.9 and 4.10. All the results obtained for the model mixtures studied are given in Figs. 4.1-4.19. Each figure shows local composition as a function of r and as a function of temperature. It is seen that LBH LC is evidently different from overall composition, and is a very weak function of temperature and cutoff distance. This applies to all the models studied here. We choose cutoff distance Lij at around 1.2CTij for the calculation of Xij, the results thus obtained are summarized in Secs. 4.4 and 4.5 and compared with the local compositions calculated using the Wilson and NRTL equations. Since LBH local compositions are either unobtainable or inaccurate for models D-1, D-2, E-1, and E-2, MC results are used in the comparison. Selected comparisons of MC results and with LBH results are given in Tables 4.10.1 and 4.10.2 respecively for D-1 and D-2 models. It is found the MC results are different from LBH results to some degree (the magnitude of the difference varies for different systems), and are generally more sensitive to temperature and to cutoff distance than the LBH results. At cutoff distance Lij around 1.2crij, the MC Xij change slightly with Lij, but MC Xij change more appreciably at the other cutoff distances. For model D-1, the MC X11, X22 is, respectively, greater than LBH XII, X22; and MC X11 change more appreciably with 36 Table 4.9 Local compositons of nearest neighbor particles around a central particle. These results are obtained by LBH theory T* = 0.7, 0.9 for B-1 model at X1 = 0.5, T* = 0.7 T* - 0.9 v/a X11 X22 v/a X11 X22 1.08 0.686 0.285 1.08 0.685 0.285 1.18 0.672 0.295 1.17 0.674 0.293 1.27 0.659 0.304 1.27 0.664 0.300 1.37 0.650 0.312 1.36 0.656 0.306 1.47 0.645 0.318 1.46 0.652 0.311 1.57 0.646 0.322 1.56 0.650 0.314 1.66 0.651 0.322 1.66 0.652 0.316 1.76 0.656 0.319 1.75 0.654 0.315 1.86 0.660 0.314 1.85 0.657 0.313 1.96 0.665 0.301 1.95 0.661 0.310 37 Table 4.10 As in Table 4.9, but for D-2 model at X1 = 0.5, T* = 0.7, 0.9, 1.1 r/<7 T* = X11 0.7 X22 r/ o T* X11 = 0.9 X22 r / o T* X11 = 1.1 X22 1.08 0.679 0.287 1.07 0.679 0.287 1.07 0.675 0.287 1.18 0.661 0.301 1.17 0.664 0.299 1.17 0.665 0.297 1.27 0.645 0.314 1.27 0.649 0.310 1.26 0.653 0.306 1.37 0.634 0.324 ‘ 1.36 0.639 0.319 1.3& 0.644 0.315 1.47 0.632 0.331 1.46 0.636 0.326 1.46 0.639 0.321 1.57 0.638 0.333 1.56 0.639 0.328 1.55 0.641 0.324 1.66 0.650 0.329 1.66 0.647 0.327 1.65 0.646 0.324 1.76 0.657 0.321 1.75 0.654 0.322 1.75 0.652 0.321 1.86 0.662 0.312 1.85 0.659 0.315 1.85 0.657 0.316 1.96 0.666 0.305 1.95 0.664 0.309 1.94 0.661 0.311 I 38 Table 4.10.1 Local compositions for D-1 model at equimolar ratio: comparison of MC results with LBH results. r / a X11 X22 r / a X11 X22 T* = 0.7 T* = 0.7 LBH data MC data 1.08 0.679 0.287 1.08 0.866 0.472 1.18 0.667 0.296 1.17 0.870 0.498 1.27 0.656 0.304 1.24 0.866 0.495 1.37 0.647 0.311 1.33 0.861 0.478 1.47 0.643 0017 1.49 0.850 0.432 1.57 0.642 0.320 1.57 0.846 0.406 1.66 0.646 0.321 1.65 0.842 0.378 1.76 0.651 0.319 1.73 0.837 0.348 1.86 0.655 0.315 1.89 0.834 0.291 T* = : 0.9 T* = 0.9 LBH data MC data 1.07 0.676 0.289 1.09 0.701 0.474 1.17 0.668 0.294 1.17 0.712 0.509 1.27 0.662 0.299 1.26 0.709 0.511 1.36 0.656 0.303 1.35 0.702 0.498 1.46 0.652 0.307 1.44 0.696 0.479 1.56 0.650 0.309 1.52 0.691 0.485 1.66 0.651 0.311 1.70 0.683 0.413 1.75 0.652 0.311 1.78 0.681 0.388 1.85 0.653 0.311 1.87 0.679 0.365 Table *1.10.2 Local compositions for D-2 model at equimolar ratio: comparison of MC results with LBH results. r / a X11 X22 r / a X11 X22 T* : = 0.7 T* = 0.7 LBH data MC data 1.08 0.679 0.287 1.08 0.671 0.357 1.18 0.661 0.301 1.16 0.626 0.344 1.27 0.645 0.314 1.23 0.612 0.338 1.37 0.634 0.324 1.38 0.615 0.326 1.47 0.632 0.331 1.46 0.621 0.318 1.57 0.638 0.333 1.53 0.632 0.309 1.66 0.650 0.329 1.68 0.669 0.331 1.76 0.657 0.321 1.75 0.678 0.257 1.86 0.662 0.312 1.90 0.676 0.220 T* = : 0.9 T* = 0.9 LBH data MC data 1.07 0.679 0.287 1.10 0.650 0.297 1.17 0.664 0.299 1.18 0.616 0.283 1.27 0.649 0.310 1.25 0.605 0.275 1.36 0.639 0.319 1.33 0.603 0.267 1.46 0.636 0.326 1.48 0.612 0.251 1.56 0.639 0.328 1.56 0.624 0.242 1.66 0.647 0.327 1.63 0.642 0.229 1-75 0.654 0.322 1.78 0.666 0.198 1.85 0.659 0.315 1.86 0.668 0.182 40 CM CM I I I i .2 - 1.8 1.4 1.6 1 . 2 Figure 4.1 Local compositions of X11 and X22 as a function of r for A-1 model at X1 = 0.10, T* = 0.7, 1.0 The solid and dashed curves give the LBH results at T* = 0.7 and 1.0 respectively. The distance from a central particle R in r/a. CYl CvJ X X !.8 1 .4 1 .6 1 .2 Figure 4.2 As in Fig. 4.2, but at XI = 0.25 42 C\J CM X X 1.6 1 . 8 1.2 .4 Figure 4.3 As in Fig. 4.1, but at XI = 0.50 *+3 OJ f \J X X I I X22 1.6 1 . 8 1.2 1.4 Figure 4.4 As in Fig. 4.1, but at XI = 0.75 kk X Figure 4.5 Local compositions of X11 and X22 as a function of The solid and dashed curves give the LBH results at T* = 0.7 and 0.9 respectively. The distance from a central particle R in r/a. . 8 . G ou f\j X X .4 2 1.2 ' 4 R ' - 6 .8 Figure 4.6 As in Fig. 4.5, but at X I = 0.25 46 OJ C\J X 1.8 1.6 1.4 1.2 Figure 4.7 As in Fig. 4.5, but at XI = 0.50 r \ j r \ j X X 1 . 2 1 . 8 1.6 1-4 Figure 4.8 As in Fig. 4.5, but at XI = 0.75 48 . 8 .6 X .4 .2 1 . 8 1.4 1 .6 1.2 Figure *1.9 Local compositions of X11 and X22 as a function of r for C-1 model at X1 = 0.10 and T* = 0.7, the solid curves give the LBH results. The distance from a central particle R in r / a . . 8 .6 C \J C \J X 1 X 4 2 Figure 4.10 As in Fig. 4.9, but at XI 0.25 50 . 8 .6 C \J C \J X X .4 2 1 .2 1 -4 ^ 1 .6 1 -8 Figure 4.1;) As in Fig. 4,9, but at X I * = 0,50 51 r v l c \ j X .2 - X22 1 .8 1 . 6 1 .2 1 .4 Figure 4.12 As in Fig. 4.9, but at XI = 0.75 52 8 .6 CM CM X X .4 . 2 J________ I ________ ! ________ L 1 .2 |.4 _ 1.6 1 -8 Figure ^.13 Local compositions of X11 and X22 as a function of r for B-1 model at X1 = 0.10, T* = 0.7, 0.9 The solid and dashed curves give the LBH results at T* = 0.7 and 0.9 respectively. The distance from a central particle R in r / a . .8 6 OJ c a j X X -4 .2 1.4 Figure 4.14 As in Fig. 4,13, but at XI = Q.50 5k . 8 X .4 .2 1 . 6 1.8 1 .4 1 . 2 Figure 4.15 As in Fig. 4.13, but at XI = 0.75 55 8 X22 .6 OJ >c x 4 Figure *1.16 Local compositions of X11 and X22 as a function of r for D-1 model at X1 = 0.10, T* = 0.7, 0.9, 1.1 The solid, dashed, and dash-dot curves give the LBH results at T* = 0.7, 0.9, and 1.1 respectively. The distance from a central particle R in r / a . 56 . 6 CM CM X X 4 .2 1.2 \A 1 . 8 1 . 6 Figure 4.17 As in Fig. 4.16, but at X1 = 0.25 57 X X22 Figure 4.18, As in Fig. 4.16, but at XI = 0.50 I 58 .8 .6 r\j oj X X .4 2 X22 Figure 4.19 As in Fig. 4.16, but at XI = 0.75 59 temperature than X22. For model D-2, at T# = 0.7, the local compositions X11, X22 from MC and LBH do not differ much. At higher temperature (T# = 0.9), the MC X11 does not change much with cutoff distance, but X22 change appreciably. 4.4 Testing the Wilson Equation Wilson equation is applied to the excess free energy results for mixtures, A—1, B-1, C-1, D-1, D-2, E-1, and E-2, First, We examine if the equation can represent the excess Gibbs energy of model mixtures well. This is done by fitting the LBH excess free energy data to the equation. Next, we compare the local compositions calculated by the equation (Eqs. 3.11 and 3*12, see Sec. 3.2) with those calculated directly from LBH theory or MC simulation. The excess free energy data in Tables 4.2-4.8 are fitted to Wilson equation. The fitting is carried out using four LBH data points of each model at each temperature. The fitted excess free energies of each model are shown in Figs. 4.20-4.26, those figures clearly show how well Wilson equation can represent data. The fitted curves in Figs. 4.20-4.26 show that Wilson equation can represent excess free energies very well for models A-1, B-1, C-1 , moderately well for models D-2, E-2, very poor for models, D-1, E-1. Models A-1, B-1, and C-1 are almost ideal solutions (excess free energies are small). In models D-2 and E-2, C > 1, therefore the mixture is more stable than the pure components. Such 60 so-called associated mixtures though quite nonideal (relatively large excess free energy) are adequately represented by the Wilson equation. In models E-1 and D-1, C < 1, and the mixture is therefore less stable than the pure components. Such mixtures have a tendency to phase separate and are known to be poorly represented by the Wilson equation. In fact, the form of the Wilson equation do not allow the prediction of phase separation or partial miscibility. As seen in Table 4.11, Wilson parameters, (f12 — f11) and (f12 - f22) are temperature, dependent. Wilson parameters for nearly ideal solutions (A-1,B-1) seem to be more sensitive to temperature than the the more nonideal solutions (D-2, E-2). Figs. 4.20 and 4.21 show that temperature does not appreciably affect the ability of Wilson equation in representing excess free energies for nearly ideal solutions (A-1, B-1), but for moderately nonideal solutions (D-2, E-2) temperature has appreciable effect. The Wilson equation apparently works better at higher temperatures. Comparisons of "experimental" local compositions (LBH or MC data) with Wilson equation are shown in Tables. 4.12-4.25 for models A-1, B-1, C-1, D-2, and E-2 over a range of temperatures and compositions. The results show that local compositions calculated using Wilson equation are much more temperature sensitive than those from LBH theory, and the agreement between Wilson and LBH results becomes better as temperature increases for all the models except B-1. Tables 4.12-4.15 show the comparisons of LBH local compositions 61 with Wilson equation for model A-1. At X1 = 0.10, the environments of molecules 1 and 2 predicted by the Wilson equation are qualitatively correct, i.e., X21 > X11 and X22 > X12 (the same as given by LBH). However, at T* = 1.0 the errors reach -59$> 13%, 17%, and —65%« respectively, for X11, X21, X22, and X12. At XI = 0.25, the LBH local compositions Xij are also qualitatively correct; however, at T* = 1.0 the errors reach -52%, 34$, 45$, and -56%, respectively, for XII, X21, X22, and X12. At X1 = 0.50, the environments of, molecules 1 and 2 given by the Wilson equation are qualitatively incorrect, i.e., XII < X21 and X22 >X12 (but LBH gives XII > X21 and X22 < X12). At X1 = 0.75, the environments of molecules 1 and 2 given by the Wilson equation are qualitatively correct; however, at T* = 1.0, the errors reach —21%, 125$, 16$, and -22$, respectively, for XII, X21, X22, and X12. The results also indicate that the agreement for A-1 at T* = 1.0 is better than at T* = 0.7. For example, at X1 = 0.75 and T* = 0.7, the erros for X11, X21, X22, and X12, respectively, are -32$, 192$, 24$, and -34$ (worse than at T* = 1.0) The comparisons of LBH local compositions with Wilson equation for model B-1 are given in Tables 4.16-4.19. The environments of molecules 1 and 2 predicted by the Wilson equation are qualitatively correct over all the compositions and temperatures studied except at the state X1 = 0.50, T* = 0.7. At X1 = 0.10 and T* = 0.7, the errors reach -38$, 8$, 12$, and -45$, respectively, for XII, X21, X22, and X12. At X1 = 0.25 and T* = 0.7, the errors reach -31$, 62 21%, 29%, and -37%, respectively, for XII, X21, X22, and X12. At X1 = 0.50 and T* = 0.7, the errors reach -20%, 41%, 57%, and -38%, respectively, for X11, X21, X22, and X12. At X1 = 0.75 and T* = 0.7, the errors reach -10%, 61%, 84%, and -12%, respectively, for X11, X21, X22, and X12. We also notice that, in contrast to the other models, increased temperature has an unfavorable effect on Xij predicted by Wilson equation for model B—1; e.g., at X1 = 0.10 and T* = 0.9, the errors reach -69%, 16%, 15%, and -57%, respectively, for XI1, X21, X22, and X12 (worse than at T* = 0.7). Table 4.20 gives the comparison of LBH local compositions with Wilson equation for model C-1 at T* = 0.7. AT X1 = 0.10 and 0.25, the environments of molecules 1 and 2 predicted by the Wilson equation are qualitatively incorrect, i.e., the relative magnitudes of X11 and X21, and of X22 and X21 are not consistent with those given by LBH. At X1 = 0.10, the errors reach -81%, 12%, 15%, and -58%, respectively, for XII, X21, X22, and X12. At X1 = 0.25, the errors reach -76%, 52%, 39%, and -49%, respectively, for X11, X21, X22, and X12. At X1 = 0.50, all the Xij given by Wilson equation are qualitatively incorrect, i.e., Wilson equation gives X11 < X21 and X22 > X12 but LBH theory gives X11 > X21 and X22 < X12. At X1 = 0.75, only the environments of molecules 2 given by Wilson equation are qualitatively correct; however the errors reach 128% and -18%, respectively, for X22 and X12. The comparisons of "experimental” local compositions (LBH and MC data) with Wilson equation for model D-2 are given in 63 Tables 4.21-4.24. The MC data are taken for this testing. The results show that only at the state X1 = 0.10 and T* = 1.0, the Wilson equation can qualitatively describe the environments of both molecules 1 and 2; at the other states only the environments of molecules 1 or 2 given by Wilson equation are found to be qualitatively correct. Moreover, the quantitative agreement is poor, and the qualitative agreement at some particular states seems to be fortuitous. At X1 = 0.10 and T* = 0.9, the errors reach -97% and 11% , respectively, for X11 and X21. At X1 = 0.25 and T* = 0.9, the errors reach -97% and 52% , respectively, for XI1 and X21. At X1 = 0.50 and T* = 0.9, the errors reach -48% and 19% , respectively, for X22 and X21. At XI = 0.75 and T* = 0.9, the errors reach -85% and 11% , respectively, for X22 and X12. It is found that the agreement in local compositions is better at higher temperatures for D-2. Table 4.25 gives the comparisons of MC local compositions with Wilson equation for model E-2. The qualitative and quantitative agreements given by Wilson equation are poor over the states studied. The qualitative agreement found for some cases seems to be fortuitous. X1 = 0.5 and T* = 0.9, is the only state for which the environments of molecules 1 and 2 given by Wilson equation are qualitatively correct, but the errors reach -95%, 86%, -90%, and 82%, respectively, for XI1, X21, X22, and X12. It is found that the agreement in local compositions is better at higher temperatures for E-2. 64 .3 0 -G .20 T*=I.O r=.7 Figure 4.20 Excess Gibbs free energy of A-1 model fitted to Wilson equation. The curves give the results of the fitted Wilson equations, and the points marked -f- and • give the LBH results at T* = 0.7 and 1.0 respectively. Figure 4.21 Excess Gibbs free energy of B-1 model fitted to Wilson equation. The curves give the results of the fitted Wilson equations, and the points marked -f and • give the LBH results at T* = 0.7 and 0.9 respectively. 66 .30 .20 .10 Figure 4.22 Excess Gibbs free energy of C-1 model fitted to Wilson equation. The curve gives the results of the fitted Wilson equation, and the points marked -b give the LBH results at T* = 0.7 67 1 . 0 T=.9 T=.7 5 » .4 .8 .6 2 Figure 4,23 As in Fig. 4,21, but of D-l model, 68 1 .0 T=.9 E I I T = .7 .5 .4 .8 2 .6 Figure 4.24 As in Fig. 4.21, but of E-l model. 69 I \ X, -6 .8 Figure 4.25 Excess Gibbs free energy of D-2 model fitted to Wilson equation. The curves give the results of the fitted Wilson equations, and the points marked _j_ , • , and X give the LBH results at T* = 0.7, 0.9, and 1.1 respectively. 70 Figure A,26 As in Fig, A,25, but of E-2 model. 71 Table *1.11 Parameters of Wilson Eq. fitted to excess free energy of model mixtures. Model T* (f12 - f11)/€l1 (f12-- f22)/€l1 A-1 0.7 -0.5390 0.5385 1.0 -0.3385 0.3404 B-1 0.7 0.9994 -0.0996 0.9 -0.5455 0.1061 C-1 0.7 -0.7893 0.1006 D-2 0.7 -3.133 -2.678 0.9 -3.288 -2.581 1.1 -3-301 -2.388 E-2 0.7 -3-467 -2.584 0.9 -3.298 -2.649 1.1 -3.103 -2.602 Table *1.12 Comparison of "experimental" local compositions with Wilson Eq. for A-1 model at X1 = 0.10 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.049 0.951 0.951 0.049 1.0 0.073 0.927 0.927 0.073 LBH data 0.7 0.176 0.824 0.797 0.203 1.0 0.180 0.820 0.793 0.207 72 Table 4.13 As in Table 4.12, but at X1 = 0.25 x# X11 X21 X22 X12 Wilson Eq. 0.7 0.134 0.866 0.866 0.134 1.0 0.192 0.808 0.808 0.192 LBH data 0.7 0.394 0.606 0.563 0.437 1.0 0.399 0.601 0.558 0.442 Table 4.14 As in Table 4.12, but at X1 = 0.50 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.317 0.683 0.683 0.317 1.0 0.416 0.584 0.584 0.416 LBH data 0.7 0.664 0.336 0.298 0.702 1.0 0.668 0.332 0.295 0.705 Table 4.15 As in Table 4.12, but at X1 = 0.75 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.582 0.418 0.418 0.582 1.0 0.681 0.319 0.319 0.681 LBH data 0.7 0.857 0.143 0.123 0.877 1.0 0.858 0.142 0.122 0.878 73 Table 4.16 Comparison of "experimental" local compositions with Wilson Eq. for B-1 model at X1 = 0.10 T# X11 X21 X22 X12 Wilson Eq. 0.7 0.114 0.886 0.886 0.114 0.9 0.057 0.943 0.910 0.090 LBH data 0.7 0.183 0.817 0.792 0.208 0.9 0.185 0.815 0.790 0.210 Table 4.17 As in Table 4.16, but at X1 = 0.25 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.278 0.722 0.722 0.278 0.9 0.153 0.847 0.772 0.228 LBH data 0.7 0.404 0.596 0.558 0.443 0.9 0.407 0.593 0.555 0.445 Table 4.18 As in Table 4.16, but at X1 = 0.50 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.536 0.464 0.464 0.536 0.9 0.352 0.648 0.530 0.470 LBH data 0.7 0.672 0.328 0.295 0.705 0.9 0.674 0.326 0.293 0.707 7k Table 14.19 As in Table 4.16, but at X1 = 0.75 T# X11 X21 X22 X12 Wilson Eq. 0.7 0.776 0.224 0.224 0.776 0.9 0.619 0.381 0.273 0.727 LBH data 0.7 0.861 0.139 0.122 0.878 0.9 0.862 0.138 0.121 0.879 Table 4.20 Comparison of "experimental" local with Wilson Eq. for C-1 model at XI 0.50, 0.75, and T* = 0.7 compositions = 0.1, 0.25, X11 X21 X22 X12 XI = 0.10 Wilson Eq. LBH data 0.035 0.185 0.965 0.815 0.912 0.790 0.088 0.210 X1 = 0.25 Wilson Eq. LBH data 0.098 0.406 0.902 0.594 0.776 0.556 0.224 0.444 X1 r 0.50 Wilson Eq. LBH data 0.245 0.673 0.755 0.327 0.536 0.294 0.464 0.706 X1 = 0.75 Wilson Eq. LBH data 0.494 0.861 0.506 0.139 0.278 0.122 0.722 0.878 75 Table 4.21 Comparison of ’ ’experimental” local compositions with Wilson Eq. for D-2 model at X1 = 0.10 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.001 0.999 0.164 0.836 0.9 0.003 0.997 0.338 0.662 1.0 0.006 0.994 0.506 0.494 LBH data 0.7 0.173 0.827 0.799 0.201 0.9 r 0.176 0.824 0.796 0.204 1.0 0.178 0.822 0.794 0.206 MC data 0.7 0.161 0.839 0.813 0.187 0.9 0.100 0.900 0.788 0.212 Table 4.22 As in Table 4.21, but at X1 = 0.25 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.004 0.996 0.061 0.939 0.9 0.009 0.991 0.146 0.854 1.0 0.016 0.984 0.255 0.745 LBH data 0.7 0.389 0.611 0.568 0.432 0.9 0.393 0.607 0.564 0.436 1.0 0.396 0.604 0.561 0.439 MC data 0.7 0.364 0.636 0.608 0.392 0.9 0.348 0.652 0.572 0.428 76 Table 4.23 As in Table 4.21, but at X1 = 0.50 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.011 0.989 0.021 0.979 0.9 0.009 0.991 0.146 0.854 1.0 0.047 0.953 0.102 0.898 LBH data 0.7 0.661 0.339 0.301 0.699 0.9 0.664 0.336 0.299 0.702 1.0 0.666 0.335 0.297 0.703 MC data 0.7 0.626 0.374 0.344 0.656 0.9 0.616 0.384 0.283 0.717 Table 4.24 As in Table 4.21, but at X1 = 0.75 T* X11 X21 X22 X12 Wilson Eq. 0.7 0.021 0.979 0.008 0.992 0.9 0.071 0.929 0.017 0.983 1.0 0.130 0.870 0.037 0.963 LBH data 0.7 0.856 0.144 0.124 0.876 0.9 0.857 0.143 0.123 0.877 1.0 0.858 0.142 0.122 0.878 MC data 0.7 0.832 0.168 0.181 0.819 0.9 0.833 0.167 0.112 0.888 77 Table *1.25 Comparison of "experimental" local compositions with Wilson Eq. for E-2 model. T* X11 X21 X22 X12 X1 = 0.10 Wilson Eq. 0.7 0.0008 0.9992 0.183 0.817 0.9, 0.0028 0.9972 0.322 0.678 1.1 0.0066 0.9934 0.458 0.542 MC data 0.7 0.401 0.599 0.931 0.069 0.9 0.068 0.932 0.887 0.113 X1 = 0.25 Wilson Eq,. 0.7 0.002 0.997 . 0.070 0.930 0.9 0.008 0.991 0.136 0.864 1.1 0.020 0.980 0.220 0.780 MC data 0.7 0.561 0.439 0.857 0.143 0.9 0.187 0.813 0.718 0.282 X1 = 0.50 Wilson Eq. 0.7 0.007 0.993 0.024 0.976 0.9 0.025 0.975 0.050 0.950 1.1 0.056 0.944 0.086 0.914 MC data 0.7 0.710 0.290 0.744 0.256 0.9 0.477 0.523 0.479 0.522 X1 = 0.75 Wilson Eq. 0.7 0.021 0.979 0.008 0.992 0.9 0.071 0.929 0.017 0.983 1.1 0.152 0.848 0.030 0.970 MC data 0.7 0.857 0.143 0.561 0.439 0.9 0.718 0.282 0.187 0.813 78 We can summarize the testing results in this section as follows. For A-1, B-1 at X1 - 0.10, 0*25, 0.75, and for C-1 at X1 = 0.10, 0.25, the environments of molecules 1 and 2 given by Wilson equation are qualitatively correct over the temperature range tested. These results indicate that the Wilson equation can give reasonable, though not accurate, estimates of local compositions Xij. Thus the Wilson equation shows partial success in predicting LC Xij, the situations, more favorable than the conclusions of Nakanishi and Toukubo [19] and Nakinishi et al. [20]. For models D-2 and E-2 (phase separating solutions), the Wilson equation cannot reproduce excess free energy or LC Xij in reasonable agreement with results obtained from perturbation theory or computer simulation. 4.5 Testing the NRTL Equation The NRTL equation is applied to the results obtained for model mixtures B-1, D-1, and E-1. We first fit the LBH excess free energy data of the model mixture to the NRTL equation to determine if the NRTL equation can represent data well. The fitted parameters are then used to calculate local compositions Xij using the NRTL equation (see Sec. 3*3)* The results are compared with local compositions from LBH or MC. The NRTL equation has three parameters, the fitting is done in two steps. The first fitting is carried out using two sets of LBH data at two temperatures simultaneously. Three parameters at , 79 (g12 - g11), and (g12 - g22) are fitted in this step. The fitted parameters are given in Table 4.26, and the fitted excess free energies are shown in Figs. 4.27, 4.29, and 4-31* The Figures show that one set of NRTL parameters cannot represent the excess free energies at both temperatures for each model. Therefore, it is necessary to fit the data at each temperature separately. So in the second fitting step LBH data at one temperature are fitted to two adjustable parameters, (g12 — g11) and (g12 - g22), while the third parameter a is set at the value determined from the first fitting step. The fitted parameters are shown in Table 4.27, and the fitted excess free energies are shown in Figs. 4.28, 4.30, and 4.32. Those Figures show that the NRTL equation can represent excess free energies very well for models B-1, D-1, and E-1, and the variation in temperature does not appreciably affect the ability of the NRTL equation in representing excess free energies. Table 4.27 shows that the NRTL parameters, (g12 - g22) and (g12 - g22), are temperature dependent. In addition, the parameters are more sensitive to temperature for highly nonideal solutions (such as models D-1 and E-1). Figs. 4.30 and 4.32 show that unlike the Wilson equation the NRTL equation can give an excellent representation of excess free energies for solutions which show possible partial miscibility (models D-1 and E-1). Comparisons of "experimental" LC with NRTL equation are given in Tables 4.28-4.31 for model B-1. The environments of molecules 2 predicted by the NRTL equation are in good agreement with LBH data 80 over the temperature and composition range studied, both qualitatively and quantitatively. At Xt = 0.10, T* = 0.9, the errors are -0.9$ and 3%, respectively, for X22 and X12. At X1 = 0.25, T* = 0.9, the errors are -1% and 2$, respectively, for X22 and X12. At X1 = 0.50, T* = 0.9, the errors are -2% and 1%, respectively, for X22 and X12. At X1 = 0.75, T* = 0.9, the errors are -2% and 0.3$, respectively, for X22 and X12. However, the environments of molecules 1 predicted by the NRTL equation are not in good agreement with LBH , results, it always predicts X11 = 1, X21 = 0 over the states studied. We also notice that for this model the agreement in LC is better at higher temperatures. Tables 4.32-4.35 give the comparisons of "experimental” LC (MC data) with those from NRTL equation for model D-1. At X1 = 0.10, 0.25, and 0.75, the environments of molecules 1 and 2 predicted by the NRTL equation are qualitatively correct at T* = 0.7, 0.9; however, at X1 = 0.50, the environments of molecules 2 predicted by the NRTL equation are qualitatively incorrect, i.e., the relative magnitude of X22 and X12 is not consistent with those given by MC. At XI = 0.10, T* = 0.7, the errors are -14$, 9$, 1$ and -11$, respectively, for XI1, X21, X22 and X12. At X1 = 0.25, T* = 0.7, the errors are -8$, 15$, -10$ and 57$, respectively, for X11, X21, X22 and X12. At X1 = 0.50, T* = 0.9, the errors are 4$, -10$, -11$ and 11$, respectively, for XII, X21, X22 and X12 (but the NRTL X22 and X11 are not qualitatively correct). At X1 = 0.75, T* = 0.9, the errors are 4.4$, -27$, -11$ and 3-4$, respectively, for X11, X21, 81 X22 and X12. The NRTL equation shows generally good agreement at the above states. For model D-1, the temperature effect on LC as predicted by NRTL equation is smaller - seems to be different at different compositions. Table 4.36 gives the comparison of MC local compositions with those from NRTL equation for model E-1. The NRTL equation predicts that the local mole fractions are the same as the overall mole fractions over the states tested. For example, at X1 = 0.10, T* = 0.7, 0.9, the NRTL equation gives X11 = 0.10 and X22 = 0.90. At X1 = 0.10, 0.25, 0.75, T* = 0.9, and at X1 = 0.10, T* = 0.7, the environments of molecules 1 and 2 given by the NRTL equation are qualitatively correct; however the quantitative agreement is not good. At X1 = 0.10, T* = 0.9, the errors reach -58% , 18$, -2% and 23$, respectively, for XII, X21, X22 and X12. At XI = 0.25, T* = 0.9, the errors reach -46$, 40$, -10$ and 20$, respectively, for XII, X21, X22 and X12. At X1 = 0.75, T* = 0.9, the errors reach -10$, 46$, -46$ and 40$, respectively, for X11, X21, X22 and X12. We notice that for this model the agreement in LC is better at higher temperatures. We can summarize the comparisons in this section as follows. The NRLT equation can give an excellent representation of excess free energies for models B-1, D-1, and E-1 (D-1 and E-1 show possible partial miscibility), but the agreement in local compositions given by the NRTL equation and perturbaton theory or MC simulation varies for each model. For model B-1, the NRTL equation 82 can give very good descriptions of the environments of molecules 2, both qualitatively and quantitatively. However, the NRTL equation cannot give good predictions of the environments of molecules 1, it always predicts X11 = 1, X21 = 0. For D-1 model,'the environments of molecules 1 and 2 given by the NRTL equation are qualitatively correct over the temperature and composition range studied except at equimolar ratio. Moreover, the local compositions Xij predicted by the NRTL equation can show generally good agreement with MC data at a few states. Such as at X1 = 0.10, T* = 0.7; X1 = 0.25, T* = 0.7 (only X12 value is not satisfactory, the error is 57%); X1 = 0.50, T* = 0.9; and at X1 = 0.75, T* = 0.9 (only X21cvalue is not very good, the error is -27%). For model E-1, the NRTL equation does not work well. Only at a few states (not most) tested are the environments of molecules 1 and 2 qualitatively correct, and the quantitative agreement is not good. At higher temperatures, the agreement in LC given by the NRTL equation and LBH theory or MC simulation is better for models B-1 and E-1. For model D-1, the temperature effect on LC given by the NRTL equation is smaller and seems to be different at different compositions. It should be reiterated that the above comparisons are made using values of a obtained in the first step of a two-step fitting (procedure as mentioned at the beginning of this section) and some of these at values are outside the recommended range (0.1 - 0.4). Therefore, it is possible that some of the conclusions would be revisesd if different ot's are used. 83 mo .3 E / T=.9 .2 +\ .1 . 6 .2 .4 . 8 Figure 4.27 Excess Gibbs free energy of B-1 model fitted to NRTL equation. The fitting is carried out at two temperatures simultaneously. The curves give the results of the fitted NRTL equations, and the points marked -f- and • give the LBH results at T* = 0.7 and 0.9 respectively. 84 4,28 As in Fig, A.27f but the fitting is carried out at each temperature separately. 85 \ Figure 4.29 As in Fig. 4.27, but of D-1 model. 86 1 . 0 0.5 .4 .8 2 . 6 Figure 4.30 As in Fig, 4.29, but the fitting is carried out at each temperature separately. 87 1 . 0 0.5 . 8 .2 .6 .4 Figure 4,31 As in Fig. 4,27, but of E-l model. 88 T =.7 I 1 . 0 i i i .8 .4 2 .6 Figure 4,32 As in Fig.. 4,31? but the fitting is carried out at each temperature separately. Table 4.26 Parameters of NRTL Eq. fitted to excess free energy of model mixtures. The fitting is carried out for each model at T* = 0.7, 0.9 simultaneously Model a (g12 - g11)/€l1 (g12 - g22)/€l1 B-1 1.1027 22.740 -0.675 D-1 0.0920 7.839 -0.410 E-1 0.00027 4.750 -0.064 Table 4.27 As in Table 4.26, but the fitting is carried out for each model at each temperature separately. Model T* i a (g 12 - g11 )/€l 1 (g12 - g22)/€l1 B-1 0.7 1.1027 22.740 -0.622 0.9 1.1027 22.740 -0.742 D-1 0.7 0.0920 11.190 0.206 0.9 0.0920 10.327 -1.843 E-1 0.7 0.00027 20.890 -15.639 0.9 0.00027 53-755 -48.737 90 Table 4.28 Comparison of , 'experimental, , local compositions with NRTL Eq. for B-1 model at xl = 0.10 T* . X11 X21 X22 X12 NRTL Eq. 0.7 1 0 0.772 0.228 0.9 1 0 0.783 0.216 LBH data 0.7 0.183 0.817 0.792 0.208 0.9 0.185 0.815 0.790 0.210 Table 4.29 As in Table 4.28, \ but at X1 = 0.25 x# X11 X21 X22 X12 NRTL Eq. 0.7 1 0 0.530 0.470 0.9 1 0 0.547 0.453 LBH data 0.7 0.404 0.596 0.558 0.442 0.9 0.407 0.593 0.555 0.445 Table 4.30 As in Table 4.28, but at X1 = 0.50 T* X11 X21 X22 X12 NRTL Eq. 0.7 1 0 0.273 0.727 0.9 1 0 0.287 0.717 LBH data 0.7 0.672 0.328 0.294 0.706 0.9 0.674 0.326 0.293 0.707 91 Table 4.31 As in Table 4.28, but at X1 = 0.75 T* X11 X21 X22 X12 NRTL Eq. 0.7 1 0 0.111 0.889 0.9 1 0 0.118 0.882 LBH data 0.7 0.861 0.139 0.123 0.878 0.9 0.862 0.138 0.121 0.879 Table 4.32 Comparison of ’ 'experimental with NRTL Eq. for D-1 model " local compositions at X1 = 0.10 T* X11 X21 X22 X12 NRTL Eq. 0.7 0.9 0.326 0.242 0.674 0.758 0.902 0.882 0.098 0.118 LBH data 0.7 0.9 0.177 0.180 0.823 0.820 0.796 0.792 0.204 0.208 MC data 0.7 0.9 0.380 0.363 0.620 0.637 0.890 O.883 0.110 0.117 Table 4.33 As in Table 4.32, but at X1 = 0.20, 0.25 T* X11 X21 X22 X12 X1 = 0.20 NRTL Eq. LBH data 0.9 0.9 0.418 0.334 0.582 0.664 0.768 0.626 0.232 0.374 XI = 0.25 NRTL Eq. LBH data 0.7 0.7 0.592 0.397 0.408 0.603 0.755 0.560 0.245 0.440 MC data 0.7 0.645 0.355 0.844 0.156 92 Table 4.34 As in Table 4.32, but at X1 = 0.50 T* X11 X21 X22 X12 NRTL Eq. 0.7 0.813 0.187 0.507 0.493 0.9 0.742 0.258 0.453 0.547 LBH data 0.7 0.667 0.333 0.296 0.704 0.9 0.668 0.332 0.294 0.706 MC data 0.7 0.870 0.130 0.498 0.502 0.9 0.712 0.288 0.509 0.491 Table 4.35 As in Table 4.32, but at X1 = 0.75 T* X11 X21 X22 X12 NRTL Eq. 0.7 0.929 0.071 0.255 0.745 0.9 0.897 0.103 0.216 0.784 LBH data 0.7 0.858 0.142 0.123 0.877 0.9 0.858 0.142 0.122 0.878 MC data 0.7 0.882 0.118 0.455 0.545 0.9 0.859 0.141 0.242 0.758 93 Table 4.36 Comparison of "experimental” local compositions with NRTL Eq. for E-1 model. T* X11 X21 X22 X12 X1 = 0.10 NRTL eq. 0.7 0.1001 0.8999 0.9000 0.1000 0.9 0.1001 0.8999 0.9000 0.1000 MC data 0.7 0.330 0.670 0.928 0.072 0.9 0.240 0.760 0.919 0.081 X1 = 0.25 NRTL Eq. 0.7 0.2502 0.7498 0.7500 0.2500 0.9 0.2501 0.7499 0.7500 0.2500 MC data 0.7 0.614 0.386 0.878 0.122 0.9 0.463 0.537 0.829 0.171 X1 = 0.50 NRTL Eq. 0.7 0.5002 0.4998 0.5000 0.5000 0.9 0.5002 0.4998 0.5000 0.5000 MC data 0.7 0.851 0.149 0.594 0.406 0.9 0.812 0.188 0.494 0.506 X1 = 0.75 NRTL Eq. 0.7 0.7502 0.2498 0.2500 0.7500 0.9 0.7500 0.2500 0.2500 0.7500 MC data 0.7 0.878 0.122 0.614 0.386 0.9 0.829 0.171 0.463 0.537 *1.6 Helmholtz Free Energy and Excess Free Energy - vdW1 Results In this section we present the vdW1 results for Helmholtz free energy and excess Gibbs free energy of the seven model mixtures studied. Tables 4.37—4-43 give these results for the model mixtures over a range of temperatures and compositions. All the results listed are obtained at zero pressure. Though the vdwl results are different from the LBH results to some degree, but the qualitative results from both theories are consistent for each model mixture. Such as the deviation from the ideal solution, the symmetry of excess free energy with repect to mole fraction, and the temperature efffect on the magnitude of excess free energy. 95 Table 4.37 The Helmholtz free energy and excess Gibbs free energy of A-1 model as determined by vdW1 theory at zero pressure, T* = 0.7, 1.0 X1 T* = 0.7 T* = 1.0 A / €11 0E / €11 A / €11 GE / €11 0 -4.3900 0 -3.9019 0 0.10 -4.5655 -0.0681 -4.1516 -0.0964 0.25 -4.7777 -0.1193 -4.4569 -0.1718 0.50 -5.0566 -0.1298 -4.8549 -0.1866 0.75 -5.2774 -0.0822 -5.1705 -0.1189 1 -5.4636 0 -5.4348 0 Table 4.38 As in Table 4.37, but of B-1 model at T* = 0.7, 0.9 X1 T* = 0.7 T* = 0.9 A / €11 GE / €11 A / €11 ( f / €11 0 -4.0565 0 -3.8136 0 0.10 -4.5375 -0.2038 -4.3080 -0.2008 0.25 -5.1030 -0.3535 -4.9019 -0.3544 0.50 -5.8197 -0.3771 -5.6622 -0.3807 0.75 -6.3741 -0.2385 -6.2578 -0.2424 1 -6.8287 0 -6.7494 0 l I 96 Table 4.39 As in Table 4.37, but of C-1 model at T* r 0.7 X1 T* = 0.7 A / €11 GE / €11 0 -3.8004 0 0.10 -4.6236 -0.2926 0.25 -5.6663 -0.5394 0.50 -7.0526 -0.5993 0.75 -8.1683 -0.3885 1 -9.1063 0 Table 4.40 As in Table 4.38, but of D-1 model. T* = 0.7 T* = 0.9 XI A /€11 GE / € 11 A / €11 (f / €11 0 -4.3900 0 -4.0201 0 0.10 -3.6030 0.8944 -3.4240 0.7340 0.25 -3.3^30 1.3155 -3.3270 1.0378 0.50 -3-7590 1.1680 -3-7890 0.9206 0.75 -4.5460 0.6495 -4.5330 0.5213 1 -5.4636 0 -5.3990 0 Table 4.41 As in Table 4.37, but of D-2 model at T* = 0.7, 0.9, 1.1 X1 T* = 0.7 T* = 0.9 A / €11 ge / €11 A / €11 GE / €11 0 -4.3900 0 -4.0201 0 0.10 -6.6867 -2.1893 -6.1530 -1.9950 0.25 -8.1028 -3.4440 -7.5440 -3-1790 0.50 -8.0160 -3-0890 -7.5620 -2.8525 0.75 -6.8566 -1.6611 -6.5720 -1.5177 1 -5.4640 0 -5.3990 0 T* = 1.1 X1 A / € 11 GE / €11 / 0 -3.8260 0 0.10 -5.7857 -1.7911 0.25 -7.1456 -2.8982 0.50 -7.2694 -2.6006 0.75 -6.4560 -1.3658 1 -5.5116 0 98 Table 4.42 As in Table 4.38, but of E-1 model. X1 T* = 0.7 T* = 0.9 A / €11 ge / €11 A / €11 Ge / €11 0 -4.3900 0 -4.0201 0 0.10 -3-8587 0.5315 -3.5608 0.4593 0.25 -3.3125 1.0775 -3.1052 0.9149 0.50 -2.9824 1.4076 -2.8478 1.1723 0.75 -3.3125 1.0775 -3.1052 0.9149 1 -4.3900 0 -4.3900 0 99 Table 4.43 As in Table 4.41, but of E-2 model. XI T* = 0 .7 T* i t i O 1 • | V O 1 1 * t 1 1 1 1 I » 1 1 A / €11 G E / €11 A / €11 CE / €11 0 -4-390 0 -4.0201 0 0 .1 0 -5 .5 1 2 5 -1.1225 -5 .0 1 8 7 -0.9986 0 .2 5 -6 .7 9 3 7 -2 .4040 -6 .1 9 1 9 -2.1718 0 .5 0 -7 .6 4 1 0 -3 .2510 -6 .9 7 8 2 -2.9581 0 .7 5 -6 .7 9 3 7 -2.4040 -6 .1 9 1 9 -2.1718 1 -4 .3 9 0 0 -4.0201 0 T* = 1.1 X1 A / €11 GE / €11 0 -3 .8 2 6 0 0 0 .1 0 -4 .6 9 3 3 -0.8673 0 .25 -5 .7 4 8 6 -1.9226 0.5 0 -6 .4706 -2 .6446 0 .75 -5 .7 4 8 6 -1.9226 1 -3 .8 2 6 0 0 100 CHAPTER 5 CONCLUSIONS f The Wilson equation gives an excellent representation of excess free energies for nearly ideal solutions (A—1, B-1, and C-1), moderately good representation for associated soutions (D-2 and E-2), and poor representation for phase separating solutions (D-1 and E-1). Temperature does not appreciably affect the ability of Wilson equation in representing excess free energies for nearly ideal solutions (A—1 and B-1). The temperature effect is appreciable for associated solutions (D-2 and E-2) - the Wilson equation works better at higher temperatures. Our comparisons also show that the Wilson parameters are temperature dependent, being more sensitive to temperature for more ideal solutions (A-1 and B-1) than for less ideal solutions (D-2 and E-2). The comparisons of "experimental” local compositions (LBH and MC data) with Wilson equation LC show that the Wilson equation can give reasonable, though not accurate, estimates of local compositons Xij. For A-1 and B-1 models, at X1 = 0.10, 0.25, 0.75, and for C-1 101 model at X1 = 0.10, 0.25, the environments of molecules 1 and 2 predicted by the Wilson equation are qualitatively correct over the temperature range tested. Thus the Wilson equation shows partial success in predicting LC Xij, the situations, more favorable than the conclusions of Nakanishi and Toukubo and Nakanishi et al. For phase separating solutions (D-1, E-1), the local compositions given by Wilson equation show generally poor agreement with experimental data. Only at some particular states tested is the Wilson equation able to predict qualitatively the environments of molecules 1 and 2. This, however, seems to be fortuitous. The temperature effect on the LC predicted by Wilson equation is indefinite, being favorable for A-1, D-2, E-2 models - better agreement is obtained at higher temperatures, but unfavorable for B-1 model - worse agreement is obtained at higher temperatures. The NRTL equation can give an excellent representation of excess free energies for B-1 model (nearly ideal solution), as well as D-1 and E-1 models (phase separating solutions). Temperature does not seem to have appreciable effect on its ability to represent data for B-1, D-1, and E-1 models. The NRTL parameters, a is taken to be independent of temperature, but the other two parameters (g12 — g11) and (g12 - g22) are appreciably dependent on temperature. The dependency seemed to be less pronounced for more ideal solutions. Our comparisons show that the NRTL equation can also give reasonable estimates of the local compositions Xij, and at some 102 situations the predictions are quantitatively good too. For D-1 model, the environments of molecules 1 and 2 given by the NRTL equation are qualitatively correct over the temperature and composition range tested except at equimolar ratio. Moreover, at the states X1 = 0.10, T* = 0.7; X1 = 0.50, T* = 0.9; X1 = 0.75, T* = 0.9, the local compositions predicted by NRTL equation for D-1 model show generally good agreement with MC data. For B-1 model, the NRTL equation gives very good descriptions of the environments of molecules 2; however, the predictions of the environments of molecules 1 are not good, it always predicts X11 = 1, X21 = 0 over the composition and temperature range studied. For E-1 model, the local compositions given by NRTL equation are not in good agreement with MC data. At a few states tested the NRTL equation can qualitatively describe the environments of molecules 1 and 2, but the errors are appreciable. At higher temperatures, the agreement in LC is better for B-1 and E-1 models, and may be better or worse for D-1 model depending on composition. Based on this study, we see that the NRTL equation is better able to describe the microscopic sturcture of mixtures than the Wilson equation. This is a direct consequence of the more rigorous derivation of the NRTL equation itself and results in its ability to represent a greater variety of mixtures, including partially miscible systems. 103 REFERENCES [1] Wilson, G.M., 1964, J. Am. Chem. Soc., 86, 127. [2] Renon, H. and Prausnitz, J.M., 1968, AIChE J., 14, 135. [3] Leonard, P.J., Henderson, D. and Barker, J.A., 1970, Trans. Faraday Soc., 6_6, 2439 • [4] Perram, J.W., 1975, Molec. Phys., j|0, 1505. [5] Baxter, R.J., 1968, Aust. J. Phys., 21, 563. [6] Mansoori, G.A., Carnahan, N.F., Starling, K.E. and Leland, T.W., Jr., 1971, J. Chem. Phys., 54, 1523- [7] McQuarrie, D.A., Statistical Mechanics , p236 (Chap. 12). [8] Barker, J.A. and Henderson, D., 1967, J. Chem. Phys., 47, 2856; 1967, ibid.,_47, 4714. [9] Barker, J.A., Henderson, D. and Smith, W.R., 1968, Phys. i Rev. Letters, 21, 134. [10] Verlet, L. and Weis, J.J., 1972, Phys. Rev. A,J^, 939. j [11] Scott, R.L., 1956, J. Chem. Phys., 25, 193- I 104 [12] Prausnitz, J.M., Molecular Thermodynamics of Fluid-Phase Equilibria , Chap. 7. [13] Longuet-Higgins, H.C., 1951, Proc. Roy. Soc. A 205, 2*17. [1*1] Henderson, D. and Leonard, P.J., in Physical Chemistry. An Advanced Treatise. Vol. VIIIB. Liquid State , ed. by D. Henderson, Academic, New York (1971). t [15J McDonald, I.R. in Statistical Mechanics , Vol. 1, ed. V ' ? K. Singer, Specialist Periodical Reports, Chemical Society, London (1973)* [163 Leland, T.W., Rowlinson, J.S. and Sather, G.A., 1968, Trans. Faraday Soc., 6*1, 14*17. ( t [17] Nicolas, J.J., Gubbins, K.E. Streett, W.B. and Tidesley, D.J., 1979, Molec. Phys., 37, 1429. [18] Barker, J.A. and Henderson, D., 1976, Rev. Mod. Phys., 48, 596. [19] Nakanishi, K. and Toukubo, K., 1979, J. Chem. Phys., 70, 5848. [20] Nakanishi, K., Okazaki, S., Ikari., K., Higuchi, T. and Tanaka H., 1980, J. Chem. Phys., 76. 105 [21] Lee, L.L., Chung, T.H., and Starling, K.E., 1983, Fluid Phase Equilibria, 12, 105. [22] Mcdermott, C. and Ashton N., 1977, Fluid Phase Equilibria, _1, 33. 106

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