THE DIFFUSION THERMOEFFECT IN BINARY LIQUID MIXTURES By Richard L. Rowley A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1978 Avfif§j\ ABSTRACT THE DIFFUSION THERMOEFFECT IN BINARY LIQUID MIXTURES By Richard L. Rowley The heat flow induced by a composition gradient is known as the diffusion thermoeffect or Dufour effect. It is characterized by the heat of transport, formally defined as the ratio of the heat flux to the mass flux under isothermal conditions. Although theoretical treat- ments allow calculation of heats of transport from thermal diffusion experiments on the basis of Onsager heat-mass and mass-heat reciprocity, no direct, quantitative, experi- mental determinations of the heat of transport in liquid mixtures have previously been reported. The direct experi- mental determination of the heat of transport for carbon tetrachloride-cyclohexane mixtures reported here has provided the first experimental verification of the Onsager heat-mass reciprocal relation. Also reported here are the first measurements of the behavior of the heat of transport in a mixture (isobutyric acid-water) near its consolute temperature. Richard L. Rowley The equations of nonequilibrium thermodynamics and the hydrodynamic conservation equations have been used to formulate coupled, nonlinear, nonhomogeneous partial differential equations which when solved subject to ap- propriate initial and boundary conditions yield time and space distributions for the barycentric velocity, composi- tion, and temperature. These equations are solved with a Crank-Nicholson implicit numerical scheme which allows inclusion of the composition and temperature dependence of the thermodynamic and transport parameters. The heat of transport for carbon tetrachloride-cyclo- hexane liquid mixtures has been determined directly by diffusion thermoeffect experiments. The technique employs a withdrawable "liquid gate" to create a nonturbulent, sharp, diffusional interface. The heat of transport is obtained from nonlinear least squares fitting of numerically predicted values to actual temperature differences meas- ured about the interface. The agreement of these direct heat of transport measurements with values calculated from thermal diffusion experiments constitutes the first experi- mental verification of Onsager heat-mass and mass-heat reciprocity in binary liquid mixtures. The temperature dependence of the heat of transport has also been measured, for isobutyric acid-water mixtures near the critical solution temperature. A microwave oven was used to jump the temperature of the initially two—phase Richard L. Rowley system from Just below to Just above the consolute tempera- ture. Above the consolute temperature, a uniform, one- phase system is the equilibrium state. Consequently, as soon as the temperature of the two-phase system is raised above the consolute temperature, diffusion com— mences and induces a temperature gradient. Temperature differences about the interface obtained as a function of nearness to the consolute temperature yield the critical exponent for the heat of transport. The heat of transport vanishes with a +2/3 critical exponent as the critical solution temperature is approached. There thus exists a previously unsuspected critical anomaly in the heat of transport, which can be traced to a diverging Onsager co- efficient. Because current kinetic theories are incon- sistent with the critical behavior of the heat of transport, a new molecular interpretation of the heat of transport is proposed to explain the nature of coupling between molecu- lar heat and mass transport as well as its critical be- havior. To Vickie ii ACKNOWLEDGMENTS I wish to thank the Department of Chemistry for the financial support I have received from teaching assistant- ships. I also wish to thank the Department of Chemistry and the General Electric Foundation for awarding me a 1978 summer fellowship. I am indebted to Professor Larry Dawson of the Food Science Department for use of his microwave oven which enabled completion of some of the experimental work. I also appreciate the work of Mr. Andrew Seer, master glass- blower, who was able to construct the assortment of glass apparatus used in this project. The time and photographic talents of Dr. Daniel and Kathleen Bradley are appreciated. The photographs contained in this thesis are the results of their help. Thanks are due to Professor Alexander Popov for use of his Karl Fischer Titrator. My gratitude is extended to Professor Frederick R. Horne whose teachings of nonequilibrium thermodynamics, thermodynamics, and analytical thinking have been the foun- dation of this work. I sincerely appreciate his encourage- ment and his confidence in my capabilities. Above all, I thank my wife Vickie for her support, encouragement, and optimism. Her unselfish support of this work has been a large driving force in its completion. iii TABLE OF CONTENTS Chapter LIST OF TABLES. LIST OF FIGURES 1. INTRODUCTION. . A. Phenomenology of the Diffusion Thermoeffect. B. Objectives. . . C. Plan of the Dissertation. MATHEMATICAL FORMULATION OF THE DIFFUSION THERMOEFFECT. A. Introduction. B. Hydrodynamic Equations. C. Nonequilibrium Thermodynamics Equations . . . . . . . . . D. Transport Parameters and Equations . . . . . . . E. Mole Fraction Equations for Carbon Tetrachloride-Cyclohexane Mixtures. . . . . . NUMERICAL SOLUTION OF THE DIFFUSION THERMOEFFECT EQUATIONS. . . A. General Scheme. B. Solutions for the Carbon Tetrachloride- Cyclohexane System. . . . . . . . . . DIFFUSION THERMOEFFECT EXPERIMENTS ON CARBON TETRACHLORIDE-CYCLOHEXANE MIXTURES A. Experimental Design B. Analysis of Technique iv Page . vii 13 15 23 28 28 Al 50 50 63 Chapter Page C. Literature Values for the Physico- chemical Properties of the Carbon Tetrachloride - Cyclohexane System. . . . . . . . . . . . . . . . . . 76 D. Experimental Results for Thermal Conductivity. . . . . . . . . . . 80 E. Experimental Results for Heat of Transport . . . . . . . . . . . . 8A 5. LIQUID-LIQUID CRITICAL PHENOMENA. . . . . . . 97 A. Classical Thermodynamics of Liquid-Liquid Critical Phenomena. . . . . 97 B. Critical Exponents. . . . . . . . . . . . 102 1. Definitions . . . . . . . . . . . . . 102 2. Universality. . . . . . . . . . . . . 111 3. Scaling . . . . . . . . . . . . . . . 113 C. Transport Properties in the Critical Region . . . . . . . . . . . . . 113 D. Predicted Liquid-Liquid Critical Exponents for Transport Phenomena . . . . 116 E. Experimental Liquid-Liquid Critical Exponents for Transport Parameters. . . . 117 1. Techniques. . . . . . . . . . . . . . 117 2. Thermal Conductivity. . . . . . . . . 120 3. Mutual Diffusivity. . . . . . . . . . 122 A. Thermal Diffusion . . . . . . . . . . 127 5. Heat of Transport . . . . . . . . . . 131 6. THE HEAT OF TRANSPORT IN THE CRITICAL SOLUTION REGION OF ISOBUTYRIC ACID- WATER MIXTURES. . . . . . . . . . . 133 A. Transport Equations . . . . . . . . . . . 133 B. Experimental. . . . . . . . . . . . . . . 138 Chapter 1. Cell Considerations . . . . Critical Temperature Mea- surements . . . . . 3. Experimental Procedure. A. Data Analysis . . . . . . . . . C. The Temperature Jump Technique. D. Literature Parameters for IBW . E. Experimental Results. 7. CONCLUSIONS . . . . . . . . A. Interpretations of the Heat of Transport. . . . . . . . B. A New Interpretation of the Heat of Transport . . . . . . . . C. Summary and Future Work Needed. . . . . APPENDIX A - TRANSFORMATION RELATIONS AND IDENTITIES. . . . . . . . . . . . . APPENDIX B - DIFFUSION THERMOEFFECT DATA FOR THE CARBON TETRACHLORIDE-CYCLOHEXANE SYSTEM. 0 O O O O O O O C O O O O 0 APPENDIX C - DIFFUSION THERMOEFFECT DATA FOR THE ISOBUTYRIC ACID-WATER SYSTEM IN THE CRITICAL REGION. . . . . REFERENCES. vi Page 138 1A3 1A8 150 155 167 173 191 191 198 206 210 21A 218 225 Table “.2 “.3 A.A A.5 5.1 LIST OF TABLES Page Literature values for the physico- chemical properties of the carbon tetra- chloride - cyclohexane system at 1 atm. The properties are expressed in the general form L = LO[1+LX(x1-O.5) + LT(T-298.15) + LXT(xl-O.5)(T-298.15) + 1/2 Lxx (x1-0.5)2] to include the temperature and composition dependence . . 77 Thermal conductivity of CC1u-ggC6Hl2 mixtures at 20°C and 1 atm . . . . . . . . 83 Values of the heat of transport and Onsager coefficients in carbon tetra- chloride-cyclohexane mixtures at 25°C. . . 86 Heat-mass transport coefficients for carbon tetrachloride—cyclohexane mix- tures at 25°C and 1 atm. . . . . . . . . . 91 Values for the difference in thermal conductivity between the equilibrium and steady states in thermal diffusion experiments. . . . . . . . . . . . . . . . 95 Critical exponents for some equilibrium thermodynamic properties. References are cited in Scott's [1972] review . . . . 98 vii Table Page 5.2 The behavior of transport coefficients in the critical region . . . . . . . . . . 100 5.3 Theoretical predictions for critical exponents of transport properties. . . . . 118 5.“ Light scattering results for the mutual diffusion critical exponent . . . . 126 5.5 Literature transport parameters and their critical exponents . . . . . . . . . 132 6.1 Expressions used for IBW transport and thermodynamic parameters in analysis of diffusion thermoeffect experiments in the critical region . . . . 168 6.2 Experimental conditions for the diffusion thermoeffect experiments performed on IBW in the critical region . . . . . . . . . . . . . . . . . . 17“ 6.3 Results of diffusion thermoeffect experiments in the IBW critical region . . . . . . . . . . . . . . . . . . 175 6.“ Transport parameters and their critical exponents . . . . . . . . . . . . 187 B.l Initial Conditions . . . . . . . . . . . . 215 B.2 Temperature differences. . . . . . . . . . 216 0.1 Initial conditions . . . . . . . . . . . . 220 viii Table Page .2 Run I. O O O O O O O O O O I O O O I O O O 221 Run II . . . . . . . . . . . . . . . . . . 222 000 er Run III. . . . . . . . . . . . . . . . . . 222 Run IV . . . . . . . . . . . . . . . . . . 223 Run V. . . . . . . . . . . . . . . . . . . 223 Run VI . . . . . . . . . . . . . . . . . . 22“ O CDNOU'I Run VII. . . . . . . . . . . . . . . . . . 22“ ix Figure 1.1 3.1 3.2 3-3 3.“ 3-5 LIST OF FIGURES Page Schematic of the diffusion thermo- effect . . . . . . . . . . . . . . . . . 2 Crank-Nicholson grid scheme for finite difference equations. Properties are evaluated at the i,n positions, 0. Derivatives are evaluated at the i, n + 1/2 positions, I . . . . . . . . . . . . . 29 Graphical comparison of the first and second order correct analogs for the first derivative as illustrated by Rosengren [1969]. a. Actual derivative. b. Second order correct analog. 0. First order forward difference. d. First order backward difference. . . . . . . . . . . . 32 Flow diagram for simultaneous numeri- cal solution of the composition and temperature equations. . . . . . . . . . . 38 Barycentric velocity surface for the carbon tetrachloride-cyclohexane system . . . . . . . . . . . . . . . . . . “2 Composition surface for the carbon tetrachloride-cyclohexane system . . . . . “3 Figure Page 3.6 Temperature surface for the carbon tetrachloride-cyclohexane system. AT represents the local temperature minus the initial temperature. . . . . . . “5 3.7 Effect of excess enthalpy on the time dependence of temperature distribution. ~ Plots are for HB = xlx2(A+Bxl) with , A=B=O. ---, A=67O J/mol; B=O. .., A=670 J/mol; B=67 J/mol. Upper curves are for (z/a)=0.6 and lower curves are for (z/a)=0.“ . . . . . . . . . “7 3.8 Effect of excess enthalpy on the time behavior of the temperature difference for the thermocouple pair (z/a)=0.“ and (z/a)=0.6. The curve is identical for HE=xlx2(A+Bxl) and (l)A=B=O, (2) A=67O J/mol; B=O, (3) A=3350 J/mol; B=O, (“) A=67O J/mol; B=67 J/mol, and (5) A=67O J/mol; B=167 J/mol . . . . . . . . . . . . “8 “.1 Withdrawable "liquid gate" diffusion thermoeffect cell. . . . . . . . . . . . . 55 “.2 Schematic diagram of withdrawable "liquid gate" diffusion thermoeffect xi Figure Page cell. A. Upper phase storage reservoir. B. Cell jacket for thermostatting or adiabatically insulating. C. Diffusion thermo- effect chamber. D. Thermocouple banks. E. Equatorial water entrap- ment rim. F. "Liquid gate" with- drawal spout. G. Ground glass fit- tings for thermocouple leads and cell drainage. H. T-connector to vacuum line and thermostat. I. Filling tubes. J. Glass-syringe . . . . . . . . . 57 “.3 Comparison of experimental AT data to predicted values for adiabatic walls (———) and diathermic walls (---) . . . . . 69 “.“ Nonlinear, weighted, least squares fit of (Q:/M) for run I. The solid line represents calculated Q: values based ~* ~ upon the fit value of (Ql/M) while the solid circles are experimental data. . . . 73 ~* “-5 Test of the stability of Q1 obtained as a function of the time range for which AT data were input. The dashed line indicates 1% deviation from the long- time value . . . . . . . . . . . . . . . . 7“ xii Figure Page “.6 Thermal diffusion factor at 25°C as a function of composition. Method of measurement is in parentheses. A, this work (Dufour effect); -+, Anderson and Horne [1971] (pure thermal diffu— sion); -I--, Stanford and Beyerlein [1973] (thermogravitation); o, Turner, g§_a1. [1967] (flow cell); I, Kor- chinsky and Emery [1967] (thermo- gravitation). (Rowley and Horne [1978]). . . . . . . . . . . . . . . . . . 88 “.7 The Dufour coefficient 8T as cal- ~* culated from the experimental Q1 values. I, experimental data;-———, least squares fit. . . . . . . . . . . . . 92 ~* “.8 Heat of transport Q1 as a function of mole fraction x1. I, experimental data; , least squares fit . . . . . . . 93 5.1 Free energy of mixing vs.mole frac- tion of component 1 as illustrated by Moore [1972]. A. Complete mis— cibility. B. Two phase system of compositions x' and x". C. Phase 1 l stability limit. . . . . . . . . . . . . . 98 xiii Figure Page 5.2 Behavior of the chemical potential of component 1 vs.mole fraction of com- ponent 2 for a critical system as depicted by Prigogine and Defay [195“]. The dashed line indicates metastable regions. . . . . . . . . . . . . . . . . .100 5.3 Coexistence curve for the n-hexane - nitrobenzene system as depicted by Prigogine and Defay [195“] . . . . . . . 101 5.“ Photographic sequence of liquid—liquid critical phenomena as the temperature is lowered to the consolute temperature of the isobutyric acid-water system. (a) T >>Tc. (b) T-TC~3°C, mixture becomes a hazy blue hue as opalescence begins. (c) T-Tcml°C, blue hue becomes a white fog as T is lowered and opales- cence increases. (d) T-Tc50.01°C, critical opalescence intensifies as dense white cloud. (e) T-Tc2-0.01, onset of phase separation as marked by dense turbidity. (f) T~< -VP (2.3) where smev is the symmetric part of the tensor 2y, n is 11 shear viscosity, o is bulk viscosity, Ki are external forces, and P is pressure. The equation of energy transport with temperature and pressure as independent variables is 96p 85" TB gg" °1'Y'3'11'Y( i'fié) (2.“) Where 5? is specific heat, T is temperature, 8 is thermal expansivity, ¢1 is the entropy source term for bulk flow, 9 is the heat flux, and Hi is partial specific enthalpy of component i (prime indicates inclusion of any neces- sary work terms due to external forces). The entropy source term ¢l is ¢1 E (g+Pl):Yy (2.5) where g is the stress tensor. Equations (2.1) - (2.“) are formulated in general terms for pedagogical reasons. Considerable simplifica- tion occurs in the preceding equations for the experimental arrangements necessary to measure the diffusion thermo- effect. If the width/height ratio of the fluid slab in Figure 1.1 is large, wall effects can be excluded, and the above equations need only be written for the z-direction taken perpendicular to the interfacial plane. No external fields are present - gravity effects are extremely small since the cell height is only one or two centimeters. 12 Pressure terms are very small for this experimental ar- rangement (Anderson and Horne [1970]). Because liquid densities are usually quite similar, the barycentric vel- ocity will be small enough that all terms of order (av/32)2 can be safely neglected. These simplifications eliminate the Navier-Stokes equation - there is no convection in the cell unless temperature gradients cause density inver- sions. Likewise, the entropy source for bulk flow ¢l is negligible due to its dependence on the square of the velocity gradient. With the above restrictions and the relation between substantial and local time derivatives, d/dt = (a/at) + Y'Y’ Equations (2.1), (2.2), and (2.“) become (Bo/3t) + (apV/Bz) = O, (2.6) p(3wl/3t) + (le/az) + pv(3wl/az) = 0, (2.7) and pCp(3T/3t) = (sq/32) - Jl[a(fil-fi2)/az] — pva(3T/Bz) (2.8) respectively. Before these three equations can be solved for v, w and T; expressions for the heat and mass fluxes 1’ must be introduced. These are deduced from the theories of nonequilibrium thermodynamics. 13 C. Nonequilibrium Thermodynamics Equations The framework of nonequilibrium thermodynamics rests on the foundation of "local states". This simply requires that all thermodynamic functions of state exist for each microscopic volume element of the system. Furthermore, these thermodynamic quantities, in the case of nonequilib- rium systems, are the same functions of the local state variables as the corresponding equilibrium thermodynamic quantities (Fitts [1962]). This permits the concepts of temperature and entropy in nonequilibrium systems even though their definitions evolved from thermostatic states. Likewise, the Gibbsian equations are valid and, therefore, _ g.__ _.____._ _. ‘ T dt ' T dt T X "1 YET" (2'9) where S is specific entropy, U is specific internal energy, and U: is the specific chemical potential of component i. Entirely from balance techniques for the entropy of a local volume element (similar to the method by which the hydrodynamic equations are often derived), an entropy equa- tion can be written in the form pdEth = ¢/T - V'j ~ ,8 (2.10) where is is the entropy flux due to mass and heat flows and the semidefinite positive quantity ¢/T is the rate l“ per unit volume of the internal entropy production. Sub- stitution of the hydrodynamic equations for mass and energy balance into Equation (2.9) and subsequent comparison to Equation (2.10) allows identification of ¢ after con- siderable rearrangement. For the system at hand (a binary, field—free, isotropic, nonreacting, nonelectrolyte liquid mixture), 6- ll p1 + $2 (2.11) where and 1’2 = - 111551-52) with yTfiisyfii+§in. It is important to note that e is of the form ¢ = :ZiJiXi where the J1 and X1 represent fluxes and driving forces respectively. The fluxes and forces in $1 are ten— sors of rank 2 while those of $2 are vectors. Centuries of experimental work have shown linear coupling of fluxes and forces J1 = gflijxj° For the isotropic liquids con- sidered here, Curie's theorem, based on spacial symmetry 15 arguments, allows coupling only between those fluxes and forces which do not differ in tensorial character by an odd integer. In Equation (2.11), the fluxes and forces of $1 and ¢2 cannot interact. Therefore, heat and mass fluxes are -g = nooyznT + 901YT(“1‘“2) (2.12) '11 = Q01‘3”? + 911YT(“1' "2) (2'13) where the 013 are called Onsager coefficients. The utility and indeed the present reason for the nonequilibrium thermo- dynamic approach is the identification of correct fluxes and driving forces and their proper coupling as required by the entropy production equation. For many years, the driving force for diffusion was thought to be a composi- tion gradient (Fick's original 1aws),but the equations of nonequilibrium thermodynamics readily identify it as a gradient of chemical potential. D. Transpprt Parameters and Equations The Onsager coefficients which appear in Equations (2.12) and (2.13) are related to experimentally observed transport coefficients. In fact, their identification is made by comparison with their phenomenological counter— parts (Fick's law, Fourier's heat conduction law, etc.). 16 The 013 for this binary nonelectrolyte mixture are _* _ 9 = KT (A) 901=DDQ1W2/ull (B) 3 u 10 DDT (C) Qll=ODW2/Eil (D) (2.1“) where K is thermal conductivity, D is mutual diffusivity, 5: is specific heat of transport (the commonly used measure of heat transported by diffusion in a diffusion thermo- effect arrangement), 5115(3E1/3w1)T,P’ and DT is the ther- mal diffusion coefficient. Often experimental thermal diffusion results are expressed in terms of the thermal diffusion factor a1 or the thermal diffusion ratio K T rather than the thermal diffusion coefficient DT' These three coefficients are related by KT 5 DT/D (2.15) and -al 2 KT/wlw2 . (2.16) Likewise, it is sometimes desirable to retain the form of Equation (2.12) in which only a single transport co- efficient appears but written in terms of mole fraction rather than chemical potential. If Equation (2.12) is written 17 —g = KYT + B Yxl , (2.17) a new coefficient 8T is defined known as the Dufour co— efficient. From Equation (2.l“B) and the relationship between mass fraction and mole fraction, 8T is related _* to Q1 by _* ~ 3T = pDQlMlM2/M2 (2.18) where M is the mean molecular weight defined by M = XlMl + x2M2. From Equations (2.12), (2.13), (2.1“B), and (2.1“D), _* the defining equation for the heat of transport Ql is seen to be (9/11)AT = 0 ° (2°19) The heat of transport can therefore be thought of as a heat flux produced by an isothermal mass flux. If the isothermal conditions of Equation (2.19) are relaxed, then q = §1Q1 - KYT (2.20) (de Groot and Mazur [1969]). This relationship shows the two effects which determine the magnitude of the temperature 18 difference measured between two points in a diffusion thermoeffect cell. The heat transported via the mass flux builds up a temperature gradient while thermal con- duction tends to diminish it. The relative magnitudes of Q: and K, for a given diffusional flux, determine the magnitude of the ensuing temperature gradient. Note also that the transient nature of the diffusion thermoeffect is due to a nonconstant mass flux. When q = 0 the heat flow transported by diffusion identically balances the con- duction heat flow and VT = le:/K . (2.21) If Q1 remained constant throughout the experiment, a steady state YT would be measured. However, as diffusion de— creases the composition gradient, 11 decreases. This lowers ET and a time dependent behavior is observed. Not all of the set{K;D,Q:,DT} are independent. On- sager [1931], applying microscopic reversibility concepts, showed that the matrix of coefficients involved in the flux-force relations must be symmetric. Though experi- mental evidence accrues constantly in support of the On- sager reciprocal relations (Miller [1960] and [1975]), the results reported herein constitute the first experi- mental evidence of the heat-mass ORR. For the system des— cribed by Equations (2.12) and (2.13), ORR implies 19 901 = 910 . (2.22) Substitution of Equations (2.1“) for the Onsager co- efficients in Equations (2.12) and (2.13) yields for the applicable one dimensional flux equations _* —q = K(3T/3Z) + pDQl(3w1/Bz) (2.23) - -1 —jl - pD(3wl/3z) - delleZT (ET/32) (2.2“) where the Gibbs-Duhem equation has been invoked to help transform chemical potential gradients to single mass fraction gradients. Ingle and Horne [1973] argue on the basis of numerical values for common liquid systems that [alwleT-ll is of the order 10'3 deg"1 and that therefore thermal diffusion is at most 0.01% of diffusion, assuming that composition gradients are an order of magnitude larger than tempera- ture gradients for diffusion thermoeffect experiments. Neglect of the thermal diffusion term in Equation (2.2“) and substitution of Equations (2.23) and (2.2“) into Equa- tions (2.7) and (2.8), yields partial differential equa— tions which completely (with appropriate initial and boun- dary conditions) define v, wl, and T as functions of t and z: 2O (ap/at) + (apv/az) = o (2.25) p(3w1/3t) = {3[pD(8wl/az)]/Bz} - pv(3wl/Bz) (2.26) pGfi(3T/3t) pD[3(Hl-H2)/Bz](3Wl/3Z) + {8[pDQ:(3wl/az)]/az} 95§v(3T/8z) + {8[K(8T/3Z)]/BZ}. (2.27) These equations are identical to the starting equations used by Ingle and Horne [1973] in their analytical double perturbation solution of the diffusion thermoeffect problem. Their perturbation scheme, while allowing solution even with composition and temperature dependent parameters, results in solutions which are extremely bulky and complex. The number of terms required and the rapidly increasing complexity of successively higher order terms limit the practical application of this technique to those liquid systems whose properties are only slightly temperature and composition dependent. This unfortunately is not the case for the systems of interest here. To avoid these dif- ficulties the numerical scheme discussed in Chapter 3 is used. The initial condition for the composition equation is a step function 21 I 2 Wl(z/a > 0059 O) ‘ u (2.28) wl(z/a < 0.5, 0) where w: and W% are the mass fractions of component 1 at which the upper and lower phases respectively are prepared. Exactly at the interface wl is an arithmetic average of the two phases but it need not be defined unless a grid point of the numerical scheme is located at that position. The measured temperature distribution just prior to interface formation becomes the initial condition for the tempera- ture equation. It should roughly correspond to isothermal conditions so that thermal diffusion can be neglected. Thus, the initial condition is T(z/a, 0) = T(z/a) (2.29) where T(z/a) is a constant for isothermal conditions. Boundary conditions can be imposed from the physical aspects of the experimental design. Because the walls are impermeable to matter, v(0,t) = 0 = v(l,t) (2.30) for the barycentric velocity and jl = 0 at (z/a) = 0 & 1 for the mass flux. Fick's law restates the vanishing 22 mass flux boundary condition as (awl/Bz)0’t = 0 = (awl/az)l,t (2.31) since D never vanishes. The boundary conditions for the temperature equation depend on the experimental arrange- ment desired. If the walls are adiabatically insulated the heat flux vanishes at the walls and, from Fourier's heat conduction law, (aT/az)O t = 0 = (ST/az)l,t . (2.32) Although Equation (2.32) is the boundary condition used in these experiments (the cell was adiabatically insulated to maximize induced temperature inequalities), it is not the only boundary condition which can be used. Before limiting discussion to the carbon tetrachloride- cyclohexane system (which provides a convenient system for study of the diffusion thermoeffect away from critical regions), it is appropriate to list the assumptions involved in the derivation of Equations (2.25) — (2.27) for they will also serve as the starting point in the analysis of systems exhibiting critical mixing. The assumptions employed are: (l) The linear hydrodynamic equations for conservation of mass and energy are valid. (2) (3) (“) (5) (6) (7) (8) 23 The binary system is isotropic, nonreacting, and field free. Local states are assumed, 1:3,, the equations of thermostatics apply for local regions. Fluxes are linear combinations of these forces which appear in the entropy production equation and which have the same tensorial rank. Pressure terms are negligible. The bulk flow entropy source term is small. The thermal diffusion portion of the mass flux is small relative to the diffusional contribution. The phenomenon takes place entirely in one dimen- sion, so that wall effects are unimportant. E. Mole Fraction Equations for Carbon Tetrachloride- Cyclohexane Mixtures The properties of the carbon tetrachloride-cyclohexane system are much more nearly constant in molal rather than in specific quantities. A transformation is therefore use- ful, with respect to numerical step sizes and to possible simplifications, from mass fractions and specific proper- ties to mole fractions and molal properties. Transformation of Equations (2.25) - (2.27) with the aid of the trans- formation identities in Appendix A results in (av/az) = (M2v —Mlv2){a[(D/vn)(axl/az)l/az}, (2.33) 1 2“ (axl/at) = VM{3[(D/VM)(axl/az)]/Bz} - v(axl/Bz) ,(2.3u) and (aT/at) (V/Cp){3[K(8T/Bz)]/Bz} + (M2V/Cp){3[(DQ:/VM)(Bxl/az)1/az} ~ 2~E 2 2 + (D/Cp)(3 H /3X1)T’P(3Xl/Bz) -v(3T/Bz) (2.35) ~ where x1 is mole fraction of component 1, V is molar vol- ~ ~ ume, Cp is molar constant pressure heat capacity, Q1 is the molar heat of transport, and HE is the molar excess enthalpy. Equations (2.33) - (2.35) are the mole fraction- molar property versions of the mass fraction-specific property Equations (2.25) - (2.27) which Ingle and Horne [1973] used. The excess molar enthalpy HE is related to the difference in partial specific enthalpies by [am -H)/321= (Fa/M M )(BZHE/axz) (8x mm (2 36) l 2 1 2 l T,P 1 ' as derived in Appendix A. For carbon tetrachloride-cyclohexane mixtures, the excess volume of mixing is very small (Wilhelm and Sack- mann [197“] indicate it to be everywhere less than 0.2% of the total molar volume) and V = V? O i where V1 is the 25 pure component molar volume. Integration of Equation (2.33) subject to the boundary conditions v(0,t) = o = v(l,t) (2.37) and (3X1/32)0,t = 0 = (3X1/32)1,t , (2.38) yields for the barycentric velocity v = (MZVE-Mlvg)(D/VM)(3xl/Bz) . (2-39) Substitution of this expression into Equation (2.3“) pro- duces upon rearrangement (ax /at) = D(32x /Bz2) + [(aD/ax ) l l 1 T,P ~ ~ 2 - 2(D/V)(EV/3x1)T’P](3xl/Bz) . (2.“0) For the experiments reported in this dissertation, the composition and temperature dependencies of D and V in Equation (2.“0) do not measurably contribute to the observed temperature difference produced by the diffusion thermoeffect. Numerical verification of this statement was made by determination of the heat of transport (from 26 experimental temperature differences) both with and without the composition and temperature dependencies of D and V. No detectable effect was found. There were of course small differences in the composition as a function of time and position because of the dependence of the parameters D and V on composition. Nevertheless, small errors in the composition profile due to relatively good assumptions in the diffusion equation had a negligible effect on the solu- tion of the temperature equation. It suffices therefore to use (axl/at) = DO(32Xl/322) (2.“1) instead of Equation (2.“0) to describe composition in time and space where the subscript 0 is used to denote evaluation of the parameter D at x1 = 0.5 and T = 298.15 °K. Explicit formulas for calculating directly the effect of the composi- tion and temperature dependence of D and V on the experi- mentally observed temperature differences may be found in the paper by Ingle and Horne [1973]. As emphasized there, these dependencies do not contribute to the temperature difference measured symmetrically about the interface be- cause they involve terms of only even symmetry about the center. The solution of Equation (2.“1), subject to the experi- mental boundary conditions, is 27 00 x1 = (x1> + 2(Axl/w) 2E0 (-1)“(22+1)'l X{exp[-(22+l)2(t/6)1}Cos[(2£+1)wz/a] (2.u2) where is the initial arithmetic mean x Ax is the 1’ 1 difference in X1 between the initial two phases, and 6 E a2/W2DO. It is important to note that the numerical technique described in the next chapter allows solution of Equation (2.“0) in its entirety when Equation (2.“1) is not satisfactory for the desired system. In actual practice, numerical solutions for both the composition and temperature equations were used in the determination of the heat of transport. This allowed development of a computer program using the more general equation which could then be quickly simplified to Equation (2.“1) for appropriate systems such as carbon tetrachloride-cyclo- hexane. CHAPTER 3 NUMERICAL SOLUTION OF THE DIFFUSION THERMOEFFECT EQUATIONS A. General Scheme Equations (2.“0) and (2.35) or Equations (2.“1) and (2.35) with the initial and boundary conditions of Equa- tions (2.28), (2.29), (2.32), and (2.38) completely des- cribe the diffusion thermoeffect for the conditions of experimental interest. Explicit solution of these equa- tions is not easy because they are not only nonhomogen- eous but are also coupled and nonlinear with nonconstant coefficients. This type of problem is, in general, un- solvable without recourse to numerical techniques. The general presentation discussed here is due to Rosenberg [1969]. To obtain a numerical solution, continuous variables are replaced by their discrete counterparts. Partial derivatives are represented by finite differences so that the partial differential equations become finite dif— ference equations - algebraic rather than differential. To obtain discrete variables, the continuous time-space domain of the problem is subdivided as shown in Figure 3.1. The time domain is divided into rows labeled with the 28 29 O O O O o O O O 0 n+1 o o o o o o o o 0 I Eff] o o o o o o o o 4- n-1 o o o o o o o o o O O O O O O O O O _ 1 1 1 l l I I l 1 o H i M position Figure 3.1. Crank-Nicholson grid scheme for finite dif- ference equations. Properties are evaluated at the i,n positions, 0. Derivatives are evaluated at the i,n+1/2 positions, I. I] 30 index n. The spatial domain is divided into R columns each labeled with an index denoted by i. Any dependent variable U can be specified in time and space with its appropriate indices Ui,n' If U is expanded in a Taylor's series about the i+l,n point Ui (at constant n), ,n U = U 2 2 2 , i+l,n + (BU/32)i,nAz + (8 U/az )i,n(Az) /2. i,n + (33U/BZ3)i,n(AZ)3/3! + ..., (3.1) finite difference representations for spatial derivatives can be obtained in terms of the distance between two consecutive spatial grid points “2. This is done by writ- ing the Taylor's expansion for Ui-l about U ,n i,n’ _ 2 2 2 , - Ui,n - (BU/az)i,nAz + (a U/az )i,n(Az) /2. Ui-1,n - (a3U/az3)i n(Az)3/3! + ..., (3.2) and then by comparing Equations (3.1) and (3.2). For instance, the forward and backward difference equations for the first derivative are obtained by rearranging Equations (3.1) and (3.2) respectively and then truncating them to obtain (BU/az)i,n = (Ui+l’n — Ui n)/Az , (3.3) 3 31 and (BU/az)i,n = (Ui,n - Ui-l,n)/AZ . (3.“) Notice that the first term omitted in the truncation is (32U/322)1’nAz/2!. The truncation error is first order in Az, and the finite difference expressions are therefore first order correct analogs. A better approximation for the first derivative is obtained by subtracting Equation (3.2) from Equation (3.1), (BU/az)i,n = (Ui+l,n — Ui_l,n)/2Az - (83U/8z3)1,n(Az)2/3! - ..., and then truncating terms of order (Az)2 and higher, with the result (BU/82),”r1 = (Ui+l,n - Ui-l,n)/2Az . (3.5) Equation (3.5) is a second order correct representation of the first spatial derivative. A graphical comparison of first and second order correct finite difference analogs is reproduced in Figure 3.2 (Rosenberg [1969]) where line "a" represents the true slope of the curve at a point. Line "b" is the second order correct analog and approximates 32 a b Um ------------------ I Ui """""""""" I I I I I . l I I I I I I I b ------- ' {JPI : I I I : l . : . : I l ' I ' | I ' I I ' I I ' I I I I | ' I . I ' I I ' I I l I Zi-1 Zi 2M Figure 3.2. Graphical comparison of first and second order correct analogs for the first derivative as illustrated by Rosengren [1969]. a. Actual derivative. b. Second order correct analog. c. First order forward difference. d. First order backward difference. 33 the slope much more closely than the forward and backward first order correct analogs shown as lines "c" and "d", respectively. A second order correct approximation for the second derivative can be obtained by adding equations (3.1) and (3.2) and truncating terms of order (auU/azu)i’n(Az)2/“! and higher, 2U (3211/3225,n = (U + Ui_1,n)/(Az)2- (3.6) i+l,n ' i,n The finite difference expression for the first time derivative can also be made second order correct by using the Crank-Nicholson method. Finite difference expressions are centered about the points zi’tn+l/2 which are half- way between the known and unknown time levels. Dependent variables U are evaluated at grid points, represented by open circles in Figure 3.1, while derivatives are cal- culated at center points such as the one designated with the black square. The time derivative is (BU/3t)i,n+l/2 = (Ui,n+l - Ui,n)/At (3'7) which is second order correct. The spatial derivatives in this scheme become 3“ (BU/32)i,n+l/2 = 1/“[(Ui+l,n+1 ' Ui-l,r1+1)/AZ + (01,1,r1 - Ui—l,n)/AZ] , (3.8) and (2)2U/az2)1’n+1/2 = 1/2[(Ui+1,n - 2111,n + Ui_l,n)/(AZ)2 2 + (Ui+1,n+1 ‘ 2Ui,n+l + U1-1,n+1)/(“Z) 3 ' (3'9) The Crank-Nicholson method is particularly effective for the diffusion thermoeffect problem because there is no stability restriction on At/(Az)2. Equations (2.“0) and (2.35) are of the same general forml, a(aU/at) - (82U/az2) + b(aU/az) = d (3.10) where a, b, and d are combinations of various transport and thermodynamic properties and are, in general, functions 1Although for the carbon tetrachloride-cyclohexane system Equation (2.“1) was used in place of Equation (2.“0), the solution procedure outlined here uses the more general Equation (2.“0). Both equations yield the same temperature difference result for carbon tetrachloride-cyclohexane mixtures. 35 of temperature and composition. Substitution of Equations (3.7) - (3.9) into Equation (3.10) followed by a regrouping of terms yields the algebraic expressions Ai,n+1/2Ui-l,n+l + Bi,n+l/2Ui,n+l + Ci,n+1/2Ui+l,n+1 = Di,n+1/2 (3°11) where A1,n+1/2 = 1 + (AZ)°1,h+1/2/2 ’ (3°12A) Bi,n+1/2 = ‘2“?(“Hanna/JAt ’ (3'125) C1,n+1/2 = l'(’-"Z)°1,n+1/2/2 ’ (3°12C) and _ 2 Di,n+l/2 - -Ai,n+l/2Ui_l,n + [2-2(AZ) ai,n+l/2/At]Ui,n — c - 2(Az)2d (3.12m) i,n+l/2Ui+l,n i,n+l/2 The equations are grouped in this fashion to display their recursive nature. Values of U on the right hand side of Equation (3.11) depend only on the nth time row while those on the left hand side depend only on the n+1St row. A 36 complete time row must be solved simultaneously from the previously calculated row since Ui n+1 appears in Equation 3 (3.11) with both adjacent neighbors Ui-1,n+l and Ui+l,n+l° Although the coefficients A, B, C and d are to be evaluated between the time rows, an iterative procedure can be avoided if Ai,n+1/2 = Ai,n; Bi’n+1/2 = Bi,n5 Ci,n+1/2 = C and d = d In this case, properties of i,n; i,n+1/2 the system need be evaluated only at the nth time row i,n' where temperature and composition have already been calculated. Because the composition equation is solved prior to the temperature equation along the n+1St row, the temperatures and compositions of the nth row grid points are used to evaluate the parameters at the respective n+l/2 locations. The temporarl spacing of the grid points is based on a thermal conduction time scale which is much faster than a diffusion time scale. Hence, the change in composition between n and n+l/2 is negligible. Similarly, as long as the row spacing At is not too large, the tempera- ture dependence of the parameters evaluated at tn will be essentially identical to the values at tn+1/2° For the temperature equation, the parameters are directly evaluated at n+l/2 with respect to composition by averaging the composition at tn and tn+l’ The temperature dependence of the parameters, however, is again evaluated at tr1 rather than tn+l/2' Little or no error results for At small compared to that required for significant changes in the 37 temperature. Although time step sizes were increased as the experiment progressed, care was taken to ensure that they remained small enough that essentially no error was introduced by evaluating the parameters with respect to temperature at tn rather than tn+l/2‘ This procedure is summarized in the flow diagram of Figure 3.3. the values of n+1’ Ui n must be known. The values for U1 0 are obtained from 2 3 To solve Equations (3.11) for Ui 3 the initial conditions and are therefore completely specified. For any given row of R spatial increments, the values of U and U must also be specified — these 0,n R,n are the boundary conditions. Putting Equations (2.32) and (2.38) into finite difference notation yields with the aid of Equation (3.5) U = U & UR,n = UR_2,n (3.13) where the grid points are aligned such that i=1 and i=R—1 correspond to cell walls. This reflective boundary condi- tion assigns imaginary grid points i=0 and i=R outside the cell walls, but U0 and UR are never evaluated. With the previous comments concerning the evaluation of A, B, C, and d; Equations (3.11) and (3.13) can be combined to yield a (R—l) x (R-l) tridiagonal matrix for each row in time START I I from I. C. initialize T 1 . 4 L 38 initialize all prOperties 8. parameters from I. C. initialize XII X11 parameter s for derivatives TI P=P(X1) A at n+1 at n; n.1/2 T2 P=P(T)_ at , if a = magic n; derIvatIves J, & P=P(X1) at Thomas n+1/2 . Algorithm solve for T or X1 at n+1 , does iflag=o? T values no yes X1 values at ”*1 at n+1 L step size Figure 3.3. yes pick out thermocoule Iocanns . I output & Lilots 1 I STOP increase t does next step exceed maximum? I no Flow diagram for simultaneous numerical solu- tion of the composition and temperature equa- tions. 39 [Blul 201112 0 o o ...- [.Dl A2Ul B2U2 C2U3 o o ... D2 0 A3U2 B3U3 C3Uu o ... D3 0 o AuU3 BuUu cuU5 ... = Du (3.1“) o 0 o ASUu 3505 ... D5 - ..l u .. where the displayed indices are i values. Each tridiag- onal matrix system (one for each row in time) is solved via the Thomas Algorithm (Rosengren [1969]) as illustrated in Figure 3.3. The numerical solution of the diffusion thermoeffect has certain advantages over the lengthy perturbation equa- tions of Ingle and Horne [1973]. The composition and tem- perature dependence of the parameters are fully included without involving numerous infinite summations. Ingle and Horne's solutions work well for systems whose pure com- ponent properties are similar and whose mixture properties are only slightly composition dependent. Otherwise, too many higher order terms are needed in the perturbation scheme for it to succeed. The main advantage of the numerical technique is that the boundary conditions and initial conditions can be slightly altered without necessitating an entirely new “0 analytical solution. Different solving techniques are usually required for different boundary conditions in the case of analytical solutions. The initial experimental conditions need not be isothermal to observe the diffusion thermoeffect as long as the initial temperature distribu- tion (it must be small enough to avoid thermal diffusion terms) is known in order to assign values to the first row of grid points. The adiabatic or reflective boundary condition can be changed relatively easily. For diathermal walls, the grid points located at either cell wall can be assigned a value of constant temperature equal to the out- side bath temperature. Once the computer program has been set up to evaluate numerically Equations (3.10), many other transport phenomena are readily simulated by a simple change of variables. Thus, essentially the same program models diffusion, thermal conduction, thermal diffusion, and pressure diffusion (most thermodynamic transport equations are parabolic partial differential equations), as well as the diffusion thermo- effect. The solutions, obtained as outlined above, were checked for stability by comparison of results obtained using dif- ferent step sizes. The number of spatial and temporal grid divisions were both varied by more than an order of magnitude without change in the dependent variables except at very short times. Programming was checked by comparison “1 to the solutions obtained by Ingle and Horne [1973] for a case in which their first order equations adequately des- cribed the system. B. Solutions for the Carbon Tetrachlorideegyclohexane System Using parameter values for the CClu - 37C6H12 system, numerical solutions were generated. Because solution of the corresponding system of tridiagonal matrices yields U = U(z,t), U was generated as a three dimensional sur- face. The velocity surface obtained indirectly from Xl(z,t) is shown in Figure 3.“. Note that the barycentric velocity v is essentially negligible except for very short times right at the interface. This results from the initial condition where composition (hence density) is a step function. If a simple algebraic solution for the Du- four effect is desired, a good approximation would be to neglect v. The composition surface shown in Figure 3.5 is indica- tive of why the diffusion thermoeffect is a transient phen- omenon in mixtures away from their critical solution tem- peratures. Note that as the experimental time proceeds, the gradient of composition flattens out. Heat and mass fluxes are related through the heat of transport by Equa- tion (2.20). As jl decreases in time, the measured “2 .Emumzm mcmxmnoaozo Impfisoficomppmp conpmo on» now mowmpSm mpfiooam> cappcmoznmm .z.m mmswfim wucizam >a~uo4u> “3 CORPUS I T ION BURFRCE Figure 3.5. Composition surface for the carbon tetra- chloride-cyclohexane'system. ““ transient temperature gradient also decreases since thermal conduction down the temperature gradient balances the heat of transport term. The transient nature of the tempera- ture distribution can be seen from Figure 3.6.1 Note that the upper or less dense phase rapidly increases in tempera- ture after initial boundary formation while the lower phase decreases. This conveniently eliminates density inversion possibilities. It also reflects a positive Q: because the phase rich in component 1 induces the colder temperature. The maximum AT between the top and bottom phases shown in Figure 3.6 is about 0.28°C. Rapid establishment of the maximum is due to the large initial composition gradient which then slowly decays. In confirmation of the results of Ingle and Horne[l973], numerical simulation shows that the heat of mixing contribu- tion to the local temperature distribution is symmetric about the interface. While an endothermic (or exothermic) heat of mixing lowers (or raises) the overall temperature of the cell, the difference in temperature between two points symmetric about the interface is not affected by the heat of mixing term. This is dramatically illus- trated in Figures 3.7 and 3.8. Even solutions in which ~ * 1This plot was made for HB = 0 with Q1 values evaluated from thermal diffusion factors using ORR. “5 TEHPERRTURE sunmce Figure 3.6. Temperature surface for the carbon tetra— chloride-cyclohexane system. AT represents the local temperature minus the initial temperature. “6 .:.ouAm\Nv Low mam mm>pdo LOSOH Ocm w.o u Am\NV LO% Ohm m®>hdo pogo: .HoE\h swam mHoE\m owmu< ..... .oum mHoE\h owwu< .IIII .oumu< . npfiz Aaxm+nSo one .m.ouAM\NV pcm a.ouAm\NV Lama maosooossmnp on» now mocmpmmmap mucummeEmu on» no L0H>mnmn pad» on» so madmnpcm mmmoxo no poommm .m.m madman 33 mo: m2: v m N F o a 4 q q q a q I a 0 mod nv IL 9.0 \m w a mud . de “9 HE differs greatly from regular solution theory allow determination of AT between points symmetric about the interface without interference from the heat of mixing. This fact led to the experimental design and data analysis used in the next chapter. Although heat of mixing data are fully included in the equations, only temperature differences measured equidistant from the interface are ~* necessary for the determination and evaluation of Q1' CHAPTER “ DIFFUSION THERMOEFFECT EXPERIMENTS ON CARBON TETRACHLORIDE-CYCLOHEXANE MIXTURES A. Experimental Design For a meaningful analysis of diffusion thermoeffect data using the mathematical methods developed in the preceding chapters, the cell in which the measurements are made must be of a design consistent with the condi- tions of Figure 1.1 and the assumptions outlined in Chap- ters 2 and 3. Although Rastogi et a1. [1965], [1969], and [1970] reported the first attempted measurements of the diffusion thermoeffect in liquid mixtures, their ex- perimental design was not amenable to theoretical analysis as Ingle and Horne [1973] and Rowley and Horne [1978] point out. Most of the diffusion thermoeffect induced temperature gradient was in fact eliminated by their cell design. The cell used by Rastogi 32 a1. had two vacuum— jacketed half cells into which the initial phases were introduced. These half cells were separated by a con- stricted region which was not insulated with a vacuum jacket. The interface was formed in the constricted region by opening a stopcock. With an interfacial diameter only half that of the bulk cell, radial diffusion and 50 51 radial heat conduction must have occurred in the two half cells, thereby vitiating the one dimensional transport equations. The temperature directly above the inter- facial area was subject to change not only by the diffu- sion thermoeffect but also by thermal conduction into the concentric ring of diathermal fluid outside the interfacial area. Understandably, no quantitative analysis or veri— fication of the Onsager reciprocal relations were obtain— able from these results. The design and use of a diffusion thermoeffect cell consistent with the conditions required in the preceding chapters therefore constitutes the first direct measurement of heats of transport in binary non- electrolyte liquid mixtures as well as the first test of Onsager reciprocity between 001 and 910 in such systems. Traditional diffusion cells use mechanical methods of interface creation often followed by siphon boundary sharp- ening. Although diffusion thermoeffect experiments require a distinct, sharp interface like that of diffusion experi- ments, the creation technique is more important in the former case since the response monitored is temperature rather than composition. Characteristic times are much shorter for thermal diffusivity than for diffusion,and boundary sharpening techniques are too slow to prevent heat conduction. Mechanical interfacial formation such as slide withdrawal or cell rotation can introduce turbulence aswell as obscure the initial time of the experiment (Bryngdahl 52 [1958]). Initial times in cells employing these tech- niques are obscured by the finite time required for slide withdrawal or cell rotation during which only part of the interface has been formed. Turbulence and initial time problems are coupled. If the mechanical motion is ac- celerated to reduce time errors, interfacial turbulence is enhanced (Bryngdahl [1958]). To allay these problems, the diffusion thermoeffect cell used here creates a sharp interface by the slow with- drawal of a third component from between the upper and lower layers. No ambiguity of the initial time is intro— duced since the third component is immiscible in the other two layers. Thus, diffusion is prevented until the middle layer has been completely withdrawn allowing contact between the two layers of interest. Interfacial turbulence is minimized since there are no moving surfaces. Furthermore, there are no seals or possible leakages in the inter- facial region. Unfortunately, the binary systems amenable to investigation with this cell are those for which a third component can be found possessing the necessary properties: (1) a density intermediate to the densities of the upper and lower mixtures, and (2) insignificant solubility in either of the other two components. Distilled water satisfies these requirements for the carbon tetra- chloride-cyclohexane system and therefore served as the withdrawable "liquid gate" for the experiments reported 53 in this chapter. The glass cell shown in Figure “.1 has two sections internally separated by an 8.5 cm length of glass tubing approximately 35 mm I.D. containing a stopcock. The upper and lower sections are jacketed for either thermostatting or vacuum insulating the containers. A stopcock in the tube connecting these jackets allows a vacuum to be creat- ed around onLythe lower cell. The bottom container in Figure “.1 (or in the schematic diagram of Figure “.2) is the actual diffusion thermoeffect cell. The upper con- tainer serves only as a reservoir for the less dense phase during displacement of the withdrawable "liquid gate". Inside dimensions are height: 2.0 cm diameter: 6.0 cm rim opening: m1 mm rim depth: ml mm where the rim is the bulge shown encircling the cell at half-height in Figure “.1. In preparation for each experimental run, the carbon tetrachloride—cyclohexane mixtures were gravimetrically prepared in two stoppered weighing erlenmeyer flasks of 50 mL capacity. Both components were "Baker analyzed" spectrophotochemical reagent grade of 99.0% guaranteed purity and were used without further purification. Horne 5“ Figure “.1. Withdrawable "liquid gate" diffusion thermo- effect cell. 55 Figure “.2. 56 Schematic diagram of withdrawable "liquid gate" diffusion thermoeffect cell. (A) (B) (C) (D) (E) (F) (G) (H) (I) (J) Upper phase storage reservoir. Cell jacket for thermostatting or adia- batically insulating. Diffusion thermoeffect chamber. Thermocouple banks. Equatorial water entrapment rim. "Liquid gate" withdrawal spout. Ground glass fittings for thermocouple leads and cell drainage. T-connector to vacuum line and thermo- stat. Filling tubes. Glass syringe. 58 [1962] has reported an in-depth error analysis for dif- ferent techniques of gravimetrically preparing carbon tetrachloride-cyclohexane mixtures. The more volatile cyclohexane was added to the already weighed carbon tetra- chloride and the flask was immediately sealed with a ground glass stopper lubricated with Fisher "Nonaq" grease, which is inert to both carbon tetrachloride and cyclohexane. Horne's [1962] discussion indicates carbon tetrachloride weight decrease via vapor loss during the addition of the cyclohexane to be less than 0.02%. No change of weight in time was detected for the filled flask once stoppered as described above. The lower or diffusion thermoeffect chamber was filled in a two step process. First, distilled water was intro- duced from the bottom until it half filled the cell. Second, the more dense carbon tetrachloride-cyclohexane mixture was layered beneath the water layer by injection from below with 21 syringe pump purchased from the Harvard Apparatus Company. This technique prevented evaporational changes in composition during cell infusion. Introduction of the carbon tetrachloride rich layer raised the water level into the storage cell. Care was taken to remove any trapped air bubbles from the lower cell after which the stopcock between it and the reservoir was closed. The cyclohexane rich or less dense layer was then quickly introduced into the storage reservoir with a 100 mL syringe. 59 The syringe was left connected to the storage reservoir during withdrawal of the "liquid gate". It served as an enclosed piston for volume displacement as the distilled water was withdrawn. After filling, the cell and storage reservoir were thermally equilibrated with the thermostatting jacket surrounding them. To maintain the initial and boundary conditions used in the mathematical description of the effect, the initial nearly isothermal conditions (complete temperature uniformity is not required if the initial temperature distribution is known) must be quickly changed to adiabatic conditions upon interface formation. Adia- batic walls were imposed by evacuating the jacket with a vacuum pump. It was found that the above conditions could not be met if a liquid was circulated in the circumam- bient jacket as the thermostatting fluid. Wetting of the cell walls by a thermostatting liquid left residual drop- lets when the jacket was drained. Adiabaticity could not then be imposed due to vaporization (and associated heat effects) of the droplets as the jacket was evacuated. Consequently, room temperature air served as the thermo- statting fluid. No temperature effects were noticed when the jacket was evacuated for a trial run in which the cell contained only pure water. Nevertheless, some runs were performed by evacuating the jacket immediately after fill- ing the cell and allowing internal thermal equilibrium to 60 take place before starting the experiment. No discrep- ancy was noted between the results obtained via the two different procedures. In all cases, temperatures at each thermocouple location were continuously monitored to ob- tain the temperature distribution at the time of interface formation. This measured initial temperature distribution served as the mathematical initial condition. The "liquid gate" (distilled water) was withdrawn via the syringe pump at a rate of 0.76“ mL/min until only a small phase separated the two carbon tetrachloride-cyclo- hexane layers. From this point until contact of the two layers, withdrawal rates were slowed to 0.0206 mL/min or 0.0382 mL/min to eliminate possible convection currents. Faster withdrawal rates slightly altered the initial tem- perature distribution in time even though the mixture dis- placing the water had been co-thermostatted in the reser- voir with the cell itself. Smooth interface creation oc- curred uniformly and isochronously throughout the cell except within the equatorial rim where the meniscii were curved by preferential wetting. However, due to this wetting, any residual water at the time of contact between the upper and lower layers was contained within the rim. Preliminary experiments indicated that constriction of interfacial diameter relative to cell bulk diameter reduced the ensuing temperature gradient. This is due to radial thermal conduction as mentioned earlier with respect to 61 the cell used by Rastogi _3 a1. Therefore, care was taken to ensure that any residual water at the time of inter- face formation was contained within the equatorial rim from which the withdrawal spouts extended. Immediately upon interface formation, a Precision Scientific Co. "time it" digital timer (0.1 second read- out) was activated, the syringe pump was disengaged, the stopcock between the reservoir and the cell was closed, and the vacuum jacket was evacuated. Temperature changes were monitored with “0 gage copper-constantan thermocouples placed equidistant above and below the interface as shownixi Figure “.2. Each thermocouple comb cbntained four thermo- couples spaced 2.0 mm from each other, the outside walls, and the interface. Welded junctions (0.2 mm in diameter) were spaced 2.0 mm from the surface of the 1.5 mm thick Delrin ® (K = 0.23 J-m'lK-l) comb. Thermocouple potentials were monitored with a Leeds and Northrup Co. K-3 poten- tiometer facility provided with 16 thermocouple stations. Temperatures at the eight thermocouple locations were made at approximately 12 second intervals. Readings were taken alternately about the interface such that differences in temperature AT between symmetrically located thermo- couples had uncertainties in time of about :6 seconds. For these experiments, differences in temperature were obtained by subtraction of two absolute temperatures because it was felt important to observe actual temperatures 62 everywhere during these pioneering measurements. In the experiments performed later on critical mixtures, enhanced precision was obtained by monitoring temperature differ- ences directly rather than referencing each thermo- couple to the ice bath. Exact thermocouple locations were measured in_§itu_with a Beck Vernier Measuring Micro— scope. Accuracy in exact thermocouple location was limited by the finite size of the welded thermocouple junction (us- ually 0.2 mm in diameter). Monolayers of water were assumed not to be present im- mediately following interface formation due to the hydro- phobic character of both layers. Furthermore, since the sharp interface initially formed becomes indistinct as diffusion occurs, a monolayer cannot exist more than instantaneously. The system becomes continuous as soon as the initial step function in composition has vanished due to diffusion, and any interfacially adsorbed water must have previously been removed. The maximum temperature difference between symmetric thermocouples was obtained after 500 - 800 seconds. The maximum temperature difference was found to be dependent upon the difference in initial compositions in accord with theory (Ingle and Horne [1973]). The starting mole fraction differences varied between 0.59 and 0.82 with corresponding temperature difference maxima from 0.21 °K to 0.30 °K respectively. Pure components were not used 63 for two reasons: (1) Preliminary experiments revealed an onset of turbulence at the newly formed interface when pure components were used, presumably due to large sur- face tension shock between two different pure components. (2) Heat of mixing effects are dependent upon the square of the initial composition difference and are thus lowered relative to the diffusion thermoeffect for smaller initial composition differences (Ingle and Horne [1973]). The second reason above is not very important for the experi- mental design used here because heat of mixing effects do not contribute to temperature differences taken at points symmetric to the interface. B. Analysis of Technique As alluded to in the previous section, the initial condition for the temperature equation was obtained from measurements of cell temperature prior to contact of the two layers. Temperatures at all eight thermocouple loca— tions were continuously monitored as a function of time. The exact time of contact was recorded,and an extrapola- tion from the previous temperature readings to the contact time yielded temperatures for each thermocouple at t = 0. No extrapolation between a previous reading and t = 0 extended over 180 seconds,and no extrapolated temperature change exceeded 0.010 °K. The initial condition data are recorded in Table 8.1 of Appendix B. For the five runs 6“ performed, three were isothermal and two had essentially linear temperature distributions. Neither of these dis- tributions included differences larger than 0.068 °K be- tween the top and bottom surfaces of the cell. Presumably, the nonisothermal distribution in these two runs was due to faster withdrawal rates and hence to faster intake rates of reservoir thermostatted liquid. It should be emphasized again that isothermal initial conditions are not required as long as the actual temperature distribution is known, and the initial gradient in temperature is small enough that thermal diffusion terms are still negligible. More- over, computer simulation using the equations developed in Chapters 2 and 3 indicates that small initial tempera- ture distributions do not markedly affect the difference in temperatures between two symmetric points for times measured after the maximum temperature difference has been reached. This is because the magnitude of AT is fixed by a balance between the opposing effects of thermal conduction and the heat of transport and not by the pre- vious temperature history of the mixture. Care was taken to ensure that no vapor or air bubbles remained in the cell. Until they were removed, air bubbles aided in leveling the cell. Residual air pockets were removed with a filling needle connected to a syringe. Entrance to the filled cell could be made by dislodging the ground glass fitting (through which the upper 65 thermocouple leads entered) with the stopcock to the reservoir closed. By carefully opening the stopcock, hydrostatic pressure allowed displacement of the last air bubbles. The ground glass fitting, coated with a thin layer of Fisher "Nonaq" grease, was then firmly reinstated. A preliminary experiment with only 97C6H12 showed that no vapor loss occurred around this fitting if coated with the inert lubricant. However, with a dry glass fitting, the upper portion of the cell persisted to be 0.05 °K colder than the lower region due to the endother- mic vaporization of ng6Hl2 around the joint. A fresh seal of the "Nonaq" grease was applied before each experi- mental run. Although temperature differences were monitored for all four thermocouple pairs, only data from the pair clos- est to the interface [(z/a) = 0.“0 and (z/a) = 0.60] are reported in Appendix B Table B.2. Only data from this pair were used in the calculation of Q: because the innermost thermocouple pair is least prone to the possible errors discussed below. Deviations from the mathematically pre- dicted temperature differences can arise from: (1) Pertur- bations due to the presence and finite size of the thermo- couple holder and the other thermocouples. (2) Thermal conduction through the thermocouple holders. (3) Side wall effects. (“) Heat losses through the cell ends; LLQL, nonconformity to the prescribed adiabatic boundary 66 conditions. The first problem was minimized by allowing the thermocouple junctions to protrude 1.5 mm from the holders. In addition, the innermost thermocouple of each group (closest to the interface) extended 1.0 mm below the holders. Any effects due to the presence of other thermocouples and/or the thermocouple holders, would not be felt by this pair of thermocouples. For similar reasons, the difference in thermal conductivity between Delrin ® holders and the system would not affect the temperature differences of the innermost pair. Side wall effects were probably negligible because of the relatively large diam— eter/height ratio. Furthermore, the main region of dif- fusion is the interfacial region and wall effects would be less important for those thermocouples closest to the interface. The fourth problem warrants more concern. Obviously, it is impossible to have perfectly adiabatic walls, yet the mathematical boundary condition used implies perfect adiabaticity. Computer simulation shows that a change in boundary conditions from adiabatic to diathermal af- fects the innermost thermocouples the least. That is, small deviations from temperatures described by adiabatic boundary conditions due to imperfect adiabaticity of the walls are absorbed by the bulk fluid before they are felt near the interface. Therefore, any heat conduction through the cell ends where the thermocouple holder is connected 67 or where other tubes enter the cell may affect the outer thermocouple pairs. The innermost pair appears to be the most accurate and reliable set with respect to each of the above four sources of error. Therefore, only data from this pair were used in computing Q:. In view of the above discussion, computer simulation was used to check the validity of the adiabatic boundary conditions. Figure “.3 compares the time dependent shapes of AT values, expected for adiabatic and diathermic boundary conditions for a given value of Q:, to the experi- mental values obtained from the innermost thermocouples. As can be seen, the boundary conditions change the time dependent behavior of the predicted AT values consider- ably. Note that the actual AT time dependence clearly corresponds to that predicted on the basis of adiabatic walls. Data from thermocouple pairs further from the interface also agreed with the behavior predicted by the adiabatic model at shorter times but deviated at inter- mediate times. The length of time during which the AT behavior was consistent with the adiabatic model was inversely proportional to the distance from the inter- face at which the particular thermocouple pair was located. Figure “.3 is obvious verification that the innermost thermocouples provide accurate readings for the experi- mental time scale (%“000 seconds) when adiabatic boundary conditions are used. A comparison run was also made in 68 .AIIIIV maamz UHEpmnpmwp can A V mafia: oapmnmfipm Log modam> Umpofipmpa op pump B< Hmpcmefipoaxo mo COmfipmoEoo .m.: osswfim 69 NV m.a mnzmfim aomw No: vw mp 92...: NF . IRCO ood mod . wwo m mo vmd omd .LV (Oo) 70 which the innermost thermocouples were each moved 10% further from the interface. The Q: obtained was unchanged. As previously mentioned, only temperature differences between thermocouples positioned equidistant from the inter- face were used in the parameter estimation procedure. More sensitivity is obtained in computing the heat of trans- port by this technique because the large background heat of mixing with accompanying uncertainties is eliminated. As Ingle and Horne [1973] have shown, the principal heat of mixing contribution is symmetric about the interface. Computer simulation using the previously described numeri- cal routine substantiates their conclusions. In fact, the symmetry of the heat of mixing term, even for mixtures which deviate substantially from regular solution theory, allows calculation of the antisymmetric heat of transport term without including the excess enthalpy provided sym- metric temperature differences are used as input data. The sacrifice made in using only AT data rather than indi- vidual T values is that only one rather than two parameters can be accurately determined for a given run. Because only one parameter was to be obtained from the nonlinear, weighted, least squares fit of theoretical to measured AT values, the most composition independent form of the heat of transport was desired. As shown in Appendix A, the relationship between Q: and al, assuming Onsager reciprocity, involves various factors of which 71 ...fi ... Q1 and M are the only strongly composition dependent terms. In fact, a is used for reporting thermal diffusion re- 1 sults because of its relative constancy with respect to composition. Consequently, (OX/M) is fairly composition independent and was used as the adjustable parameter in the fitting procedure. Program "KINFIT“" (the 1977 version of "KINFIT" as published by Dye and Nicely [1971]), a generalized, weighted, nonlinear, least squares fitting routine extensively used in fitting chemical kinetics data at Michigan State University, was used as the param- eter estimating routine into which the previously des- cribed numerical partial differential equation solver was introduced. An example of the fit obtained using this procedure is shown in Figure “.“. To test the stability of parameter estimates obtained for (Q:/M) as a function of the time range over which data were input, numerous fits of the first run (depicted in Figure “.“) were made as a function of data truncation. Thus, only data out to t = 1500 seconds were included for obtaining a value of (OI/M), then data out to t = 1700 seconds were included and the value of (Qi/M) was again computed; etc. The results of this data truncation test are shown in Figure “.5. Note that the parameter estimate remains unchanged within 1% for inclusion of data past 2800 seconds. When data past 3“00 seconds are in— * cluded, essentially no change in Q1 occurs as more data 72 .mpmp Hmpcmsfimmoxm A€\mov mo mSHm> paw one so 3 comma modam> oaaam the .H cam hos Aa\*@v so has amass H m a m mam mmaosfio pfiaom one mafins pmpmHSono mucommsomm mcHH pmmma .Uounwfimz .pmocfificoz .:.: mhswfim 73 NV 0m 00 3.: mpswfim somm Nos. «N 9 A m:2:. NP a _ B 00.0 00.0 vwd 00.0 7“ .m:am> mefipiwcoa on» Sony coapma>mc RH mopmoapca mafia cmnmmp one .uSQGH who: pump B< scan: sow swamp mafia on» no coauocsm m mm pwcfimuno ”a mo szHHnmpm on» no pmoe 3m... mo: mi: 2. mm on «N .m.: onswfim 3 u — 4 u — u _ 208:4: .. m ”w 75 points are included. The effect shown in Figure “.5 has essentially two causes: (1) The value of the fit parameter becomes increasingly stable and (presumably) more accurate as statistically more points are included. (2) Although the parameter value for the perfect model would oscillate randomly about a mean value as more points are added, the assumed model, that (OI/M) is a constant independent of composition and temperature, is not strictly true. There- fore, inclusion of data taken at the relatively longer times is important in finding the appropriate (Oi/M) for the mean composition reported for each run because at short times two very different compositions are located on either side of the interface. As diffusion levels the initially sharp composition gradient at (z/a) = 0.5, the compositions in the regions near the interface become more nearly the mean value, and the fit (Q:/M) becomes the value for that composition. All experimental runs were analyzed with respect to (Q:/M) by inclusion of data up to “200 seconds. This not only provided enough points forzastatistically "good" esti- mate of (Q:/M) but also eliminated the problems discussed in the previous paragraph. Analysis of the diffusion thermoeffect in gaseous mixtures has often relied entirely on the maximum temperature difference measured (Bousheri and Afrashtehfar [1975]). Inclusion of points spaced in time and use of the integrated equations provides a 76 statistically better estimate of the heat of transport, particularly since the maximum AT occurs so early that very few data points can be obtained up to that time. For thermocouples 2 mm from the interface, AT reaches its maximum value in 500 - 800 seconds. C. Literature Values for the Physicochemical Properties of the Carbon Tetrachloride-Cyclohexane System Before Equation (2.35) can be used in conjunction with experimental data to obtain values for Q:, all other parameters appearing in (2.35) must be available. Table “.1 contains a synopsis of the values used for the system carbon tetrachloride—cyclohexane. The parameters of Table “.1 are based on the expansion L = LO[1+Lx(xl-0.5) + LT(T-298.15) + LxT(xl—0.5)(T-298.l5) + 1/2Lxx(xl-0.5)2] (“.1) where L is the property in question, L0 is the value of L for an equimolar mixture at 298.15 °K, and Lx, LT, LXT; Lxx are corresponding composition and temperature co— efficients. The parameters are viewed as expansions about X1 = 0.5 rather than about pure component values (LS and LS) because the mean composition of each run was nearly 50 mole percent. In the sections below, further discussion 77 nmuaflu comoHEwh Haemau sauna“: new mmmmHH coo: HaemHH saunaaz HaemHL wcazm HONQHH COmh00C< moemHL swam: mocohoumm nmwmau comoaewh Haemau saunas: can mmmmHH poo: Haemau saunas: Hoemau wcazm HONmHu comhmvc< HaemHH use»: mocohouom 000.0 ~m0.0 A Humaumaue.a Hiaosms ”HIHOEI h. race.nx Hiums Hos.mx HI H H..- mafia: H000.0I mm00.0I NH00.0I hH00.0 &o\BKA a- on oaxemmo.a =H.==H mmm.HI mioaxmmm.a Homaa.o maoo.ou III--- a maoo.o maa.o- a» maoo.o mea.ou . m ~moo.o- mem.ou a eawxaxmmmav mmflo.o mma.ou a IIIIIII mam.o m xo\aq an a .QHCOHOHMhflOO Ohfivflhfl 0» DC“ COHUHmO ECU .m mHH.o moa.o g oaxemmo.a auoaxmoem.o a» we.mma mm.Hma . m IIIIIII IIIIIII m a H Amxa\mm-0 muoaxmmz.a m-oaxmm~.a a ma:wo.o =omma.o m MI MI I as as a .xomH.mmm as uohnuxus huwoaazdo he can nucocomEoo manning» ho uoaphuQOHm .< .mocoucoamp coauauonsoo 0cm unsuppoQEop hams» oczaoca o» x Hmflm.oiax0xxa~\a + Ama.mmmuavam.oiaxvaxa + Ama.mmmievaq + Am.oIHx0 q+auoa I a Show Hapocmw on» :« .Euw H um Emuuzm ocaxonoaoao I ocuhoaco Impumu conpmo 0:» mo modupmaopa Hwo«EonoooHumsn onu.hou mosaw> ohsumpoaaq .H.= magma commouaxo who moapnoaona one 78 of the literature values is given. (1) Diffusion coefficient - The diffusion coefficient as reported by Anderson and Horne [1970] agrees well with other literature values cited therein. The composition and temperature dependencies are included in their report and were used in the numerical fitting routine. (2) Excess enthalpy - Ewing and Marsh [1970] report the composition dependence of the excess molar enthalpy at three different temperatures. The temperature dependence was obtained by a fit of their three temperature indepen- dent equations for HE [1968]). using program "MULTREG" (Anderson (3) Constant pressure heat capacity - Values for molar heat capacities were obtained from Wilhelm and Sackmann ~0 lCP,l + «.0 ~ - .- x2CP,2 where ACP — -0.6xlx2, adequately describes the com [197“]. They find that (GP/J'K'lmolIl) = ACP + x position behavior of CP. As must be the case for thermo- dynamic consistency, the constant pressure temperature derivative of the excess enthalpy agrees well with the excess heat capacity. Due to the smallness of the excess heat capacity, the temperature dependent behavior of C? is contained entirely in the temperature dependencies of the pure component heat capacities. Upon rearrangement, the temperature dependence with respect to the mixture becomes that shown in Table “.1. (“) Molar volumes - Molar volumes were calculated 79 from the density data of Wood and Gray [1952]. The form (V/cm3mol-l) = leg + x2Vg + VE yields excellent agreement E 0 with their results when V = 0 and V0 and V l 2 pure component temperature dependencies. Program "MULTREG" contain the provided best fits of their temperature dependent data which were then rearranged into the form required for Table “.1. The values obtained by Wilhelm and Sackmann [197“] also agree well with these equations. (5) Thermal conductivity - Thermal conductivity data for liquid mixtures are scarce due to experimental convec- tion problems. For the same reason, the uncertainty in good experimental data is between 3% and 7% depending on tech- niques and equipment used. The only experimental data reported in the literature for carbon tetrachloride—cyclo- hexane mixtures are those of Venart [1968]. His results show a most peculiar cusp at x1 = 0.5 when K is plotted against x1. No other nonelectrolyte mixture displays this behavior. Furthermore, the scatter in data for this system relative to that of analogous systems studied by Venart indicates a pecularity and/or difficulty in obtain- ing accurate thermal conductivity data for carbon tetra— chloride-cyclohexane mixtures. Jamieson, 33 a1. [1975] and Jamieson and Hastings [1969] have recommended the NEL (National Engineering Laboratories) equation, = o o_ O_ _ K le1 + w2K2 C(K2 Ki)(l /w2)w2, (“.2) 80 for predictive estimates of the thermal conductivity of binary liquid mixtures, where K3 is the thermal conductivity of pure component i and C is an adjustable parameter. Component 2 is assumed to have the largest thermal conduc- tivity in using Equation “.2. Thermal conductivity predic- tions based on this equation have been shown by Jamieson gt al. to agree with experimental values over the entire composition range to within 5% if a fit value obtained at a single composition is used for C and to within 7% if C is defined by C E 1.0 (most nonelectrolyte mixtures are best represented by this value). Rather than from Venart's data or from the NEL equation, the composition dependence of the thermal conductivity was obtained from the diffusion thermoeffect experiments them- selves by an iterative technique. The method and the results obtained for the thermal conductivity are presented in the next section. D. Experimental Results for Thermal Conductivity The composition dependence of K was obtained from the diffusion thermoeffect experiments. An iterative procedure was followed in which the composition dependence of (Q:/M) was first neglected and then its experimental value was included in order to determine the composition dependence of the thermal conductivity. To illustrate the approach, 81 consider the temperature equation without the very small term due to the barycentric velocity [obtained from Equa- tion (2.35) for D/V independent of xi], CP(3T/8t) vn(a2T/az2) + V(3K/3X1)T(3xl/3z)(ET/dz) + ~*~ ~ DM2(Ql/M)(32xl/322) + D(a2HE/ax§)T(axl/az)2 + ~*~ 2 DM2[8(Ql/M)/3xl]T(3xl/az) . (“.3) In fitting the experimental data by the numerical scheme described in Chapter 2, the last term on the right-hand side of Equation (“.3) was first omitted. The calculations then gave Q: and a first approximation for (BK/axl)T. Once Q: was obtained for all the experimental runs, its observed composition dependence was included and Equation (“.3) was then used, in full, to obtain an improved esti- mate of (BK/3x1)T. Initial analysis of the 5 diffusion thermoeffect ex- periments reported herein was done using the NEL equation for the thermal conductivity with the adjustable parameter C = 0.6. This value was obtained by fitting both (OI/M) and C simultaneously to the data of Run I. However, be- cause Q: and K at the mean mole fraction primarily determine AT, C and (Q:/M) were largely coupled. Sensitivity co- efficients for these parameters indicate coupling. 82 Nevertheless, the thermocouple pair was sufficiently re- moved from the interface that the magnitudes of K at these thermocouple locations (hence compositions) sufficiently decoupled C and (Q:/M) to obtain a unique but shallow minimum in the residual search. Once the parameter C had been identified (C = 0.6), Runs I through V were analyzed with the fixed value for C leaving only (OI/M) as an adjust- able parameter. Since (Qi/M) is not a constant as assumed in this fitting procedure, the composition dependence of (Q:/M) was absorbed in the fit value of the NEL parameter 0. The composition dependence of (Q:/M) obtained from the five runs at different compositions was then introduced into the fitting routine in an iterative fashion to yield the improved value of C = 1.05. The value C = 0.6 is seen to be an "effective" value into which the residual composi- tion dependence of (Oi/M) was absorbed. Note that the improved value obtained for C in the NEL equation from diffusion thermoeffect data is in excellent agreement with the recommended value of C = 1.0 for nonelectrolyte mix- tures. Absolute values of the thermal conductivity, as predicted by the NEL equation with C = 1.05, are tabulated in Table “.2. Also shown in Table “.2 are values obtained from Venart's data by interpolation to the appropriate mole fraction. Note that the values obtained from this itera- tive treatment of the diffusion thermoeffect agree within 83 Table “.2. Thermal conductivity of CCI“‘EEC6H12 mixtures at 20°C and 1 atm. KD/w.s'1x'1(a) KV/W°S-1K-1(b) (KD—KV)/KVX100% 0.3“69 0.1089 0.107 1.8 0.“112 0.1078 0.10“ 3.6 0.“295 0.107“ 0.10“ 3.3 0.“8“3 0.1066 0.10“ 2.5 0.551“ 0.1056 0.103 2.6 (a) Values obtained from diffusion thermoeffect experiments. (b) Values interpolated from the data of Venart [1968]. 8“ experimental error (ca. 3%) with Venart's data in the composition range applicable to the measurements reported here. This range of compositions involved the previously noted cusp in Venart's data; i;g;, actual data in this region were higher than expected on the basis of smooth composition behavior with no inflection points for the entire mole fraction range. Thermal conductivities pre- dicted for compositions outside the experimental range of interest deviated more than 3% but are irrelevant to the analysis of data. Although using AT data rather than values for T itself did not allow accurate simultaneous determination of two parameters, inclusion of multiple run information in the above described iterative fashion decoupled the two param- eters allowing accurate determination of both the heat of transport and the thermal conductivity. The thermal conductivity values obtained agree, within the experimental uncertainties involved in measuring liquid mixture thermal conductivities, with those measured by Venart [1968]. The values obtained for the heat of transport are discussed in the following section. E. Experimental Results for Heat of Transport As previously indicated, the experimental data con- sisted of numerous temperature differences between thermo- couples located at (z/a) = 0.“ and (z/a) = 0.6. Appendix 85 B contains the raw data obtained for the 5 experimental runs and the initial run conditions associated with each. Table “.3 shows the results obtained for Q: using the previously described fitting procedure. Initial composi- tions and temperatures are also included in this table. From the resultant Q: values, the Onsager coefficient 001 is calculated on the basis of Equation (2.1“B). Literature data for the thermal diffusion coefficient a1 (Anderson and Horne [1971], Stanford and Beyerlein [1973]; and Turner, §t_al, [1967]) provide values for 010 after averaging, adjusting to the given temperature via the equation reported by Anderson and Horne, and using Equations (2.1“C), (2.15), and (2.16). A comparison of these two Onsager coefficients, obtained independently of each other, is shown in Table “.3 along with the actual values of 001 and 010. As required by the Onsager heat-mass reciprocal relation, 001 = 010 to within 3%. This constitutes the first experimental verification of the Onsager heat-mass and mass-heat recipro- cal relation in liquid systems (Rowley and Horne [1978]). The verification of the heat—mass reciprocal relation now allows transformation of experimental heats of transport to thermal diffusion factors by way of Equation (2.22) with the definitions of Equations (2.1“) — (2.16). Thermal dif- fusion factors obtained in this manner are adjusted to 25 °C using the temperature dependence -d1 = -l.827 + 0.18lxl + 0.010“ (T—298.15) - 0.0008xl (T-298.15) reported by 86 .H6 pom mosam> shapmsoufia wcfim: A03H.mv coapmzom Soap popmHSOHmo .Amza.mv coapmsvm Eopm pmpmHSOHmo A00 A00 .aaoo so coasemse made he mecmsdaeae HaasacH Ahv .:H00 00 Cofipomsm mHoE cmoz Amv 0m.0 mm.m 0H.0 mH.0 mm.mm 00H>.0 :Hmm.0 0H.0 wm.m mm.m aw.m em.mm mmaw.0 m:m:.0 0m.0l :m.m :H.w m:.m mm.am >wmm.0 mmm:.0 m0.0 m:.m w:.w mm.m H0.mm mmww.0 NHH:.0 00.0 0m.0 mm.0 5:.m Hm.mm mmmm.0 002m.0 a H Hiemmaiswxeioa “areamaiswx Aarommauswx AHIHosex cos 9 aa xv OH 8 7022.080 7023.0“: 3.0». e \A 0| 00 aloomfliewx HIoomHIwa HIHthx\*w OH HO eioa\e a eioaxe d .00mm 00 mopspxfie mcmxmzoaozoIocahoano stpop coopmo CH mucofiofimmooO sommmco 0cm psoamcmsp mo pawn esp mo monam> .m.: magma 87 Anderson and Horne [1970]. A comparison of these thermal diffusion factors (obtained from diffusion thermoeffect experiments) to those obtained from various thermal dif- fusion experiments is shown in Figure “.6. In particular, the solid triangles are the results of these diffusion thermoeffect experiments, the solid line and solid circles represent the pure thermal diffusion results of Anderson and Horne-[1971], the dashed line represents the thermo- gravitational results of Stanford and Beyerlein [19731. the open circles are the flow cell data of Turner, Butler, and Story [1967]; and the solid squares represent thermo- gravitational results obtained by Korchinsky and Emery [1967]. The various techniques for obtaining thermal dif- fusion factors all yield consistent results within the experimental uncertainties. This comparison confirms the diffusion thermoeffect as a valid and accurate method for obtaining heats of transport and thermal diffusion factors. The advantages of performing diffusion thermoeffect measure- ments are perhaps manifest most strongly in the liquid- 1iquid critical region as will be shown in Chapters 5 and 6. As shown in Appendix A, the relationship between Q: and a1 is ~* Q 1 = -a1MRT(l+F)/M1M2. (“.“) 88 .Aaeeaaa mates paw amazomv.ACOfiwmuMszonpmSpv mummag macaw new mxmcanopox .I ”AHHmo onmV mummau .Hm pm avenue .0 ”Acoaumpfi>mpwoenmnuv mmnmau Camapomom 0cm chomc0pm .IHI mAGOHmSMMHU Hosanna essay mammau mahom new compmcc¢ .II ”Apommmm nsousmv xsoz was» .4 .mmmmnpcmpmo :H ma acmsmnsmmme no vogue: .cofipfimanoo mo cofipocsm 0 mm 00 mm um souomm coamsmmac anemone ¢ .00X 0.. 0.0 0.0 NO 0.0 0.0 Q0 m0 N0 ..0 0.0 _ _ _ d a _ _ _ _ .m.a mhswfim 89 The transformation from heats of transport to thermal dif- fusion factors therefore involves the "thermodynamic factor" (1+?) defined by (1+F) E (1+32nyl/alnxl)T’P (“.5) where Y1 is the activity coefficient of component 1 (usually based on the pure component standard state for nonelectro- lyte mixtures). Although (1+F) is close to unity for this system at all mole fractions, a least squares fit of the activity coefficient data reported by Turner 33 a1. [1967] was used to calculate “1 values from corresponding Q: values. It is straightforward (but tedious) to show from the excess enthalpy of Table “.1 that the temperature de- pendence of (1+?) is negligible over the experimental range. Nevertheless, in critical mixtures, the "thermodynamic fac- tor" plays an important role in the behavior of properties very near the consolute temperature. Because the "thermo- dynamic factor" is often close to unity for nearly ideal mixtures, early formulations of diffusion assigned composi- tion gradients as mass flux driving forces. Systems and regions (such as the critical region) where activity cor- rections are important have been invaluable in clearly identifying chemical potential gradients as the correct diffusional driving forces. From the experimentally obtained heats of transport and the fundamental relationships between the heat-mass 90 cross coefficients, the other commonly used transport co- efficients can be evaluated. From Equation (2.18), the Dufour coefficient 8T can be directly calculated. Similarly the thermal diffusion coefficient DT can be obtained from Equation (2.1“C) by using the now proven Onsager relation 910 = 001. These dependent coefficients along with Q: and d1 are tabulated in Table “.“. A comparison of Figures “.7 and “.8 reveal the main reason for the multiplicity of coefficients. Note that the Dufour coefficient 8T appears to be nearly independent of composition for this system at 25 °C. On the other hand, Q: is quite dependent upon composition. A similar relationship holds between a1 and D 1 being more composition independent. T’ a The diffusion thermoeffect results allow calculation of another interesting quantity. There is some ambiguity in the thermal conductivity which appears in thermal dif- fusion equations. Before the temperature gradient is applied in a thermal diffusion experiment, the isothermal equilibrium mixture has a definite thermal conductivity KO. After the temperature gradient has been applied, a steady state is reached when the temperature gradient- induced mass flux identically balances the mass flux caused by the propensity for diffusion down a chemical potential gradient. This steady state mixture also has a definite but different thermal conductivity Kw. As Horne and Bearman [1967] show, these two properties are 91 Table “.“. Heat-mass transport coefficients for carbon tetrachloride-cyclohexane mixtures at 25 °C and 1 atm. 91/k3°mOl-l BT/lO-ZJ-m'ls'l -mfi_ DT/lo'lomzs';l 0.3“69 5.“2 5.71 1.79 6.3“ O.“112 5.56 5.60 1.77 6.12 0.“295 5.“o 5.39 1.70 5.80 0.“8u3 5.80 5.59 1.77 5.7“ 0.551“ 6.10 5.6“ 1.79 5.69 .pfim mmnmsvm umwma . .mmsaw> H0 Hmpcmefisooxm on» Eopm cmpmasoamo mm 90 encasemmooo ssomsa 029 .~.: msswam *2 mouse Hopcoefinmoxo .I 92 Fx me me. To me i — - a i I sod AU .. I mod N P I . l.i I I . w. I L 00.0 le.» .. - Be _ _ _ _ 93 «mama amazes .pam mmnwsdm pmmma . Ifihmaxm . I .H% Coapvmefiw mHOE .HO COHDOCSM .0 mm ”a phoamcmhu ho pmmm .w.= mhdwfim FX 00 0.0 «.0 0.0 _ _ _ q I v o... I m It I I M / w m. I o ( I .i. 9“ related by _* _* “ 6 where Q: is the specific heat of transport of component 1. The relation 001 = 010 has been used to obtain the second equality shown in Equation (“.6). From diffusion thermo- effect experiments, Q: and 001 are directly obtained. Therefore, the difference between the two thermal conduc— tivities KO-Km is readily calculable from diffusion thermo- effect experiments. Table “.5 shows the difference as obtained from the experimental results reported in this chapter. Note that current uncertainties in experimentally determined thermal conductivities are much larger than the difference between K0 and Km. The two may therefore be used interchangeably without sacrifice of numerical accuracy until very much improved thermal conductivity measurements can be made. Unlike the AT versus time profiles, measured T versus time profiles were quite asymmetric about (z/a) = 0.5. In all cases, the increase from the uniform initial temperature was much less for the thermocouple located above the inter- face than was the decrease in temperature for the lower thermocouple. This asymmetric effect is analogous to that observed by Mason, Miller, and Spurling [1967], by Waldmann [19“7], and by Miller [19“9] for gaseous diffusion 95 Table “.5. Values for the difference in thermal conductiv- ity between the equilibrium and steady states in thermal diffusion experiments. (KO-Km)/ T/OK KO/w-s"1K‘l 10'5 w-s'lx‘l % (a) (b) Difference 0.3“69 295.36 0.108“ 7.6“ 0.07 0.“112 296.16 0.1072 7.95 0.07 0.“295 295.13 0.1070 7.3“ 0.07 0.“8“3 296.02 0.1060 8.18 0.08 0.551“ 296.“3 0.10“9 8.28 0.08 (a) Taken from Table “.2 and adjusted to prOper tempera- ture using Table “.1. (b) Calculated from Equation (“.6). 96 thermoeffect experiments. These investigators all noted that the temperature effect was greater below the diffu- sion interface than above it. Our computer simulation concurs with the hypothesis for this effect advanced by Mason, Miller, and Spurling - the asymmetry in the tempera- ture effect is primarily due to composition dependencies of the transport parameters, particularly the thermal conductivity. Individual thermocouples were not used to fit the composition dependence of the thermal conductivity, however, because the theoretical T vs. t behavior at a given location, predicted with the inclusion of the large heat of mixing term, does not agree very well with observed behavior at long times. Reasons for this are not known, but may be due to wall effects or thermocouple effects. In addition to its intrinsic importance for liquid mixture transport theory and behavior, the diffusion thermo- effect can also be useful for exploring the critical solu- tion region. Anomalous behavior is often noted for trans- port prOperties near the consolute temperature. Attempts to measure thermal diffusion factors very near the critical temperature have been hampered by the large temperature gradients required to observe the effect. The diffusion thermoeffect should provide a valuable tool in this region since only very small temperature gradients are induced by the moderate composition gradients associated with critical coexistence curves. CHAPTER 5 LIQUID-LIQUID CRITICAL PHENOMENA A. Classical Thermodynamics of Liquid-Liquid Critical Phenomena At uniform temperature and pressure, the tendency of a liquid mixture to separate into two phases is governed by the requirement that the Gibbs free energy be a mini- mum at equilibrium. That is, the criterion for phase stability in a binary liquid system is a downward convexity of the free energy G (or the free energy of mixing GM) as a function of mole fraction at a given T and P (see for example Prigogine and Defay [195“] and Moore [1972]). Curve "A" of Figure 5.1 (Moore [1972]) depicts a system for which (32GM/3XI)T,P > 0 (GM is convex downward) over the entire composition range. This corresponds to complete miscibility of both components at all concentrations. If, however, the free energy of mixing for a binary mixture is similar to curve "B" of Figure 5.1, GM can be minimized (for those overall compositions between xi and x3) by a separation into two distinct liquid phases of compositions xi and xi. Curve "C" represents a system at the stability limit (critical solution temperature or consolute tempera- ture) where the two inflections of curve "B" have merged. 97 AGM/RT Figure 5.1. 98 *0 l , u T X. ’ _ XI 0- ’/ B \\ // \\ l -O.| \ C -O.21e - -O.3 - '- -0.4 " 0 -0.5 — "1 ~06 I- ~ ‘ A -07 _ _ - l 8 I I I O 0.2 0.4 06 08 IO X. Free energy of mixing vs. mole fraction of com- ponent l as illustrated by Moore [1972]. A. Complete miscibility. B. Two phase system of compositions x' and x". C. Phase stability limit. 1 1 99 The stability criteria at this point are 2 2 _ _ 3 3 (a GM/axl)T,P - 0 - (a GM/axl). (5.1) Liquid-liquid phase coexistence criteria could be written equally well as _ _ 2 2 (aUl/3X2)T’P - O ‘ (3 “1/3X2)T,P (5.2) with the additional restriction (a3ul/ax3) < o (5.3) at the critical point (Prigogine and Defay [195“]). The two-phase region corresponds to a horizontal line in a “1 vs x2 plot (Figure 5.2, Prigogine and Defay [l95“]); 1:9,, a region of two coexisting phases of compositions xi and xi with pi = pi. A coexistence curve (at constant P) for the system Q- hexane—nitrobenzene (Figure 5.3, Prigogine and Defay [195“] illustrates that the critical solution temperature (CST) is the maximum temperature at which two phases can coexist at a given pressure. The critical composition (xlc) istflmacom- position locus at which the CST occurs. Above the critical temperature TC a mixture prepared at any composition forms a homogeneous fluid phase. However, at 10 °C and 0.5 over- all mole fraction, for the system shown in Figure 5.3, 100 Figure 5.2. Behavior of the chemical potential of com- ponent 1 vs. mole fraction of component 2 for a critical system as depicted by Prigogine and Defay [195“]. The dashed line indicates metastable regions. 101 25— One Phase 20— _______ c _________ Tc I5— T°C Two Phases '01. ___________________ ESL- (> L l l 0-25 0-50 075 X Figure 5.3. Coexistence curve for the nehexane-nitrobenzene fystfm as depicted by Prigogine and Defay 195 l. 102 two phases of compositions xC H = 0.18 and xC 6 5 = 0.82 coexist. B. Critical Exponents 1. Definitions - Figure 5.“ captures a time sequence of the physicochemical response of the isobutyric acid (IBA)-water system to a slow decrease in temperature from T > TC to T < Tc along an isobar at the critical IBA mole fraction. Notice that even several degrees above the CST a change begins to occur. On a molecular level, A-A interactions relative to A-B interactions adjust rapidly. Local concentration fluctuations of dimension 5 increase dramatically, giving rise to a Tyndall-like light scatter- ing effect known as critical opalescence. This occurs when E acquires lateral dimensions on the order of the wavelength of light. Figure 5.“ (b), (c), (d), and (e) show the light scattering associated with an increasing E. An understanding of how macroscopic transport properties are affected by this molecular commencement of phase separation promises to yield valuable information about the relationship between molecular and macroscopic phenomena. Since the properties of the system obviously begin to adjust several degrees above phase separation (Figure 5.“), a set of indices known as critical exponents (CE) are used to correlate the temperature dependent behavior of proper- ties as the CST is approached (Stanley [l97l]). The 103 .cofiumsmamm mmmno mo coaumHoEoo .oeve A00 .mpfipansSp mmcoc an omxsms mm coapmsmaom omega mo pomco .H0.0I N oBIE A00 .Usoao moan: mmcmp mm mmfimflwcopca mocmomwflmqo HmOHuHso .06 H0.0 w eels A00 .mmmmopocH cocoommamao 0cm Umsmzoa mm 9 mm mom mean: 0 moEoowQ 0:: mafia .06 H a oBIB on .mcfiwon mocmomoamoo mm 0:: moan and: m moEooon manpxfie .0o m a oBIB ADV .09 AA 9 Amv .msduxHE anewpwso smpmzlvfiom ofiszusnoma on» no mszpmpoQEmp mpsaomcoo one on UmLmZOH we magpmpmo IEmp one mm mcoEocmza Hmoapfiso wasUHHIUfiSUfiH mo mucoswmm canomswouosm .:.m madman 10“ 105 limiting behavior of a property f(€) in the critical region is denoted A f(€) '9 e (5.“) where T-T e E C (5.5) T0 and = £nf(e) 1 _ :13 _7EET_ (5.6) It is important to realize that Equation (5.“) does not imply f(8) = Ask. In general, there will be correction terms which vanish as T + TC; i.e., f(e) = AeA(1 + Bey + . . .) (5.7) where y > 0. Figure 5.5 shows the results of a light scattering determination of the mutual diffusion coef- ficient D by Chu, Lee, and Tscharnuter [1973]. They plot log D vs.log e to obtain the CE as the slope of the resultant line. Note the contribution of the higher order terms of Equation (5.7) as illustrated by the devia- tion from linearity when T - TC 2 5 °C. 106 .mmwmau schnapmnome 0:0 .004 .5:0 as Umpzmmms mm pcmHOHmmmoo coamSMMHU Hmsuze can no coaumcHELopov wcfisopumom pzwaa was as 06 m z A O BIB pom oo .m.m magmas 107 .m.m assess Gov ._.I._. .O. OO— - TO. WIC. IO. — _ q _ _ _ _ _ D _ _7a a _ _ _ _ q _ with. i \e. I \\ .1 l O\ .1 HI. \\ l O \\ wro- O\ I o\ I 0\\ ll \ oxo 1380.83 I O\ I] I \0\ I. O I \o\ I l \Aqu .1 n \o I I o 0. 108 Although experimental determination of the critical exponent A does not provide the entire 5 dependence of f(s), the CE depicts the essential behavior sufficiently close to the CST. As Figure 5.6 shows, a negative CE characterizes a diverging function while a positive CE represents a vanishing function as 5+0. The larger |A| is, the further away from T0 the anomalous behavior appears. The use of the word "anomalous" in reference to the e-functionality of a property in the critical region indicates a deviation from the behavior predicted by extrapolation of the T-dependent behavior exhibited far from the CST. Some of the more common thermostatic properties which exhibit anomalous critical behavior have been experi- mentally characterized quite well and have specific sym- bols reserved for their critical exponents. Thus (X5 - xé) m IslB (constant P) (5.8) defines 8, 7,. t- (aul/axl)T’P m Isl (cons.ant P,X2c) (5.9) defines v+ where + indicates 6+0 from the positive side, and 109 Figure 5.6. Behavior of properties as functions of s for various values of the critical exponent A. 110 (A) f(‘) e (B) A>O Figure 5.6 111 EP,x m |€|-a+ (constant P; ch) (5.10) defines 0+. Table 5.1 lists some typical values for these thermostatic critical exponents as reviewed by Scott [19721. In addition to anomalous behavior representation, critical exponents are themselves fundamentally important. Recent emphasis on experimental determination of CE's has had a two-fold incentive: (1) the value of a particular CE transcends the system under investigation (universality) and (2) equalities between several exponents allow pre- diction of unknown CE's from known ones (scaling). 2. Universality - The theory of universality states that when allowance is made for any extra variables and when the properly analogous quantities are compared, the CE's for different systems are identical. An il- lustrative example is the exponent B which characterizes the temperature behavior of the order parameterl. Within experimental error, the order parameter for liquid-liquid systems (xi - xi), gas-liquid systems (pV - 0L), and magnetic systems (magnetization M) all exhibit the same critical exponent B = 0.33. 1An order parameter is the property that is nonzero for T 00 20x00 HHHmsm: I I080 camcoH COHpmHohpoo pom pcocoqxo HmoHpHso n > > .mwcmp mason u 0 A00 .cmpmHH mH Locusm pmsHm >Hco A00 2 mpHch flmmeH memmzmx was: >0 e 0 mommHg HHoppmm 00:2 >0 e 0 mopme memwzmx cowp0> ma 0 we AtaHsHEV HQOHL seazm 22 Hmemz pmoe pm 0 60 a 9 >0 8 0 0:0 mammHH mmocmvmx z m\H0 e 0 HmmmHg maosmvmx 000 .mmmmHu chuczoz .HemmHL £60560 9 z: Hiam\Hud e a Hiam\HI0 a y Huam\H6 2 a HammHL assesses z: and 2 ex HmemHL assume: 9 0 p0 soam m m\:I a m mzmeg H m coast: as 0 em a a mecmhdemm A60 A60 0 A60 Ahv A60 0 Apv A60 .mmesooosa phOdmcmpu mo mucmcoaxm HasHpHpo pow mCOHuoHcmsa Honumsomce .m.m 0Hpme 119 in a system in macroscopic equilibrium. The former method constitutes a "thermodynamic" experiment, while the latter usually involves light scattering techniques. Although macroscopic gradients lead to experimental dif- ficulties in systems near the consolute temperature, thermodynamic measurements are extremely important for understanding force-flux behavior in this region. Determination of transport coefficients from non- equilibrium thermodynamics involves linear hydrodynamic equations; $421: the coefficients are not gradient de- pendent. This is certainly valid for very small grad- ients. However, near the CST where the correlation length 5 diverges, coefficients may be nonconstant over distances on the order of g even for moderately small gradients. For this reason, several authors have suggest— ed that nonlinearities are to be expected sufficiently close to the critical point (e.g., Fixman [1962], Kawasaki [1966], and Grossmann [1969]). Nevertheless, experiments have failed to show any gradient dependence in measured coefficients. To the contrary, Woermann and Sarholz [1965] and Tsai [1970] have shown the shear viscosity to be constant for a change in shear rate of “ and 5 orders of magnitude respectively. Similarly, Michels and Sengers [1962]luuneshown the thermal conductivity near the gas—liquid critical point to be independent of AT. These results justify the assumption of linear 120 laws in the experimentally accessible neighborhood of the critical point - especially if gradient driving forces are kept small. 2. Thermal Conductivity — Thermal conductivity experiments (and pure thermal diffusion experiments) are plagued with convection problems. Convection is especially enhanced in the consolute region where thermal gradients can produce large density fluctuations. These difficulties have kept data scarce. Gerts and Filippov [1956] and Filippov [1968] meas- ured the thermal conductivity of nitrobenzene-nehexane, nitrobenzene-nyheptane, methanol-nrhexane, and triethyl- amine-water mixtures as T + Tc' The absence of convec- tion was demonstrated by independence of results on AT (this also further justifies the linear flux-force laws). The results for two of the investigated systems as re- ported by Gerts and Filippov are shown in Figure 5.7. As is the case for the mixtures depicted in Figure 5.7, none of the four systems evidenced any anomaly. Osipova [1957] did report an anomaly in the thermal conductivity of a phenol-water mixture. However, most reviewers (Sengers [1971]) suggest that the reported large scatter in data and large errors in measurement of AT are indicative of unreliable results. Although further experiments are desirable, thermal 121 .mcoucmnoanc nl.pz m.mm .m “0:00:0n0p0Hc R .03 mm .H unuHR 0coucmnoppHCI0CMpamnlc Empmmm Hmv .mcmucmnoppHc u .03 um + 0cmxonic u .03 m0_H<0 .mmmmHu >oaaHHHm 0:0 mppmo Ha umupoamp mm COmep HmOHano 020 CH >0H>Hposccoo Haemmne .>.m wastm 8.: 8oz. CON Hm. ow. now OdN am. on H q A 0 1m“ — _ i x x 10.010. L. .J n. 0.. ... a m in m. n w .6 m. w 0 ma? Cw I\ o 0 N0 mm 60 03 122 conductivity evidently remains finite as Tc is approached from the homogeneous fluid side. Thus, K = moo/T N 6°. 3. Mutual Diffusivity - Thermodynamic measurements of mutual diffusion coefficients near the liquid-liquid critical point (Kricheviskii 22 al. [195“], Claesson and Sundelaf [1957], Lorentzen and Hansen [1957] and [1958], Kricheviskii gt al.[1960], Haase and Siry [1968], and Balzarini [197“]) show unequivocably that D vanishes as T+TC. With the exception of Balzarini [197“], none of the above experimentalists report a critical exponent. The prominent feature of these more qualitative works is especially noticeable in Figure 5.8 where the represen- tative results of Haase and Siry [1968], for the water- triethylamine system exhibiting a lower consolute tempera- ture, and of Claesson and Sundelfif [1957], for the n: hexane-nitrobenzene system exhibiting an upper consolute temperature, clearly indicate that (aD/BT)xlc becomes infinite as the CST is approached. Differentiating with respect to T the expression for D in Table 5.2 yields A2+1/3 (3D/3T)X m e . (5.12) lc Recall from the discussion of critical exponents that in order for (SD/3T)X to diverge as 5+0, the critical lc exponent must be negative. With respect to Equation (5.12) Figure 5.8. 123 Mutual diffusion in the critical region. (A) Results of Haase and Siry [1968] for the water- triethylamine system exhibiting a lower con- solute temperature at 91.26 mol % water and 18.3 °C. (B) Results of Claesson and Sunde16f [1957] for the n—hexane-nitrobenzene system at equal mole fractions. 12“ (A) (B) D-Io’ (cmzsed') 20> Iol-l °2o 2'6 3‘0 3'6 T(°C) Figure 5.8 125 this means that A2 < -l/3. Reasoning along these lines constituted the first evidence that 011 diverged at the critical point; 1,3,, that the observed anomaly in D was not strictly dependent on the thermodynamic factor Ell“ Balzarini's more recent thermodynamic experiments yield 0.7“i0.08 for the CE of D. Recently, values for A2 have been obtained much nearer the critical point by light scattering experiments. Light scattering measures the decay rate of concentration fluc- tuations and therefore eliminates the need for macroscopic gradients and system-perturbing response measurement devices (for example, the thermal lens effect associated with interferometry, Giglio and Vendramini [197“1). As Table 5.“ illustrates, the diffusion coefficient is un- I questionably represented by D 0 ex with A' = A2 + “/3 = 2/3. This value (and the values in Table 5.“) compares favorably with the exponent for thermal diffusivity in gas-liquid systems, K/OCP m e)' where 0.61 g 1' i 0.69 (Sengers [1973]). As was mentioned earlier, universality requires that comparison of like modes in different systems yield identical exponents. Mutual diffusivity in liquid— liquid systems corresponds to the thermal diffusivity mode in gas-liquid one-component systems. The kinetic A2 contribution 0 must therefore diverge as 011 m e 11 with A2 = -2/3 as T + To. 126 Table 5.“. Light scattering results for the mutual dif- fusion critical exponent. System Exponent Reference(b) Isobutyric acid-water n-hexane-nitro- benzene 3-methylpentane- nitroethane cyclohexane-aniline perfluoromethylcyclo- hexane-carbon tetra- chloride lutidine-water phenol-water methane-tetrafluoro- methane 0.68:0.0“ 0.62:0.02 (c) 0.66:0.02 0.63 (c) 0.62 (c) 0.61:0.01 0.66(5):0.o3 0.63:0.005 (c) 0.55“:0.015 0.68:0.03 0.67:0.02 Chu [1968] Chu [1972] Chen [1969] Chu [1972] Chang [1972] Berge [19701 Chu [1968] Chu [1972] Gfilary [1972] Goldburg [19721 and Bak [19691 Blagoi [19701 (a) Exponent refers to A' in Dmex' (b) Only first author is listed. where A' = A2+“/3. (0) Indicates renormalized values after taking into account the "regular part" and temperature dependence of the viscosity. 127 “. Thermal Diffusion — Much like thermal conductivity experiments, thermal diffusion measurements require sub- stantial macroscopic temperature gradients which may induce convection. Furthermore, the consolute tempera- ture cannot be approached very closely with tempera- ture gradients present. For these reasons, early meas- urements cd‘ DT in the liquid—liquid critical region yielded, at best, qualitative results (Thomaes [1956], Tichacek and Drickamer [1956], and Haase and Bienert [1967]). It is instructive to plot the data of Thomaes and those of Tichacek and Drickamer in a typical log- log plot of the thermal diffusion ratio KT versus 6 so as to obtain from the slope an effective value for the CE. This is done in Figure 5.9. Note the large scatter in data and more importantly the large discrepancy in the critical exponents or slopes obtained. The data of Haase and Bienert [1967] on the water-triethylamine system are not plotted here for two reasons: (1) the data are not given at the critical composition, and (2) the consolute temperature is not approached sufficiently closely for comparison purposes. Although all three sets of thermal diffusion data indicate that KT diverges as 5+0, Haase and Thomaes qualitatively argue that DT also diverges while Tichacek indicates that DT vanishes. That is, 1 Thomaes's results indicate DT % a“ while Tichacek and Drickamer's data yield DT N 6+l/3. 128 4. I I T I ' slope —1.7 Thomaes 3 " d F- 'o x E 2 - ' - slope -O.3 Tichacek 8. _ Drickamer a 1 1 I I l -5 -4 -3 In 6 Figure 5.9. Log-log plot of the thermal diffusion data of Thomaes [1956] for the gehexane-nitrobenzene system and of Tichacek and Drickamer [1956] for the perfluoro-nrheptane+2,2,“-trimethyl- pentane system. 129 Giglio and Vendramini [1975] claim the first "ac- curate measurements of the thermal diffusion ratio KT in the neighborhood of the consolute critical point of the mixture aniline-cyclohexane." Their measurements were performed using a classical Soret cell and a steady- state beam-deflection technique. By evaluation of the time evolution of beam deflection, they also obtained the temperature dependence of the mutual diffusion co- efficient. From these data, the behavior of DT was ob- tained. Giglio and Vendramini's [1975] log-log plots of KT, D, and DT are shown in Figure 5.10. In this figure, the line through the diffusion data are the light scat- tering results of Berge gt il- [1971]. Values for DT are calculated from DT = KTD. The relatively good agree- ment for D with the light scattering experiments of Berge gt_al, [1971] seem indicative of reliable results. The "best-fit" CE value for KT is A" = -0.7310.02 in the expres- II A . However, Giglio and Vendramini [1975] 1 sion KT m 6 conclude that KT m D- % 8-2/3 because calculation of DT shows it to be temperature independent; 142;, DT N 6°. The slightly larger exponent determined from the least squares fit of KT is attributed to a deviation of the II A in the region where numerous data points relation KT = A6 are located. That 15, KT is expected to behave more like Equation (5.7) further away from the critical point. Inclusion of points in this region leads to an effective 130 5x05 ..mn . mum. r so _ ’//9’4(’.q - /° ’8 Ida:— \.\. D/D 1 IO ii - ‘- / - k I— \ .1 T N\ : f’K : LE) _ D‘DD)’ \(k‘r H 2;; [ D’B/l ‘>\‘ d _ D)’ \- 0 I057 I IIJJJl I I I,IiIIIIl I | 0.3 I.O I00 30.0 A IO’5 ’ —.-.-O'.'0.0‘O‘O O-.—O—.-—.—.—.— ' CQEE C) I I II III I, I I I llilll I 3 03 IO no.0 30.0 at T-TC co Figure 5.10. Thermal diffusion and mutual diffusion results of Giglio and Vendramini [1975] for the critical region of aniline-cyclo- hexane mixtures. 131 exponent of larger magnitude than the true exponent (of. Figure 5.5). 5. Heat of Transport - The diffusion thermoeffect measurements discussed in Chapter 6 are believed to be the first direct evaluation of the temperature dependence for Q: near the consolute point. Although Haase and Bienert [1967] calculated 0: from Thomaes's [1956] thermal diffusion data, the values obtained were meaningless be- cause (1) Thomaes's data are inconsistent with the more accurate work of Giglio and Vendramini [1975], and (2) no verification of the Onsager reciprocal relation in the critical region has ever been made. To indicate further the need for a direct study of the heat of transport, note that the critical exponent obtained for 0: on the basis of ORR is positive (+2/3) if the data of Giglio and Vendramini are used, positive (+1) if the data of Ticha- cek and Drickamer are used, but negative (-l/3) if Thom- aes's data are used. Table 5.5 summarizes the results of the preceding sections for the four transport properties of interest to the present discussion near the critical demixing point of a binary liquid mixture. The indicated ignorance of the temperature behavior of Q: as T + TC gives impetus for the measurements described in the next chapter. 132 Table 5.5. Literature transport parameters and their critical exponents. Behavior Critical Property Definition Near CST Exponents K K = GOO/T K N 900 K N 8° 9 N 5° “11:11 — 00 2/3 D D = "Bfi“ D ” S211u D m E 2 ll _ “11 ” Eu/B Q _ lO (a) -Q Q a = ___i0_ 0. q, ____]_-2_ C! "b 8-2/3 (3) l wlw2pD l a — l 11”11 K m 5‘2/3 (a) T 9 w 9 = 10 2 10 (a) KT ET'ET" KT m allfill 910 m so 11 11 Q Q '16 _* _* .96 o 61 Q1 = 69$ Q1 “ 5%% Q1 “ 8' ll Q01 “ 5? (a) Results of Giglio and Vendramini [1975] are used. CHAPTER 6 THE HEAT OF TRANSPORT IN THE CRITICAL SOLUTION REGION OF ISOBUTYRIC ACID-WATER MIXTURES A. Transport Equations To evaluate the temperature dependence of the heat of transport Q: as the CST is approached, the partial dif- ferential equations describing the diffusion thermoeffect must be solved and the solution fitted to the experi- mental points. Although this technique was introduced in Chapters 3 and A for the CClu‘Efcsng system, each equation with its underlying assumptions must be checked for correctness in the critical region before use. The starting partial differential equations of Chap— ter 2 can be written (Bo/3t) + (Bov/az) = O , (6.1) 0(3Wl/3t) = {8[oD(8wl/az)1/Bz} - pv(8wl/az), (6.2) and pCfi(3T/3t) = pD[8(Hl-Hé)/Bz](3wl/Bz) - onv(3T/Bz) + {3[oD§:(3wl/Sz)]/Bz} + {3[K(8T/Bz)1/Bz . (6.3) 133 134 The assumptions made in obtaining Equations (6.1) - (6.3) are 3 (l) (2) (3) (A) (5) (6) (7) The linear hydrodynamic equations for conservation of mass and energy are valid. The binary system is isotropic, nonreacting, and field free. Local states are assumed; i;g;, the equations of thermostatics apply for local regions. Fluxes are linear combinations of those forces which appear in the entropy production equation and which have the same tensorial rank. Pressure terms are negligible. The bulk flow entropy source term is small. The termal diffusion portion of the mass flux is small compared to the diffusion portion. Assumptions (2), (3), and (5) are obviously as cor— rect near the CST as away from it. Assumptions (1) and (A) were discussed in Chapter 5. The demonstration that n, Kg-l (thermal conductivity near the gas-liquid critical point), and Kl_l (thermal conductivity near the liquid- liquid critical point) are independent of their respective driving forces is indicative of linearity in the critical region. Assumption (6) is also valid near the consolute point because the bulk flow entropy source term is pro- portional to the square of the barycentric velocity which is itself small (especially for the system to be 135 investigated here). Assumption (7) must be dealt with somewhat more care- fully. The mass flux, on the basis of the above assump- tions, can be written -Jl = oD(8wl/az) - pDolwlng’lCBT/az). (6.u) The previous chapter (cf. Table 5.5) demonstrated that the most accurately determined temperature behaviors of D and al in the critical region are D m e2/3 and a m 1 8-2/3, respectively. Thus, while D vanishes in the criti— cal region, al becomes large at about the same rate. Al— though the first term in Equation (6.A) vanishes, Dal in the second term remains finite. At first sight, it appears that the thermal diffusion term could be impor- tant sufficiently close to the CST. However, (awl/Bz) is always much larger than (aT/az). Away from the CST, Ingle and Horne [1973] estimate the maximum contribution of the thermal diffusion term to be 0.01%. With this estimate for temperatures away from the CST, calcula- tions show that the critical point must be approached to within about 0.0l°C before the decrease in D allows a 1% contribution to J1 from the thermal diffusion term [at constant (aT/Bz) and (awl/az)]. The experiments described herein show that (ST/82) also vanishes as the CST is approached while (awl/Bz) remains finite. Hence, 136 the thermal diffusion contribution to the mass flux will never reach 1% even for temperatures very near Tc and can be safely neglected. With all assumptions thus verified, Equations (6.1) - (6.3) can be used for critical mixtures. The isobutyric acid-water system (IBW) is particularly convenient for measurement of Q: in the critical region because of the very similar densities of the pure com- o a 3. o = ponents (pIBA,20°C 0.958 g/cm , pH20,20°C 0.9989 g/cm3). Regardless of the compositions in the initial layers, the density of the system will be essentially invariant with respect to position and time. Equation (6.1) then simplifies to (av/az) = o . (6.5) Integration of Equation (6.5) and application of the physically imposed boundary condition that the velocity vanish at the wall yields the trivial solution for the barycentric velocity v E 0 . (6.6) Large concentration fluctuations characterize the liquid-liquid critical region. For systems in which the components have very dissimilar densities, density fluc- tuations result. The gravitational field will produce 137 density gradients in such a system (Mistura [1971]). Gravitationally induced density gradients have been meas- ured fin» a few systems very near the CST where the sedimentation (pressure diffusion) coefficient diverges (Giglio and Vendramini [1975] and Greer et a1. [1975]). As Morrison and Knobler [1976] indicate, the presence of a gravitational field poses no problems for this system because of the nearly equal pure component densities. As was done in Chapter 2, Equations (6.2) and (6.3) can be transformed into equations involving molal param- eters and mole fractions. This transformation with the use of Equation (6.6) yields analogous equations for both composition and temperature: -D'l(3xl/at) + (32x1/822) + {a[2n(D/vw)J/az}(ax1/az)=o (6.7) and -Ep/vn(aT/at) + (32T/az2) + (aInK/az)(aT/az) = K'1{a[M2Dé:(axl/az)/vM]/az} + DK‘lv'1(32fiE/axi)(axl/az)2 . (6.8) The initial and boundary conditions are the same as before 138 (of. Chapter 2), L xl(0.5 z/a:l,0) = x3; x1(0:z/a 0.5,0) = x1 T(z,0) = constant (6.9) and (3X1/32)z/a=0,t = O = (axl/az)z/a=1,t (6.10) (aT/az) = o = (aT/az) z/a=0,t z/a=l,t The solutions of Equations (6.7) and (6.8) subject to Equations (6.9) and (6.10) for a known set of parameters can now be numerically obtained using the previously described program based on the Crank-Nicholson finite dif- ference scheme. B. Experimental 1. Cell Considerations - Measurements of the diffu- sion thermoeffect near the consolute temperature cannot be performed in the "liquid gate" withdrawal cell des- cribed in Chapter A because there appears to be no liquid which is both (1) immiscible with both components and (2) of intermediate density. However, liquid phase 139 behavior in the critical demixing region allows design of a much simpler cell. The "liquid gate" withdrawal cell used a third com- ponent to create the sharp diffusional interface. Systems that exhibit partial miscibility regions near room tem- perature are usually dissimilar enough that finding a third mutually insoluable component is virtually impos- sible. Even could such a liquid be found, it is undesir- able to introduce a third component because of the large effect minute concentrations of impurities have on the absolute consolute temperature. The consolute tempera- ture for IBW is known to be particularly sensitive to ionic impurities (Gammell and Angell [197“] and Greer [1976]) which tend to lower the CST dramatically. Al- though small amounts of impurities may affect the absolute TC by several degrees, critical exponents and temperature dependencies of properties relative to the measured CST are not influenced by the presence of impurities (Sengers [1975], Hocken and Moldover [1976], Bak and Goldburg [1969], and Fisher and Seesney [1970]). Instead of a mechanical (or fluid) technique to create the initially sharp interface, the natural, stationary, and unperturbed interface present in a binary mixture for T < TC can be utilized. With the binary mixture thermo- statted at T < T two phases characterized by ui = pi C, are in equilibrium. If the temperature of the entire 1A0 diphasic fluid slab is suddenly jumped via microwave absorption such that T > Tc’ then pi # pi and diffusion accompanied by the diffusion thermoeffect must begin. Very gradual relaxation of T back toward Tc allows meas— urement cd‘ the difference in temperature AT, caused by the diffusion thermoeffect, between symmetrically placed thermocouples as a function of e. Thermostatting of the mixture at T < Tc allows reuse of the same mixture in subsequent runs (time must be allowed for phase equilib- rium to occur). The cell designed to perform the above experiment is shown in Figure 6.1. This cell, constructed of 2.0 mm thick glass, has an inside height of 1.2 cm and an inside diameter of A.8 cm. The relatively large ratio of diameter to height minimizes wall effects. As shown in Figure 6.1, two tiny thin-walled glass, closed, conical tubes project into the radial center off the cell from opposite walls. These tubes, which serve as thermocouple wells, extend from the wall attachment site at haltheight [(z/a)=0.5] to positions (z/a) = 0.80 and (z/a) = 0.20 equidistant from the cell half-height (interface formation was at half- height). The average inside tube diameter is 0.A mm. A small stopcock atop the cell prevents vapor loss (hence concentration changes) throughout the experiment but still allows pressure equilibration during the temperature jump. Large pressures result when liquid systems 1H1 Figure 6.1. Temperature jump cell for diffusion thermo- effect experiments in the critical region. 1112 1A3 encapsulated in a closed container are temperature jumped due to the thermal expansivity of the liquid. Thermal insulation for the cell is a composite wall of 1.3 cm ® ® Styrofoam , 0.67 cm Acrolyte (to eliminate air cur- rents through the Styrofoam), and 2.0 cm Styrofoam. Small holes on opposite sides of this assembly allow thermo- couple insertion. Thermocouples were made from calibrated copper-con— stantan A0 gage thermocouple wire by welding a small junction. Response time of the thermocouple wells was enhanced by insertion of a small drop of mercury into each closed tube. Thermal equilibration times, checked for similar thermocouple wells by monitoring the mean time required for the potential to relax to 0.0 pV when the probe was suddenly introduced into the thermocouple reference bath, were about 2.5 seconds. Thermocouple connections were made to the potentiometer facility such that relative temperatures (the upper thermocouple ref- erenced to the lower) could be directly measured in addi- tion to absolute temperatures. This allowed relative temperatures or temperature differences to be measured to :0.1 “V (g_a_. o.oo2°c). 2. Critical Temperature Measurements - Each experi- mental determination of the critical exponent for the heat of transport also involved measurement of the critical iuu solution temperature. As previously mentioned, consolute temperatures vary significantly with small impurity con- * centrations. In order to obtain Ql as a function of T-T the CST determination was made on a portion of the 0’ same mixture immediately before and after the T—jump experiment. The literature value for the critical com- position (xlc = 0.111 - 0.11“) was used without further verification. The cell shown in Figure 6.2, constructed of 1.5 mm glass, was used for CST measurements. After filling with the homogeneous critical fluid (x1 = xlc’ T > Tc), the cell was sealed against vapor loss with the small glass stopcocks shown. The consolute temperature drifted less than 0.03°C during a period of over a week, indicating essentially no change in concentration from vapor loss. The outer water jacket depicted in Figure 6.2 con- trolled cell temperatures with circulating water from a Neslab Tamson T-9 (10 L capacity) constant temperature bath. Cooling water to the T—9 bath was from a Lab- Line Tempmobile (90 L capacity) while current to the heating element in the T-9 was maintained by a model 2156 Versa-Therm Proportional Electronic Temperature Controller. Determination of T0 was visual. The onsets of both phase separation and phase disappearance were ascertained by careful temperature adjustment. Phase separation 145 Figure 6.2. Critical solution temperature cell. 1U? temperatures agreed with phase disappearance tempera- tures to within 0.010°C. The visual technique of de- termining TC is illustrated in Figure 5.U. Photographs (d) and (e) show that with T slightly above Tc’ critical opalescence deepens from a light white fog to a dense white cloud. The onset (or disappearance) of a turbid cloud in the stirred opalescent mixture marks the phase separation (or disappearance) point. Figure 5.A (e) depicts the turbid dense cloud observed for tempera- tures just below Tc' Transition between states depicted by photographs (d) and (e) is rapid with respect to tempera- ture change, allowing determination of TO to about t0.005°C. Maintenance of constant temperature without stirring for several minutes to observe meniscus formation was periodically used as a check on the stirred visual technique of CST determination. Figure 5.4 (f) illustrates meniscus formation for T < Tc‘ All temperatures were measured with copper-constantan thermocouples similar to those used in the diffusion thermoeffect cell as des- cribed in the preceding section. The thermocouple well visible in Figure 6.2 was a small mercury-filled glass capillary tube into which the welded junction was inserted. The length of this well positioned the welded thermocouple junction at cell half height. 1A8 3. Experimental Procedure — Fisher Certified Reagent Grade isobutyric acid was used without further purifica- tion. However, Karl Fischer analysis of water content in the isobutyric acid yielded 0.070 wt. % water. This was accounted for in preparing the mixtures at the critical composition. Distilled, deionized water was used for the second component. Mixtures were prepared by additive weighing of pure components in separate two-armed 50 mL bottles equipped with stopcocks on each arm to prevent vapor loss. Excess vapor space was minimized, and no vapor loss with time could be noticed gravimetrically. The two pure component weighing bottles were then connected with a short piece of tygon tubing and thermostatted above Tc' Subsequent transfer of the pure components (through the connected sidearms) back and forth between the two weighing bottles served to mix the components while maintaining a sealed environment. With the mixture prepared and located entirely in one of the two bottles, the side arm stopcock was closed and the second bottle was removed. Transfer of the homogeneous mixture through the top arm of the filling bottle fitted with a short piece of narrow tygon tubing, into the T-jump cell of Figure 6.1, was completed quickly with T 3 Tc' To exclude any vapor space, the cell was filled above the stopcock region and the stopcock was then closed. The cell of Figure 6.2 1H9 used for CST determinations was immediately thereafter filled in an analogous manner except that a small vapor space was left. All glassware was thoroughly washed, rinsed in deionized water, oven dried for several hours, and filled immediately upon cooling before each use to eliminate adsorbed water and ionic impurities. Phase equilibrium was established with the cell sitting un- perturbed. Occasionally it was necessary to rotate the cell carefully to dislodge "droplets" of discontinuous phase from cell walls. A few days were assumed sufficient for equilibrium to be established. Actual experimental runs were made in the following manner . (l) Filling and stirring of the ice point thermo- couple reference bath with distilled water and finely ground ice. (2) Observation of the initial temperatures of both thermocouples and any difference in reading between them. (3) Removal of the thermocouples (microwaves were absorbed by the coatings and insulation on the wires). (A) Simultaneous activation of the Litton industrial microwave oven and the digital, 0.1 second readout timer. (5) Disengagement of the microwave temperature jump after a 1.0 to 2.2 second heating pulse. (6) Careful insertion of thermocouple leads into appropriate wells. Temperature readings as a function of 150 time were begun. (7) Acquisition of temperature data. Temperature difference readings were obtained at 20 to H0 second intervals and absolute readings were taken about every 100 seconds. Absolute readings of the two thermocouples were taken within about 10 seconds of each other. (8) Reestablishment of phase equilibrium after the temperature had relaxed below Tc‘ The cell was set aside for future runs. (9) Measurement of TC in the critical solution tempera- ture cell. This was done both immediately before and im- mediately after each run. Values obtained for phase separation and phase disappearance were averaged. A. Data Analysis - Any one run consisted of absolute temperatures at each thermocouple as a function of time, temperature differences between the thermocouples as a function of time, the consolute temperature Tc, and the initial temperature of the cell. The required data to )4 ~* fit Q1 8 A6 are AT vs. (T -TC) and the initial phase cell compositions. Data reduction thus involved: (1) Fit of absolute cell temperatures to a poly- nomial in time. Program "MULTREG" (Anderson [1968]) yielded a cubic equation for each thermocouple. These were averaged to give the instantaneous interfacial temperature. From the measured TC, (Tcell-Tc) was 151 available at any time. (2) Determination of the upper and lower phase con- centrations x3 and x§, respectively. The concentrations were calculated from knowledge of the initial temperature. Figure 6.3 shows the coexistence curve (temperature-com- position relation) for IBW determined by Woermann and Sarholz [1965]. A more accurate method of obtaining phase concentrations for a given temperature uses the known critical exponent B for the order parameter; iLQL, xi - xi = C(Tc-T)l/3. A plot of (Tc-T)1/3 vs. xl contain- ing the data of Woermann and Sarholz [1965] and Chu g3 a1. [1968] is shown in Figure 6.“. A least squares fit of the data yielded ‘ ll X10 + (Tc-T)l/3/l“.518 (6.11) >4 ll 3 x1e - (Tc-T)1/3/25.680 for the isobutyric acid rich (upper) layer and the water rich (lower) layer, respectively. These equations are the solid lines in Figure 6.“. Initial phase compositions were thereby readily calculable from the initial tempera- ture. (3) Provision of AT vs. (Tcell’Tc) data from directly determined AT vs. time data and from fit (Tcell-Tc) vs. time data. From these, coupled with the initial phase 152 l l I I I T 00 I— —J — -1 -1.0 b - 8 .— — S. 1"- -2.0 '."' - F0 -30 - -I I— I q _4.0 J J I l J l 6.0 8.0 10.0 12.0 14.0 ’ 16.0 18.0 MOLE PERCE NT IBA Figure 6.3. Coexistence curve for IBW as determined by Woermann and Sarholz [1965]. 153 Figure 6.“. The function (Tc-T)l/3 vs. mole fraction of isobutyric acid. a, data of Woermann and Sarholz [1965]; 0, data of Chu £3 31. [1968]; -——3 least squares fit. 15“ I I I T I IND- .~E’ . A 'T o t D oa— ' , D D O o '3 O I I I I I (M03 (MC) CH2 (1H3 0W6 MOLE FRACTION OF IBA Figure 6.“ 155 A ~* compositions, Q1 = A6 “ was numerically fit by weighted, nonlinear least squares regression of AT as calculated from Equations (6.7) - (6.10). As before, program "KIN- FIT“" was used ("KINFIT“" is the 1977 version of the orig- inal "KINFIT" published by Dye and Nicely [1971]) inter- meshed unifii the numerical integration routine. C. The Temperature Jump Technique The advantages of the temperature jump technique are apparent: (l) The natural, stable interface between the two coexisting phases is undisturbed by the shearing action associated with mechanical formation techniques. Impuri- ties are not introduced as they would be with a liquid extraction technique. (2) There are no moving parts susceptible to leakage and vapor loss. (3) There is no ambiguity in mixture preparation. Half-cell techniques require two phases of different com- position which, when completely mixed, are at the critical composition. Use of the T-jump cell allows filling of the cell at the critical composition. For temperatures moderately below Tc, the difference in composition between the coexisting phases Axl is not large, and a distinct interface with no preferential wetting of the walls (no 156 curvature in the meniscus) is formed. (“) No special thermostatting is required to main- tain separate phases at a prescribed temperature (T > To) during interface formation. It is desirable to make the temperature jumps as short-lived as possible. Temperature jumps of long dura- tion obscure the initial time to. As an example, consider a very long duration heating input, say by conduction. Even before the cell temperature reaches TC, the concen- trations of the two phases begin to change via diffusion. Although the two phases are not yet completely miscible, they are no longer at their equilibrium concentrations and some diffusion will occur. Such behavior cannot be des- cribed by the previous equations. This problem was avoided by use of a commercial Litton Industries microwave oven which supplied a short duration, moderate intensity heat- ing pulse. Moreover, the pulse supplied uniform bulk heating rather than surface conduction heating. Heating constants for the previously described T-jump cell filled with the critical mixture were about 7°C/S. Total heat- ing time was between 1.5 and 2.2 seconds. Despite bulk heating by the microwave oven, some nonuniformities are to be expected because of geometrical asymmetries with respect to the microwave source within the oven. Shortly after heating, it was found that the upper thermocouple often read 0.6°C higher than the lower thermocouple. 157 Although these temperature nonuniformities are not pre- dictable, the time behavior of the cell temperature dis- tribution after they have been measured is calculable using the diffusion thermoeffect program developed in Chapter 3. Fortunately, temperature nonuniformities relax via thermal conduction to the correct Dufour-effect- caused AT within 500 to 800 seconds. This is because the temperature gradients diminish by conduction until con- duction just balances the heat transported by diffusion. Figure 6.5 illustrates this behavior for various initial temperature nonuniformities. All curves in Figure 6.5 were obtained by computer simulation for nonuniformities symmetric about (z/a) = 0.5. Although all curves refer to a mean cell temperature “°C above the critical point, each individual curve corresponds to a different initial temperature nonuniformity. Curve "a" corresponds to the normal AT induced by the diffusion thermoeffect from initially isothermal conditions. Curves "b" and "c" correspond to the AT induced when initially the top (b) or bottom (c) 5% of the fluid is 2°C warmer than the bulk liquid while the bottom (b) or top (0) 5% is 2°C colder. This might physically correspond to a surface effect in the temperature jump. Curve "d" directly corresponds to two different initial temperature nonuniformities: (l) the initial temperature distribution in the cell varies continuously and linearly from the upper surface to the 0.4 0.3 0.2 0.1 AT 0.0 -O.1 -0.2 -0.3 158 I I T r l d - ,. rc F A L l l I l o 4 8 12 16 20 t (102 sec) Figure 6.5. Predicted decay of initial temperature nonuniformities in a temperature jump dif- fusion thermoeffect experiment. 159 “°C colder lower surface, and (2) the entire upper phase is initially 2°C warmer than the entire lower phase. This latter nonuniformity might occur for preferential absorption of microwaves by one of the components. Curve "e" corresponds to an initial linear and continuous grad- ient of 8°C from top to bottom. The point of Figure 6.5 is the coalescence of AT for all these nonuniform initial temperature conditions into the identical AT produced by the diffusion thermoeffect with isothermal initial condi- tions. This occurs in each case within 500 to 800 seconds. Thus, in spite of moderate initial tempera- ture distributions produced by the T-jump technique, the AT measured after 800 seconds is dependent upon only the heat of transport, not the initial conditions. For moderate T-jumps, AT values obtained at times longer than 800 seconds can therefore be used without knowledge of the actual T-distribution immediately following the heating pulse. Since relaxation to the mean temperature is quick— est near the interface where diffusion occurs, no ambi- guities in the composition distribution produced by heat- ing nonuniformities are expected even though D has a significant temperature dependence in this region. No experimental data for times shorter than 1050 seconds were used in the calculation of 0 To verify further * 1' these computer simulations, pure water was T-jumped with the microwave oven. With T-jumps comparable to those 160 used for IBW, similar initial temperature nonuniformi- ties were noticed shortly after perturbation. However, the measured AT vanished completely after about 500 seconds in the case of pure water. Obviously, the AT measured for the binary IBW system, persistent through- out our measurement region (1050 seconds i t i 5000 seconds) depends solely on the diffusion thermoeffect. The T—jump technique is useful near the CST basically because the diffusion coefficient diminishes in this region while the thermal conductivity coefficient remains finite. This changes the diffusion thermoeffect from a transient phenomenon (Figure “.“) to essentially a steady state phenomenon for times on the order of these experi— ments. To see how this happens, compare the composition sur- faces shown in Figures 6.6 and 3.5. Note that Figure 6.6 shows that the gradient of composition, the main driv- ing force for the diffusional process, remains almost constant throughout the experiment except for an initial blurring of the sharp step function at the interface. The mass flux j1 = -pD(3wl/Bz) therefore remains prac- tically constant in time for a given temperature. Heat conduction down the produced temperature gradient opposes the heat carried by the mass flux and will reach a point where it counter balances production of the gradient by the heat of transport. The transient phenomenon observed 161 COHPOSITION SURFRCE RS T HPPRDRCHES TCRIT Figure 6.6. Predicted composition surface for IBW in a temperature jump diffusion thermoeffect experiment “°C above Tc' 162 away from the CST is due to the constantly diminishing mass flux. To quantify this point, the equation gnet = gHT T Scond = 116: ’ KAT (6°12) (analogous to Equation 2.20) where gHT is the heat flux due to the heat of transport and Scond is that due to thermal conduction, shows that in the steady state gnet = 0 and j1§:/K=AT. Because in the critical region at any given temperature jl§:/K changes only very slowly, aT remains essentially constant. In terms of the actual experiment, the effect of this steady state is that perturbations from the AT produced by the diffusion thermoeffect, will relax back to the correct value. The establishment of the steady state is rapid since thermal conduction remains large and finite in the critical region while diffusion dim- inishes. Because near the critical point a steady state is established between thermal conduction and the heat of transport term, the cell temperature may be allowed to relax gradually toward TO. The assumption is that the steady state is established more quickly than the finite drop in cell temperature. That is, the measured AT at any instant is the appropriate steady state AT produced 163 ~* by Q1 based on the instantaneous cell temperature; i.e., ~* Au Q1 = A[(Tcell-TC)/TC] . This implies that measured AT values are essentially uncorrelated. They depend only on the composition profile and the immediate deviation of the cell temperature from the critical temperature and not directly on any past history of AT or T The cell' measured AT as Tcell changes is always the appropriate AT relative to the instantaneous cell temperature because in the critical region the thermal conductivity is always much larger than the diffusion coefficient thereby rapidly establishing the steady state for slow changes in abso- lute cell temperature. Verification of this assumption was checked numeri- cally by comparing simulated AT's for two kinds of systems. In system 1 Tcell decreases to T0 at a rate of about 10"3 °C/s (comparable to the experimental situation). System 2, following the T-jump, remains at a fixed mean tempera— ture.1 A comparison of the induced AT in system 1, when the decreasing cell temperature corresponded to that of system 2, to the induced AT in system 2 was made after correcting for small composition differences due to the temperature dependent diffusion coefficient. The results of this comparison are shown in Figure 6.7. In each case, the AT expected in system 1 as Tcell + T0 * 1For these simulations, Ql values from Table 6.3 were used. 16“ .m Empmmm mo was» on Hmofipcowfi mfi HHooB :mnz H Empmmm cfi B< Umpofipopo .I mm mamummm Hmfihmnp Iowa CH B< .IIII ma Empmmw CH E< popOprpQ . .09 I HHooB mm pommmm IoEpmcp cofimSMMHU pom mocwcwpcfime mumpm zpmmpm mo cowumOHmem> pmpSQEoo .~.m mhswfim 165 >.m mhswfim . 8.: 07* mo mé mN md m6 — _ . 1 _ - — J 3mm No: a mm «a on mm mm . w. v_ o. q . q _ _ _ 11. _ OOAV L No.0 I v0.0 I mod ._.< 4 mod I 08 I N5 _ 166 is identical to that predicted for system 2 when compared at the same mean cell temperature. In Figure 6.7, the dashed lines represent steady state AT values for systems 2, the solid line represents system 1 AT values, and the -black squares are system 1 AT values at each of the system 2 cell temperatures. Note that when T l of system 1 cel reaches Tcell of each system 2 as indicated by the black squares, the expected AT values are indeed identical. Experimental evidence that measured AT values are the appropriate steady state values at the instantaneous temperature was obtained by performing a similar T-jump experiment on a pure component - water. After the T- jump, no difference in temperature was measured between symmetric thermocouples (aT = 0) throughout the time region of the measurements (500 seconds i t i 5000 sec— onds) during which the cell temperature dropped “°C. No effects on the measured AT were due to the small heat losses through the walls required to allow the decrease in Tcell' Measured AT values for mixtures are therefore due entirely to the diffusion thermoeffect. An obvious advantage of allowing Tcell to approach Tc is that each experiment contains the entire e—behavior of 0: from which the CE 0“ can be obtained. Notice also that AT becomes very small as the consolute temperature is approached, allowing measurements very near Tc' How close measurements can be made to Tc is limited by the 167 cell temperature distribution. No meaning can be attached to any result for which T < Tc in a portion of the cell. However, AT decreases as e-+0, allowing closer and closer approach to the consolute temperature. A few measure- ments were obtained within 0.010°C of the consolute tem- perature. D. Literature Parameters for IBW Fitting O: and its temperature dependence from mea- sured temperature differences requires fitting of the values calculated using the previously described numerical scheme. Literature values of the equilibrium and trans- port properties of the IBW system were used in this pro- cess. Composition dependencies of the parameters were included as polynomial expansions in mole fraction by fitting literature data using "MULTREG" (Anderson [1968]). The temperature dependence was included via critical ex- ponents where known and applicable, and by polynomial fitting for properties with no anomaly in the critical region. Table 6.1 summarizes the actual expressions used for properties discussed below. (1) Critical properties - The reported values of the critical mole fraction vary from 0.110 to 0.115. (Greer [1976], Woermann and Sarholz [1965], Chu gp'al. [1968], Allegra a; a1. [1971], and Friedlander [1901]). Because the consolute temperature is lowered by 1638 .mmsHm> HmucoEHpoaxo ccw oopmHonmo Ho :oHumH>oc mumscn came uoop Hwy IxHIe. 2 mIo Summm. mIIMu mI m NIxHIE. .3 mIOHxHHm. Hum Ix Is.3 cmoo.ouoyo Ix Is.3 I Saw. HIIN Hm H H H m m N m m HIxHIe. .3 whom. cum m:mH. ammuav m+HmH. wmmIev m+ HHo=ocoo o.HIo mmez IHv_oquu_oI~xm H swung Hasnmnu onuHx Lou ouxa HonVHxv HInme mmm.o. OHKAHxv HImNE mm=.MIU u HoHHVHxV HIome mmHo.0Iu ImmEHHIonmm.Hu as HonAHxv HlamE memo.oum HoHKVHxv HImNE HIOHx~o~.~I HonAHxV HImmE HIOmewo.mu< . N 0H IH 0H IH x no on H NH x xvo+H x xvm+HasHHHc a HIxHwa.ax MIonsm.~Io , . u o . . u HIxHImx.nx moo o o HIxHwa ex mmm m m o o a HuHomawo HIxHIxx.hx FMHo.OIu< HA Elevo+m+H EIBVCH mIE.wxmonmmoo.Hn< a\zl> LmHOE mmHH.ouon x HonVHxV m\onomw.mmI-< 0 OH H coHuomLH 0:00 on o HonAHxV m\onmHm.=Hu< <\m\HHBI Bv+ xi x 0H0: . N HIHoe.mx HHmmo.ou z szx+ :qum uaHaooHoe ccHumH>mu .m.E.L mucmumcoo Ho nosHm> coHumHox choHuwuaaeoo auLOQOHm Hmv .conoL HonuHLo ecu :H mucoEHpoaxo poouuoOELocu consddHu do meszcm cH wLonEmpwa oHEmczcoenocu can whoawcanu BmH Lou com: acoHnnopaxm .H.w oHnwe 169 impurities (especially ionic impurities), values tend to vary somewhat from laboratory to laboratory. Most recent experiments indicate the critical temperature to be between 25.988°C and 26.385°C (Greer [1976], Woermann and Sarholz [1965], Gammell and Angell [197“], and Allegra 33 a1. [1971]). The apparatus previously described al- lowed measurement of relative temperatures in this labora- tory to 0.002°C. Because only relative departures from TC were needed for analysis, no elaborate calibration was made in an attempt to obtain absolute temperatures. However, the measured TC appeared to be slightly higher than the best literature values. Only relative tempera- tures were used in data analysis. The small value of ch/dP (—o.055°K-atm’l reported by Morrison and Knobler [1976]) indicates that TC is essentially independent of barometric pressure. (2) Density and molar volume - Woermann and Sarholz [1965] and Greer [1976] report very accurately measured densities in the critical region as a function of composi- tion and temperature. The best "MULTREG" fit of their data is shown in Table 6.1 for IT-298.l5|515°K. This equation fits the reported values to within 0.1% for all values of WI. The polynomial expression for p fits well very near Tc because thermal expansivity has such a small CE - Morrison and Knobler [1976] report it as 0.08 - 0.1“ with an uncertainty of 0.1. Molar volumes were obtained 170 from V = M/p. (3) Heat capacity - The temperature dependence of the specific heat at the critical composition can be well represented very near the critical point with a logarithmic singularity (Klein and Woermann [1975]). Klein and Woer- mann found that correction terms to the logarithmic singu- larity could not be neglected for deviations from TC larger than 0.5°K. The fit of their data for 0°KST-TC: 3.5°K is shown in Table 6.1. This logarithmic singularity is in agreement with the very small critical exponents reported by investigators of other systems. For example, Pelger gp‘al. [1977] report a = 0.55 [see Equation (5.10) and Table 5.1 for the definition of a], Voronel and Ovodova [1969] and Cope gt 11. [1972] report a 2 0.0; and Gambhir QEIQI. [1971] and Viswanathan 33 al.[l973] report 03010-1- Although the results of Klein and Woermann were obtained only at x1 = x1e, the data of Davies [1935] and, to a lesser extent, those of Kresheck and Benjamin [196“] indicate that C? is relatively composition independent for the relevant range of interest for the experiments reported herein. (“) Diffusion coefficient - Light scattering measure- ments of the diffusion coefficient for this system have been performed by Chu and coworkers [1968], [1969], and [1973]. The best "MULTREG" (Anderson [1968]) fit of their data is also given in Table 6.1. Data used in the fit 171 included the self diffusion coefficient of water for the point x1 = 0.0 in addition to the concentrations reported by Chu 23 l. The values for D were measured by Chu t l. at only two compositions in addition to the critical composition. (5) Thermal conductivity - No data exist for the thermal conductivity of IBW mixtures. Fortunately, the critical exponent of K is well defined at the critical composition. As shown in Chapter 5, 81 = 0 and the thermal conductivity shows no anomaly; iLQL, K exhibits the same temperature behavior near the consolute temperature as it does further away from the CST. Consequently, the NEL equation (Jamieson [1975] and Chapter “ of this thesis) was used for the composition dependence. The temperature dependence was included via the temperature dependencies of the pure component thermal conductivities. A linear interpolation of data reviewed by Jamieson [1975] defined the temperature dependence of K3. ture behavior of K3 was obtained from a "MULTREG" fit of The tempera- the data reviewed by McLaughlin [196“]. The NEL equation shown in Equation (“.2) was used with Jamieson's recom- mended value of C = 1.0 for the adjustable parameter C. (6) Excess enthalpy - As discussed in Chapter “, the heat of mixing effect is symmetric about the interface even for very nonideal mixtures. The excess enthalpy is ~§ therefore not required for determinations of Q1 based on 172 AT data taken at symmetric positions with respect to the interface. No data have been reported for HE in the IBW critical region although Daoust and Lajoie [1976] have reported some heats of dilution. (7) Heats of transport - The experimental results presented here are the first determinations of the criti- cal exponent for the heat of transport in liquid mixtures in the critical region. From the preceding discussion, it is apparent that the composition dependence of most of the parameters is not well known. Actual values of 0: calculated from measured AT data would reflect this uncertainty and would certainly be no more accurate than the total uncertainty of the properties used. However, in determining the critical exponent of 0:, composition changes very little in time. This is illustrated well by Figure 6.6. Therefore, the composition contribution to the value of any property remains the same when the consolute temperature is ap- proached. That is to say, all of the compositional dependencies and uncertainties in the input properties contribute a constant amount to 0: regardless of e and are thus grouped together into the pre-e factor A in the expression 0: = Ask“. Since the temperature dependence of all the input properties was well known, the critical exponent A“ can be calculated with good certainty. A fit of experimental data yields the true value of the CE 173 of the heat of transport Au. The pre-e factor A, however, will be an effective value for each run. E. Experimental Results A “ were obtained from the Best estimates for O: = A: experimental AT data using nonlinear least squares and Gauss-Markov regression. The values of various properties used in the calculations are listed in Table 6.1. Table 6.2 contains the initial conditions for the seven runs that were performed on two independently prepared mixtures. Further experimental conditions are available from Table C.l of Appendix C. The overall mole fraction of iso- butyric acid at which the mixtures were prepared is de- noted by in Table 6.2. Ti'Tc represents the initial temperature from which initial phase compositions x3 and x§ were calculated using Equations (6.11). The initial difference in composition between the upper and lower phases is given in the column labeled Axl. Tmax'Ti represents the temperature jump range. The results obtained for the critical exponent A“ are shown in Table 6.3. Also listed in this table are values obtained for the pre-e factor A. Negative values for 0: indicate that the temperature of the phase rich in isobutyric acid increases while it decreases in the water rich phase. As mentioned above, the composition 17“ :NmH.o mm:o.o mmmm.o Hmm.= Hmm.m omHH.o HH> HmmH.o s::o.o wmmm.o HHH.m Hmm.m omHH.o H> wmmH.o Hmzo.o mmmm.o Hmm.HH oms.m omHH.o > mmom.o ammo.o HHHm.o mHo.s Hom.m mmHH.o >H m:om.o ammo.o om:m.o HHm.m How.o mmHH.o HHH SHmH.o mmno.o m:mm.o mmm.m mHo.m mmHH.o HH moom.o mo:o.o Hosm.o mmo.w mo:.m mmHH.o H Hxa Mx wx mo\HoeIxmeev HoxHoeIHevI AHxv cam .COHme HQOHHHHO on» CH 3mH co cosmomnma mpcmEHpooxo poommmoEHonp COHmsmmfip map How mCOHpfipcoo prcmefinoaxm .N.w manma 175 .cmmE on» no COHpmH>mU ppmocmum H5 .Empwopa wcfipuHm mmHmSUm ammoH map an pmpmHSono mm 20HpmH>op Unmvcwpm Hmv anzo.o mm.o Hz No.0 mm.o H.o 3.0 QMHH.o H> no.0 m~.o m.o m.o omHH.o > No.0 mm.o H.o m.o mmHH.o >H mo.o mm.o H.o m.o mmHH.o HHH No.0 mm.o H.o H.H mmHH.o HH No.0 mm.o H.o m.H mmHH.o H HmvHo :H HmVHIHoE.hHHOH\ 5.0°C were included in the data analysis. Inclusion of data outside this range would yield an effective critical exponent rather than the true value corresponding to the definition of Equation 5.3. 178 0.15 r T I I I I 0.12 .1 'r 009 _ O 3. I- ca 006 H 003 _ 0-00 1 l I I I I 10 14 18 22 26 30 34 38 TIME (102 SEC) 1 l l l l l I 1 l l 4 3 2 1 O T-Tc 1°C) MI ~u Figure 6.8. Run I. Experimental results for Q1 a As with A and A“ as adjustable parameters. 179 I l I F I l 012» ESIJOQ 8. I'- <‘ 0.06 006 l l J l l 1 1O 18 26 34 TIME (102 sec) _L J l l J l J J 3 2 1 O T-Tc 1°C) ~§ A“ Figure 6.9. Run II. Experimental results for Q1 - As with A and A“ as adjustable parameters. 180 008 _o 8 AT (°C) 002 p L J I 10 15 20 25 TIME (102 sec) 1 4 1 l l J J 3 2 1 O T_TC (°C) ~* Figure 6.10. Run III. Experimental results for Q1 A A: “ with A and A“ as adjustable param- eters. 181 008 006 (°C) AT 004 002 l l 1 l 10 15 20 25 30 TIME (102 sec) 1 1 1 L L l l l 3 2 1 0 T-Tc 1°C) A ~* Figure 6.11. Run IV. Experimental results for Q1 - Ae “ with A and A“ as adjustable parameters. 182 I I I I I I 008- a 0.061- 0 I— ‘Q 004- 002- I J I I I I 15 25 35 45 TIME (102 sec) I I I I I 5 4 3 2 1 0 ...-Tc (DC) A ~a Figure 6.12. Run V. Experimental results for Q1 = A8 “ with A and A“ as adjustable parameters. 183 (108 (106 (°C) 002 J I J I J I 15 25 .35 45 TIME (102 sec) I J I I I I J L J I 4 3 2 1 o T‘TC (°C) ~« Figure 6.13. Run VI. Experimental results for Q1 = A A: “ with A and A“ as adjustable param- eters. 18“ l I I I I I (108- F‘<106—- L) 8. P . ‘3 OCH 0 0.02 - I I I I I I 10 14 18 22 TIME (102 sec) J I I I 15 1 C15 0 T-Tc (°C) ~* Figure 6.1“. Run VII. Experimental results for Q1 A - Ae “ with A and A“ as adjustable param- eters. 185 From Figures 6.8 - 6.1“ note that even when T ap- proaches Tc, the measured AT does not identically vanish. There also appear to be larger discrepencies between experiment and calculation in this region. These effects are primarily attributable to the composition dependence ~* ~* of DQl' AS T approaches T DQl becomes small near the C, interface where x1 = ch‘ However, on either side of the interface, the composition differs from x1 and the phase 0 separation temperatures for those compositions are con- siderably lower than TC. In these regions (x1 # x1e), O: and D are still finite even when T = Tc because of the lower phase separation temperature at these compositions. The measured AT is related to DO: and will therefore be nonzero when T = Tc because of the contribution from diffusion occurring in regions slightly removed from the interface. This contribution should also vanish if the cell temperature is lowered to the local phase separation temperature. Experimentally, AT did vanish at tempera- tures below TC when diffusion entirely ceased. As previously mentioned, this compositional contribu- tion from various properties will be essentially constant as T + To and will not affect the determination of A“ except very near the CST when the main contribution from diffusion at the interface vanishes. To include the com- position dependence in the predictive treatment, an em- pirical correction for AT as a function of e was included 186 in the numerical integration program for the very near CST region. This empirical relation was obtained from a fit of Run 1. This same relation was then applied equally to the other six runs which, as can be seen from Figures 6.9 - 6.1“, gave good results in each case. This empirical fitting procedure affected the fit of experimental to predicted values only in the region of the last few data points. Furthermore, because all known composition dependencies were already included in the equations, the empirical correction was at most 0.008°C. Fits obtained with and without the data points of this region, where the empirical composition correction was used, yielded A“ values which agreed within 2%. Table 6.“ is a reproduction of Table 5.5 with the now known critical exponent for the heat of transport included. Notice that Q: vanishes with a +2/3 exponent. Also notice that 5: is identical to 001/011. Since no thermostatic properties (such as U11) are involved in the behavior of 0:, the entire observed anomaly is due to the behavior of the Onsager coefficients. There is no am- biguity in attributing anomalous behavior to the kinetic or Onsager effects in the case of the heat of transport. Since recent light scattering investigations of D in the critical region reveal that mutual diffusion vanishes with a +2/3 exponent and since 311 % 6+u/3, 011 must diverge with a -2/3 exponent. Insertion of this value 187 Table 6.“. Transport parameters and their critical ex- ponents. Behavior Critical Property Definition Near CST Exponents K K = QOO/T K m 000 K N 8° — Qoo ” 8° 0- u = 11 11 — 2/3 D D ——E§E— D m Qllull D m e — “/3 ull ” 8 Qll “ 8-2/3 Qlo _ 0 DT DT _-—3— DT m 010 DT N e (a) -0 0 al=__l_0_ a «...—11— 0. ”8-2/3 (a) wlw2oD l 9 — l llull w KT - 01° 2 KT t __319_ KT % 6-2/3 (a) Qllull Qllull 910 N 6° (a) 0 ‘1 GI = ——§°1 ‘1 ~ r“ 6'1” .2/3 (b) 11 ll 0 (b) 901 m 5 (a) Results of Giglio and Vendramini [1975] are used. (b) Results of this work. 188 into the definition of 0: indicates that 001 N 8°. It is interesting to note from Table 6.“ that all is therefore the only Onsager coefficient for a binary liquid mixture (without pressure gradients and external fields) with a nonzero critical exponent. This fact is, however, con— sistent with the general criterion for the direction of an irreversible process as derived by Haase [1969]: 0 0 00 01 > o . (6.12) Q10 911 Equation (6.12) implies QOOQll-'901910 > 0. Since this expression is true away from the critical region and only 9 l diverges (while the other coefficients remain l finite), the relation is certainly still valid in the critical region. Note that in qualitative support of Onsager reci- W 8° N 0 procity in the critical region 00 That is, l 10' the reciprocal effects have equal critical exponents. Actual verification of an identity between 001 and 010 must wait until composition dependencies of the various transport properties and thermodynamic properties have been accurately determined. Figure 6.15 shows dramatically the manner in which the measured AT vanishes as the consolute temperature * is approached. Since AT is related to DQl’ where D 189 4 ; t (102 sec) 40 TEHPERRTURE RELRTIVE T0 CELL HERN TEHPERHTURE. Figure 6.15. Simulated temperatures relative to the mean cell temperature in temperature jump dif- fusion thermoeffect experiments as T ap- proaches Tc' 190 vanishes as T + Tc’ the qualitative behavior of Figure 6.15 is to be expected. The feature unexpected a priori is that AT vanishes more quickly than D; i;§;J that Q: (the heat "carried" by a diffusing molecule) itself vanishes. There is a critical decrease in Q: as well as in D. This observed behavior obviously contains in- formation relevant to the microscopic mechanism of the diffusion thermoeffect and is discussed in Chapter 7 as it pertains to current theories and molecular interpre- tations of the heat of transport. CHAPTER 7 CONCLUSIONS A. Interpretations of the Heat of Transport Although the work described in this dissertation was the first quantitative measurement of the heat of trans- port in binary liquid mixtures, numerous papers on the theory of thermal diffusion and the heat of transport have been published during the last 50 years. Two main approaches can be identified, (1) the kinetic approach and (2) the statistical mechanical approach. The kinetic interpretation of the heat of transport has developed from a model for diffusion akin to Eyring's significant structure theory. The basic reasoning fol- lows that proposed by Wirtz [1939], Wirtz and Hiby [19“3], Denbigh [1952], and Prigogine EE.§l- [1950]. Some ex— tensions have been made by Dougherty and Drickamer [1955] and Rutherford and Drickamer [195“]. If a particle is to leave its position on the quasi-crystalline liquid lattice and move to a new location, the activation energy can be divided into two parts: (1) qH, the "Hemmungsenergie" required to break free from the attraction of the neigh- boring molecules, and (2) qL, the "Lochbildungsenergie" required to form the hole into which the diffusing mole- cule passes. Thus the activation energy is 191 192 a qH + qL. (7.1) The diffusion coefficient and the mass flux can then be written in a typical Arrhenius fashion with the above activation energy. Consideration of a nonisothermal system in which a molecule passes from a temperature Ta to a temperature Tb requires qH at Ta and qL at Tb' Opposing rates for the flux can be written which when balanced for the case of the thermal diffusion steady state yields 2 q ’q -(ia-rTlg) = ié—L (7.2) 8.8. RT where C is molarity. The definition can then be made * = _ Q1 - qH,l qL,l (7.3) where k0: is the heat of transport based on the kinetic model. This is different from the phenomenological defini— tion of Equation (2.19). Denbigh's presentation is slightly different. He defines the "two energy terms involved in this process (the jumping of a molecule from one site to the next): (a) the energy of detaching the molecule from its neigh- bors; (b) the energy of creation or filling of the hole." With WH and WL representing these two energies respect- ively, the energy associated with the transfer of a 193 molecule of component 1 is WH l - WL, which prompts the 9 definition k x = Consideration of the regular solution theory facilitates a representation of WH and WL in terms of configuration or interchange energies for the case in which molecules of both components are about equal in size. The excess ~ molar free energy GE in regular solution theory is = Nwalx2 = w'xlx2 (7.5) in Which NA is the Avogadro number, w is the interchange energy, and w' = NAw. Because the two components of the mixture are perfectly randomly arranged, the excess entropy of mixing is zero - the entropy of mixing corresponds to that of an ideal mixture. The interchange energy can be thought of as the change in potential energy when 2 dis— similar 1-2 molecular pairs are formed from z/2 1—1 and z/2 2-2 molecular pairs. The interchange energy is related to the pair potential energies relative to infinite sepa- ration Wij by w = z[w12 - %(wll + W22)] (7.6) 19“ where z is the coordination number (Prausnitz [1969]). In terms of wij’ Denbigh found 81 = ’ZNAfX2/2EX1(W11 ' "12) ‘ x2("22 ’ "12)3 (7.7) where f is a numerical factor less than unity which physi- cally corresponds to the fraction of nearest neighbor "bonds" broken during the jump. Dougherty and Drickamer [1955] have made some compari- sons of experimental values for 0:, obtained from thermal diffusion experiments on the assumption of Onsager reci- procity for 001 and 010, with values calculated from Equation (7.7). In this comparison, the Wij were related to physical properties such as latent heats of vaporiza- tion. Good qualitative agreement was found in the com- parison but the quantitative agreement was poor. As Tyrrell [1961] indicates, in and Q: are not necessarily the same. The heat of transport depends on the reference plane. Denbigh's work assumes a volume fixed reference frame while the phenomenological defini- tion of Equation (2.19) is for a barycentric or center of mass reference system. Although related, the two heats of transport are subtly different. Of course the heat of transport obtained from the kinetic theory is different from the excess enthalpy, which for a regular solution is 195 RE = NAZXlX2/2(2W12 - wll - w22). (7.8) The heat of transport must be due to the very mechanism of diffusion itself. The statistical mechanical approach removes the restrictive assumptions about the structure of the liquid and the mechanism of diffusion. In so doing, the equations are difficult to evaluate for real systems because they contain integrals over pair correlation functions. Based on Kirkwood's Brownian motion method, Bearman, Kirkwood, and Fixman [1958] developed an expression for the heat of transport for a system in which: (1) particles react with central forces only, (2) intermolecular potentials can be written as sums of pair potentials; and (3) both components possess only translational energy. The heat of transport for such a system in the absence of ex- ternal forces can be split into two terms * x x where Q: is the heat of transport defined by Equation (2.19), Q11 is a term involving averages over equilibrium ensembles, and Q:2 involves perturbations of the equilibrium distribution due to the flow of heat and matter. Bearman 33 El: derived a general form for Q11 and Q:2. In the case of regular solutions they are _ _ f f * _ l m1X1 V1V2 2 l Q11 - §(-ME— + X2) (:r-- z“) (7.10) V2 V1 D -D m x L L * _ 1. 2 1 1 l 1 2 — — 2 l 2 v1 v2 dw g 21 I (2.0) 3 -L2V2) - 3fI'(-—d?— - 1)V2lg2l d E} (7.11) + 2x2(L2V2 where v is the mean molecular volume, V1 and V2 are the partial molecular volumes of components 1 and 2 respec- tively, v1 and v2 are the molecular volumes of the pure components, L1 and L2 are the negatives of the latent heats of vaporization of components 1 and 2, respectively, from the solution to the ideal gas state, L1 and L2 are the negatives of the latent heats of vaporization of the pure component to the ideal gas state, m1 and m2 are mol- ecular masses of components 1 and 2, D1 and D2 are self diffusion coefficients for each species in the mixture, and r and r are the magnitude and vector distances between two molecules, respectively. The integral term in Equation (7.11) cannot yet be evaluated for real systems because it contains the radial distribution function gig’o), the intermolecular potential V21 and a term involving $21 which is related to the nonequilibrium radial distribu- tion function. The above functions are unknown for real liquids. Notice that the equations from the kinetic theory 197 are analogous to Equation (7.10), but neglect completely the nonequilibrium portion Q:2 of the phenomenon. Bearman and Horne [1965] have compared experimental thermal diffusion factors with (1) thermal diffusion factors calculated from Equations (7.10) and (7.11) on the assumption of ORR, and (2) thermal diffusion factors calculated from similar statistical mechanical equations derived directly for thermal diffusion in terms of mole- cular properties. The integral terms involving radial distribution functions were left out. This corresponded to a hard-sphere assumption. Because these integrals were left out, the thermal diffusion factors calculated from the thermal diffusion theory were somewhat lower than those obtained from the heat of transport theory. The values obtained from the heat of transport theory for the thermal diffusion factors in carbon tetrachloride-cyclo- hexane mixtures agreed quite well with the experimental results. An important result of their calculations, was that the Q:2 term contributed over 50% of the abso- lute value of Q:. Thus, the nonequilibrium term in the Bearman-Kirkwood—Fixman theory, which the kinetic theory completely neglects, is in fact the predominant term. Story [1967] and Story and Turner [1969] have ex- amined experimental thermal diffusion factors for carbon tetrachloride—benzene and cyclohexane-benzene mixtures with respect to both the kinetic theory and the statistical 198 mechanical theory. They find that the kinetic theory is not only in error with respect to magnitude, but often yields the wrong sign. They found similar difficulties in magnitude and sign using the statistical mechanical theory with the integral of Equation (7.11) neglected. B. A New Interppetation of the Heat of Transport The kinetic approach results in an expression for the heat of transport obtained entirely from equilibrium properties of mixtures. The Bearman—Kirkwood-Fixman theory indicates that this cannot be done. In order for the heat of tranSport to be nonzero, the molar energy transported by diffusion must be different from the partial molar enthalpy contribution due to the mass flux. This is readily seen from Equations (2.“) and (2.19). It there— fore seems likely that the heat of transport is not just a difference in potential energies experienced by the diffusing particle, but should depend on the kinetics of transport - the very mechanism of diffusion itself. The results of Chapter 6 clearly indicate that Q: vanishes as the consolute temperature is approached and does so with a +2/3 critical exponent. An implicit goal throughout the evaluation of the critical behavior of Q: has been that the results would provide insight into the microscopic nature of heat and matter coupling and its relationship to the diverging correlation length 199 associated with critical mixtures. Clearly, any consistent * * model for Q1 must also explain the observed behavior Q1 W 52/3 in the near critical region. The kinetic theory of 0* l ings. In the regular solution theory, the configuration appears to be inconsistent with these experimental find- energy w is at most a weak function of temperature, and therefore Equation (7.7) does not exhibit the required behavior in the critical region. Even the basic defini- tion given in Equation (7.“), where regular solution theory has not been invoked, does not display the experi- mentally observed behavior. Note that Equation (7.“) is consistent with the observed decrease of the diffusion coefficient in the critical region if the increased cor- relation length is assumed to enhance the diffusional activation energy. The heat of transport defined in Equa- tion (7.“) does not depend on the activation energy. It depends only on the difference in energy required to remove a molecule and the energy released when its hole is filled. It would seem that this difference would depend on relative potential energies rather than lengths of correlation and therefore this model does not ade- quately describe the critical behavior of Qi. To formulate a new kinetic theory for the heat of transport which is consistent with the experimental behavior in the critical region, the lattice model for liquid structure must be discarded in favor of the more 2OO intuitive idea of randomness due to molecular thermal motions. Hildebrand [1977] has shown that "changes of viscosity and diffusivity with temperature can be ac- curately and more simply expressed in nonexponential for- mulas than by plotting their logarithms against reciprocal temperatures." "Activation energy" is therefore not a necessary construct. The mechanism of diffusion in this formulation is a succession of small displacements due to random molecular thermal motions rather than to actual "jumping" from one lattice site to the next. The tem- perature dependence of the diffusion coefficient is simply due to increased thermal motion, which decreases the time needed for a net transference of molecules from one loca- tion to another. The thermal motions of each molecule vary but pre- sumably obey a maxwellian or normal distribution. In fact, the mode or expectation value of this distribution of energies defines the thermodynamic temperature as kT, where k is Boltzmann's constant. Because of this dis- tribution of energies, some molecules are more energetic than others at any given time. For convenience of dis- cussion, define the "excess energy" of a particle or molecule as that amount of energy which it possesses at a given time in excess of the expectation or kT amount of energy. Thus, the further out in the leading wing of the distribution, the more "excess energy" the molecule 201 has relative to the average value. Now bring two iso- thermal subsystems (of different pure components for the moment) into contact and allow mutual diffusion to begin. Which molecules from the energy distribution for subsystem 1 will be more likely to be found in sub- system 2 shortly after initial contact of the phases has been made? The conclusion that the more energetic mole— cules diffuse more rapidly than their "average" energy counterparts is inescapable. Although collisions are energy randomizing events, molecules at any one time possessing "excess energy" move faster through the solu- tion than their lower energy counterparts and for any given period of time will move further through the mixture. The heat of transport of component 1 is simply the "excess energy" transported by molecules which undergo diffusion. From this picture of the heat of transport, several concepts, vague in the previous kinetic theory, become clear. Notice that the difference between the heat of transport and the heat of mixing is evident. The heat of mixing is a state function dependent only on the states of the initial pure components and the final mix- ture. The heat of transport cannot be separated from the diffusional mixing process. The heat of transport is thus dependent upon the nonequilibrium movement of mole- cules as in the Bearman-Kirkwood-Fixman theory and cannot be calculated merely from equilibrium properties and/or 202 equilibrium intermolecular potentials. The heat of transport is a property of the system because the distri- bution of energies is certainly dependent upon the com- ponents (mass, vibrational degrees of freedom, rotational degrees of freedom, etc.) and the relative amounts of each present. 0: is specific to the mixture and retains its value even if two mixtures of equal chemical poten- tial are brought into contact. For this case 0: is nonzero, but no temperature change occurs in the system because a forward diffusional event is as likely to occur as a reverse event. The previous kinetic theory pre- dicts kQ: = 0 in this case. The temperature changes in a diffusion thermoeffect experiment are explicable from this model of the heat of transport. When a molecule of component 1 migrates from a particular region carrying with it "excess energy", the molecules behind are lowered in energy relative to the previous kT value by an amount equal to the "excess energy" transported. However, other high energy mole- cules are finding their way into that region carrying "excess energy" which tends to raise the distribution of energies. Because there is a competing effect between the net diffusion of component 1 in one direction (into the lower chemical potential region) and the net diffusion of component 2 in the other direction, the temperature change in a particular location is related to the dif— ference between the two distributions of energies in 203 the initial phases. It is dependent upon the "book—keep- ing" of "excess energies" carried into and out of the region. The molecular distributions of energy for the two initial phases are themselves dependent upon the masses, intermolecular potentials, and complexity of the molecules. The same statistical nature of molecular thermal motion gives rise to thermal diffusion. Consider a uni- form, isothermal, binary liquid mixture between two parallel plates. Because the system is isothermal, both components have the same expectation value for their thermal energy distributions. However, the breadth of the distributions need not be the same and is dependent upon the properties of the components. If a temperature gradient is now imposed, the energy distributions of both components very near the hot wall are shifted up in energy while the distributions near the cold wall are shifted down. Because both components are more energetic near the hot wall, there is a net random migration of both components toward the cold wall (thermal expansion). However, the component with the broader distribution of energies will tend to move faster, i;§;, its more ener- getic molecules, on the average, move through the fluid faster than those of the component with the narrower distribution. Unlike isothermal conditions where the faster migration of the component with the broader 20“ distribution occurs equally in both directions, under the influence of a temperature gradient this statistical "excess migration" is predominately toward the cold wall since the distribution of energies was lowered in that region. There is a net accumulation of this component in the cold region (as thermal energy is now absorbed into the cold plate) hence a relative accumulation of the other component near the warm wall where heat is continuously supplied. Finally, a steady state is reached when the above accumulation process balances diffusion in the reverse direction caused by unequal populations. The critical behavior of the heat of transport for this model is closely tied to the critical behavior of diffusion. Both are kinetic processes (as opposed to thermal processes such as thermal conduction). The Stokes-Einstein-Kawasaki equation (Kawasaki [1970]) D = kT/(snng) w e2/3 , (7.12) where n is shear viscosity, describes the critical de- crease of the diffusion coefficient near the consolute point in terms of a rapidly diverging size effect as more and more particles become correlated. Thus, D m 5'1 m 82/3, and D vanishes as groups of molecules become correlated. It should be mentioned that the critical exponent for n 205 is still not known exactly, but appears to be zero or slightly negative (the small anomaly observed may be a logarithmic singularity). For a related reason, Q: in the proposed model must also vanish as correlation lengths increase. As the consolute temperature is approached, correlation lengths increase rapidly. As correlations increase, the max— wellian distribution of energies must necessarily narrow. This can be viewed as an effect due to increased mass per diffusing particle or as a decrease in large magnitude fluctuations due to increased correlations. Since the heat of transport is viewed as the "excess energy" or energy above the expectation value carried by a diffusing molecule, 0: must vanish as the distribution of energies narrows about the kT or expectation value. In the limit of perfect correlation between the molecules, each dif- fusing species has exactly kT of energy associated with it, Q: is identically zero, and no change in local temperature is produced by the diffusional event. This can be viewed as the limiting case when T + To“ As T approaches To the diversity of energies associated with particles de- creases and appears to do so inversely proportional to the correlation length 5. Therefore Q, w a w e (7.13) in agreement with the experimental measurements of Chapter 6. 206 C. Summary and Future Work Needed As stated in the Introduction, the objectives of this work have been fourfold: (l) to measure quantitatively for the first time the diffusion thermoeffect in liquid mixtures, (2) to test experimentally the Onsager heat- mass reciprocal relation, (3) to study the behavior of the diffusion thermoeffect in the consolute region of a binary mixture in order to understand how it may relate to microscopic phenomena; and (“) to examine experi- mentally and compare the behavior of the Onsager heat- mass and mass-heat coefficients in the consolute region. It was felt that the accomplishment of these four goals would contribute to the overall objective of transport investigations: to understand the microscopic causes of observed macroscopic phenomena sufficiently enough to make accurate a priori predictions. In fulfillment of the above objectives, a new cell was designed which enabled quantitative observation of the diffusion thermoeffect. This cell used a withdraw- able third component to create an initially sharp dif- fusional interface. The equations of nonequilibrium thermodynamics and hydrodynamics were solved numerically for the conditions involved in the actual experiments, leaving (OI/M) as an adjustable parameter. Nonlinear least squares fitting of these solutions to the measured temperature differences between thermocouples placed 207 symmetrically about the interface led to the first direct determination of the heats of transport in carbon tetra- chloride-cyclohexane mixtures. These values compared well with those obtained via thermal diffusion techniques, which led to the first experimental verification of the Onsager reciprocal relation for the heat-mass and mass- heat coefficients. The temperature dependence of the heat of transport was determined for isobutyric acid-water mixtures as a function of distance from the critical temperature via the diffusion thermoeffect technique. A temperature jump cell was employed. The local temperature could be rapidly jumped (with a microwave oven) to temperatures above the consolute temperature. The difference in tem- perature between two points symmetric about the inter- face was monitored as a function of cell mean tempera- ture. The cell temperature was allowed to slowly relax toward Tc permitting determination of the critical ex- ponent for the heat of transport by nonlinear least squares fitting. It was found that 0: vanishes rapidly as T0 is approached and can be represented by O: m e2/3. From these results, the critical exponent for the Onsager co- efficient 001 is 0. Very recent thermal diffusion experi- ments indicate 0 m 5°. The reciprocal heat-mass and 10 mass-heat Onsager coefficients therefore have identical critical behavior. Furthermore, existing kinetic theories 208 of 0: do not explain its critical behavior. A new model based on the thermal motion picture for diffusion not only explains the critical behavior of 0:, but also explains the basic features of the heat of transport left vague in existing models. More research is needed to verify and quantify the model for 0:. The mathematical formulation of the model is the next step. Of further interest would be diffusion thermoeffect experiments in multicomponent systems, especially near critical points of higher order. The more areas in which the heat of transport can be evaluated to give special criteria which must be met by any consistent theory, the better the model will become as well as our understanding of the processes involved. With respect to the experiments themselves, it would be desirable to perform similar studies on other systems near their consolute temperature. Further studies on the composition dependencies of thermodynamic and transport properties in this region need to be performed so that absolute values of heats of transport can be calculated in this region. This would then establish a basis for examination of 001 and 010 to test Onsager reciprocity in the critical region. Additional measurements of the temperature and pressure dependence of Q: away from the critical temperature are desirable to facilitate empirical predictive capabilities 209 and to elucidate the microscopic coupling of heat and mass transport. These first experiments on the diffusion thermo- effect in liquids have opened an area of investigation in the study of transport processes which for decades was discounted as unfeasible. Much information can be gained from study of this cross transport coefficient. It is hoped that other cross coefficients can also be tapped and used as tools in the study of transport phenomena. APPENDICES APPENDIX A TRANSFORMATION RELATIONS AND IDENTITIES The transformation of mass fraction-specific property Equations (2.25) — (2.27) to mole fraction-molar property Equations (2.33) - (2.35) involves the following defini- tions, identities, and procedures: A.3. Mass fraction-mole fraction. The relationship between mass fraction w1 and mole fraction x1 is wi = XiMi/M (A.l) where M1 is the molecular weight of component i and M E lel + sz2 is the mean molecular weight. Specific property—molar property. If T is any specific property either of the mixture or of the pure component and 3 is the corresponding molar property, then LT= 3/I7I. (A.2) Mass fraction derivatives—mole fraction derivatives. The following identity is easily shown from the 210 A.“. 211 chain rule: _ ~2 (dwl/d) - MlMZ/M (dxl/d). (A.3) Partial specific enthalpies. The derivative of the difference in partial specific enthalpies contained:h1Equation (2.27) can be related to the excess molar enthalpy HE. From the chain rule, Equation (A.2), and the Euler relation x dH ll+X the derivative of the difference in partial specific enthalpies can be written as mal-fievaz] = (M/M1M2X2)(3Til/3x1)T,P(3xl/8z). The definition of partial molar enthalpy implies —' - _ ~ 2” 2 [3(Hl-H2)/Bz] - (M/M1M2)(3 H/3x1)(3xl/Bz) where the total molar enthalpy H is usually split into ideal and excess contributions 212 with H? and Hg representing pure component molar enthalpies. These two equations combine to yield [3(Hi-H2)/azl = (M/Mle)(BZHE/axi)T,P(axl/az) (A.“) which is the desired result identical to Equation (2.36). Thermodynamic factor. The transformation from heats of transport to thermal diffusion factors involves the thermodynamic fac- tor (1+F) defined as (1+r) s [1+(32nyl/alnxl)T,P] (A.5) where Y1 is the activity coefficient for component ~* 1. The relationship between 01 and 01 is ~* ~ ~ _1 as is easily shown from Equations (2.1“), (2.15), and (2.16). The molar quantities 0: and fil are related to their specific quantities by Equation (A-2) where N becomes Ml' From thermostatics, the molar chemical potential is related to the activity coefficient Y1 and the standard state chemical potential pi of component 1 by 213 ~ n1 6 + RTan y "1 l 1' Obviously, Y1 is the activity coefficient based on the same standard state chosen for pi. Therefore, (cpl/Binxl)T,P RT(1+32nY1/3£nxl)T,P = RT(1+P) and -0lMRT(l+P)/M1M (A.6) 2 as required in Equation (“.1). APPENDIX B DIFFUSION THERMOEFFECT DATA FOR THE CARBON TETRACHLORIDE-CYCLOHEXANE SYSTEM Using the withdrawable "liquid gate" cell described in Chapter “, diffusion thermoeffect measurements were made on five mixtures of CClu-EHC6H12. The boundary conditions used were for adiabatic, impermeable walls at @/a) = 0 and (z/a) = 1. Initial conditions for each run are listed in Table B.l where x? is the initial mole fraction of carbon tetrachloride in the upper phase and XE is the initial mole fraction of carbon tetrachloride in the lower phase. The initial temperature distribution TO obtained from thermocouple readings just prior to interface crea- tion is given by To = TOO-TZ(z/a-0.5) with T00 and TZ listed in Table B.l. Isothermal initial conditions correspond to TZ = 0. Temperature differences between thermocouples lo- cated at (z/a) = 0.“ and (z/a) = 0.6 are listed as functions of time in Table B.2. The initial contact of the upper and lower phases established the diffusion interface and was assigned the time t = 0. 21“ 215 Table B.l. Initial Conditions. Run # xu xL T /°K T l l 00 z I 0.0179 .80““ 296.160 0 II 0.0503 .6“36 295.355 0.068 III 0.0951 .7638 295.131 0 IV 0.0750 .8935 296.020 0 V 0.193“ .909“ 296.“29 0.062 216 :mH.o mwmm mow.o 0H0: HwH.o mmmm 02H.o mmHm um me.o mzmm mmH.o wmwm omH.o Noam mmH.o wuom mm mmH.o mHHm an.o mmnm mmH.o mmmm mNH.o Foam mm mmH.o mwmm wom.o oumm mmH.o mmHm HNH.o mmwm :m :om.o mmwm mmm.o mHam mmH.o wmmm HNH.o omwm mmH.o NJH: mm mmH.o mowm :mm.o mmmm mmH.o munm owH.o Hmmm mmH.o mmmm mm me.o mmmm :mm.o nwom HNH.o Hmwm owH.o omzm mmH.o :Hnm Hm nom.o ommm HHm.o mHmm mmH.o nmzm me.o mHmm mmH.o ummm om mom.o mem mHm.o mmwm mmH.o wzmm mmH.o Hme mmH.o Hmmm mH mom.o mmom mmm.o mmzm mmH.o :Hmm omH.o mzom msH.o wmmm wH mom.o momH mmm.o mmmm HFH.o owom mmH.o :mmH 05H.o ommm pH mHm.o mme Hmm.o mem owH.o ommH omH.o mowH wuH.o mmom mH me.o mme Hzm.o mmmH me.o mNNH mmH.o zme wwH.o Foam mH mom.o mzmH Ham.o mzmH 55H.o mmmH omH.o oan omH.o ommm :H mom.o OHHH mmm.o ommH HmH.o wnzH mnH.o mzmH me.o nmom mH mom.o ome :mm.o mmmH me.o HmmH zmH.o mmHH mom.o zomH NH HHm.o NHHH mam.o meH me.o mmmH omH.o mmOH mHm.o mme HH mHm.o mNOH nmm.o ommH mmH.o omHH zmH.o 2mm :Hm.o HamH 0H mHm.o mom Hmm.o wwOH mom.o zwm mom.o mmm mHm.o HumH m mHm.o Hm» amm.o wmm mmH.o How mom.o mm» me.o mmHH w mHm.o mom wmw.o won mNH.o omn oom.o mmm nmm.o 0mm n mom.o cam mom.o ozo mmH.o mom mNH.o aHm mmm.o Hmw m oom.o om: mmm.o mHm 05H.o mu: mmH.o co: omm.o How m mmH.o mHm mem.o Hem omH.o mam mmH.o New smm.o em: a NEH.o mmm mmm.o mom mEH.o :mm mmH.o me mmm.o mom m omH.o umH HHm.o owH HHH.o me mHH.o mHH umm.o mmm m mmo.o HHH me.o mOH mOH.o om nmo.o 2m zHH.o mOH H xo\9< m\o xo\E< n\p xo\e< n\p xo\e< m\p xo\e< n\p * 53000 > cam >H com HHH com HH cam H com .moocopomHHp chapmmoQEoB .m.m oHnme 217 mmH.o mmo: mm HHH.o mmoa mmH.o aHmm mm HwH.o Hmmm mmH.o mmsm Hm HHH.o swam mmH.o moo: meH.o OHmm om mmH.o memm mmH.o comm mHH.o Hmam mm HmH.o mmem moH.o mHHm NHH.o oemm mm Hoxee n\o Ho\H< n\o Hoxee n\o Ho\H< n\o Ho\ee m\o a Empma > com >H cam HHH cam HHH com H com .ooacHocoo .m.m oHooH APPENDIX C DIFFUSION THERMOEFFECT DATA FOR THE ISOBUTYRIC ACID-WATER SYSTEM IN THE CRITICAL REGION Using the temperature jump cell described in Chapter 6, diffusion thermoeffect measurements were performed on two different mixtures of isobutyric acid and water prepared at the critical composition. The initial conditions of each run are listed in Table C.l along with temperature jump data. After the temperature jump, the mean cell tempera- ture relaxed toward the critical temperature as monitored by thermocouples located at (z/a) = 0.2 and (z/a) = 0.8. A polynomial fit of mean cell temperature as a function of time was obtained for each run. The polynomial equations of the form t+Tt2+Tt3 +T1 2 3 T - T0 = TO fit the individual data smoothly where T-TC is the cell temperature minus the measured critical temperature. The T T are listed in Table C.l. coefficients T , and T 0’ l’ 2 3 Also listed in Table C.l are the critical mole fractions of isobutyric acid at which mixtures were prepared xlc’ initial temperatures relative to consolute temperatures (just prior to the T-jump) Ti'Tc’ and the duration of the heating pulse 218 219 Temperature differences between the two thermocouples were measured directly and are listed adjacent to the time at which they were observed in Tables 0.2-0.8. All times are relative to initiation of the heating pulse. As dis- cussed in Chapter 6, data obtained for times less than 980 seconds are not included because of possible heating non- uniformities. 220 :0w.ml mw>.: mam.ml Hmm.z m.H www.ml OMHH.O HH> mmm.ml mmz.m www.ml zaz.m o.m :mm.ml OMHH.O H> mom.ml me.m ma=.:l mbm.HH m.m omw.ml OMHH.O > omm.m- Hmm.m mma.m- mfio.a :.m aom.w- mmHH.o >H mmo.ml mmm.w mo>.ml Ham.m o.m aow.ml mNHH.O HHH maa.m- mmfi.m :Hm.m- mom.m a.H mam.mu mNHH.o HH Ham.z- Hmm.m mmm.m- mmo.m m.H mo:.m- mmHH.o H m-m.gwmw-ofi m-m.WmM-OH H-m.Wmm-OH xo\oe m\mp eo\Aoe-Hev oax * cam .mcofipficcoo HmfipficH .H.o magma 221 Table C.2. Run I. No t/s AT/°K No. t/s AT/°K No. t/s AT/°K 1 1060 0.126 21 1753 0.067 41 2566 0.031 2 1091 0.123 22 1766 0.067 42 2585 0.029 3 1112 0.121 23 1819 0.062 43 2648 0.026 4 1135 0.118 24 1845 0.062 44 2705 0.026 5 1200 0.113 25 1888 0.059 45 2769 0.023 6 1260 0.103 26 1937 0.055 46 2786 0.023 7 1279 0.100 27 1980 0.054 47 2830 0.023 8 1331 0.098 28 2049 0.052 48 2890 0.023 9 1383 0.093 29 2078 0.049 49 2957 0.021 10 1401 0.087 30 2096 0.049 50 2974 0.021 11 1414 0.087 31 2163 0.045 51 3035 0.018 12 1476 0.082 32 2183 0.044 52 3118 0.018 13 1495 0.080 33 2241 0.044 53 3169 0.016 14 1518 0.080 34 2269 0.041 54 3233 0.018 15 1535 0.080 35 2343 0.039 55 3296 0.018 16 1578 0.077 36 2360 0.039 56 3370 0.016 17 1642 0.072 37 2432 0.034 57 3382 0.016 18 1655 0.072 38 2448 0.034 58 3439 0.013 19 1670 0.072 39 2458 0.034 59 3489 0.013 20 1730 0.070 40 2518 0.031 60 3554 0.013 222 Table 0.3. Run II. No. t/s AT/°K No. t/s AT/°K No. t/s AT/°K 1 1104 0.103 18 1681 0.057 35 2294 0.031 2 1158 0.098 19 1734 0.054 36 2374 0.029 3 1176 0.095 20 1804 0.050 37 2398 0.029 4 1195 0.090 21 1823 0.049 38 2464 0.027 5 1207 0.090 22 1841 0.054 39 2552 0.026 6 1214 0.090 23 1857 0.054 40 2587 0.024 7 1225 0.087 24 1873 0.047 41 2623 0.024 8 1238 0.087 25 1933 0.045 42 2687 0.023 9 1294 0.082 26 1952 0.044 43 2761 0.021 10 1376 0.077 27 2051 0.039 44 2809 0.021 11 1394 0.075 28 2068 0.039 45 2832 0.021 12 1404 0.075 29 2080 0.039 46 2888 0.019 13 1475 0.070 30 2095 0.041 47 2946 0.018 14 1556 0.064 31 2115 0.036 48 3006 0.018 15 1571 0.062 32 2193 0.036 49 3078 0.017 16 1586 0.062 33 2216 0.036 50 3154 0.016 17 1648 0.060 34 2244 0.034 51 3198 0.016 Table C.4. Run III. No t/s AT/°K No. t/s AT/°K No. t/s AT/°K 1 1059 0.079 14 1608 0.046 27 2215 0.023 2 1073 0.077 15 1638 0.043 28 2247 0.020 3 1085 0.077 16 1715 0.041 29 2299 0.020 4 1193 0.066 17 1753 0.038 30 2318 0.019 5 1210 0.066 18 1810 0.036 31 2341 0.018 6 1236 0.061 19 1875 0.033 32 2390 0.018 7 1249 0.059 20 1936 0.031 33 2459 0.015 8 1276 0.059 21 1949 0.031 34 2547 0.010 9 1320 0.056 22 1960 0.028 35 2566 0.013 10 1380 0.049 23 2016 0.028 36 2578 0.013 11 1425 0.051 24 2071 0.028 37 2635 0.013 12 1459 0.046 25 2136 0.025 38 2655 0.010 13 1553 0.046 26 2149 0.025 39 2713 0.010 223 Table C.5. Run IV. No t/S AT/oK NO. t/S AT/OK NO. t/S AT/oK 1 1060 0.105 16 1719 0.054 31 2373 0.028 2 1122 0.098 17 1731 0.054 32 2448 0.026 3 1140 0.095 18 1755 0.052 33 2502 0.026 4 1160 0.092 19 1814 0.049 34 2535 0.023 5 1224 0.087 20 1895 0.046 35 2592 0.023 6 1296 0.082 21 1954 0.044 36 2607 0.023 7 1322 0.077 22 1971 0.041 37 2659 0.021 8 1387 0.072 23 1980 0.041 38 2690 0.021 9 1429 0.072 24 2000 0.041 39 2741 0.018 10 1489 0.067 25 2051 0.039 40 2764 0.018 11 1505 0.067 26 2110 0.039 41 2824 0.016 12 1555 0.062 27 2180 0.036 42 2834 0.016 13 1622 0.059 28 2251 0.034 43 2910 0.016 14 1652 0.057 29 2285 0.031 44 2951 0.016 15 1667 0.057 30 2354 0.031 45 3014 0.013 Table C.6. Run V. No. t/s AT/°K No. t/s AT/°K No. t/s AT/°K 1 1810 0.087 13 2934 0.038 25 3830 0.023 2 1848 0.084 14 3015 0.036 26 3877 0.022 3 1946 0.077 15 3151 0.033 27 3954 0.022 4 2065 0.074 16 3242 0.033 28 4047 0.020 5 2149 0.069 17 3349 0.028 29 4159 0.020 6 2245 0.061 18 3376 0.028 30 4290 0.020 7 2381 0.056 19 3431 0.028 31 4334 0.018 8 2442 0.051 20 3495 0.025 32 4393 0.018 9 2530 0.051 21 3589 0.028 33 4424 0.018 10 2619 0.043 22 3620 0.026 34 4731 0.013 11 2725 0.041 23 3713 0.024 35 4853 0.013 12 2833 0.041 24 3740 0.023 36 4937 0.013 224 Table C.7. Run VI. No t/s AT/°K No. t/s AT/°K No t/s AT/°K 1 1446 0.093 18 2437 0.046 35 3527 0.021 2 1512 0.090 19 2507 0.044 36 3568 0.021 3 1550 0.085 20 2555 0.044 37 3655 0.021 4 1618 0.080 21 2624 0.041 38 3678 0.018 5 1648 0.082 22 2650 0.039 39 3727 0.018 6 1707 0.077 23 2714 0.039 40 3794 0.018 7 1762 0.070 24 2753 0.038 41 3836 0.018 8 1832 0.070 25 2825 0.036 42 3913 0.018 9 1913 0.067 26 2870 0.034 43 3933 0.018 10 1929 0.067 27 2958 0.034 44 3990 0.016 11 2027 0.062 28 3036 0.031 45 4076 0.016 12 2051 0.062 29 3102 0.029 46 4142 0.016 13 2106 0.059 30 3187 0.026 47 4196 0.015 14 2132 0.059 31 3246 0.026 48 4257 0.013 15 2224 0.056 32 3302 0.023 49 4330 0.016 16 2250 0.054 33 3392 0.022 50 4427 0.013 17 2317 0.052 34 3429 0.023 51 4485 0.013 Table C.8. Run VII. No t/s AT/°K No. t/s AT/°K No. t/s AT/°K 1 1053 0.059 9 1482 0.033 17 1879 0.018 2 1081 0.057 10 1506 0.031 18 1934 0.016 3 1137 0.054 11 1568 0.028 19 2003 0.016 4 1203 0.049 12 1623 0.026 20 2036 0.016 5 1244 0.046 13 1693 0.024 21 2125 0.013 6 1290 0.041 14 1718 0.023 22 2148 0.013 7 1353 0.039 15 1780 0.021 23 2208 0.013 8 1403 0.036 16 1838 0.021 REFERENCES REFERENCES Allegra, J. C., A. Stein, and G. P. Allen, J. Chem. Phys. :2. 1716 (1971). Anderson, T. G., Ph.D. Dissertation, Michigan State Univer- sity, 1968. Anderson, T. G., and F. H. Horne, J. Chem. Phys. 53, 2332 (1970). —" Anderson, T. C., and F. H. Horne, J. Chem. Phys. 55, 2831 (1971). '_— Bak, C. S., and W. I. Goldburg, Phys. Rev. Lett.23, 1218 Balzarini, D. A., Can. J. Phys. 52, 499 (1974). Bearman, R. J., and F. H. Horne, J. Chem. Phys. 42, 2015 (1965). —— Bearman, R. J., J. G. Kirkwood, and M. Fixman, Advances in Chemical Physics, Vol. 1, l (1958). Berge, P., P. Calmettes, C. Lai, M. Toumarie, and B. Voloshine, Phys. Rev. Lett. 24, 1223 (1970). Blagoi, Yu. P., Doctoral Dissertation, Physico-tech. Inst. of Low Temp., Ukr. Acad. Sci., 1970. Boushehri, A., and S. Afrashtehfar, Bull. Chem. Soc. Jap. 48, 2372 (1975). Bryngdahl, 0., Acta Chem. Scand. 12, 648 (1958). Chu, B., and F. J. Schoenes, Phys. Rev. Lett. 21, 6 (1968). Chu, B., F. J. Schoenes, and W. P. Kao, J. Amer. Chem. Soc. 2g, 3ou2 (1968). Chu, B, and F. J. Schoenes, Phys. Rev. 185, 219 (1969). Chu, B., Ber. Bunsenges. Phys. Chem. 16, 202 (1972). Chu, B.,D. Thiel, W. Tscharnuter, and D. V. Fenby, J. Phys. Paris 133, SUppl. Nr. 2/3, Cl-lll (1972). Chu, B., S. P. Lee, and W. Tscharnuter, Phys. Rev. A Z, 353 (1973). 225 226 Chang, R. F., P. H. Keyes, J. V. Sengers, and C. 0. Alley, Ber. Bunsenges. Phys. Chem. 16, 260 (1972). Chen, S. H., and P. Polonsky-Ostrovsky, Opt. Comm. 2, 64 (1969). Claesson, 3., and L. o. Sunde16f, J. Chim. Phys. Physico. Chim. Biol. 34, 91a (1957). Cope, A. F. G., H. H. Reamer, and C. J. Pings, Ber. Bun- senges. Physik. Chem. 16, 318 (1972). Daoust, H., and A. Lajoie, Can. J. Chem. 64, 1853 (1976). Davies, D. G., J. Chem. Soc. London 1935, 1166 (1935). Denbigh, K. G., Trans. Faraday Soc. 46, 1 (1952). de Groot,S. R., and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Co., Amsterdam, 1969). Deutch, J. M., and R. Zwanzig, J. Chem. Phys. 46, 1612 (1967). Dougherty, E. L., and H. G. Drickamer, J. Chem. Phys. 23, 295 (1955). Dougherty, E. L., and H. G. Drickamer, J. Phys. Chem. _9, 443 (1955). Dufour, L., Poggend. Ann. Physik. 246, 490 (1873). Dye, J. L., and V. A. Nicely, J. Chem. Ed. 46, 443 (1971). Ewing, M. B., and K. N. Marsh, J. Chem. Thermodyn. 2, 351 (1970). Ferrell, R. A., Phys. Rev. Lett. 24, 1169 (1970). Filippov, L. P., Int. J. Heat Mass Transfer 2;, 331 (1968). Fisher, M. E., and P. E. Seesney, Phys. Rev. A. 2, 825 Fitts, D. Nonequilibrium Thermodynamics (McGraw-Hill, New York, 19627? Fixman, M., J. Chem. Phys. 36, 310 (1962). Friedlander, J., Z. Phys. Chem. Leipzig 26, 385 (1901). 227 Gambhir, R. D., B. Viswanathan, and E. S. R. Gopal, Indian J. Pure Appl. Phys. 6, 787 (1971). Gammill, P. M., and c. A. Angell, J. Chem. Phys. 60, 584 (197 ). ‘— Gertz, L. G., and L. P. Filippov, Zh. Fiz. Khim. 36, 2424 (195 ). Giglio, M., and A. Vendramini, Appl. Phys. Lett. 26, 556 (1974). Giglio, M., and A. Vendramini, Phys. Rev. Lett. 34, 561 (1975). Giglio, M., and A. Vendramini, Phys. Rev. Lett. _6, 168 (1975). Goldburg, W. I., and P. N. Pusey, Phys. Rev; Agi,766 (1971). Greer, S. C., T. E. Block, and C. M. Knobler, Phys. Rev. Lett. 34, 250 (1975). Greer, S. C., Phys. Rev. A 14, 1770 (1976). Grossmann, 8., Phys. Lett. A 36, 374 (1969). Gfilary, E., A. F. Collings, R. L. Schmidt, and C. J. Pings, J. Chem. Phys. 66, 6169 (1972). Haase, R., and B. K. Bienert, Ber. Bunsenges. Phys. Chem. 11: 392 (1967). Haazg, R., and M. Siry, Z. Phys. Chem. Frankfurt 61, 56 (19 ). Haase, R., Thermodynamics 62 Irreversible Processes (Addi- son-Wesley, Mass., 1969). Hildebrand, J. H., Viscosity and Diffusivity: 4 Predictive Treatment (John Wiley and Sons, New York, 1977). Hocken, R., and M. R. Moldover, Phys. Rev. Lett. _1, 29 (1976). Horne, F. H., Ph.D. Thesis, University of Kansas, 1962. Horne, F. H., J. Chem. Phys. 46, 3069 (1966). Horne, F. H., and R. J. Bearman, J. Chem. Phys. 46, 4128 (1967). 228 Ingle, S. E., and F. H. Horne, J. Chem. Phys. _6, 5882 (1973). Jamieson, D. T., and E. H. Hastings, Thermal Conductivity, Proc. 8th Conf., C. Y. Ho and R. E. Taylor, Eds. (Plenum Press, New York, 1969). Jamieson, D. T., J. B. Irving, and J. S. Tudhope, Eds., Liggid Thermal Conductivity, a data survey to 1973 (H.M.S., Edinburgh, 1975). Kadanoff, L. P., and J. Swift, Phys. Rev. 166, 89 (1968). Kadanoff, L. P., J. Phys. Soc. Jap., 26, suppl. 122 (1969). Kawasaki, K., Phys. Rev. 166, 291 (1966). Kawasaki, K., Progr. Theor. Phys. 41, 119 (1969). Kawasaki, K., Ann. Phys. 61, l (1970). Kawasaki, K., Phys. Rev. A19 1750 (1970). Klein, H., and D. Woermann, Ber. Bunsenges. Phys. Chem. Kregfieck, G. C., and L. Benjamin, J. Phys. Chem. 66, 2476 19 ). Kricheviskii, I. R., N. E. Khazanova, and L. R. Linshits, Dokl. Akad. Nauk. SSSR 66, 113 (1954). Kricheviskii, I. R., N. E. Khazanova, and I. V. Tsekhanskaia, Russ. J. Phys. Chem. 34, 598 (1960). Lorentzen, H. L., and B. B. Hansen, Acta Chem. Scand. 11, 893 (1957). Lorentzen, H. L., and B. B. Hansen, Acta Chem. Scand. 12, 139 (1958)- Mason, E. A., L. Miller, and T. H. Spurling, J. Chem. Phys. McLaughlin, E., Chem. Rev. 64, 389 (1964). Michels, A., and J. V. Sengers, Physica 26, 1239 (1962). Miller, L., Z. Naturforsch A 4, 262 (1949). Miller, D. G., Chem. Rev. 66, 15 (1960). 229 Miller, D. G. , Found. Continuum Thermodyn. Proc. In t. Symg. 1973, J. J. D. Domingos, M. N. R. Nina, and J. H. Whitelaw, Eds. , (Wiley, New York, 1975). Mistura, L., J. Chem. Phys. 66, 2375 (1971). Mistura, L., J. Chem. Phys. 62, 4571 (1975). Moore, W. J., Physical Chemistry (Prentice-Hall, New Jersey 1972). Morrison, G., and C. M. Knobler, J. Chem. Phys. 66, 5507 (1976). Mountain, R. D., and R. Zwanzig, J. Chem. Phys. 46, 1451 (1968). Onsager, L., Phys. Rev. 31, 405 (1931). Onsager, L., Phys. Rev. 36, 2265 (1931). Osipova, V. A., Dokl. Akad. Nauk. Azerb. SSR 13, 3 (1957). Papoular, M., J. Chem. Phys. 66, 86 (1974). Pelger, M., H. Klein, and D. Woermann, J. Chem. Phys. 61, 5362 (1977). Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria (Prentice-Hall, New Jersey, 19697—' Prigogine I., L. deBrouckere, and R. Armand, Physica 16, 577 (19503 Prigogine, I., and R. Defay, Chemical Thermod namics, D. H. Everett, trans. (Longmans Green and Co., London, 1954). Rastogi, R. P., and G. L. Madan, J. Chem. Phys. 43, 4179 (1965). Rastogi, R. P., and B. L. S. Yadava, J. Chem. Phys. 61, 2826 (1969). Rastogi, R. P., and B. L. S. Yadava, J. Chem. Phys. 52 2791 (1970). Rosenberg, D. U. , Methods for the Numerical Solution of Partial Differential EquationsI(American Elsevier, New York, 1969). 230 Rowley, R. L., and F. H. Horne, J. Chem. Phys. 66, 325 (1978). Rutherford, W. M., and H. G. Drickamer, J. Chem. Phys. 22, 1157 (1954). Scott, R. L., Ber. Bunsenges. Phys. Chem. 16, 296 (1972). Sengers, J. V., Critical Phenomena, Proc. Intern. School "Enrico Fermi", Course LI, M. S. Green, Ed. (Academic Press, New York, 1971). Sengers, J. V., Transport Phenomena, 1973 A.I.P. Conf. Proc. No. 11, J. Kestin, Ed. (A.I.P., New York, 1973). Sengers, J. M., and H. L. Sengers, Expgrimental Thermo- dynamics 62 Non-reactingFluids, Vol. II, Butterworths, London, 1975. Stanford, D. J., and A. Beyerlein, J. Chem. Phys. 66, 4338 (1973). Stanley, H. E., Introduction 66_Critica1 Phenomena (0x— ford, New York, 1971). Story, M. J., Ph. D. Thesis (University of Cambridge, 1967). Story, M. J., and J. C. R. Turner, Trans. Faraday Soc. £2. 349 (1969). Sundele, L., Ark. Kemi 1;, 317 (1960). Swift, J., Phys. Rev. 113, 257 (1968). Thomaes, G., J. Chem. Phys. 26, 32 (1956). Tichacek, L. J., and H. G. Drickamer, J. Phys. Chem. 66, 820 (1956). Tsai, R. C., Masters Thesis, University of Akron, 1970. Turner, J. C. R., B. D. butler, and M. J. Story, Trans. Faraday Soc. 63, 1906 (1967). Tyrrell, H. J. V., Diffusion and Heat Flow 16 Liguids (Butterworths, London, 1961). Venart, J. E. S., Thermophysical Prgperties, 4th Symp. (Univ. Delaware, Delaware, 1968). Viswanathan, B., K. Govindarajan, and E. S. R. Gopal, Indian J. Pure Appl. Phys. 11, 157 (1973). 230 Rowley, R. L., and F. H. Horne, J. Chem. Phys. 66, 325 (1978). Rutherford, W. M., and H. G. Drickamer, J. Chem. Phys. 22, 1157 (1954). Scott, R. L., Ber. Bunsenges. Phys. Chem. 16, 296 (1972). Sengers, J. V., Critical Phenomena, Proc. Intern. School "Enrico Fermi", Course LI, M. S. Green, Ed. (Academic Press, New York, 1971). Sengers, J. V., Transport Phenomena, 1973 A.I.P. Conf. Proc. No. 11, J. Kestin, Ed. (A.I.P., New York, 1973). Sengers, J. M., and H. L. Sengers, Experimental Thermo- dynamics 61 Non-reacting Fluids, Vol. II, Butterworths, London, 1975. Stanford, D. J., and A. Beyerlein, J. Chem. Phys. 66, 4338 (1973). Stanley, H. E., Introduction 66 Critical Phenomena (0x- ford, New York, 1971). Story, M. J., Ph. D. Thesis (University of Cambridge, 1967). Story, M. J., and J. C. R. Turner, Trans. Faraday Soc. £11. 349 (1969). Sundele, L., Ark. Kemi 16, 317 (1960). Swift, J., Phys. Rev. 113, 257 (1968). Thomaes, G., J. Chem. Phys. 26, 32 (1956). Tichacek, L. J., and H. G. Drickamer, J. Phys. Chem. 66, 820 (1956). Tsai, R. C., Masters Thesis, University of Akron, 1970. Turner, J. C. R., B. D. butler, and M. J. Story, Trans. Faraday Soc. 63, 1906 (1967). Tyrrell, H. J. V., Diffusion and Heat Flow 16_Liguids (Butterworths, London, 1961). Venart, J. E. S., Thermophysical Prgperties, 4th Symp. (Univ. Delaware, Delaware, 1966). Viswanathan, B., K. Govindarajan, and E. S. R. Gopal, Indian J. Pure Appl. Phys. 11, 157 (1973). 231 Voronel, A. V., and T. 0vodova, Sov. Phys. JETP 9, 169 (1969). — Waldmann, L., Z. Phys. 114, 53 (1939). ., Z. Phys. 21, 501 (1943). Waldmann, ., Z. Phys. 124, 2 (1947). Waldmann, ., Z. Naturforsch. A 2, 358 (1947). L L Waldmann, L., Z. Phys. 124, 30 (1947). Waldmann, L L ., Z. Naturforsch. A 4, 105 (1949). Waldmann, Weasts, R. C., Ed., Handbook 62 Chemistry and Physics, Slst Edition (The Chemical Rubber Co., Ohio, 1970). Wilhelm, V. E., M. Zettler, and H. Sackmann, Ber. Bunsenges. Phys. Chem. 19: 795 (1974). Wirtz, K., Ann. Physik.,Leipzig, 36, 295 (1939). Wirtz, K., and J. W. Hiby, Physik. Z. Liepzig 44, 369 (1943). Woermann, D., and W. Sarholz, Ber. Bunsenges. Phys. Chem. 6_9_. 319 (1965). Wood, S. E., and J. A. Gray III, J. Amer. Chem. Soc. 14, 3729 (1952). W m u '41 Em “ 3168 9817 I1 'él3illIHIIWIIWIIHIH 3 1293