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FLUCTUATION THERMODYNAMIC PROPERTIES OF NONELECTROLYTE LIQUID MIXTURES BY ROBERT JOSEPH WOOLEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987 to Sue, for her patience and encouragement. ACKNOWLEDGEMENTS I would like to thank Dr. J. P. O'Connell for his help and guidance throughout this study. Further, I wish to thank Dr. Robley C. Williams, Jr., of Vanderbilt University for his enthusiastic help and many thoughtful suggestions on how to install a laser on an old ultracentrifuge. His sincere interest in a project not actually in his field was inspiring. I also wish to thank Dr. Fricke who had his differential refractometer fixed so that I could use it and Dr. Mike Young and his lab workers for tolerating my presence and allowing me unlimited free use of their microcomparator for many long hours of measurements. I am grateful to Stewart Goldfarb for making his Dortmund VLE database available to me and to the Valdosta State College Library who allowed me to check out VLE data books that were not available anywhere in the State of Florida. I would also like to thank Sue Wooley for her assistance in typing, proofreading and formatting this final document. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ............................. ........ iii KEY TO SYM BOLS ........................................... vii A BSTR A CT ................................................... xii CHAPTERS 1 INTRODUCTION .................................. 1 2 DETERMINATION OF SOLUTION THERMODYNAMIC PROPERTIES BY RAYLEIGH LIGHT SCATTERING ............................. 6 Introduction ....................................... 6 Background ....................................... 7 Rigorous Thermodynamic Fluctuation Development ...... 11 Light Scattering in Relaxing Solutions .................. 15 Calculation of Thermodynamic Properties .............. 18 Fluctuation Properties ............................ 18 Activity Coefficients ............................ 18 Activity Coefficients from Equation (246) .......... 19 Comparisons with Phase Equilibrium Measurements .. 23 Sensitivity of Activity Coefficient Calculations .......... 25 Sum m ary .......................................... 30 3 CONCENTRATION DERIVATIVES OF DILUTE SOLUTION ACTIVITY COEFFICIENTS USING AN ANALYTICAL ULTRACENTRIFUGE .............. 32 Introduction ....................................... 32 Equipm ent ........................................ 40 A pparatus ..................................... 40 Composition Detection ........................... 41 System Selection .................................... 45 Error A analysis ..................................... 49 Sum m ary .......................................... 52 4 EXPERIMENTAL MEASUREMENTS OF DILUTE SOLUTION ACTIVITY COEFFIEIENT DERIVATIVES ..................... 53 Introduction ....................................... 53 Carbon Disulfide/Acetone System ..................... 54 Chloroform/Acetone System .......................... 60 Carbon Tetrachloride/Acetone System .................. 64 Benzene/Acetonitrile System ........................ 68 Carbon Tetrachloride/Methanol System ................. 72 Summary of Experimental Results ..................... 76 Pressure Effects on the Results ...................... 77 The Effect of Imputities on Ultracentrifugation ......... 81 Sum m ary .......................................... 82 5 DIRECT CORRELATION FUNCTION INTEGRAL DATABASE AND MODELING .................... 83 Introduction ....................................... 83 DCFI Database .................................... 87 Isothermal Compressibility Data ................... 91 Thermodynamic Property Modeling .................... 99 DCFI M odeling ................................. 99 Activity Coefficient Modeling from DCFIs .......... 100 Sum m ary .......................................... 115 6 SUM M ARY ....................................... 119 APPENDICIES A ULTRACENTRIFUGE OPERATION ................ 124 B LASER LIGHT SOURCE FOR MODEL E ULTRACENTRIFUGE INSTALLATION AND ALIGNMENT ............................... 138 C FORTRAN PROGRAM FOR CONVERSION OF FRINGE MEASUREMENTS TO COMPOSITION PROFILES IN THE ULTRACENTRIFUGE FRNGCNV AND FRNGCNVP ..................... 149 D EXCESS VOLUME AND PARTIAL MOLAR VOLUME CALCULATIONS ....................... 165 E CALCULATION OF INITIAL ULTRACENTRIFUGE SPEED ..................... 167 F RESULTS OF EXPERIMENTAL ULTRACENTRIFUGE MEASUREMENTS OF FIVE BINARY SYSTEMS ...................... 169 G FORTRAN PROGRAMS FOR THE CALCULATION FOR PVT CALCULATIONS AND DATA REGRESSION ..................................... 228 H DIRECT CORRELATION FUNCTION INTEGRAL DATABASE ........................... 247 I FORTRAN PROGRAMS FOR REGRESSION OF ACTIVITY COEFFICIENT DATA TO EQUATIONS 539a AND 539b ........................ 306 BIBLIOGRAPHY .............................................. 311 BIOGRAPHICAL SKETCH ..................................... 319 KEY TO SYMBOLS A Composition profile in ultracentrifuge uncertainty analysis. A(k) Relaxation correction to J(k). A12 Margules Binary Parameter. A21 Margules Binary Parameter. B(k) Relaxation correction to J(k). Bij Matrix Element, Eq. 218. Bk Characteristic parameter, Eq. 226. Cp Heat capacity at constant P. Cv Heat capacity at constant V. Cn Direct Correlation Function Integral between two type 1 compounds. C12 Direct Correlation Function Integral between a type 1 and a type 2 compound. C22 Direct Correlation Function Integral between two type 2 compounds. C' Characteristic parameter, Eq. 519. CM Characteristic parameter of a mixture, Eq. 519. Ck Characteristic parameter, Eq. 226. AC Grouping of direct correlation function integrals. D Activity coefficient derivative in ultracentrifuge uncertainty analysis. D Diffusion coefficient. F Factor in Eq. 248. G12 NRTL binary parameter. G21 NRTL binary parameter. J Ratio of Rayleigh peak to the Brillouin peak. J(0) J corrected to 0 wave vector. J(k) J as measured. JID J ideal. K Grouping of variables, Eq. 244b. MI Molecular weight, component 1. NR Random number. P Pressure. R Gas constant. R90 Rayleigh ratio measured at 90. Rc Composition contribution to Rayleigh Ratio. Rc,D Composition and density cross contribution to Rayleigh Ratio. RD Density contribution to Rayleigh Ratio. RIs Isotropic portion of Rayleigh ratio. AST Total entropy change. S Entropy. T Temperature. T* Characteristic parameter, Eq. 519. TL Characteristic parameter of a mixture, Eq. 519. Ui Uncertainty of variable i. V Volume. Vi Partial molar volume of component i. a Binary parameter, Eq. 539. ai Typical variables in uncertainty analysis. b Binary parameter, Eq. 539. c Speed of light. cij Direct correlation function. d Approach to equilibrium. f Parameter in Eq. 233. gij Radial distribution function. hij Total correlation function. h7 Excess enthalpy. ic Intensity of Rayleigh peak. iB Intensity of Brillouin peak. j Fringe number. Ajp Change in fringe number due to pressure. k Boltzman constant. kv Wave vector. k Compression. n Refractive Index. Anp Refractive Index change due to pressure. r Parameter in Eq. 243, Chapter 2. r Radius from center of Rotor, Chapter 3. t 8 Time to Equilibrium. v Infinite frequency velocity of sound. Vo Zero frequency velocity of sound. vE Excess Volume. vs Velocity of sound. w Width of centrifuge cell. xi Mole fraction, component i. xO() Parameter in Eq. 26. y() Parameter in Eq. 25. y(2) Parameter in Eq. 29. ap Coefficient of thermal expansion. Ai Portion of scattered light transferred from side peak to the central peak. 8 Approach to equilibrium. e Dielectric constant. ^Yi Activity coefficient of component i. 'Yc Cp / Cv. ^y Infinite dilution activity coefficient. Ks Adiabatic compressibility. KT Isothermal compressibility K( Excess isothermal compressibility. X or Xo Wave length of incident light. Ili Chemical potential of component i. Av Frequency of peak separation. v Frequency of incident light. p Density. Pi Concentration of component i. pU Depolarization ratio. p. Characteristic parameter, Eq. 519. pL Characteristic parameter of a mixture, Eq. 519. T 0 (0 v x ((Ae)2} ((Ae)C) <(A)DIAB) ((Ap)2) ((AX2)2) Superscripts 0 NRTL parameter. NRTL parameter. Relaxation time. Scattering angle. Rotational speed. Volume of centrifuge cell. Reduced bulk modulus. Total mean square fluctuation in dielectric constant. Central peak contribution to the dielectric fluctuation. Adiabatic contribution to the dielectric fluctuation. Mean square density fluctuation. Mean square composition fluctuation. Pure component property, or initial concentration. Ideal property. Excess property. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FLUCTUATION THERMODYNAMIC PROPERTIES OF NONELECTROLYTE LIQUID MIXTURES By Robert Joseph Wooley December 1987 Chairman: John P. O'Connell Major Department: Chemical Engineering The behavior of the concentration derivatives of the pressure and component activity coefficients has been studied via their connection to integrals of the statistical mechanical direct correlation function (DCFI). Consideration of their direct experimental accessibility with light scattering and equilibrium sedimentation in an ultracentrifuge was made and dilute solution data have been taken with the latter method. A new binary solution model, based on simplified forms of the DCFI and reduced bulk modulus, has achieved good agreement with a newly compiled database of DCFI and fluctuation properties for nonelectrolytes. The data needed to establish a new theory based on fluctuation solution properties do not include reliable values of activity coefficient derivatives at low concentrations. These are usually obtained by differentiation of a specific model with parameters fitted to vaporliquid equilibrium data with results greatly affected by the model chosen. As alternatives, two experimental methods for direct determination of the activity coefficient derivative have been explored. It has been concluded that measurement of concentration fluctuations of binary systems with light scattering techniques is not useful for activity coefficients because of sensitivity to experimental error and excessive requirements of ancillary data. Equilibrium sedimentation in an analytical ultracentrifuge has been judged more promising and measurements have been made with a Beckman Model E apparatus with a newly installed laser optical system for Rayleigh fringe detection. Measurements have been made of five systems that meet welldefined criteria for refractive index and density differences as well as solution nonideality. The measured chemical potential derivatives have been found to be lower than those calculated from VLE models. However, in each system a single multiplicative factor generally brings the results to within experimental error of the VLE models. These values support the low concentration extrapolations of the Wilson and NRTL Equations. It was not possible to distinguish the better of these models for activity coefficient derivatives. A database of DCFI values for 28 strongly nonideal binary systems has been assembled using liquid compression and excess volume data along with Wilson and NRTL parameters fitted to VLE data. Modeling of the DCFIs and activity coefficients was attempted. While the density dependence is extremely complex, a successful approach for activity coefficients is to separate the activity coefficient derivative into a function of DCFIs, 1/AC, and the reduced bulk modulus, x. The results are generally insensitive to the mathematical form chosen for 1/AC so a linear expression has been used along with an idealized variation of x to obtain a new binary activity coefficient model. Least squares fits to all systems in the database yielded excellent results. CHAPTER 1 INTRODUCTION With continually increasing energy, raw material, and capital costs, chemical process design engineers are constantly called upon to lower the costs associated with the design and operation of chemical plants. Recent advances in computer simulation technology allow the process engineer to rapidly assess the feasibility and economics of new processes and process modifications, thus helping reduce capital and operating costs with more optimum designs. However, while these computer simulators can easily describe unit operations and converge flowsheets with complex recycles, they are limited by their ability to model and predict the physical properties needed for a reliable design. Further, physical property calculations, especially phase equilibria properties, usually consume a considerable amount of the computer time in process simulation (O'Connell, 1983). Finally, most empirical models for phase equilibrium calculations can require exhaustive amounts of binary and sometimes multicomponent data, especially if the compounds are highly nonideal or unusual. As a result, new and different processes, such as those found in synthetic fuels and bioprocessing, continue to tax the currently available models. The consequence of all of this is that computer simulation requires simple and reliable models for process design of current and future systems. To meet this demand, new thermodynamic property models, free of common simplifying assumptions, such as pairwise additivity of intermolecular forces, rigid molecules, etc. are needed. The fluctuation solution approach (Kirkwood and Buff, 1951; O'Connell, 1971; 1981) offers these advantages. Kirkwood and Buff (1951) formulated the density derivatives of the chemical potential and total system pressure in terms of integrals of the radial distribution function. O'Connell (1971) used these formulations and the Ornstein and Zernike (1914) equations to give the thermodynamic properties in terms of the direct correlation function integrals (DCFI). These direct correlation functions appear relatively insensitive to the details of intermolecular interactions and may be modeled with simple functions. Additionally, because the thermodynamic properties are obtained by integration, the results can be much less sensitive to their parameterization. For example, Mathias and O'Connell (1979, 1981) found that this approach worked very well for gases dissolved in liquids. With models for the direct correlation function integrals, the solution density and the chemical potential, essential in phase equilibrium calculations, can be calculated or predicted from the pressure, temperature and composition. In particular, such a model may allow for accurate modeling of the activity coefficient of highly nonideal systems. The DCFIs modeled by Mathias and O'Connell (1979, 1981) work well for liquids containing supercritical components (dissolved gases) but apparently not as well for condensedphase systems (Campanella, 1984). Campanella made an attempt to model the DCFI for vaporliquid equilibria (VLE) and liquidliquid equilibria (LLE) systems in a fashion similar to that of Mathias (1978), using a hard sphere term and a perturbation term. He found in VLE systems that the hard sphere term did not show adequate compositional variation. Further, at low concentrations the data used to calculate DCFIs were not accurate enough to properly determine the compositional dependence in that region. His work was hindered by having to calculate activity coefficient derivatives indirectly by differentiating either tabulated phase equilibria data or data from a model, such as Wilson (1964). This has lead to uncertainty in the actual compositional behavior of DCFIs. The objective of the present work has been to explore the compositional behavior of the DCFI in more detail. The relationships between DCFIs and mole fractions and measurable binary quantities of isothermal compressibility, KT, partial molar volume, Vi, volume, V, and activity coefficient derivatives, lead to DCFI values from experiment. Binary data have been collected (aided by the recent compilation of isothermal compressibilities by Huang (1986)) on nonideal systems and a database of DCFIs for a wide variety of systems has been created. A problem in developing an accurate DCFI database is the collection of accurate activity coefficient derivative data. Typically, this quantity would be calculated by differentiating a specific activity coefficient model whose parameters are fitted to vaporliquid equilibrium data. However, as Campanella (1984) has pointed out, there can be a significant difference in the activity coefficient derivative depending on the model chosen to fit the VLE data. This effect is especially apparent in the low concentration regions where the various activity coefficient models can give significantly different values. Also, extrapolations of finite concentration VLE data do not necessarily agree with directly measured infinite dilution activity coefficients (Loblen and Prausnitz, 1982; Schreiber and Eckert, 1971), suggesting further uncertainty in reliability of the activity coefficient models fitted to VLE data in this region. To eliminate the need for an intervening activity coefficient model between experimental VLE data and the activity coefficient derivative, methods have been explored for experimental measurements that directly yield the activity coefficient derivative, especially in dilute regions. In addition to its use in DCFI model development, such data could be used by others in establishing new activity coefficient models. Two methods for determining the activity coefficient derivatives directly have been considered. The potential use of Rayleigh light scattering is explored in Chapter 2. Several authors (Coumou and Makor, 1964; Brown et al., 1978; Miller, 1967; Miller and Lee, 1973; Maguire et al., 1981) have described procedures for finding thermodynamic properties from light scattering. These involve measuring the contribution of concentration fluctuations to the light scattering either with the ratio of the total light scattered at an angle of 900 to the incident light or the ratio of the scattered central Rayleigh peak to Brillouin side peaks. This leads to several different methods for connecting light scattering measurements to thermodynamic properties, many involving approximations that are subject to question. Therefore, a thorough analysis of the procedure and errors involved in obtaining activity coefficients using light scattering data has been performed. The method requires a considerable amount of auxiliary data and appears to be limited in its accuracy while requiring a very high refractive index difference for the chemicals being measured. The second method considered for measuring activity coefficient derivatives is by ultracentrifugation as described in Chapters 3 and 4. This procedure has been used previously to measure activity coefficients (Cullinan and Lenczyk, 1969; Rau, 1975; Johnson et al., 1959). The composition profile established during centrifugation readily leads to the activity coefficient derivative. Chapter 3 describes the theoretical basis and expected errors involved in determining activity coefficients from ultracentrifugation. The method has the advantage over light scattering techniques of only needing mixture refractive index and density data to support the centrifuge measurements. It does require that the binary pair have a suitable difference in density, but the refractive index difference need not be extremely high. Chapter 4 describes the experimental measurements that have been made on five systems: carbon disulfide in acetone, chloroform in acetone, carbon tetrachloride in acetone, carbon tetrachloride in methanol and benzene in acetonitrile. Ultracentrifuge measurements have been made in the 1.5% to 10% (mole) range at temperatures from 100 C to 350 C. In support of those experiments, mixture refractive index measurements have been taken in a differential refractometer on all systems. Chemical potential derivatives and activity coefficient derivatives from these measurements have been compared with vaporliquid equilibrium data fit to the Wilson (1964) and NRTL (Renon and Prausnitz, 1968) models. Once activity coefficient derivatives have been obtained, a database of DCFIs can be developed. As described in Chapter 5 sixteen nonideal systems selected by Campanella (1984) for study plus twelve other nonideal systems have been examined. For each of these systems, excess volume data, pure component compressibility data and vaporliquid equilibrium data were known. For nineteen of the systems, mixture compressibility data are also available. The DCFI can be conveniently divided into volumetric and activity coefficient derivative terms. To calculate an "ideal" DCFI, an ideal mixing rule is assumed for the volume and isothermal compressibility while the activity coefficient derivative term is set to zero. By subtracting this ideal DCFI from the real DCFI, determined from experimental quantities, an excess quantity, resembling the activity coefficient derivative term, remains. Models for this excess DCFI can lead to the compositional variation of these theoretically based quantities as well as aid in future modeling efforts. Campanella (1984) pointed out that most of the activity coefficient models have a common form when written in terms of the composition derivative. Chapter 5 explores the possibilities of modeling the excess DCFI with empirical forms. The constant pressure activity coefficient derivative of a binary solution can be written in terms of the three pairwise DCFIs. This grouping of DCFIs has suggested a new empirical relationship for activity coefficients and excess Gibbs energies. CHAPTER 2 DETERMINATION OF SOLUTION THERMODYNAMIC PROPERTIES BY RAYLEIGH LIGHT SCATTERING Introduction The scattering of light by a fluid medium is caused by the electric field associated with the light inducing periodic oscillations of the electrons in the sample. This leads to energy being reradiated as light, scattered at different frequencies and angles relative to the incident (Oster, 1948). By observing this scattered light as a function of the scattering angle, information about the sample can be deduced. In particular, a contribution to the light scattering by solutions is due to concentration fluctuations which can be related to the concentration derivative of the chemical potential or, equivalently, the activity coefficient (Miller, 1967). The major advantage of such an analysis is that the material is not affected by the experiment and only very small amounts of the sample are required. Such measurements are attractive as an aid for the development of models for solution behavior. In fluctuation solution theory (Kirkwood and Buff, 1951; O'Connell, 1971; 1981) the concentration derivative of the chemical potential, along with the isothermal compressibility, and the partial molar volume are the properties needed to determine the three paircorrelation function integrals of a binary solution. While there are often sufficient data from vapor pressure measurements to accurately calculate the composition derivative of the chemical potential at midrange concentrations of the components, the accuracy deteriorates at low concentrations. To develop a solution model based on correlation function integrals, nontraditional means of getting this derivative must be considered. An attractive candidate appears to be light scattering, since it leads directly to the derivative. Previous work in this area has focused on using the technique to obtain excess Gibbs energies by double integration of the concentration derivative obtained from the scattering (Brown et al., 1978; Maguire et al., 1981). The conclusion was that phaseequilibrium measurements were easier and more accurate for giving this quantity. In particular Miller and Lee (1973) pointed out that a refractive index difference of at least 0.2 between the pure components is required in a system to produce reliable results by light scattering. If that is truly the case, most common organic systems of interest would be eliminated. Several different analyses connecting the lightscattering measurements to the desired quantities have appeared (Brown et al., 1978; Maguire et al., 1981; Coumou and Mackor, 1964; Miller, 1967; Miller and Lee, 1973), many involving approximations that are subject to question. To quantitatively test whether the method can be used, the system cyclohexane/benzene, a typical system of lower refractive index difference (An = 0.07) for which all of the auxiliary properties are available, was examined here. This chapter describes the basis and procedures necessary to obtain information on excess Gibbs energies, activity coefficients and their derivatives. Included is a sensitivity analysis to decide on the method's utility as well as delineation of all the thermodynamic property information required for the technique. Background When light is scattered from a solution, there exists a central, unshifted peak of highest intensity, known as the Rayleigh peak, and two shifted (Brillouin) peaks of lesser intensity on either side. The concentration fluctuations contribute only to the central Rayleigh peak (Miller, 1967). Einstein (1908) originally developed the Rayleigh Ratio, the ratio of the intensity of the Rayleigh Peak to that of the incident light. From Einstein's theory, the Rayleigh Ratio measured at a scattering angle of 90 can be written as a function of the mean squared fluctuation of the dielectric constant, e. r2 Rs 24 V((AE)2) (21 where R1s is the measurable isotropic portion of the Rayleigh Ratio at 900. If the scattered light is not all depolarized, the isotropic portion must be extracted from the total scattering with the appropriate Cabannes (1929) factor. RIS= Ro90 "7p (22 ( 6 + 6 pu where p, is the depolarization ratio. For pure components ((Ae)2) was originally approximated as a function of density only. <(AE)') = ((Ap)2) (23 Coumou et al. (1964) pointed out that another thermodynamic variable (temperature or pressure) would be required to describe this fluctuation properly. To determine which of these variables would be most appropriate, they wrote ((Ae)2) as a function of density and temperature as well as of density and pressure. The fluctuation in E as a function of temperature and density is V((AE)2) =kT (] (24 where y() [KT 1 ][x(1)]2 (25 KS x)= 11 +1 + r + !S (26 ap aT aP ap paT aP with KT and Ks are the isothermal and adiabatic compressibilities, respectively, and n is the refractive index. This treatment assumes that S(de 1 aE 2n ( an (27 Idp K aP KT I aP T It is also possible to write ((Ae)2) as a function of density and pressure V((Ae)2 kTKT [1+ y(2)] (28 where y(2) = + [x)]2 [1 x()]2 (29 X KS and the approximation pdE 1 E 2n an (210 SIp J aT p ap IT p has been used. With data for the partial derivatives of refractive index and compressibilities, Coumou et al. (1964) evaluated values of y0) and y(2) and compared them to V((Ae)2) for representative substances. They found that y(2) was 2% to 10% of V((Ae)2), whereas y(1) was 0.01% to 0.1%. Therefore, the fluctuations of pure component dielectric constants are best described as a function of density and temperature because the corrections can be ignored. Combining equations (24) and (21), neglecting y(1) and changing variables (de = 2n dn) gives an expression for scattering by a pure component: SRD = 22kTn2 ( an (2 Rs R X4KT (211 For solutions, Coumou and Mackor (1964) described the isotropic Rayleigh scattering as a function of density and concentration. Ris = RD + Rc (212 Kirkwood and Goldberg (1950) gave the concentration contribution as 2w 2kT ( On )n/ (/0p12 Rc = 4 V x, n J (213 However, Brown et al. (1978) concluded that a cross term, RCD, is required to represent the interaction of the density and the concentration effects. Such a term had previously been written by Dezelic (1973), R IT 2kTKTXlX2 )T (OE )( RcD = X4 p (214 2\ Iap aX2P Using the same logic as Coumou et al. (1964), the density derivative of the dielectric constant can be replaced with the pressure derivative of the refractive index, yielding S2rr2kTXlX2n2 On n (215 CD 4 aP X2J Thus the full connection of experiment to chemical potential derivative can be made from equations (211), (213), (215) and (216). RIs = RD + Rc + RC,D (216 The value for Rc yields the desired concentration derivative. While the added cross term of Brown et al. (1978) improved results over the earlier work of Coumou and Mackor (1964) for some systems, their general contention was that the method was not as accurate as standard phase equilibrium methods for activity coefficient determination. This is true because of the many ancillary measurements that must be made along with the light scattering intensity to determine the activity coefficient. These other properties include Rgo, the Rayleigh Ratio at scattering angle of 900, pU, the depolarization ratio, calculated from polarizabilities, p, the solution density, KT, the isothermal compressibility, n, the refractive index, the composition derivative of the refractive index, ax2 Ap 8an2 the pressure derivative of the refractive index. Equation (216) gives a relationship between light scattering measurements and thermodynamic properties which uses approximations in the form of the crossterm and of the density derivative of the refractive index. A parallel, but more general development has also been done. Rigorous Thermodynamic Fluctuation Development The rigorous treatment purposely bypasses the density and concentration crossterms (Miller, 1967). Instead, fluctuations in total entropy from variations in the temperature, volume, entropy, pressure, chemical potential of the solute and the number of moles of the solute are considered (Landau and Liftshitz, 1980). 1 AST = 1 (ATAS APAV + Aj.2Ap2) (217 2T This fluctuation in total entropy can be written as a general function of any three of the state variables listed above, 13 3 AS 1 i1 Bi xi xj 2 i=1 i=1 where (xi xj) = k B (218 (219 Choosing the three independent variables as T, P, and P2 = x2 p, gives the others as AS AT + AP+ Ap (220 SaT 2 aP Tp aP2 )PT AV= AT+ 8 AP + Ap2 (221 laT p,P2 aP Jp2 ap2 PIT AiL2 L) AT+ 1v AP + fL2 Ap2 (222 I T pp a, 9P p aP2 P T By inserting equations (220)(222) into equation (217) and using elementary thermodynamics, we find l[Cp 2 T 2 (AT)2 2VATAP + T (AP)2 + (Ap2)2 ST T T lP2 Using equations (218) and (223), the following fluctuations can be identified, kT2 ((AT)2)  (ATAP)  KTCV ((AP)2) = VKS (223 (224a (224b (224c ((Ap,)2) = k T (224d (ATAp2) = (APAp2) = 0 (224e The same connection between this fluctuation theory and light scattering can be made through equations (21) and (22). It is known that adiabatic pressure fluctuations do not mix with entropy and composition fluctuations; they contribute only to the Brillouin peaks. These are given by ((AE)1DIAB)= f ((AP)2) (225 Substituting =a( aJ TVtp + a( (226 SE 2 2 2 results in C ^/YC a P'P2 +2kT2 ( ) + kV (227 KTCV 'T Pp2 I KsV aP 2T Using an equation of the form of (21) yields the ratio of the intensity of the Brillouin peaks to the incident light from equation (227). However, a more useful form exists. The entire dielectric fluctuation is, in terms of fluctuations of the state variables, T, P, and p2, ((AE)2) )2 ((AT)2)) + 2 (ATAP) aT 2 T (P tp + ( ((AP)2) + ((A)2) (228 dP JT,p P2 T,P Here, only one of the three cross terms is nonzero. From equations (224ae), this can be converted to kT2 a + 2kTT P9(e aE Cv T p,p2 KTCV IT Jpp 2 P2 kT & E P2 )TP + )2 + kT (229 VKs aP T, p2 aP2T P The contribution to the central or Rayleigh peak is given by ((Ae)2) = ((Ae)2) ((Ae)2DIAB) (230 Subtracting (227) from (229) leads to C) kT2 + k nT P2 (231 ( ) V aJ k 1 JP,p 2/ aP 2 )TP , Because the ratio of the Rayleigh to the Brillouin peaks allows the use of two intensities that are more similar than those of scattered to incident light, their ratio, J, is used. J = ic / 2 iB = ((Ae)2) / ((AE)2DIAB) (232 Defining the variable f in terms of x() from equation (26) (Miller and Lee, 1968) gives 2x_() yc [x(112 1 x(+) (1 x0())2 Equation (232) can be rewritten to include both the temperature and pressure derivatives of the refractive index /c 1 Cp (an 2 / n 2( J + f (c 1) x1 x(1 +fTc) (234 1l+ fyc T Tx2 aT 8x2TP where Cp T/c (235 Cv Often (Miller, 1967; Coumou and Mackor, 1964), it is assumed that x'(1 = 0, giving C=Ycl +n 2c/ [( xn i2( a 2 ]( JT= c x/1 + [[c 1) xTP] (236 T I dxa )T[,] The equation connecting light scattering data to thermodynamic properties is (234) or (236). It assumes that there are no internal molecular relaxation processes that contribute to the scattering. Unfortunately, this is not normally the case. Light Scattering in Relaxing Solutions Miller and Lee (1968) noted that internal molecular relaxations in a liquid can transfer energy from the central to the outer peaks. Fishman and Mountain (1970) showed how to correct the dynamic changes in pressure, temperature and concentration for this effect using linearized hydrodynamic equations for momentum, energy and diffusion transport. Miller and Lee (1973) implemented the Fishman and Mountain expressions to find the portion of the scattered light transferred from the side peak to the central peak, (Ai). Then, the measured ratio of the Rayleigh to the Brillouin peaks, J(k), becomes J(0) + A, J(k) = (237 1 Ai where J(0) is the ratio to be used in equations (234) and (236), and Ai is A, = A(k) J(0) + B(k) (238 with A(k) = [(v2 v2) k T2 + ()4 ( )2] / [ (vskVT)2 + (! 4j (239a B(k) = [(v v) k) 72 + ( 2 [(vskvT)2 + ()] (239b Here, the physical quantities are vo: infinite frequency velocity of sound vo: zero frequency velocity of sound vs: velocity of sound in solution T: single relaxation time for the internal degrees of freedom kv: change in wave vector by the relaxation The value of vs can be determined from the frequency shift, or the separation, Av, between the Rayleigh and Brillouin peaks. cAu VS 2v sin(0/2) (240 where the additional physical properties are v: frequency of incident light c: speed of light 0: scattering angle The value of kv can be obtained from the incident light wave length Xo and 0. kv = ) sin(o/2) (241 The ratio of scattered peaks, J(0), for use in equations (234) or (236) can be written in terms of the measured ratio, J(k) and the relaxation functions, A(k) and B(k): J) = J(k) [1 B(k)] B(k) J(0) = (242 1 + A(k) + J(k) A(k) Combination of equations (242) and (234) gives a completely rigorous treatment. Fishman and Mountain (1970) point out that A(k) is small and can sometimes be neglected, allowing measurement of J(k) at several angles to be used to establish B(k) and J(0) simultaneously. This eliminates the need for relaxation data but requires a light scattering apparatus that can measure scattering at different angles. An alternative simplification was attempted by Miller and Lee (1973). They ignore A(k) and force T, v0 and vo to vary linearly with concentration. They then obtain J(k) = c r1 + (yc 1) rKxi x2 1 + x2 a ] (243 where 2 r 2 v2 (244a CK an an This more general treatment arrives at equations (242) and (234) without approximations. There are four different treatments available to relate light scattering to thermodynamic properties. Three of the four methods involve approximations which can lead to different results, but require less input data. Our objective was to do the most complete analysis possible and determine what quantitative information can be obtained while listing the total amount of input information required for such a treatment. Calculation of Thermodynamic Properties Fluctuation Properties Recently, several authors (Iwasaka et al., 1976; Kato et al., 1982; Kato and Fujiyama, 1976; Kato, 1984) have used the early theory of Miller (1967) to describe the concentration fluctuations in solutions from light scattering measurements. Iwasaka, et al. (1976) calculated the concentration fluctuations, V((Ax2)2) by solving for ((Ap2)2) in equation (228) for the system, CCI4 / CS2. Miller and Lee (1968; 1973) indicated that both these substances are relaxing fluids, an effect ignored by Iwasaka and Kato. Kato (1984) calculated the KirkwoodBuff (1951) integrals over the total correlation function from chemical potential derivatives and other fluctuation properties. As we establish in more detail later, the stated accuracy of about 10% in the light scattering measurement is not sufficient for determination of reliable chemical potential derivatives, even when relaxation is included. In fact, without consideration of relaxation effects, the results are expected to be seriously in error. Activity Coefficients Most investigators ultimately use a formulation to calculate solution activity coefficients. The relationship of most generality and utility for the activity coefficient derivative is ( alny2 c 1 Cpx an /ran r [,r [nc 1 l (245 lax,] =lif1 RT2 [[x// "Tjj/ l['J(0)1 (245 ax2 1+ f11 c RT' ax2 aT 1+ ffycJ X2 where J(0) is defined by equation (242). The most recent use of this complete theory of Miller and Lee and of Fishman and Mountain is that of Maguire et al. (1981), who used J(0) values from light scattering to obtain activity coefficients and excess Gibbs energies to compare with those from experimental vaporliquid equilibrium data. The activity coefficient can be calculated from Ix ( +fJID/ J(0 ]) + f'c /x2 dx2 (246 The quantity, JID, is a convenient grouping of variables that resembles J(0) for an ideal solution: TYc 1 + CpX1X2 an an ]2 (247 1 + fyc RT2 aX2 aT where f is defined by equation (233). This is not the actual J(0) for an ideal solution because there are still many data in JID that are dependent on the composition of the real solution. For comparison to vaporliquid equilibria data, equation (234) can be used with chemical potential derivatives calculated from VLE data to compare with J(0), (JTHERM)* Before discussing the results of the calculations by Maguire et al. (1981) and their consequences, we recapitulate, in detail the elaborate procedure and extensive set of property values required to obtain activity coefficients from light scattering data. We also duplicate their calculations for the cyclohexane/benzene system, which was chosen because most of the data have been measured. In other systems, approximations would be required, leading to greater errors, we believe. Activity Coefficients from Equation (246) Listed below are the calculational steps used by Maguire et al. (1981) to obtain the activity coefficients and excess Gibbs energies for the cyclohexane/benzene system. In cases of ambiguity in the publication, we state our assumptions about what was done. 1. J(k) from measured light scattering, at T, P, x (Table 21, Column 2) From Table II of Maguire et al. (1981). Assumed P = 1 bar, T = 296 K as in Brown et al. (1978). 2. kv from equation (241) (Table 21, Column 3) a. Refractive index, n, at T, x (Table 21, Column 4). Interpolated from Table 4 of Brown et al.(1978) b. Wavelength of incident light, \X. Assumed to be that of Brown et al. (1978), 5.46x107 m. c. Scattering angle, 0. Assumed to be 90 degrees. 3. Vs, from equation (240) (Table 21, Column 5) a. Frequency of Brillouin Shift, Au not reported. Used Figure 4 for vs(k) vs. composition of Maguire et al.(1981). Table 22 lists values read off graph. Values linearly interpolated to Table 21 compositions. 4. A(k) and B(k) from equations (239a,b) (Table 21, Columns 6 and 7) a. vo at T, x. (Table 21, Column 8) Pure benzene from measurements of Eastman et al. (1969). Pure cyclohexane calculated using A(k) from Figure 3, J(0) and J(k) from Table I, all from Maguire et al. (1981), and B(k) calculated using equation (242). Final value calculated from equation (239b). Linear composition dependence assumed. b. Vo at T, x (Table 21, column 9) Pure benzene from measurements of Eastman et al. (1969). Pure cyclohexane from measurements of Dorfmuller et al. (1976). Table 21 Physical Properties Needed for Light Scattering Analysis of Cyclohexane(1)/Benzene(2) Solutions A(k) B(k) x2 J(k) kv m1 x 107 V0 Vo T m/s m/s sec x 101 1 2 3 4 5 6 7 8 9 10 0.000 0.55 2.32 1.4285 1345.0 0.0166 0.0795 1395.7 1280.0 4.10 0.102 0.68 2.33 1.4321 1345.0 0.0391 0.0995 1411.5 1284.5 4.30 0.202 0.92 2.34 1.4366 1347.1 0.0588 0.1184 1426.9 1288.9 4.49 0.296 1.07 2.35 1.4412 1351.8 0.0736 0.1352 1441.4 1293.0 4.68 0.304 1.09 2.35 1.4417 1352.5 0.0744 0.1365 1442.6 1293.4 4.69 0.350 1.23 2.35 1.4442 1358.5 0.0764 0.1434 1449.7 1295.4 4.78 0.488 1.38 2.36 1.4532 1382.6 0.0738 0.1623 1471.0 1301.5 5.05 0.651 1.46 2.39 1.4656 1420.2 0.0584 0.1824 1496.2 1308.6 5.37 0.747 1.41 2.40 1.4739 1444.1 0.0474 0.1939 1511.0 1312.9 5.56 0.850 1.29 2.41 1.4836 1480.0 0.0222 0.2031 1526.9 1317.4 5.76 0.900 1.16 2.42 1.4887 1500.0 0.0071 0.2068 1534.6 1319.6 5.86 1.000 0.90 2.44 1.4990 1540.0 0.0219 0.2140 1550.0 1324.0 6.05 Table 21 Continued X2 J(0) J(0)t JID f CPr (a (al} c1 (an) KT PP J/molK Pa' K1 Pa1 K1 x 102 x l10 x 104 x 109 x 103 1 11 12 13 14 15 16 17 18 19 20 21 0.000 0.416 0.378 0.378 1.47 155.0 0.0314 5.08 0.3700 5.38 1.15 1.21 0.102 0.481 0.436 0.410 0.48 153.1 0.0394 5.12 0.3751 5.46 1.13 1.21 0.202 0.623 0.571 0.458 2.24 151.2 0.0472 5.16 0.3801 5.54 1.12 1.21 0.296 0.686 0.628 0.510 4.26 149.4 0.0546 5.20 0.3848 5.61 1.10 1.21 0.304 0.696 0.638 0.513 4.64 149.2 0.0552 5.21 0.3852 5.62 1.10 1.21 0.350 0.778 0.717 0.530 7.01 148.4 0.0588 5.24 0.3875 5.66 1.08 1.21 0.488 0.845 0.776 0.616 7.55 145.7 0.0696 5.27 0.3944 5.77 1.07 1.21 0.651 0.884 0.804 0.673 7.99 142.6 0.0823 5.29 0.4026 5.93 1.04 1.21 0.747 0.846 0.760 0.669 7.38 140.8 0.0898 5.27 0.4074 6.02 1.02 1.21 0.850 0.785 0.692 0.622 5.22 138.9 0.0979 5.24 0.4125 6.14 1.01 1.21 0.900 0.703 0.606 0.571 4.24 137.9 0.1018 5.24 0.4150 6.21 1.00 1.21 1.000 0.515 0.412 0.412 1.35 136.0 0.1096 5.23 0.4200 6.38 0.99 1.21 tJ(0) Adjusted to Match JID at Pure Components Linear composition dependence assumed. c. T at T, x (Table 21, column 10). Pure benzene calculated using A(k) from Figure 3 and J(0) and J(k) from Table I, all from Maguire et al. (1981), and B(k) was calculated using equation (242). Final value calculated from equation (239b). Pure cyclohexane from measurements of Dorfmuller et al. (1976). Linear composition dependence assumed. 5. f from equation (233) (Table 21, Column 14) x) from equation (25). 6. J(0) from equation (242) (Table 21, Columns 11 & 12) Calculated values, Column 11. Linearly adjusted to match pure component JD, Column 12 7. JID from equation (247) (Table 21, Column 13) a. Cp at T, x (Table 21, Column 15) Pure component values from Table III, Maguire et al. (1981) Linear composition dependence assumed. b. a (Table 21, column 16) 8.X2 JT Analytical equation of Brown et al. (1978) c. a (Table 21, Column 17). Interpolated from Table 4, Brown et al. (1978) d. (yc 1) or (Cp Cv)/Cv at T, x (Table 21, column 18) Pure component values from Table I, Maguire et al. (1981). Linear composition dependence assumed. (a n e. ) (Table 21, Column 19) Interpolated from Table 2, Coumou and Mackor (1964). f. KT (Table 21, Column 20) Pure component data from Table III, Maguire et al. (1981). Linear composition dependence assumed. g. ap at T, x (Table 21, Column 21) Pure component data from Table III, Maguire et al. (1981) Linear composition dependence assumed. 8. Activity coefficients from equation (246) (Table 23, Column 3) The integral was calculated using a trapezoidal rule. Experimental data of Nagata (1962) (Table 23, Column 2) Table 22 Speed of Sound in Cyclohexane/Benzene Mixtures at 296 K and 1 atm X2 VS X2 Vs m/s m/s 0.0 1345 0.6 1410 0.1 1345 0.7 1430 0.2 1347 0.8 1460 0.3 1352 0.9 1500 0.4 1365 1.0 1540 0.5 1385 Comparisons with Phase Equilibrium Measurements After duplicating the calculations of Maguire et al. (1981) in this way, a quantitative comparison can be made of this method with vaporliquid equilibrium measurements. The values of Iny, are shown in Figure 21 and listed in Table 23. The absolute average error is 0.25 or 25% in the activity coefficient. However, much greater errors occur at low concentrations because the integration in equation (246) accumulates errors. We expect that the results are not within the experimental accuracy of the vaporliquid equilibrium measurements, which should be correct to within 2%. Figure 22 shows the scattering J(0) compared with JTHERM from analysis 1.4 .3  From Light 1.2Scattering 1.2  ..... ....... From VLE Data 1.1  1 0.9 0.8 In 2 0.7 0.6 0.5 0.4 0.3  0.2 " 0.1 "'" . 0 0.2 0.4 0.6 0.8 Mole Fraction, x2 Figure 21 Activity Coefficients Calculated From VaporLiquid Equilibrium Data of Nagata (1962) and From Light Scattering Data For the Cyclohexane(1) / Benzene(2) System at 23 C. of the vaporliquid equilibrium measurements (Table 23, Column 6). These have an average absolute difference of 0.035 or 5% which leads to the 0.25 average difference in Iny2. Table 23 Comparison of Activity Coefficients From Light Scattering and VaporLiquid Equilibrium Data For the System Cyclohexane(1)/Benzene(2) at 296 K X2 ln/y2 from VLEt ln Y2 from light scatter lny2 abs. diff L OlnY2) from VLEt JTHERM from VLEt data 1 2 3 4 5 6 0.000 0.102 0.202 0.296 0.304 0.350 0.488 0.651 0.747 0.850 0.900 1.000 0.335 0.275 0.222 0.176 0.172 0.151 0.096 0.046 0.025 0.009 0.004 0.000 1.378 0.965 0.624 0.424 0.412 0.344 0.188 0.087 0.050 0.021 0.009 0.000 1.043 0.689 0.403 0.248 0.240 0.192 0.092 0.041 0.026 0.012 0.005 0.000 0.607 0.561 0.513 0.465 0.461 0.436 0.358 0.256 0.191 0.117 0.079 0.000 0.378 0.412 0.469 0.533 0.538 0.562 0.670 0.735 0.719 0.648 0.584 0.412 t Nagata (1962) Sensitivity of Activity Coefficient Calculations To determine the effect of uncertainties in the light scattering measurements, an additive random error was introduced into the J(k) values and these were used in a second set of calculations J(k)ERROR = J(k)EXPT + F x NR (248 where J(k)EXPT is the measured J(k) and NR is a random number between 0.5 and 0.5. The factor, F, was adjusted until the standard deviation of the error between the 1.4 . U / \ / \ / \ 1.3 / / \ 1.2 / \ / \ 1. / 1 / \ J / 0.9 ... ... S...+" '... 0.8  0.4 .0 A JTMERM 0.37 Mole Fraction, x2 Figure 22 Various Rayleigh to Brillouin Scattering Ratios (J) For the Cyclohexane(l)0.6 / / Benzene(2) System at 23 C. Cyclohexane(1) I Benzene(2) System at 230 C. J(k)ERROR and J(k)EXPT was either 0.05 or 0.10. The 12 random number values used had an average of 0.113. This means that we introduced both random and systematic errors. The results of this numerical experiment are given in Figures 23 and 24 for the calculated J(0) and Iny2 while Table 24 summarizes the results for the compositions of Table 21. Part of the errors in J(k) are passed on to J(0) and, subsequently to ln(y2). Since other input parameters, such as the speed of sound in the solution or the relaxation parameters, may be difficult to find, we vary these individually by 520% so their impact on J(0) and In(y2) can be detected. Table 25 shows that rough approximations to these parameters, particularly for v T and B(k), will not be good enough for obtaining activity coefficients as accurately as from vaporliquid equilibrium data. Table 24 Errors in J(0) and Iny2 Caused by Random Errors in J(k) Standard Deviation Standard Deviation Average Absolute of in Error Error Random Error in J(k) J(0) Calculated Iny2 0.00 0.041 0.249 0.05 0.046 0.243 0.10 0.092 0.598 0.9 0.8 0.7 0 0.6 0.5 J 0.4 ..  0 0.3 0.2 ......... JTHERM J(0)* 0.1 + 5% Error 0 10% Error o I I I I I I 0 0.2 0.4 0.6 0.8 1 Mole Fraction, X2 Figure 23 Values of J(0)* Calculated From Measured Values of J(k) For the System Cyclohexane(1) / Benzene(2) With Various Random Errors Added at 230 C. .......... ....... e  4   4.  Light Scatter VLE Data 10 % Error 5% Error / I .... .7.  I I I I I I I I I 0.2 I I 0.4 I I 0.6 I I 0.8 Mole Fraction, x2 Activity Coefficient in the Cyclohexane(1) / Benzene(2) System Calculated from Light Scattering, With Various Random Errors Added to J(0)*. In 12 Figure 24 I Table 25 Error in Iny2 Caused by Bad Input Values Standard Average Deviation Absolute Errored Table 1 In Error of Error Inputs (%t) Values J(0) Iny2 All Original Values 0.033 0.249 Cyclohexane v,, 1100 (15) 1280 0.063 0.330 Cyclohexane v oCyclohexane v 1600 (+15) 1395 0.047 0.366 Cyclohexane T 1x101 4.1x10" 0.160 2.02 Cyclohexane T 1x101 4.1x10" 0.036 0.215 Benzene B(k) 0.205 (4) 0.214 0.032 0.260 Benzene B(k) 0.225 (+4) 0.214 0.035 0.229 Cyclohexane B(k) 0.0625(20) 0.0795 0.037 0.275 Cyclohexane B(k) 0.095 (+20) 0.0795 0.030 0.224 t Deviations from Table 1 Values Summary The measurement of concentration fluctuations from light scattering is a possible route for obtaining activity coefficients. An investigation was made of cyclohexane/benzene, a typical organic system, for which all necessary data for the rigorous analysis, including relaxation information, were available. The results are seriously in error from reliable vaporliquid equilibrium data. Miller and Lee (1973) state that large excess scattering (solution scattering over pure component values) is needed to obtain activity coefficients. They point out that systems with a pure component refractive index greater than 0.2 are required for reliable results. For pure components here, J(0) is approximately TYc 1 (the LandauPlaczek formula). The difference between the maximum J(0) observed for the cyclohexane/benzene system and ic 1 is approximately 0.40, evidently not great enough to reliably determine activity coefficients. Miller and Lee (1973) indicate that in systems with large pure component refractive index differences (An > 0.2) the excess scattering can be adequate. The results of this analysis, where An = 0.07, are consistent with the prediction of Miller and Lee. The activity coefficients of the cyclohexane/benzene system cannot be determined by light scattering even if all information is available. If the ancillary information is erroneous, it is clear that the sensitivity of the results makes light scattering unlikely to be viable for determination of activity coefficient derivatives in any system. Finally, it should also be noted that instrument expense and dustfree sample preparation are not insignificant factors to be considered in the lightscattering experiment. It might be appropriate to test the method with a system having An > 0.2 for which all the auxiliary properties are available in order to determine the real possibilities of the technique. However, such a high refractive index difference requirement would limit the number of real systems so severely that another measurement must be sought. Equilibrium sedimentation in an ultracentrifuge is such a technique. As discussed in the next chapter, it can be used for systems where An > 0.06 and the extra required information is much less. CHAPTER 3 CONCENTRATION DERIVATIVES OF DILUTE SOLUTION ACTIVITY COEFFICIENTS USING AN ANALYTICAL ULTRACENTRIFUGE Introduction While light scattering is apparently not accurate enough to provide useful information about solution activity coefficients, other possibilities besides traditional vaporliquid equilibrium (VLE) measurements (Gmehling et al., 1977, Wichterle et al., 1973) need to be explored. In particular, accurate measurements need to be made in the 2.5% to 10% (mole) range to determine activity coefficients and their composition derivatives more accurately. Data in this region are of interest because, as Loblen and Prausnitz (1982) and Schreiber and Eckert (1971) point out, extrapolations of finite concentration VLE data to infinite dilution do not usually agree with the direct measurements of y by differential ebulliometry and chromatography. For example, Figure 31 shows VLE data for the carbon disulfide/acetone system (Litvinov, 1952) fit to the Wilson (1964), NRTL (Renon and Prausnitz, 1968) and Margules (Van Ness and Abbott, 1982) activity coefficient models. Clearly, all of the models fit the midrange concentration data equally well. However, at low concentrations, the three equations deviate from each another, the departure being characteristic of the equation. The differences in these models are even more pronounced when examined at the derivative level. As seen in Figure 32, when the same VLE data are fit to different activity coefficient models, the compositional derivatives can be quite different. Here the Margules model is not even monotonic. Besides the sensitivity of activity coefficient derivatives to discriminate between models, such data lead directly to the theoretical quantities of fluctuation solution theory (Kirkwood and Buff, 1951; 1.5 1.4 \ 1.3 \ 1.2  Wilson 1.1 Margules NRTL 1  x A Litvinov, 1952 0.9 VLE Data 0.8 0.7  0.6 0.5  0.4 0.53 0.2 0.1  0 0.1 1 0.1 I I I 1 I I I 0 0.2 0.4 0.6 0.8 Mole Fraction, x, Figure 31 Various Activity Coefficient Models Fit to the VLE Data of Litvinov (1952) For the Carbon Disulfide(l) / Acetone(2) System at 250 C. 0 /I / 1.2 dI1.4 /" 0.6  1.2 , 1.8 6 / SWilson 2  Morgule! 2.2 / __ NRTL 2.4 / / / 2.6  0 0.2 0.4 0.6 0.8 Mole Fraction, x, Figure 32 Activity Coefficient Derivatives From Various Models Using Parameters Fit to the VLE Data of Litvinov (1952). For the Carbon Disulfide (1) / Acetone (2) System at 25 C. O'Connell, 1971; 1981; Campanella, 1984). In particular, integrals of the statistical mechanical direct correlation function (DCFI) can help future efforts toward solution models. For example, Campanella (1984) has noted that the derivatives of common activity coefficient models have a common mathematical form (ratios of polynomials in mole fraction). Such observations could lead to an accurate and general empirical activity coefficient model. Thus, an experimental method which avoids use of a model and directly produces values of activity coefficient derivatives in dilute solution would be highly desirable. This would allow the determination of which, if any, of the activity coefficient models are most accurate and provide a basis for future models. Equilibrium sedimentation in an analytical ultracentrifuge provides such an experiment. Equilibrium sedimentation distributes a heavier solute in a less dense solvent. Centrifugal force acts to push the heavier solute to the bottom of the cell. Diffusive forces tend to redistribute the solute. After a time these two forces balance each other and establish a compositional profile which is lower in the solute nearer the top of the cell. The experimental results are obtained from Rayleigh interference optics which gives composition profile while the sample is still in motion. A specially designed sample cell has two chambers or sectors, one is a reference which is filled with solvent while the other has the sample mixture. Laser light is split to shine through both chambers simultaneously. When recombined this light produces an interference pattern corresponding to the radial refractive index differences of the two cell chambers. The refractive index difference is directly related to the composition. An illustration of the sample cell and light path is given in Figure 33. Initially the solution in the cell is homogeneous and the ultracentrifuge is operated in a steady fashion until equilibrium is established. After rotation starts, the time to reach a degree of equilibrium distribution (t8) is found from Ref Cell Sample Cell  Center of Rotation Section Recombined Laser Light, Interference Pattern Window Cell  Window' Laser x7~ II Ii Ligh Figure 33 Ultracentrifuge Double Sectored Cell Showing the Location of the Sample Solution and Reference Solvent Relative to the Laser Light Path. AA A (Ar)2 2 t8 4T2D r(1 8) (31 where D is the diffusion coefficient and 8 is the degree of equilibrium. For this experiment, a 99.9% degree of equilibrium (8 = 0.999) can be considered adequate. The solute (1) activity coefficient derivative is obtained from the isothermal derivative of the chemical potential being zero at equilibrium. In the ultracentrifuge the chemical potential is a function of distance of the solution from the axis of rotation, r, the pressure, P, and the mole fraction of solute, xj. dVL dr+ f, dP+ "' dx = 0 (32 d Or TP,xl ( Trx OX TP.r By substituting in known quantities for the partial derivatives, writing the pressure derivative in a centrifugal field in terms of radius and rearranging, the activity coefficient derivative can be found. ( R1 1 (alnh1 ] =RT + J (33 ax1 T,,r Xl Ox T,P LV1 V1 (34 T rr,x1 where V1 is the solute partial molar volume. In a centrifugal field, dP = p wo2 r dr (35 where o is the angular velocity in radians per second. (Ox'1) =M, r (36 IOr JTPX1 where M, is the solute molecular weight. Substituting equations (33), (34), (35) and (36) into equation (32) and rearranging gives fOln71 _T (M pV1) w2r (dr 1 (37 ax, RT Idxj J x1 In equation (37) the term (MI pV1) is called the sedimentation parameter. It is a function only of the physical properties of the compounds used and is determined from other measurements. The angular velocity, w, and radius are measured directly. The concentration profile dr/dx1 is determined from the interference fringes. High speed centrifugation has been used in the past by Cullinan and coworkers (Cullinan, 1968; Cullinan and Lenczyk, 1969; Sethy and Cullinan, 1972 and Rau, 1975) for obtaining activity coefficient data. However, they used a preparatory ultracentrifuge, requiring that the machine be stopped and samples withdrawn by syringe. Besides the possibility of causing disturbances in the distribution, this technique limited their work to midrange concentrations where vaporliquid equilibrium (VLE) measurements are of comparable accuracy and sensitivity. In their initial work, Cullinan and Lenczyk (1969) used the method for the system hexanecarbon tetrachloride. They assumed the sedimentation factor (M1 pV) remained constant over the range of hexane compositions (35% to 65%) and obtained results within 5% of those calculated from VLE (Christian et al., 1960). Later, Sethy and Cullinan (1972) and Rau (1975) used the same method on the carbon tetrachlorideacetone system. By substituting the composition derivative of the Wilson (1964) activity coefficient model into equation (37), centrifuge data over nearly the entire composition range were regressed to obtain Wilson (1964) parameters. Figure 34 shows activity coefficient derivatives of Rau's (1975) experiments on this system. They are very similar to the results obtained using the Wilson (1964) parameters reported by Prausnitz et al. (1967) indicating that the method is reliable for midrange concentrations. The ultracentrifuge, equipped with a conventional light source such as a mercury arc lamp, has been used to measure thermodynamic properties and molecular weights of biochemicals, polymers and salt solutions (Johnson et al., 1954; Richards and Schachman, 1959; Johnson et al., 1959; Nichol and Winzor, 1976; Hwan et al., 1979; Hsu, 1981). Earlier investigators used Schlieren optics for the refractive index profile in the 0.4 0.5 0.6 0.7  dirlyi dlnX1 dx1 0.8  0.9  0.38 0 Rai, 1975 Prausnitz et al., 1967  0 0 fl D a 0.42 I I 0.46 0.5 0.54 I I 0.58 I I 0.62 Mole Fraction, x1 Figure 34 Activity Coefficient Derivative Data Calculated From the Ultracentrifuge Data of Rau (1975) For the System Carbon Tetrachloride(1) / Acetone(2). Also Shown is the Wilson Model Fit to the Ultracentrifuge Data of Rau and to VLE Data, as Given by Prausnitz (1967). I sedimented solution. More recent investigators (Richards and Schachman, 1959; Johnson et al., 1959) point out that Rayleigh interference optics is more accurate for equilibrium sedimentation experiments. Conventional light is not coherent enough to give an interference pattern with the nonelectrolyte, low molecular weight organic systems of interest here. Williams (1972; 1978) describes a laser light source which yields Rayleigh interference patterns from these systems. Equipment Apparatus A Beckman Model E analytical ultracentrifuge (serial #685), modified with a laser light source, was used in the present studies. This ultracentrifuge can spin a sample at speeds from 10,000 rpm to greater than 50,000 rpm. The rotor used in these experiments was an AND type (serial #3581). This is an aluminum rotor with a maximum safe speed of 52,000 rpm when new. The high stress of continued operation at high speed weakens the metal in the rotor, requiring that the maximum safe operating speed be lowered with age. The sample cells were double sector (chamber) type, 12 mm deep. The aluminum cell was manufactured by Beckman and the titanium cell was custom manufactured by Central Machine Products, Gainesville, FL. The sapphire cell windows were from Adolf Meller Company, Providence, RI. The window liners were originally thin (15 mils) strips of PVC as suggested by Yphantis (Ansevin et al., 1970) to cut down on the stress distortion of the windows at high speed. However, the organic solvents studied here attacked the PVC. Therefore, teflon of approximately the same thickness was substituted. All other cell parts were standard Beckman elements. Appendix A gives more details of the cell assembly and operation of the ultracentrifuge. The ultracentrifuge has a vacuum chamber which can be evacuated to about 1 Kpa to allow the rotor to spin with very little air resistance which would raise the rotor and sample temperature. The chamber is controlled at temperatures from 100 C to 350 C 0.50 C using a heating element in the bottom of the chamber and a refrigeration system. The temperature of the rotor is sensed by a thermistor in the base of the rotor. For details of the thermistor calibration, see Appendix A. The optical system used to detect the concentration profile is shown in Figure 35. Its design was based on that of Williams (1972; 1978; 1985). Alignment was accomplished by combining the procedures of several authors (Rees et al., 1974; Richards et al., 1971a; 1971b; Gropper, 1964; Dyson, 1970) with helpful suggestions of Williams (1985). The resulting alignment procedure is given in Appendix B. In short, the optics consist of a SpectraPhysics 5 milliwatt HeNe laser (Model 1051) mounted on the ultracentrifuge frame. The laser light is expanded with a spatial filter and passes through the rotating sample, just as conventional light would. The interference pattern between the solution side of the cell and the pure solvent (reference) side is established using a Rayleigh interference mask. The fringe pattern was exposed on a strip of Technical Pan 2415, EstarAH based Kodak film held in place with a custom made film holder and developed using standard darkroom techniques with Kodak D76 developer and Kodak fixer. The radial location of each shifted fringe is then determined with a Nikon Shadowgraph, Model 6C (#7244) microcomparator. Two reference marks on the image, whose radial locations are known exactly, yield the actual radial distance to each shifted fringe. Composition Detection From the photographic negative (illustrated in Figures 36 and 37) the composition profile in the cell at equilibrium is determined. Each fringe, j, along the radius, r, that is shifted from the horizontal is equivalent to a constant refractive index change, An. j(r) = An dl/ (38 where d is the cell depth and X is the light wavelength, 632.8 nm. HeNe Laser7 Cylindrical Lens Camera Lens Photo Plate I Condensing Lens S Interference Mask Sample Cell (in Rotor) Collimating Lens Mirro Mirror Figure 35 Schematic of the Laser Optics Used in the Ultracentrifuge for Composition Detection. Figure 36 Laser Light Fringe Photo From the Ultracentrifuge, Showing a Sample of Carbon Disulfide / Acetone. The Initial Concentration of Carbon Disulfide was 3.56% (Mole) and the Speed of Rotation was 21739 RPM. Figure 37 Laser Light Fringe Photo From the Ultracentrifuge, Showing a Sample of Carbon Disulfide / Acetone. The Initial Concentration of Carbon Disulfide was 3.56% (Mole) and the Speed of Rotation was 37000 RPM. By measuring the horizontal distance between each of the fringes, a relative refractive index can be determined as a function of position, r. Because the concentration at the surface is finite, the actual refractive index profile and the concentration profile must be determined by material balance. By separately measuring the refractive index as a function of solute mole fraction, n(x1), and knowing the initial mole fraction, xy', and volume of the cell, v, the actual composition profile, xl(r), can be determined. The following equations give the conversion from fringes to composition profile: rt rfw(r) d x,[n(r)] p[xl(r)] dr = v po x' (39 rb where p is the solution density at x', w(r) is the cell width normal to the radius which varies with radius in the cells used, and p[xl(r)] is the solution density at the position, r, with solute mole fraction xl(r): n(r) = j(r) X / d + n' (310 where n' is the refractive index at the top of the sample, at j=0. Equations (39) and (310) are used to iteratively solve for the unknown n'. Then, equation (310) gives the full composition profile in the cell. The analysis starting with the microcomparator measurements at each fringe and leading to xl(r) are carried out with the FORTRAN computer program FRNGCNV. This program is listed in Appendix C. It is fully documented with comments and a sample input and output file. While the program has the capability of approximating the mixture refractive index from pure component values with several mixing rules, all final calculations used new refractometer data fitted to a linear equation. These were carried out in a Cromatix differential refractometer (Milton Roy Corporation, Model KMX16). The refractometer measures the difference between a solution of known composition and the pure solvent. The light source is a HeNe laser of the same wavelength as that on the ultracentrifuge eliminating any effect of wavelength on the refractive index. The refractometer was first calibrated with known solutions of dried, reagentgrade sodium chloride and deionized water using the data at 250 C of Kuis (1936). Refractive index measurements were made on all systems used in the ultracentrifuge in the range 010 mole percent solute. For carbon disulfideacetone, literature data from Campbell and Kartzmark (1973) and Loiseleur et al. (1967) exists. The experimental data measured here compare favorably with the literature data. All refractive index data were fitted to a first or second order polynomial in x1. The data and expressions are given in the next chapter. Once the composition profile is found, equation (311), which is a finite difference approximation to equation (37), can be used directly, with the necessary physical properties, to calculate the desired activity coefficient derivative. SOln1n (M p Vr) ( r1) 1 (311 a x, .r 2RT ((x0i (x,)j,) (xOj This calculation was carried out in a Lotus 123 spreadsheet. Details of the density calculations required are given in Appendix D. The initial speed for operation of the ultracentrifuge was estimated as described in Appendix E so that about 15 total fringes would appear. In all cases each concentration was run at two speeds to check for consistency. The first speed was based on this estimate while the second speed was either 1.5 times greater or less than the initial speed, depending on the actual number of fringes found. System Selection There are three major criteria to determine if a binary pair of chemicals could be suitable for study by equilibrium sedimentation. First, because of the laser light composition profile determination, an appropriate difference in refractive index must exist between the two chemicals. Second, the sedimentation parameter (Mi pV1) must be large enough for the solute to sediment at a detectable level in the solvent under an accessible centrifugal force. Mixture density data must be available to determine this accurately. Finally, the system should be at least moderately nonideal. That is, the dilute activity coefficient derivative should be sufficiently large that the result from equation (311) yields a significant difference of the two large numbers. Additional considerations are to have components of low volatility and toxicity. Here, no compounds were chosen that have a normal boiling point below 490C. Measuring fringe separation with the microcomparator indicated that 10 to 20 shifted fringes can be accurately found in a reasonable amount of time. This, together with the range of rotational speed, gives the limits of pure component refractive index difference and sedimentation parameter. A binary pair must have a pure component refractive index difference of at least 0.06, and a sedimentation parameter of at least 35 (measured at a solute concentration of 2%). Finally, an infinite dilution activity coefficient greater than 2, or less than 1/2 insures that the system is nonideal enough. The system carbon disulfide in acetone was studied because it is highly nonideal and had been thoroughly examined by Campanella (1984). The second system, carbon tetrachloride in acetone, was studied by Cullinan and coworkers (Sethy and Cullinan, 1972; Rau, 1975) in an ultracentrifuge, which allowed a direct comparison with their method. Since both of the above systems showed a positive deviation from Raoult's law, chloroform in acetone was selected to demonstrate the effects of negative deviations from Raoult's law. While no sedimentation data exist for this system, there are many vaporliquid equilibrium literature data references. An extensive search was made to identify two additional systems for which the ultracentrifuge could supply useful dilute solution information. The following classes of systems were identified as being interesting: 1. water/nitrile 2. water/amine 3. chloroalkene, alkene or alkane/cyclic ether 4. alcohol/alkane 5. amine/chlorinated alkane 6. nitrile/alkane or cyclic alkane 7. alcohol/chlorinated alkane These were first evaluated by refractive index differences, An. In the first group, water and benzonitrile were identified as the only common system with An > 0.06 (An = 0.19). However this system is only partially miscible. The candidate system water/acetonitrile had a refractive index difference of only 0.03. For the second group, triethyl amine was selected among the lower molecular weight nonaromatic amines since refractive index generally increases with carbon number. For water/triethyl amine, An = 0.06. Aromatic amines give larger An values (e.g., for water/aniline An = 0.2) but these show immiscibility. For the third group, several cyclic ethers (furan, tetrahydrofuran, pyran, tetrahydropyran and 1,4dioxane) were considered for solution with tetrachloroethylene or trichloroethylene. However, the only pairs with a suitable refractive index difference are tetrachloroethylene with either 1,4dioxane (An = 0.09) or tetrahydrofuran (An = 0.08). With pyran the only alkane and alkene with a large enough An are heptane and hexene. The best pair from the fourth group would involve a low carbon number alcohol such as methanol, and a high carbon number alkane such as noctane (An = 0.07). For the fifth group the aliphatic amines and linear aliphatic monochlorides have very similar refractive indexes. Thus, an aromatic amine, Nmethyl aniline, was chosen to be paired with chloroform (An = 0.12) or monochlorobutane (An = 0.09). In the sixth group aliphatic nitriles and alkanes have nearly the same refractive index. However, benzonitrile can be paired with either a linear or cyclic alkane, such as with heptane An = 0.13 or cyclohexane An = 0.11. Also, the acetonitrile/benzene system has An = 0.16. Finally, chloroform and carbon tetrachloride have the highest refractive index of chlorinated alkanes. Therefore, with any alcohol, particularly methanol, either of these compounds has an adequate An. The systems above were then evaluated by their ability to sediment as measured by their sedimentation parameter. For screening purposes, the factor (M, pV1) was approximated by (M1 po0mV) where the density (Po.02) was for a solute concentration of 2% (mole). Table 31 Sedimentation Factors For Various Binary Pairs Binary Pair Sedimentation Factor (M Po.02V?) water/benzonitrile 0 water/triethyl amine 5 water/aniline 0 tetrachloroethylene/1,4dioxane 59 tetrachloroethylene/tetrahydrofuran 38 methanol/noctane 4 chloroform/methyl aniline 39 Nmethyl aniline/chlorobutane 0.5 benzonitrile/heptane 30 benzonitrile/cyclohexane 20 benzene/acetonitrile 36 chloroform/methanol 55 carbon tetrachloride/methanol 75 chloroform/acetone 54 carbondisulfide/acetone 27 carbon tetrachloride/acetone 75 From the table above it is easily seen that the sedimentation parameter is less than the desired 35 for many of the candidate systems. The only systems, not previously selected, with a high enough sedimentation factor were tetrachloroethylene with either 1,4dioxane or tetrahydrofuran, Nmethyl aniline with chloroform (or carbon tetrachloride), chloroform (or carbon tetrachloride) with methanol and benzene with acetonitrile. Final selection of benzene with acetonitrile and methanol with carbon tetrachloride was based on low toxicity, and availability of solution densities and compressibilities. Error Analysis Equation (37) describes the fundamental relationship between the derivative of the activity coefficient and measurements in the ultracentrifuge. All quantities in that equation except the derivative dr/dx1 are well defined as to their expected uncertainty. This derivative must be calculated by an iterative integral method. For that reason the analysis of error for this process is done in two steps. First, the effect of variations in dr/dxl on the activity coefficient derivative is found. Then the error to be expected in the dr/dxl term is determined. An uncertainty analysis (Holman, 1971) can determine the uncertainty in the activity coefficient derivative. This method assumes that a quantity in question, A, can be written in terms of several independent variables, aj. A = A(al, a2, a3 ..... a,) (312 If each variable ai has an uncertainty Ui, the uncertainty in A, UA can be written as UA U1 + U 2 + ... f U (313 A 8a1 J a2 9 ) ) IS.a )) In equation (37) the variables of interest are p, V1, w, r, T, x'. If the activity coefficient derivative is defined as A and (dr/dxj) as D, equation (37) becomes Saln1 A =(M pV) 2r (314 ax, J RT x, and equation (313) can be written for this specific case as UA= A )IA )U ) + I UA 2A ap PJ aV Vi J a) r J r (A ( 1 (A 2 U'A 12 (315 aT UT ax+ I aD1D ] ())5 The measurement of refractive index as a function of composition was done in an instrument of much greater accuracy than required for the ultracentrifuge experiment. Thus, the uncertainty in this quantity is ignored. Each of the derivatives in equation (315) can be evaluated from equation (314) and substituted into equation (315) giving U V[( o2 r D p 2 r 2 + (2(MrpV) r U + (M2pV) 2 (MpV) r W U )2 +U 2 + D(Ur +DUT)+ ) DU RT Ur+ RT2 T (x)2 x + (MjpVj) r W2 2D] 2(316 RT UD(316 Uncertainties were assigned to each of the quantities in question as follows: U = 0.1% UV = 0.1% UT = 0.5 Co Ux, = 0.0002 U =0.1% U = 0.001 cm W r Typical values (from run 382 x' = 0.035 and run 291, xO = 0.109) were substituted into equation (316) to illustrate the relationship of the uncertainty in D (dr/dx,) to uncertainties in A n The results are shown in Table 32. O, xt1) Table 32 Uncertainty in ) as a function of ( dr } the Uncertainty in  I dxj x' = 0.035 x = 0.109 Ut % Ui U % Ui Ui% 0.1 0.2 3.5 .02 0.3 1.0 0.3 5.5 .05 1.0 5.0 1.0 20.8 .23 4.5 t % Uncertainty in d (. dx1 t Absolute and % Uncertainty in  1 ax, ) The error to be expected in (dr/dxl) will now be examined by introducing a random error into its primary variables, xj, j and r. This error will be of an average magnitude to be expected in the experiment. As an example, the fringe measurements from run 382 (carbon disulfide/acetone) were used. The average error in r, as indicated above, should be no more than 0.001 cm, but a worst case of 0.01 cm is used here. Then the measurements on run 382, which has only 8.9 fringes, would have an error of about 0.1 fringes. A summary of the contribution of various errors in the experimental quantity (dr/dx1) and subsequently ( is given in Table 33 for the xi ) most severe case (x' = 0.035). Table 33 Errors in and  ax, dxj x' = 0.035 Cause of Error U (lnyU (f radius 5.5% 1% j (fringes) 6.5% 1.3% x4 initial solute mole frac. 3.5% 0.1% The error due to radius measurement and fringe number are redundant, since they are absorbed in the same effect, the number of fringes. The error introduced in x' could be added to the other, yielding for a total maximum error in of less than 10% at low concentrations. Summary Direct measurements of the dilute solution activity coefficient derivative would be valuable to settle discrepancies between extrapolated vaporliquid equilibrium measurements and direct measurements of infinite dilution activity coefficients, for differences between activity coefficient models and for theoretical and modeling purposes. The analytical ultracentrifuge has the capability of providing these. The experimental equipment, including the laser light source for sensitive Rayleigh Interference composition measurement, have been described. A complete thermodynamic analysis of the ultracentrifuge has been given for the solute activity coefficient derivative in terms of the measurable quantities in the ultracentrifuge. A complete uncertainty analysis was conducted for the experiment. It pointed out that with reasonable care the activity coefficient derivative can be determined to within 10% at concentrations near 2%. Not all binary systems of chemicals are amenable to this technique. There must be a moderate difference in refractive index of the pure components (An > 0.06), a sufficient tendency of the solute to sediment in the solvent, [(M1 pV1) > 35] and moderate nonideality (y > 2) so that the activity coefficient derivative is large enough to be well determined. A thorough screening was conducted to select interesting systems to meet these criteria. Five systems were chosen and both sedimentation and refractive index measurements were made on them. CHAPTER 4 EXPERIMENTAL MEASUREMENTS OF DILUTE SOLUTION ACTIVITY COEFFICIENT DERIVATIVES Introduction Experimental measurements were made to obtain the composition derivative of the solute chemical potentials and activity coefficients of five systems: carbon disulfide in acetone, carbon tetrachloride in acetone, chloroform in acetone, benzene in acetonitrile, and carbon tetrachloride in methanol. Experiments were conducted on all systems over a range of composition, usually from 1.5% (mole) to 10% (mole) in the solute (first component listed). Tests were usually conducted at three temperatures, 100, 250 and 350 C. Equation (41) shows the basic relationship between the activity coefficient derivative and experimentally accessible quantities a 1/RT I (1n, 1 (MI pV) 02 r ddn dr) ax, ax, x1 RT dx dn(41 The density, p, and partial molar volume, V1, are obtained from literature data, the refractive index/composition derivative (dn/dx1) was measured with a laser differential refractometer as described in Chapter 3, and the refractive index profile, (dr/dn), was obtained from the ultracentrifuge. This chapter describes the experiments performed and presents the specific results on each system. Comparisons with analyses from vaporliquid equilibrium (VLE) data are also made. Discussions of the influence of pressure due to cell loading and composition and of impurities are given. Tables giving detailed results for each experimental run are in Appendix F. These tables give the composition profile at each fringe, all physical properties needed and the calculated activity coefficient derivatives and chemical potential derivatives. Carbon Disulfide/Acetone System The carbon disulfide/acetone system was the initial system studied. As experience was gained, inconsistent results were found for some of the measurements; these were omitted. The final 21 runs adopted covered a range of carbon disulfide concentrations from 1.05% (mole) to 10.9% (mole). All concentrations were run at more than one speed to verify results. Two concentrations (10.9% and 2.07%) were run at 100, 25 and 300 or 330 C. All other runs were at 250 C or 26.50 C. A summary of the results for this system is in Table 41. Included are the overall solute concentration, temperature, calculated activity coefficient derivative and calculated chemical potential derivative. Also included are the activity coefficient derivatives and chemical potential derivatives calculated from parameters fitted by Gmehling et al. (1977) to the VLE data of Litvinov (1952) at 25 C for the Wilson (1964) and NRTL (Renon and Prausnitz, 1968) excess free energy models. Excess enthalpy data of Campbell et al. (1970) were used in equation (42) to calculate the activity coefficient at 350 C from the 25 C VLE data of Litvinov (1952). (In)T=, = h 1 1 1 + (lnY)T=T, (42 SR T2 T(2 Numerical composition derivatives of lny, were then calculated to verify the consistency of the activity coefficient derivatives obtain from VLE data at different temperatures contained in Table 41. The excess enthalpy data were compiled and regressed by Gmehling et al. (1977). The chemical potential derivatives from the centrifuge experiment and from the VLE are given in Figure 41. Table 41 Summary of Carbon Disulfide(1)/Acetone(2) Ultracentrifuge Experimental Results and Comparison Values Calculated from Literature VLE Data __ fa1ny_ ', 1n,, ( Iny, a' alny1 xx T )1nl J/RT 1.l (/. RT [1/JIRT al S axE I ax, E xxi xs WxsJ ax T) ( axT) l axl C Expt Expt Wlsnt Wlsnt NRTLt NRTLt Wisnt 19.03.0* 16.81.2 17.42.1 12.21.4 13.51.9' 8.93.8'  15.221. 8 7.11.2 7.01. 1 9.90.7 9.50.5 9.50.7 9.40.9 9.90.8 7.72.4' 6.60.3 6.70.3 3.60.2 4.70.1 4.40.1 3.50.2 74.6 36.0 35.5 35.5 34.2 38.9 32.5 27.7 27.9 18.1 18.5 18.4 18.5 16.4 18.7 10.6 10.4 5.5 4.4 4.7 5.6 2.466 2.440 2.433 2.409 2.387 92.73 50.87 1.828 1.825 93.36 51.48 2.599 2.567 45.78 1.824 46.39 2.560 32.72 1.821 33.31 2.532 25.71 1.818 26.28 2.506 0.01051 0.01876 0.01876 0.02074 0.02074 0.02074 0.02074 0.02846 0.02846 0.03558 0.03558 0.03558 0.03558 0.03767 0.03767 0.05806 0.05806 0.10949 0.10949 0.10949 0.10949 25 26.5 26.5 10 25 25 30 18.8 26.5 26.5 26.5 26.5 26.5 25 25 26.5 26.5 10 25 25 33 2.319 14.90 1.810 15.41 2.427 2.174 6.96 1.792 7.34 2.261 + Standard deviations from various fringes. t Calculated from VLE data of Litvinov (1952) at 250 C. t Calculated from VLE data of Zawidzki (1900) at 350 C. Appear inconsistent. The results of the ultracentrifuge experiment are numerically different from the VLE curve, though they have the same general trend with composition. Except for five runs, a factor of 1.4 times the ultracentrifuge derivatives would make them essentially coincide with the VLE values, as shown in Figure 41. 2.380 24.16 1.817 24.73 2.499 100 90  80 70 60 d _/RT dx 5o  40 30 20 10 0 0.01 0.03 0.05 0.07 0.09 Mole Fraction, x1 Figure 41 Chemical Potential Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Litvinov, 1952, at 25 C) Fitted to the Wilson and NRTL Models, for the System Carbon Disulfide(l) / Acetone(2). 0.11 Figure 42 shows derivatives of the activity coefficient. The filled rectangles are calculated from the ultracentrifuge experiment, the lines are from various models fitted to VLE data and the open diamonds are corrected centrifuge values. Again, it is clear that, except for five points, the corrected data are within experimental error of the calculations. The five points that show the greatest deviation from the others and from the VLE curve are probably in error. In one the concentration of solute was only 1.05% (mole) and there was a blur in the interference pattern at the bottom of the cell. This interrupted the fringe measurements and required extrapolation to determine the total number of fringes present. Another was an early experiment for which all systems were not necessarily operating properly. The others had less than the desired number of fringes (7 to 20). The two sets of measurements that were made at different temperatures do not show a consistent change with temperature. The values at 2.1% (mole) marginally indicate more negative activity coefficient derivatives as temperature increases, for the 10.9% (mole) system, the results at temperatures above and below do not encompass the 250 C value. Excess enthalpy and VLE data agree with the 2.1% (mole) data, indicating a slightly more negative activity coefficient derivative at higher temperatures. It should be pointed out that refractive index/composition data have previously been measured (Campbell and Kartzmark, 1973 and Loiseleur et al. 1967). As shown in Figure 43 their results agree very well with the present measurements even though their light source was the sodium line (589.3 nm) rather than the 632.8 nm laser light used here. Thus, it appears that the refractive index measurements are reliable. Measurements of mixture refractive index were made at 300 and 350 C and extrapolated to other temperatures, as shown in Table 42. 13 12 0 Centrifuge Data 0 Adiustecr Data 10 VLE Wilson S........................... VLE NRTL 8   VLE Margules 6 0 o 4 2 dlni o_  dxi 2 ..... ................. 4 6  8  10 12 14 16 18 19 20 21  22 23 i I I I, 0 0.02 0.04 0.06 0.08 0.1 Mole Fraction, xl Figure 42 Activity Coefficient Derivatives From Ultracentrifuge Data (as Measured and Adjusted) at 250 C and From VLE Data (Litvinov, 1952, at 250 C) Fitted to Various Models, for the System Carbon Disulfide(1)/Acetone(2). Error Bars are Typical of All Ultracentrifuge Runs. 1.384 1.382  1.38 1.378 1.376  1.374  1.372  1.37  1.368  1.366  1.364  1.362  1.36  6 a Expt. Data 1.358 / + Loiseleur et al. (1967) Linear Fit 1.356 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Mole Fraction, x, Figure 43 Experimental and Literature Refractive Index Data (at 25) Fit to a Linear Equation in Mole Fraction for the System Carbon Disulfide(l) / Acetone (24 Table 42 Refractive Index (n) Experimental Results for the System Carbon Disulfide(1)/Acetone(2) x, n n n 250 C 300 C 350 C Extrapt Expt Expt 0.00000 1.35600t: 1.35395: 1.35183t: 0.00947 1.35786 1.35570 1.35354 0.02671 1.36045 1.35810 1.35574 0.02864 1.36106 1.35887 1.35667 0.05624 1.36629 1.36383 1.36137 t Values at 250 C were extrapolated linearly from the experimental measurements at 300 and 350 C. t Pure component refractive index data was taken from Loiseleur et al. (1967) and Campbell and Kartzmark (1977). Chloroform/Acetone System The chloroform/acetone system is the only system examined which had negative deviations from Raoult's law. This means that the activity coefficient derivative will be positive rather than negative as in the carbon disulfide/acetone case. A summary of the results for this system is given in Table 43. A range of concentrations from 1.9% (mole) to 8.7% (mole) (chloroform in acetone) was studied. Each concentration was run at two or more different speeds to verify results. At one concentration (5.3%) runs were made at both 26.50 C and 3730 C. All other runs were made at 26.50 C. Table 43 also shows the activity coefficient derivatives and chemical potential derivatives calculated from the VLE measurements of Rabinovich and Nikolaev (1960) at 250 C using the Wilson, and NRTL model parameters from Gmehling et al. (1977). The higher temperature VLE data of Zawidzki (1900) given in Table 43, was verified as correct by scaling the VLE data of Rabinovich and Nikolaev (1960) at 250 C to 350 C using the excess enthalpy data of Chevalier and Bares (1969). Table 43 Summary of Chloroform(1)/Acetone(2) Ultracentrifuge Experimental Results and Comparison Values Calculated from Literature VLE Data ST f /RT f f RT f01 P^RT fn ax0 J I( ax JRT a9x1) J at Y RT an aln a a RT ( xny, 1, (xx ) G, !,I x ) Ox I ax, I (, xt) ( Ox, ) C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt 0.01888 26.5 14.83.0' 37.2 1.720 54.69 1.569 54.54 1.221 0.01888 26.5 15.22.2* 36.5 0.02515 26.5 7.81.2" 31.3 1.711 41.48 1.565 41.33 1.218 0.02515 26.5 8.10.7' 30.9 0.03713 26.5 6.31.6 20.4 1.694 28.63 1.559 28.49 1.210 0.03713 26.5 6.60.7 20 0.03713 26.5 6.92.6 19.8 0.05269 26.5 3.80.6 15 1.672 20.65 1.552 20.53 1.201 0.05269 26.5 3.70.5 15.1 0.05269 37.3 4.40.6 14.4 0.06357 26.5 2.70.5 12.9 1.657 17.39 1.547 17.28 1.194 0.06357 26.5 2.20.5 13.5 0.06357 26.5 2.70.6 13 0.08724 26.5 1.70.4 9.7 1.625 13.09 1.540 13.00 1.179 0.08724 26.5 1.70.4 9.6 + Standard deviations from various fringes. t Calculated from VLE data of Rabinovich and Nikolaev (1960) at 25 C. t Calculated from VLE data of Zawidzki (1900) at 350 C. Appear inconsistent. Figure 44 shows the chemical potential derivatives from the centrifuge and the VLE data. Again if all of the ultracentrifuge values are multiplied by 1.4, the results are within experimental error of those calculated from VLE data, as demonstrated in Figure 44. Figure 45 shows the derivative of the activity coefficient calculated from the ultracentrifuge experiments, from VLE data, and from the "corrected" values of dc, /RT dx1 25 20 15 10 5 0.01 0.01 0.03 0.05 0.07 Mole Fraction, x, 0.09 Figure 44 Chemical Potential Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Rabinovich and Nikolaev, 1960, at 25 C) Fitted to the Wilson and NRTL Models, for the System Chloroform(l) / Acetone(2) Centrifuge Data Adjusted Data VLE Wilson VLE NRTL VLE Margules 1 I 1 1 0.04 0.06 Mole Fraction, x1 0.08 0.08 Figure 45 Activity Coefficient Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Rabinovich and Nikolaev, 1960, at 25' C) Fitted to Various Models, for the System Chloroform(l) / Acetone(2). 2 3 4 5 6 dinI dxl 7 8  10  11  12  13  15  16 I I 0.02 0 0 0 ooo..o.......... ......,....................... ~___ ___ the chemical potential derivatives. It is clear that the "as measured" results are erroneous since they are negative. One experiment (mole fraction chloroform was 0.053) was run at both 26.50 C and 37.30 C. Ultracentrifuge results give a lower activity coefficient derivative at 37.30 C than at 26.50 C. The VLE data at 350 C (Zawidzki, 1900) and 250 C (Rabinovich and Nikolaev, 1960) indicate the same trend. Negative excess enthalpy data of Chevalier and Bares (1969) further support this behavior. The difference in activity coefficient derivatives measured in the ultracentrifuge at 26.50 and 37.3 C is about 0.8. The VLE data indicate a difference in derivatives at 250 and 350 C of 0.6. There were no mixture refractive index data for this system available in the literature. Pure acetone data were taken from Loiseleur et al. (1967) and Campbell and Kartzmark (1973), while mixture data were measured at 300, the lowest temperature for reliable results, and 35 C. Values at 250 C were extrapolated and all data are summarized in Table 44. Table 44 Refractive Index (n) Experimental Results for the System Chloroform(1)/Acetone(2) X1 n n n 250 C 300 C 350 C Extrapt Expt Expt 0.00000 1.35600t 1.35395t 1.35180t 0.04076 1.35985 1.35779 1.35562 0.10279 1.36561 1.36350 1.36130 t Values at 250 C were extrapolated linearly from the experimental measurements at 300 and 350 C. t Pure component refractive index data were taken from Loiseleur et al. (1967) and Campbell and Kartzmark (1977). Carbon Tetrachloride/Acetone System For the carbon tetrachloride/acetone system measurements were made at three compositions, and multiple speeds, from a low of 0.4% (mole) to a high of 7.3% (mole) carbon tetrachloride, at 23 C for the low concentration point and 26.5 C for the others. These are summarized in Table 45 and show the same trend as earlier experiments. Figure 46 shows that the chemical potential derivatives from the ultracentrifuge follow, but are consistently lower than, those calculated from the VLE data of Brown and Smith (1957), which were adjusted from 45 C using heat of mixing data from Brown and Fock (1957). Table 45 Summary of Carbon Tetrachloride(1)/Acetone(2) Ultracentrifuge Experimental Results and Comparison Values Calculated from Literature VLE and hE Data xaT axn__ (axJ/RT fan__ (ax1J/RT (alnxl ( 41)/RT (aln~yl x0 T RT /RT /RT 1 Ixi ) ax) Fax, ) x, 8xx) lax, Ox) 0C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt 0.00429 23 58.67.0' 161.2 1.063 231.98 0.972 232.07 1.271 0.00429 23 61.38.9' 159.7 0.02208 10.3 6.72.0 38.2 1.055 44.23 0.971 44.32 1.199 0.02208 26.5 8.71.6 36.1 0.02208 26.5 8.11.4 36.5 0.07246 26.5 2.71.1 11 1.033 12.77 0.967 12.83 1.086 0.07246 26.5 2.80.3 10.9 0.07246 37.3 3.20.8 10.5 + Standard deviations from various fringes. t Calculated from VLE data of Brown and Smith (1957) at 450 C. t The VLE values at 450 from Brown and Smith (1957) were adjust with the excess enthalpy data of Brown and Fock (1957) to obtain a result at 250 C. Appear inconsistent. As Figure 46 also shows, if all of the ultracentrifuge values are multiplied by the common factor of 1.2, the results correspond to the VLE data, particularly at the higher concentration points. Figure 47 shows the resulting activity coefficient derivatives for these experiments. Two experimental ultracentrifuge runs on this system were conducted at temperatures other than those shown in Figure 46. At a mole fraction of 0.0221 240 220 200 180 160 140 d, /RT dx1 0 0.02 0.04 0.06 Mole Fraction, x 0.08 Figure 46 Chemical Potential Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Brown and Smith, 1957, at 450) Fitted to the Wilson Model, for the System Carbon Tetrachloride(1) / Acetone(2).   I I 0.02 0.04 I I 0.06 I I 0.08 Mole Fraction, x1 Activity Coefficient Derivatives From Ultracentrifuge Data (Adjusted) and From VLE Data (Brown and Smith, 1957, at 45*) Fitted to the Wilson Model, for the System Carbon Tetrachloride(1) / Acetone(2). Adj. Centrifuge Data 0 0 VLE Wilson VLE NRTL ........... VLE Margules  0.5 0.4 0.3  0.2 0.1  0 dln d xi 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 . 0.9  1  1.1 Figure 47 carbon tetrachloride, runs at 100 C and 26.5 C were made, while at x, = 0.0725, runs were made at 26.5 and 37.30 C. The results showed activity coefficient derivatives that were statistically unchanged with temperature. Here the VLE and positive excess enthalpy data (Brown and Smith, 1957; Brown and Fock, 1957) give more negative derivatives with decreasing temperature. There were no mixture refractive index data for this system available in the literature. As before, experimental mixture data were measured at 300 and 35* C and linearly extrapolated to other temperatures. This data is summarized in Table 46. Table 46 Refractive Index (n) Experimental Results for the System Carbon Tetrachloride(1)/Acetone(2) x1 n n n 250 C 300 C 350 C Extrapt Expt Expt 0.00000 1.35600t 1.35395t 1.35180t 0.04000 1.36144 1.35930 1.35706 0.07990 1.36679 1.36459 1.36230 t Values at 250 C were extrapolated linearly from the experimental measurements at 300 and 350 C. t Pure component refractive index data was taken from Loiseleur et al. (1967) and Campbell and Kartzmark (1977). Benzene/Acetonitrile System For the benzene/acetonitrile system, measurements were made from a low of 227% (mole) to a high of 8.6% (mole) benzene at 26.50 C and are summarized in Table 47. Again, as is shown in Figure 48, the chemical potential derivatives from the ultracentrifuge follow, but are consistently lower than, those calculated from the VLE data of Werner and Schuberth (1966) at 200 C (using the Wilson and NRTL equations with parameters fit by Gmehling et al. 1977). 50 45 40 35 30 1dJ/RT dx1 25 20 15 10 0.02 0.04 0.06 Mole Fraction, x, Figure 48 Chemical Potential Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Werner and Schuberth, 1966, at 200 C) Fitted to the Wilson and NRTL Models, for the System Benzene(l) / Acetonitrile(2). 0.08 Table 47 Summary of Benzene(1)/Acetonitrile(2) Ultracentrifuge Experimental Results and Comparison Values Calculated from Literature VLE Data S(aIny, (a8Q ny___ a l( ny __ iny x( T ax1 a)/RT (l )/RT ln a_/RT ln x J x axJ ax J ax ax, J x t ax 0C Expt Expt Wlsnt Wlsnt NRTLt NRTLt Wlsnt 0.02271 26.5 8.61.1 35.1 2.274 41.78 2.166 41.89 1.9973 0.02271 26.5 8.02.0 35.8 0.03747 26.5 3.41.7* 23.2 2.221 24.45 2.097 24.57 1.9574 0.03747 26.5 8.80.5' 17.8 0.04255 26.5 5.41.3 18 2.204 21.33 2.074 21.46 1.9441 0.04255 26.5 8.31.1 15.1 0.04255 37.3 8.61.2 14.8 0.05622 26.5 6.20.5 11.5 2.156 15.64 2.015 15.78 1.9082 0.05622 26.5 5.30.5 12.4 0.05622 37.3 6.60.8 11.1 0.05821 26.5 3.72.5 13.4 2.149 15.03 2.006 15.18 1.9030 0.06427 26.5 3.92.1 11.6 2.129 13.42 1.981 13.57 1.8873 0.06427 26.5 3.20.7 12.3 0.08610 26.5 3.40.3 8.2 2.057 9.56 1.895 9.72 1.8322 + Standard deviations from various fringes. t Calculated from VLE data of Werner and Schuberth (1966) at 200 C. t Calculated from VLE data of Palmer and Smith (1972) at 450 C. Appear inconsistent. Multiplying all of the ultracentrifuge values by 1.2 gives values which are within experimental error of the VLE data (Figure 48). The data for this system show much more scatter than the others, because lower speeds and smaller samples yielded fewer fringes in most of the runs. These less satisfactory conditions were necessary to minimize the extra spinning force on the rotor due to the heavier mass of the titanium cell used for these chemicals. Activity coefficient derivatives calculated from the "corrected" chemical potential derivatives are given in Figure 49. Two compositions of this system were studied at 37.3 C. While the uncertainty is quite high, both runs appear to give more negative activity coefficient derivatives at   . . ' . r;1 4 .................... I0.02 ) 0.02 I I 0.04 Mole 0.06 Fraction, I I 0.08 0.1 Activity Coefficient Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Werner and Schuberth, 1966, at 200 C) Fitted to Various Models, for the System Benzene(l) / Acetonitrile(2). 0 Centrifuge Data Adjusted Data VLE Wilson ............. VLE NRTL  VLE Morgules 2 dinY1 dxl 4 5 6 7 8 9  Figure 49 r w w this higher temperature. The VLE data of Werner and Schuberth (1966) at 200 C and of Palmer and Smith (1972) at 450 C indicate a trend in the opposite direction, i.e. less negative derivative values at higher temperatures, consistent with the positive excess enthalpy of Absood et al. (1976). There were no mixture refractive index data for this system available in the literature. The experimental data were measured at 300 and 350 C and extrapolated to other temperatures. Pure component refractive index data for acetonitrile were not available at different temperatures. The centrifuge data analysis can be done equally well with absolute or differential mixture refractive index data. Therefore, for this system, only the differential refractive index is reported in Table 48. Table 48 Differential Refractive Index (An) Experimental Results for the System Benzene(1)/Acetonitrile(2) x, An An An 250 C 300 C 350 C Extrapt Expt Expt 0.00000 0.00000 0.00000 0.00000 0.02032 0.00503 0.00499 0.00495 0.04022 0.00970 0.00962 0.00954 0.06668 0.01595 0.01581 0.01566 0.09176 0.02161 0.02142 0.02123 t Values at 250 C were extrapolated linearly from the experimental measurements at 300 and 350 C. Carbon Tetrachloride/Methanol System For the carbon tetrachloride/methanol system, three compositions at 26.50 C were studied from a low of 1.92% (mole) to a high of 4.9% (mole) carbon tetrachloride, as summarized in Table 49. Again the results show the trend of earlier experiments. As shown in Figure 410, the chemical potential derivatives from the centrifuge follow, but are consistently below the curve calculated from VLE data using the Wilson and \ \ \ .o.o..... H\ I \ \ \\ l*. *N 1I 0.03 I I 0.05 0 Centrifuge Data + Adjusted Data  VLE Wilson ... VLE NRTL VLE Margules \ \ 0.07 0.09 Mole Fraction, > Figure 410 Chemical Potential Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Wolff and Hoeppel, 1968, at 20 C) Fitted to Various Models, for the System Carbon Tetrachloride(1) / Methanol(2). c, /RT dx1 25 20 15 10 5 0.01 NRTL equations with parameters regressed by Gmehling et al. (1977) and the data of Wolff and Hoeppel (1968). Table 49 Summary of Carbon Tetrachloride(1)/Methanol(2) Ultracentrifuge Experimental Results and Comparison Values Calculated from Literature VLE Data x0 T [l'] ft1 fRT RT ( RT ax k J xaxi(J axI J xt ax axi J (OxJ 0C Expt' Expt Wlsnt Wlsnt NRTLt NRTLt 0.01923 26.5 10.41.2 41.3 5.080 46.92 6.900 45.10 0.01923 26.5 9.71.6 41.9 0.01923 37.3 6.25.8' 45.3 0.03438 26.5 7.62.7 21.2 4.833 24.26 6.468 22.62 0.04913 26.5 5.41.1 14.9 4.611 15.74 6.082 14.27 0.04913 26.5 5.90.8 14.4 0.04913 37.3 6.20.9 14.1 + Standard deviations from various fringes. t Calculated from VLE data of Wolff and Hoeppel (1968) at 200 C. Appears inconsistent. In this case the experimental values are closer to the VLE data than in other systems studied, but are still low by a factor of 1.1 which gives a reasonable fit to the VLE results (Figure 410). The activity coefficient derivatives calculated from the "corrected" chemical potential derivatives are shown in Figure 411. One run was made at 37.30 C and 4.9% (mole) benzene. Statistically, its activity coefficient derivative was the same as the corresponding 26.50 C runs. This is consistent with the excess enthalpy which for this system is nearly zero (Abramov et al., 1973), when the solution has less than 20% carbon tetrachloride. Also, the VLE data of Wolff and Hoeppel (1968) show very little variation in the calculated activity coefficients or its derivative from 200 to 40 C. The run at 37.30 C and 1.9% (mole) benzene has a very high standard deviation and is obviously in error. 1 0 1  2 3 I .02 0.02 I I 0.04 I I 0.06 0.08 Mole Fraction, x, Activity Coefficient Derivatives From Ultracentrifuge Data (as Measured and Adjusted) and From VLE Data (Wolff and Hoeppel, 1968, at 20 C) Fitted to Various Models, for the System Carbon Tetrachloride(1) / Methanol(2). 0 X .. .. '' '' ...... ...... ...... , o.* *'"  dx1 6 7 8 9 10 11  X Centrifuge Data 0 Adjusted Data VLE Wilson ................ VLE NRTL  VLE Margules Figure 411 There were no mixture refractive index data for this system available in the literature. The experimental data were measured at 300 and 350 C and extrapolated to other temperatures. This data is summarized in Table 410. Again, pure component refractive indexes were not used, so only the differential refractive index is given in Table 410. Table 410 Differential Refractive Index (An) Experimental Results for the System Carbon Tetrachloride(1)/Methanol(2) x, An An An 250 C 30 C 350 C Extrapt Expt Expt 0.00000 0.00000 0.00000 0.00000 0.02389 0.00732 0.00720 0.00707 0.05500 0.01630 0.01606 0.01582 t Values at 250 C were extrapolated linearly from the experimental measurements at 300 and 35 C. Summary of Experimental Results Chemical potential derivatives measured in the ultracentrifuge for each of the five systems studied were lower than those calculated from VLE data. In each system a multiplicative factor would bring the results to within experimental error of the VLE curves. The factor was not the same for all systems, nor was it the same for all systems using acetone as a solvent. Experiments were made at different temperatures for all five systems. Generally the temperature effects on activity coefficient derivatives could not be distinguished for experimental uncertainties. The results obtained are surprising and disappointing. The error analysis made for the method indicated more reliable results and a systematic deviation was unexpected. An analysis was made to determine the source of such systematic deviations. This included the effect of pressure and impurities. Pressure Effects on the Results Pressure in the cell during operation is a function of radius and speed. Thus, during a run at constant speed there is a considerable pressure gradient in the sample, and runs made at different speeds can have quite different average pressures. However, it was found that the experimental runs made at different speeds with the same sample were generally consistent (see Tables 41, 43, 45, 47 and 49). This indicates that pressure, which increases greatly with speed, had little effect on the results. However, to insure that pressure is not the cause of the experimental chemical potential derivatives being low, a more complete analysis was made. Pressure is related to speed in a centrifugal field by the following expression dP = p wo2 r dr (35 Integrating equation (35) and using the values from Run 382 (see Appendix F), which are typical of the experiments run, the pressure varies from 1 atmosphere at the top to 42 atmospheres at the bottom. It is conceivable that such a pressure could affect the activity coefficient derivative calculation of equation 41 through the sedimentation parameter and the refractive index profile. To illustrate the affect of pressure on the sedimentation parameter, the correlation of compressibility data of Winnick and Powers (1966) yields densities and partial molar volumes at the bottom of the cell under 42 atmospheres of pressure. The resulting sedimentation parameter is only 0.15% less than that at the surface, clearly a negligible difference. Cullinan and coworkers also concluded this for their studies (Sethy and Cullinan, 1972; Rau, 1975). The pressure can also have an affect on the refractive index of fluids, causing the refractive index to vary differently in a spinning cell from a static composition variation. To determine the extent of the effect of pressure on the refractive index of fluids, measurements of pure fluids were made in the ultracentrifuge following the ideas of Richard (1980; Richard et al., 1979). The method entails loading a pure liquid sample in one side of the ultracentrifuge interference cell and air on the other side as a reference. When this cell is rotated in the ultracentrifuge the interference optics show a fringe pattern from the changing refractive index with increasing pressure through the cell. The pressure can be calculated at any depth in the cell using the integrated form of equation 35. The refractive index is known at the surface (1 atmosphere) and the refractive index increment to each fringe is known from equation 38. These yield the pure component refractive index as a function of pressure. As an example, the results for acetone are given in Figure 412. While the data have some scatter caused by variations in microcomparator measurements, it is well described by a quadratic equation in pressure. The other pure components studied were adequately fit with a linear equation. The pressure coefficient of the refractive index, n, dn/dP, for carbon tetrachloride was found by Richard to be 5.5x105 atm1; here a value of 5.9x105 atm1 was obtained. The agreement is probably within experimental error, indicating proper technique for this measurement. Measurements were made on three of the solvents used here to determine the potential extent of pressure on refractive index profiles. Table 411 summarizes the measured data. If a solvent and solute have the same linear change in refractive index with pressure (i.e., dn/dP constant), then the effect would not be observed since the change in the solution side of the double sector cell would be cancelled out by a change in the reference side of the cell. However, the different chemicals used do not have the same pressure variations, and acetone (the solvent in three systems) has a quadratic variation with pressure. The quadratic variation with pressure is particularly important because the reference (pure solvent) side of the cell is filled to a higher level than the sample (solution) side to insure that the interference pattern of the sample is fully covered by solvent. However, this means that the pressure throughout 0.0015 0.0014 0.0013 0.0012 0.0011 0.001 0.0009 n 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 Figure 412 6 10 14 18 22 26 31 Pressure, atm Pressure Effect on Refractive Index (n) as Measured in the Ultracentrifuge for Acetone at 25. the reference side is higher than the sample. Consequently, there can be no cancellation of effects, even when the dn/dP for the solute and solvent are the same. Table 411 Experimental dn/dP Values Compound dn/dp Pressure Range x 105 atm atm1 Acetone 5.0t 5 30 Chloroform 6.7 5 13 Chloroform 6.0 4 16 Carbon Disulfide 7.2 10 40 Carbon Tetrachloride 5.9 4 14 t Value given is at 20 atm, for acetone dn/dP is a function of pressure, dn/dP[atm"1] = 5.2x105 P[atm]x10"7. To quantitatively determine the magnitude of this effect, certain calculations were performed for carbon disulfide in acetone, and chloroform in acetone. Using the measured value of dn/dP, the change in refractive index due to pressure can be determined, Anp )P AP (43 This can be calculated as an average over the entire cell or for each fringe, as is done in the computer program FRNGCNVP (see Appendix C). This calculation must be performed for both the pure solvent and the solution. In the solution a mole fraction average of the solvent and solute values of dn/dP is used. Using the relation of fringes to refractive index difference (equation 38) the change in the number of fringes due to pressure can be determined. Ajpi = An. d / X (44 where i can be solvent or solute. The net pressure effect on the fringes is found by subtracting the sample effect from the solvent effect. Ajp = Ajp(solvent) Aji(sample) The result of this effect is summarized in Table 412 for the carbon disulfide in acetone and chloroform in acetone systems. The measured number of fringes for each run and the net effect of pressure on the number of fringes are given. In addition, the number of fringes necessary to match the ultracentrifuge activity coefficient derivatives with those derived from VLE data were computed and are listed. It is apparent that the effect of pressure on the refractive index is much smaller than the observed discrepancy. Table 412 Fringe Adjustments Due to Pressure Effects and Total Number of Fringes Required to Match VLE Data In Two Typical Cases x, Speed Fringes Fringes Percent Fringes Percent RPM Actually with P Change to Match Change Measured Correction VLE Carbon Disulfide(1)/Acetone (2) 0.03559 21739 4.85 4.68 3.5 3.83 21 0.03559 29502 8.94 8.2 8.3 7.14 20 0.03559 37000 14.02 12.12 13.6 11.27 20 Chloroform(1)/Acetone(2) 0.03713 25969 9.78 9.32 4.7 6.43 34 0.03713 29501 12.56 11.89 5.3 8.26 34 0.03713 37019 19.55 18.21 6.9 13.25 32 The Effect of Impurities on Ultracentrifugation Certified spectrographic grade chemicals from Fisher have been used here. The acetone and acetonitrile were dried with Fisher molecular sieve 3 A. Acetone was also tried without drying and no noticeable differences were found. Several factors can be considered. Impurities in the solute could not cause the discrepancy, their contribution would be too small. If an impurity only altered the solvent or solution refractive index, its effect would be taken care of by the measurement of refractive index in the differential refractometer, which used the same solvent samples as in the ultracentrifuge. Further, the effect would have been the same for all of the acetone (45 systems, but it was not. Finally, any effect of a solvent impurity would be nearly a constant number of fringes at all concentrations of every solute, rather than a relative number as it was. The only impurity known for the present chemicals was one small amount (0.75 %) of ethanol used as a stabilizer for the chloroform solute. However, since ethanol has a refractive index and density almost exactly the same as acetone (solvent), it would appear as more acetone and thus have no effect. Summary Determination of activity coefficient derivatives using laser interference optics in an analytical ultracentrifuge has been carried out on five systems. In all cases the experimental results were consistently lower than those calculated from vaporliquid equilibrium data. In each system a single factor applied to the chemical potential derivative over the entire range of compositions matched the ultracentrifuge experimental quantities to the chemical potential derivatives calculated from VLE measurements. Investigation of pressure effects on the physical properties of the systems shows an insufficient corrective effect, while impurities in the chemicals also do not appear to be the cause. Accepting a constant multiplicative factor for the ultracentrifuge data gives results which generally support the low concentration extrapolations of two equations (Wilson and NRTL) applied to VLE measurements. It was not possible to differentiate a best model for the VLE systems from the data measured here. CHAPTER 5 DIRECT CORRELATION FUNCTION INTEGRAL DATABASE AND MODELING Introduction In an effort to develop new thermodynamic property models, free of common simplifying assumptions such as pairwise additivity of intermolecular forces and rigid molecules, the fluctuation solution approach has been taken by several authors (Kirkwood and Buff, 1951; O'Connell, 1971; 1981). Kirkwood and Buff (1951) formulated the density derivatives of the chemical potential and total system pressure in terms of integrals of the radial distribution function (gi). O'Connell (1971) used these formulations and the Ornstein and Zernike (1914) equations to give the fluctuation thermodynamic properties in terms of the direct correlation function integrals (DCFI or Cij). These direct correlation functions appear relatively insensitive to the details of intermolecular interactions and may be modeled with simple functions. Additionally, because the thermodynamic properties are obtained by integration, the results can be much less sensitive to their parameterization. For example, Mathias and O'Connell (1979, 1981) found that this approach worked very well for gases dissolved in liquids. The direct correlation function is given as cij = hij k Pk f Cik hjk drk (51 where hij  gij 1, gij is the radial distribution function and Pk is the molar concentration of component k. The density derivatives of the pressure and chemical potential or activity coefficient (shown here for component 1 of a binary system) are given as P/RT (1 (p C + P2 C12) / p (52 0p p KT RT where Ciy = p f c drij, is the direct correlation function integral. P/RT = T + 2=( pp+2 C12C 2+p C2) / p2 (53 Bp J K RT (a_../RT 1 Cn11 (54 01 T,p2 P1 P (lny), = (1 Cn)/p (55 If the direct correlation function integrals can be modeled as functions of temperature and composition, integration of equations (52) and (54) can yield the pressure, solution density and the chemical potential, essential in phase equilibrium calculations. In particular, such a model may allow for accurate modeling of the activity coefficient of highly nonideal systems. The DCFIs modeled by Mathias and O'Connell (1979, 1981) work well for liquids containing supercritical components (dissolved gases) but apparently not as well for condensed phase systems (Campanella, 1984). Campanella made an attempt to model the DCFI for vaporliquid equilibria (VLE) and liquidliquid equilibria (LLE) systems in a fashion similar to that of Mathias (1978),. using a hard sphere term and a perturbation term. He found in VLE systems that the hard sphere term did not show adequate compositional variation. Further, the data used to calculate DCFIs at low concentrations were not accurate enough to properly determine the compositional dependence in that region. His work was hindered by having to calculate activity coefficient derivatives indirectly by differentiating either tabulated phase equilibria data or data from a model, such as that of Wilson (1964). This has lead to uncertainty in the actual compositional behavior of DCFIs. The objective of the present work has been to thoroughly explore the compositional behavior of the DCFI. The binary DCFIs can be written in terms of mole fraction and the measurable binary quantities of; isothermal compressibility, KT, partial molar volumes, Vi and V2, volume, V, and activity coefficient derivatives, C dln__ (alny 'I X ,and x, (1 Cu) + X2 Lny (56 VKT RT ax, )TP (1 C12) V V2 I Lxn p (57 VKT RT 8x )TP (1 C2 x + x1 (58 VKT RT I ax2 JTP Binary data on nonideal systems have been collected, aided by the recent compilation of isothermal compressibilities by Huang (1986), and a database of DCFIs for a wide variety of systems has been created. To compensate for the lack of directly measured activity coefficient derivative data, derivatives of the NRTL and Wilson models with parameters fit to VLE data will be used to cover the range of values for a given system. With these activity coefficient derivatives, a database of DCFIs can be developed. The sixteen nonideal systems selected by Campanella (1984) plus twelve other nonideal systems were chosen for study. For each of these systems, excess volume data, pure component compressibility data and vaporliquid equilibrium data was obtained. For nineteen of the systems, mixture compressibility data was also available. For modeling purposes, the DCFI can be conveniently divided into ideal and excess terms. The ideal DCFI is calculated by assuming an ideal mixing rule for the volume and isothermal compressibility. The activity coefficient derivative term is zero in the ideal case. By subtracting the ideal DCFI from the real DCFI, determined from experimental quantities, an excess quantity, resembling the activity coefficient derivative, remains. The ideal and excess DCFIs in a binary system are given as (1 C1)ID _____ ( x1 V' K + x2 V' K42) RT (59 (1 C 2)ID = Vl V2 ( x V KoI + x2 Vn zK) RT (510 (1 C22)ID = (V)2 ( x1 V K + x2 V' K) RT (511 (1 Cn)E = (1 ) ( C)ID = fn (vE, K) + x, l (512 ( ax1 )TP (1 C12)E = (1 C) (1 C12)ID = f21 (VE, KT) x1 x 2l(513 (1 C22)E (1 C22) (1 C22)ID = f22 (v KE) + x, 1 2 (514 V xa2 )TP where the superscript 0 denotes pure component properties and fni, f12 and f22 are functions of excess volume and excess compressibility. If this excess DCFI could be easily modeled it would lead to a better understanding of the compositional variation of the DCFI and aid future theoretical modeling efforts. Because the excess DCFI so closely resemble the activity coefficient derivatives, a modeling approach that uses a form similar to the derivatives of common activity coefficient models such as the Wilson (1964) or NRTL (Renon and Prausnitz, 1968) might prove beneficial. Indeed, Campanella (1984) pointed out that most of the activity coefficient models have a common form when written in terms of the composition derivative. For component 1 of a binary, three common models, Van Laar (Van Ness and Abbott, 1982), Wilson and NRTL, can be written as Van Laar lny (X2 + 2 (515 ax, )JTP (xl0 + X2F)2 (X10 + X2)3 where ( = 2 A12 Ai1 0 = A12 F = A21 T = 2 A21 A12 (A21 A2) A12 and A21 are Van Laar model binary parameters. Wilson Oln__ 2 + 2 1X2 ax,1 P i=1 x1Oi + X2ri =1 (x1i, + X2Fi)2 where 1, = 1 0, = 1 17 = A12 (2 = A21 02= A21 F2 = I (516 P = A12 (A12 1) 2 = A21 (A21 1) A12 and A21 are Wilson model binary parameters. NRTL Lny 2 ix2 2 pix2 XTP i= X21 + + x2 ax, 1) i=1 (xI~i + x2,i)2 i=1 (xEie + X2 1)3 where (1 = 2 T21 G21 (516 0,= 1 02= G12 r2 = 1 =2 = 2 T12 G12 (G12 1) T12, T21, G12 and G21 are NRTL model binary parameters. The common term here, S+ x2 may work well as a model for the excess (xIE) + X20m DCFI. The activity coefficient derivative of component 1 in a binary system can be written directly in terms of the three binary DCFIs and the reduced bulk modulus as a =x2 [(1 CuX1 C22) (1 C2)2] p KT RT (518 Examination of this grouping of DCFIs calculated from the database may suggest a new empirical relationship for the activity coefficient derivative. DCFI Database The database of DCFIs includes only significantly nonideal systems. These types of systems will have the largest and most complicated excess terms. Campanella (1984) studied many systems, both ideal and nonideal. He found the ideal systems to be "2 = 2 T12 G12 11 = G21 1, = 2 T21 G21 (G,2 1) 