A STUDY OF DIFFUSION OF BINARY NON-IDEAL NONASSOCIATING LIQUID SOLUTIONS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY Duane L. BidIack I964 {THESIS 'J LIBRARY Michigan State University This is to certify that the thesis entitled A STUDY OF DIFFUSION OF BINARY NON-IDEAL NONASSOCIATING LIQUID SOLUTIONS presented by Duane L. Bidlack has been accepted towards fulfillment of the requirements for ELL degree in W81 ne e ring » '//;£/ / .x; . ,. [I‘Vflv LL; I“; i/L/(J- ’,-1‘~.7_\ _ A A )— Majiir professor Date October 11+, 1961+ 0-169 ON LY. O ABSTRACT A STUDY OF DIFFUSION OF BINARY NON-IDEAL, NONASSOCIATING LIQUID SOLUTIONS by Duane LI Bidlack The purpose of this study is to gain some insight into the expected diffusion behavior of binary, non-ideal, non- electrolyte liquid systems where there is no instance of association. For this purpose, the study is divided into two separate parts. The first part involves mutual diffusion over the entire concentration range. Mutual diffusion data were obtained by a very accurate optical method for the systems hexane -hexadecane , heptane -hexade cane , hexane -dode cane , and hexane-carbon tetrachloride. In addition, the system cyclohexane-carbon tetrachloride, found in the literature, was included in this investigation. According to existing theories for the diffusion of non-ideal systems, the diffusion coefficient-viscosity product, divided by an activity correction, DTI/(d ln ai/d ln Ni)’ should vary linearly with mole fraction. This study shows that, like associating systems, the activity correction tends to overcorrect in nonassociating systems, DUANE L. BIDLACK sometimes by more than the original deviation from Raoult's Law. Part II involves the study of how solutes diffusing at infinite dilutionin nonassociating solvents depend on solute size and shape. The diffusion coefficients of twenty-four solutes in carbon tetrachloride and twelve solutes in hexane were collected either by experiment or from the literature. Contrary to such theoretical developments as the Stokes-Einstein equation which predicts that the diffusion coefficient is inversely proportional to the cube root of the solute molar volume, this study finds that the power of the molar volume should be 0. 77. This study also finds that the shape of the solute molecule is more important than previously anticipated, although the shape of the solvent molecules do not seem to affect diffusivity. This study does, however, as predicted by theory, show that the diffusivity is inversely proportional to solvent viscosity. A STUDY OF DIFFUSION OF BINARY NON-IDEAL NONASSOCIATING LIQUID SOLUTIONS BY r «J Duane Lf Bidla ck A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1964 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to Dr. Donald K. Anderson for his valuable guidance during the course of this work. The author is indebted to the Division of Engineering Research of the College of Engineering at Michigan State Uni- versity and to the donors of the Petroleum Research Fund, which is administered by the American Chemical Society, for providing financial support. Grateful acknowledgement is extended to William B. Clippinger who assisted in the design and fabrication of the laboratory apparatus necessary for this research. TAB LE OF CONTENTS ACKNOWLEDGEMENT LIST OF TABLES LIST OF FIGURES INTRODUCTION PART I. MUTUAL DIFFUSION OF NONASSOCIATING BINARY LIQUID SOLUTIONS PREVIOUS THEORE TICAL WORK Hydrodynamic Theory The Intrinsic Diffusivity Self— Diffusion Statistical Mechanical Theory Self— Diffusion Discussion of Previous Theoretical Work EXPERIMENTAL METHOD Inte rfe romete r Alignment ViFrations The Came ra The Wate rbath Diffusion Cell Adjustment of Equipment Procedure for Experimental Run Calculations Accuracy of Experimental Method Purith (Emponents Experimental Re sults MUTUAL DIFFUSION OF NONASSOCIATING SYSTEMS Solution Theory Regular Solutions Discussion of Previous Experimental Work Activity Data and Selection of Systems ii Page iv 12 15 21 26 28 28 32 32 33 36 38 44 47 51 55 55 58 62 62 64 67 68 TABLE OF CONTENTS (continued) Hexane - Hexade cane ' 6 8 Heptane-Hexadecane and Hexane - Dode cane 7O Cjclohexane-Carbon Tetrachloride 71 Hexane-(Tarbon TetrachIoride 72 Discussion of Results 73 PART II. DIFFUSION OF SOLUTES AT INFINITE DILUTION INTRODUCTION AND BACKGROUND 88 Previous Theoretical Studies of Liquid Diffusion at Infinite Dilution 90 Stokes-Einstein 90 Kirkwood Equation 91 Engineering Correlations 93 PRESENTATION AND DISCUSSION OF RESULTS 95 Solutes 96 Discussion of Results 102 Possible Model - Turbulent Solvent Model 110 Ki rkwood Ejuation 1 l 1 Viscosity 114 SUMMARY OF CONCLUSIONS 117 LITERATURE CITED 120 APPENDIX A: Sample Calculation 125 APPENDIX B: Nomenclature 131 iii Table II III IV VI VII VIII IX XI LIS T OF TAB LES Comparison of experimentally determined diffusivity data with previous data for aqueous sucrose solutions. Comparison of physical constants with previous recorded data for purity evaluation of mate rials used. Summary of experimental mutual diffusion coefficient, viscosity, and density data for study of mutual diffusion. Comparison of properties of systems to determine their regularity- Comparison of diffusion coefficients calculated at the average concentration of the two solutions in the diffusion cell and extrapolated to zero concentration. Experimental data of solutes diffusing in infinite dilution in carbon tetrachloride and hexane as solvents. Comparison of refractive indices of chemicals used with values in the literature. Tabulation of the values for the constant G.of equation 84 calculated from the intercepts of plots of Figures 18 and 19. Tabulation of experimental and calculated diffusion coefficients for solutes diffusing in carbon tetrachloride. Tabulation of experimental and calculated diffusion coefficients for solutes diffusing in hexane. Comparison of observed diffusion coefficients with values calculated by means of equation 79 combined with equation 81, the Kirkwood equation for irregularly shaped solutes. iv Page 56 57 59 69 97 99 101 106 107 109 ' 115 Figure 10 11 12 13 14 15 16 17 LIST OF FIGURES Schematic Diagram of Interferometer Showing Position of Mirrors- Photograph of Mach- Zehnder Interferometer Showing Components. Photograph of Diffusion Cell for Measurement of Diffusion Coefficients. Diagram of Diffusion Cell. Detailed Drawing of Diffusion Cell. Appearance of Vertical Fringe Pattern as a Function of Path Difference. Typical Set of Photographs Taken During Diffusion Run (Experimental Run No. 104). Fringe Patte rn. Experimental Viscosity and Density versus Mole Fraction for Hexane-Hexadecane. Experimental Viscosity and Density versus Mole Fraction for Heptane-Hexadecane. Experimental Viscosity and. Density versus Mole Fraction for Hexane-Dodecane. Experimental Viscosity and Density versus Mole Fraction for Hexane-Carbon Tetrachloride . D17 and Dn/(d 1n a/d 1n N) as a Function of Mole Fraction for the System Hexane-Hexadecane. Dn and Dn/(d 1n a/d 1n N) as a Function of Mole Fraction for the System Heptane-Hexadecane. Dn and. Dn/(d 1n a/d 1n N) as a Function of Mole Fraction for the System Hexane-Dodecane. D11 and Dn/(d 1n a/d In N) as a Function of Mole Fraction for the System Cyclohexane-Carbon Tetrachlo ride . Dr; and Dn/(d 1n a/d In N) as a Function of Mole Fraction for the System Hexane-Carbon Tetra chlo ride . Page 29 30 39 40 41 45 50 52 74 75 76 77 79 80 81 82 18 19 20 LIST OF FIGURES (continued) Solutes in Carbon Tetrachloride at Infinite Dilution, 1n D versus 1n v. 103 Solutes in Hexane at Infinite Dilution, In D versus 1n v. 104 Comparison of Kirkwood Equation with Experimental Data. 113 vi INTRODUCTION If two liquid solutions containing the same components but having different concentrations are placed in the same vessel, the molecules of the components will tend to migrate from the higher to the lower concentration until the solution is entirely uniform. This phenomenon, known as diffusion, was first described by Ficklzs). By means of an analogy to heat flow, he postulated that the rate of transfer at constant temperature and pressure was proportional to the concentration gradient. For one-dimensional diffusion in the x-direction, this relation may be written J = - Jag—:- (1) where J is the flux or rate of material transfer across a plane of unit area and 8c/3x is the concentration gradient. The factor D is a proportionality coefficient which was at first thought to be constant for each system at constant temperature and pressure but which is now known to be a function of concentration. The negative sign indicates that the flow is opposite to the direction of positive concentration gradient. The basic diffusion law of Fick may be generalized by T: — DVc , (7-) where Tis the vector flux and We is the vector concentration gradient. However, because of the difficulty in obtaining measurements for diffusion in more than one direction, the systems are usually simplified to the one-dimensional form as given by equation 1. A study of diffusion is of interest both to the theoretical physicist for studying the mechanics of the liquid state and the design engineer interested in a better understanding of liquid phase mass transport. Any acceptable liquid state theory must also describe the diffusion process with accuracy. Recently, many studies have been published concerning diffusion of non-electrolyte binary ideal mixtures and non-ideal solutions in which non-ideality is caused by association of the (3’ 32,40). In an associated mixture, two or more molecules molecules cluster together and move through the medium as a unit instead of as single molecules, thus complicating any diffusion study. The purpose of this research is to attempt to remove this complication by studying only non-ideal solutions in which molecular association is absent. This study is divided into two parts: Part I, a study of mutual diffusion throughout the entire concentration for several different binary solution systems, and Part II, a study of numerous solutes at infinite dilution in two different solvent systems. Part I Mutual Diffusion of Nonassociating Binary Liquid Solutions PREVIOUS THEOR ETICAL WORK Caldwell and Babbus’ 16) found that the diffusion coefficient for ideal binary mixtures is a linear function of mole fraction; however, non-ideal systems may vary quite considerably from linearity. Several approaches to the prediction of diffusion behavior with concentration and other measureable thermodynamic quantities have been offered by various investigators. Among the earliest developments was a theory proposed by Eyring and co—workersl27' 54) which was based on the absolute rate theory. According to this theory, the quantity DTI/(d ln ai/d ln Ni) is a linear function of mole fraction, where n is the solution viscosity, ai the activity, and Nithe mole fraction. Recently, two other theories have appeared which also predict the linearity of the quantity D77 /(d ln ai/d ln Ni) with mole fraction. These two theories, the hydrodynamic concept of Hartley and Crank(33) and the statistical mechanical treatment(6), vary widely in their derivation and so both will be briefly reviewed in the following pages. There is some controversy among the prOponents of these theories as to which is the most valid derivation. Since the final results of the three theories are identical, the purpose of this study is not to add to the controversy but to briefly outline the derivations of two of the concepts and examine the final result experimentally. Hydrodynamic Theory According to the hydrodynamic model, the molecules of a liquid oscillate within a cage or potential hole formed by its neighboring molecules. A molecule undergoing this oscillation will occasionally acquire enough energy to jump over the potential barrier of this hole and migrate to a neighboring hole. A molecule migrating through a liquid in this manner is said to be undergoing “intrinsic" diffusion. For a pure liquid or solution of uniform concentration, this jumping effect will be entirely random. However , when a concentration gradient exists in the solution, this molecular jumping will still be random but the overall movement, of course, will be in the direction of lower concentration. In the Hartley-Crank theory(33), the volume reference frame was used, that is, diffusion is assumed to take place in a closed system in which the volume is constant. Thus, when a component of type ”i" migrates across a reference plane in the closed volume, the increase of material must be compensated by a bulk flow in the opposite direction. The diffusion coefficient that is studied experimentally is the overall or mutual diffusion coefficient that includes both the bulk motion and intrinsic diffusion. Therefore, the purpose of this development is to obtain equations that describe the mutual diffusion in terms of the intrinsic diffusion and the bulk flow. The intrinsic diffusivity,fl/- of component i, is defined 1 as, 8c. _ 1 J1 - “/0: '5'; , (3) where I1 is the flux relative to the bulk motion, i. e. , flux due only to the molecular motion, and Sci/8x is the gradient in one-dimensional intrinsic diffusion in the x-direction. Thus, for binary diffusion , BC BC Volume flux due _ A B [ 1 ' ’VA’OZ. ‘3?- -VB’0T3 ’53;— to diffusion [Compensating bulk flow 1 '5 (I) , where v.1 is the molar volume, cm. 3/mole, of component i and the two components of the system are denoted by A and B, respectively. As previously indicated, the increase in materials must be compensated by the bulk flow. Therefore, 8c 8c 'VA’O/Ahi'é -VB’0B_8x—B “1’ = 0- ‘4’ The transport of molecules of species A by both bulk motion and intrinsic diffusion can be described with one over- all diffusion coefficient, DAB' by 0 acA JA = 'DABB‘X— (5a) where JAO is the total flux of A. Similarly, o acB JB : 'DBA’SE— ' (5b) Since the reference frame is a closed volume of liquid, there will be no net transfer of material, so that, o o _ JA VA + JB VB — 0 (6a) and BC BC A B _ ‘ VADABTX— ' VBDBAK— — 0 (6‘3) A material balance shows, VACA + vBcB = 1, which when differentiated with respect to distance is dCA + v dCB -0 (7) Adx de " V if Vi is assumed independent of concentration. When equations 6b and 7 are compared, it is found that, DAB : DBA (8) To eliminate the factor 4; of equation 4, a volume flux balance is written for component A: ac dc A __ A ' DAB VA "—ax ‘ ’Oth T: + ‘1’ CAVA (9) which is combined with equations 4 and 7 to yield D =(yd/B-zaz)c AB + AVA A’ (108.) which reduces readily to DAB = ’0/BCAVA +/0/ACBVB . (10b) The Intrinsic Diffusivity - In order for a thermodynamic system to be at equilibrium at constant temperature and pressure, Gibbs(26) has shown that the chemical potential must be uniform throughout the system. For example, a system with two phases, <1) and 8, will only be in equilibrium if the chemical potentials in the two phases are equal for each component, i. e. , Hid) = ”i9 . If the chemical potentials are unequal, for instance, Hid) > “i0 the system is in a state of non—equilibrium and there will be a general Spontaneous migration of component i from the phase of higher potential to the phase of lower potential. This spontaneous migration will persist until the potentials are the same in both phases, or more precisely, a non-equilibrium system tends to move spontaneously toward equilibrium. Of course, the degree of non-equilibrium and the rate of approach toward equilibrium will depend somewhat on the chemical potential difference. The process described above does not have to occur across a phase boundary. For instance, diffusion is a non- equilibrium thermodynamic process which may occur in a single phase. In the particular case of this study, diffusion in a liquid solution is being considered. Here the system is a liquid solution in non-equilibrium because of a concentration gradient within the liquid. The molecules of component i of the solution will diffuse spontaneously from the region of high concentration and high potential to the region of low concentration and low potential. The rate of approach toward equilibrium in this case will depend on the difference in potential as a function of distance. Therefore, the chemical potential gradient is the factor which pushes the system toward equilibrium, that is, the driving force for diffusion. In his early studies, Fick<25> suggested the concentration gradient itself might be the driving force, a conclusion that was arrived at mostly by intuition. The osmotic pressure gradient has also been proposed as the driving force. Both of these conclusions have proven to be generally unsatisfactory; however, it can be shown that the chemical potential gradient and the osmotic pressure gradient are closely related. In any case, the driving force for diffusion is considered as the negative gradient of chemical potential, thus , 10 Driving force for one- 3,11 dimensional diffusion = — 6—— . (11) . . . . x of 1 1n the x—direction This driving force is compensated by a resisting force which was shown by Einstein<24) and Sutherland(55) to be proportional to viscosity and to the radius of the diffusing molecule. While experimental evidence generally proves the proportionality of resisting force to viscosity, most investigators find it is not proportional to molecular radius. This will be further discussed in Part II. Therefore, in the present consideration , let — 0’.T) uiI (12) 1 Resisting force for diffusion _ of molecule i — where U is the solution viscosity, u-I 1 is the velocity of molecule i as it migrates through the liquid medium and (Ti is a proportionality constant which depends on molecular size and shape, and is known as the friction factor. Hartley and Crank(33) assumed it to be approximately independent of the solution viscosity and composition. The negative sign is a result of the velocity of molecule i being in the opposite direction as the resisting force . Summing the forces acting on the diffusing molecule, 311. 1 I _ __8x - O'inui —O (13) as pi 2 pic + RT 1n ai , multiplying both sides by the concentration of i, Ci’ and rearranging, _ I RT 81h a RT 3 1n 3i 8 c- J. — c. u. : - —— c. __.__l_ :-___ c. 1 1 1 1 Din 1 3x Oi T) 1 TE:- ‘33? . 8c. 2 RT d 1n a1 1 (14) -0177 a 1n bi 3x Comparing equations 14 and 3, it is found that fl: RT dln ai (15) i Oi TI 3 1n ci Thus, the intrinsic diffusion coefficient is related to elementary thermodynamic quantities. These quantities may be related to the mutual diffusivity by means of equation 15 by letting i be the components A and B of a binary solution and substituting the resulting equations for yO-A/and A}: into equation 10b. Thus, dlna W: ___RT _._._A (Isa) 12 dlna ,O/ 2 LT _____dl B (16b) B a'Bn ncB dlna dlna RT B RT A (17) __ —————-+ c v BB UAT) dlncA By using the relations CAVA + CBVB = 1, NA + NB — 1, and NA/NB = cA/CB, it is found that, dlnaA : NB dlnaA :2- dlnaA dIn'cA CBVB dlnNA VB dInNA (18) dlnaB: NA dlnaB :2. dlnaB dln CB CAVA HlnNB VA dlnNB where y- is the molar volume of the solution. As a result of the Gibbs-Duhem equation, —: 1:: =———: 111:: . n A n B Combining, equations 17, 18,and 19, results in the following equation D = 5.2 3+1]? dlnaA. (20) AB 7’) GB (TA 3 1n NA Self-Diffusion - Consider a liquid solution with uniform composition and let the molecules of component i be labelled as either type 1 or type 2. If there happens to be a higher concentration of type 2 molecules in one region of the solution 13 than in another, diffusion of the 1 molecules will take place even though the solution is uniformly mixed as far as the various components are concerned . This type of diffusion is known as ”self diffusion" of component i. Experimentally, the molecules of component i may be labelled by radioisotOpes of component i. Usually, a solution with known concentration, and with component i labelled by means of its radioisotope, is placed in a capillary tube sealed at one end and with known dimensions. This capillary is in turn placed vertically in a bulk solution of approximately the same concentration as the capillary solution except the bulk solution contains no radioisotope of i. In this way, the radioisot0pe tagged molecules are allowed to diffuse for a few days and then the concentration is again measured by a radiation detection instrument. From these data the necessary self-diffusion calculations are made. The effects of the slightly different mass and the small amount of energy emission are negligible. The self-diffusion coefficient, D7:< l # dc. 35:. 1 i'1dx' , is defined by (21) where Ji* is the flux and dciik/dx is the gradient of the labelled (19) component of i. Darken(21) and later Carmen and Stein extended equation 20 to include self-diffusion, so that 14 * * dlnaA DAB‘ [NADB +NBDA ] my, (22’ The assumptions involved in the derivation of equation 22 are that the labelled and unlabelled Species of the self- diffusing component are so similar that the friction factor- viscosity product, and the activity coefficients for the two species are identical. Statistical Mechanical Theory Bearman , Kirkwood, and co-workers14 ' 5 ’ 6 ’ 7 ’ 8 ’ 9 ’ 39) have recently published a series of papers describing transport processes in multicomponent liquid solutions from a non- equilibrium statistical mechanical viewpoint. As a consequence of the develOpment of the equation of motion, they found that (1)—. — (1)*-» _1 V ‘r’ ”at C0. ”1”} (rd—2‘; ); ar—E " {C(53) ('r’l. ‘r’) - cpéz)(?1. 3%} d3? . (23) — >:< —) F“) (r1) is the mean frictional force exerted on a molecule . . 9 of speCies 0. at location r1 by all the other molecules of a :1 component system, and is equivalent to the resisting force for diffusion of equation 12 of the hydrodynamic theory. The . (1) -’ . . concentration, c (r1) is the concentration of 0. molecules per 0. (2) " .9 9130.1, r) is the average concentration of molecules . . " of speCies a. at pOint r unit volume, c . -D . 1’ and of type (3 at a distance r relative -> (z) —) -) . . to r1, and C80. (r1,r) is the average concentration of molecules . . ‘9 . 4 . of speCies B at pOint r1, and of type Ct at a distance r relative -) '0. . . . to rl where r is the vector distance of magnitude r separating the two molecules. The term V0“3 is the mutual potential energy between the 0. molecule and the 8 molecule. 15 1 I." Ila'l‘.1 {I‘ll-I'll: . It'll, 16 According to classical mechanics, the dynamics of a molecule at any instant is completely described by its three Cartesian coordinates and the conjugate momenta. A system of N molecules will thus have 6N dimensions. An interval of classical space with such dimensions is known as "phase space". For most systems, this is an extremely large number of dimensions, so that, for simplicity, all information for the system is condensed to a single representative point in the system. It follows that the concentration of (1 molecules is C(a1)(?l), the number density of 0. molecules at point ?1 and C(é)(?2) is the number density of molecules [3 at F2 , where ?l and r; are the representative points of a and [3 respectively. _) The distance r is the separation of the two points, i. e. , —a. —> _) r I‘l-I‘Z Additional assumptions of the classical mechanical theory are (1) the molecules have only the three degrees of freedom due to translational motion, and (2) the forces between the molecules are two body forces, so that the potential energy of the system is, N 1 V v [3 V: - Z Z Z 2‘. V (r . ), (24) 3:1 a=1k=1 j=l “5 “38k aj 9! pk where Vufihajfik) is the mutual potential energy between a pair of molecules of species a and p and is a function only of the distance between the centers of masses , r The notation ojfik 17 aj 3! [3k is used to indicate that there is no term corresponding to i = j when a = [3. Na. and N are the number of a and [3 molecules in the system respectively. In this arrangement of pairs of molecules, one can define another concentration, c‘ all; (r1, r). This term is the average - . . -» concentration of molecules of type a at pOSition r1, and of type . -’ . -’ . . . [3 at a distance r relative to r The number denSity in pair 1° 2 . . . space, Chi; , is related to the Singlet concentrations by “1(ng ‘1? '3?) 2 Gym-1h)Cisl)(?2’g(zii(ry?) (25) where g(2 BM?" r)is the pair correlation function. The distance between the molecules over which the intermolecular forces are effective is relatively small for most circumstances, so that c;3 may be considered constant over this range. Therefore , (285:1, 3!) z c;1)(_I-"1) cg’G'z) g‘zgu r1. 'i’) - (7-6) Substituting equation 26 into equation 23, Q (1)->1 V a ?dVap CH? 1) F0, (3’ r=1) E {:31 Co,(r1) C1391); 173?— X g1? ("l-’1, ‘r’) - géza) (331.3%) d3? . - (27) At equilibrium Cw " Ca figafi 18 20) where gab is the radial distribution function of equilibrium statistical mechanics, an important factor for describing thermodynamic behavior. This function is related to the potential of mean force between molecules 0 and I3 , WoB by the Boltzmann distribution 2,0 gig ) = exp. (-WaB/kT) (29) where k is Boltzmanns constant and T is the absolute temperature. (2’ 0)is spherically symmetrical about molecule 0. and since by gap _ (2.0) _ g(2 0) definition WQB — WBQ’ go’B _ gfio. For systems not at equilibrium, one must use the pair correlation 2, O 2 2 function g: 0.8 ,) which is spherically unsymmetric and gill; 7fg ( ) . (2 ) However,g 0.8 may be related to the equilibrium radial distribution function by g<2) _ (2.0) + g(2.1) (3o) gafi gafi a8 (2.1) where go"3 is a perturbation function. Substituting equation 30 into equation 27 f”)* = f‘1’0)* + F(1’1)* (31a) CI. 0. O. —(1 1) _ V T” (2.1) (2 1) d» Fa =1: -7 51? (3 mi; gap — gflo: r (31b) —(1.0) _ 1 V 3" dvap (2 01g.(2 0) _ F0. * — 2 El CBS-1: 1?— (g gafi - gfia 3d” r _0 fl (31c) 19 where F(1 ’ 1*) is actually the perturbation friction forces on molecule 0. . As in the Hartley-Crank hydrodynamic theory, the friction force on o. is equated to the gradient of chemical potential, Vua = fél'l’i . (32) By using the linear phenomenological relationships of non-equilibrium thermodynamics , Laity<42) found that , V = - 2: Vila (3:1 Ce Lag u ("’ -") I3 (33) -> -> . . where ua and u are the overall veloc1ties of molecules of I3 type 0. and [3 respectively due to both diffusion and bulk motion and gap is a friction coefficient which obeys the Onsager reciprocal relation: gag = €139. Equation 33 agrees with 32 if, g (2 1) _, (1) r . —) - jI2L0Y $0 ‘5 (up 30.) ‘34) <15 where+é B) is an arbitrary parameter. Therefore , v dV vVol : "-51; (3—21 C13 (30 - 3(3)1‘TiTEgap(z 0) (#181, (1)) r (35) where the theorem(38) (F(B)(A’- ‘B’H‘a’dfi‘ = % A’ )F(B)B2dBV is used. Comparing equation 35 and 33, it is found that dV _ 1 E gIZB: 0) (1) E 1 dra (1’05” Ig‘é’) d3? (36) 20 Since V0. 2 V by definition, the Onsager reciprocal relation F3 F30 g = g , is obeyed. a8 [30. For a binary system with components A and B diffusing in the x-direction, equation 33 becomes duA ax : - CB Z"AB (uxA - uxB) (37a) and duB 3;. : - CA VAB (UXB - UXA) . (37b) Since the derivations hereafter will all be for uni-directional diffusion in the x-direction, the subscript x of the velocities will subsequently be dropped. The closed volume of liquid is still the reference frame, so that o o _ JAvA+JBvB _ 0 (8c) and J° = c u - (38a) A A A’ O z: , 8 JB cB uB (3 b) Therefore, cAuAvA + CBquB = O (39) which can be rearranged, A A u : - — —- 1.1 . (40) B VB CB A Substituting equation 40 into 37a dVA _ vA A ax— ’ ' CB (”AB (VA + V" '5‘“ DB) (413‘) 21 u I. = _ 1:32 . (41b) v13 . . _ o The chemical potential, “A — “A + RT 1n aA, so that “A _ RT dlnaA _ RlenaA 8CA :_uAt"AB . (4,) dx — dx c d In c 5x v A A B Rearranging equation 42 JO-uc :-§T_v dlnaA acA (43) A A A (”AB B d In CA 5x and comparing equation 43 with equation 5a, D .331 , dlna». z 31-: s “MA. (44.) AB éA13 B a In CA C‘AB a In NA Similarly, D zi‘lv dlnaB: RT lenaB. (44b) AB (”AB A a In CB éA13 a In “B Self-Difquion - Self-diffusion is considered as the diffusion of three components: (i) component A1, unlabelled species A having a concentration CA1; (ii) A*, labelled species A with concentration CA*5 and (iii) component B. Equation 33 also applied here and is written for the chemical potential gradient of component A* as , duA* _ _ * - dx CA éA"‘A1 (“A ‘1 A1) - cB VAIVB (uA* - uB). (45) As in mutual diffusion, the closed volume reference frame is used, so that +cuv =0 (46) c u v + CA’I‘uA’I‘VA’I‘ BBB However, VA* = VAl and since there is no concentration gradient 22 of B, its overall movement will essentially be zero, i.e. uB = 0. Therefore , = . 7 CAluAl + CA*uA* O (4) Since A1 and A)"< are identical members of species A, VA’I‘A '3 VAA 1 and VA’I‘B = (”AB , and the total concentrations of A, CA = CA + CA’i" 1 With these concentrations in mind, equation 47 becomes, duA’I‘ — _ * T—x " (9AA CA + gA13 CB) 11A (48) Again noting that uA* = p;* + RT 1n aA* , * 3k * duA — RT dlnaA ch dx — ”CASE d 1n CASE ax Then * * J as: C *u >1: z- RT+ :inaA ch . (49a) A A A CAEAA CBZ’AB ncAV‘I dx Because of no overall concentration gradient and Al and A* being identical components of species A, (1 1n a * d In 3'A A = 1 = 1 . (49b) (1 1n c 55? d In c A A1 Therefore, comparing equations 49 and 21, RT D * = . (50) A CAVAA + C13 Z"AB where D: is the self-diffusion coefficient of component A. At this point, Bearman and co-workerswl introduced an additional parameter, +08; +11%) (Da’i‘ + Dp’I‘) . (51) 23 This new parameter +043 is found to be concentration independent for ideal and the ”regular" solutions described by Hildebrand and Scott<36’ 37). Regular solutions are real solutions with the same degree of randomness as an ideal solution and hence the same entropy of mixing. Other properties of regular solutions are that the radial distribution functions are independent of composition at constant temperature and pressure and the molar volumes are additive. More will be said about regular solutions later (see discussion in Solution Theory). For self-diffusion friction coefficient, VAA , equation 36 becomes , dVA A* QAAngA’k =iS—5rrl—‘gp. (2 A9“) (+(l)*++Al’°)Al3?)d (52) After insertion of equation 51 and 50 and considering A1 and A* as components of species A, 1 deAAg (2, 0) d3}; _ z3AA RT 6 d gAA AA —CAT”AA+CB CAB C, V RT AA NA 4AA ”“13 gA13 N RT t~‘AA ( AVA NB VB) NA gAA + NB éA13 Normally, the left side of equation 54 will be a function of concentration but for regular solutions, one can see that it will be constant over the entire concentration range for iso- 24 thermal, isobaric diffusion. Therefore, at infinitely dilute concentration of A, i.e. NA") 0, 1 dVAA (2,0) I d3? _ gAA vB RT 3 Tr gAA AA ’ 2 AB and at infinitely dilute B, 1 dVAA g,(2 0) +31. _ v RT 6 dr gAA AAd — A Consequently, g v AA A = — . (55a) gAB VB With a similar argument, C. B B AB A After substitution of equation 55a, equation 50 becomes * _ RT _ VB RT DA _ v ' g, (c v ) CA A g + C ; AB A v+A cB B —vB AB B AB v RT = JE— . (56a) éAB Similarly, VA RT D * = . (56b) B gAB Remembering that ‘1; = NAVA + NBVB’ equation 56 may now be substituted into equation 44 with the result (1 1n a RT A D = —— [N v + N v ] AB gAB A A aTrTN'A" C] 1n aB = [NADB* +N EMA ] m . (57) 25 It will be noted that the relationship between mutual and self-diffusion for the hydrodynamic equation 22 and the statistical mechanical approach, equation 57 are identical. h) In CC 1h ti. tC 26 Discussion of Previous Theoretical Work It will be noted that the Hartley-Crank-Darken(19 ’ 21’ 33) hydrodynamic theory indicates 3}: D, n : fl 1 01 In this equation, since 0'i is assumed to be independent of (58) * compOSition under isothermal, isobaric circumstance, Di 1') will also be independent of composition. The general statistical mechanical theory as applied to transport processes also shows that Di*n is independent of composition but only for solutions that are considered regular. Thus, if either equations 22 or 57 are inspected, it is found that function DABn/(d 1n a/d In N) should be linear with mole fraction for binary solutions according to these two theories. As stated earlier, this behavior has already been predicted by Eyring. As stated previously, there is some confusion among the (6,50) different proponents of these three theories as to which is the most valid derivation. However, since they all show the same result, all three developments will be considered equally valid here and only the final result will be examined experimentally. The only comment that will be made here is that the statistical mechanical approach puts the additional stipulation on the result that the solution involved must be regular. This stipulation would seem to eliminate the application ass ex; 1151 go) 27 of these theories directly to systems in which molecular association is present. As a matter of fact, it has been found experimentally that the diffusion behavior of associating systems usually deviates widely from the behavior expected from the fore- going theories. EXPERIMENTAL METHOD The experimental diffusion coefficients for this study were obtained with an Optical diffusiometer. With this instrument, the diffusion took place in a glass-windowed cell and was followed with a Mach-Zehnder(44’63) type interferometer. This set-up was patterned after a similar diffusiometer described by Caldwell, Hall, and Babb(15’17). Interferometer - A diagram and photograph of the inter- ferometer are shown in Figures 1 and 2. The various components of the interferometer were supported by ordinary laboratory bench carriages stationed along a continuous rail composed of three optical benches laid end-to-end. These three benches were bolted to an I-beam and were made adjustable by slotting the bolt holes. The adjustments on the benches were necessary in order to align the components of the interferometer. The purpose of the I-beam and the alignment procedure are discussed later in this section. The heart of the interferometer is the system of four mirrors which are positioned at the four corners of a parallel- ogram. Mirrors 1 and 4 are half-silvered mirrors which reflect half of a beam of light and allowed the other half to pass through. Mirrors 2 and 3 are full reflectors. Mirrors 1, 2, and 4 each 28 .muouuflz mo coflwmonm mafia/CAM HouoEonownoacH mo Emamdwfl oSmEonom A ondmwm need mcsmfifioo a 3.33.4 condom. < «50% m H9032 9.84 muogmU N MOMS: v noun“: ocmfinm omega 29 .munocomEoO $3395 young?“ own was hovcgoNugumg mo ammumouonnmd oudwfim 30 31 have three modes of fine adjustment: (i) a vertical adjustment, (ii) a rotational adjustment with the axis of rotation perpendic- ular to the plane of the paper, and (iii) another rotational adjust- ment with the axis of rotation in the same plane as the mirror and perpendicular to the other axis. Mirror 3 had these adjustments plus an additional fine longitudinal adjustment in a direction parallel to the optical bench. Collimated, monochromatic light was split in amplitude by mirror 1, half the beam going to full reflecting mirror 2 and half to full reflector 3. The two beams were then recombined at mirror 4. Interference between the two beams would take place according to the usual laws when the path 1-2-4 very nearly equalled the path 1-3-4 in length and the beams were positioned to recombine exactly at mirror 4. The interference pattern could be changed to assume various shapes by slight adjustment of the mirrors; however, for this study, straight, vertical, parallel fringes were required. The light source for the interferometer was a Cenco quartz mercury arc lamp fitted with a combination of Corning filters which isolated the 5461 (A green mercury line. This monochromatic light beam was focused on a point source by means of a centered plano-convex condensing lens (f = 93. 0 mm.) and subsequently collimated by. an achromatic lens (f = 193. 0 mm. ). 32 Alignment - The three optical benches were bolted to the I-beam to form as near as possible one continuous straight rail. The components of the interferometer were then set on the optical benches , one at a time, and carefully adjusted to the right height using a cathetometer. The collimator was positioned by moving it back and forth on the Optical bench rail until the point source appeared sharp and clear when seen through a telescope focused at infinity. Next the mirrors of the interferometer were placed on the rail. It was important that these mirrors be aligned exactly; therefore, they were all placed on the same optical bench. It was found that the mirrors could be placed in rough position by shutting off all room lights and removing the filter from the light source. This way, the light beam could be clearly seen, making the location of the mirrors relatively easy. The height of the mirrors had to be carefully adjusted using the cathetometer and final fine adjustments of the mirrors had to be made to obtain the interference fringes. Vibrations - Probably the most difficult problem with the equipment was the removal of vibrations from the interferometer mirrors well enough to photograph the interference fringes. The length of the paths of the two light beams must be within a few wavelengths of each other so that any small vibration of the mirrors 33 will cause violent movements of the fringe pattern or disappear- ance of the fringes altogether. The vibrations came from two sources--from the building and environment of the instrument, and from within the instrument itself such as from the camera, air movement, etc. The first source was largely eliminated by bolting the instrument to a 6 inch I—beam which in turn sat on ten inverted rubber cup-like cushions sold commercially as ”Instrumounts". This whole set-up rested directly on a concrete pedestal. The internal vibrations were more difficult to remove. When the instrument was first tested, a small shock would cause movement of the fringes which might last for several seconds. The mirror holders were mounted atop 3/8 inch shafts that fitted into the Optical bench carriages. These shafts were rather non- rigid and probably amplified the shock. Because of this, the mirrors were lowered 1-1/2 inches; thereby removing much of the amplification and shortening the damping time from 6 to 10 seconds to less than 2 seconds. Additional improvement could have probably been obtained by more rigid mounting of the mirrors. The Camera - The interferometer was arranged so that the interference beam could either be reflected by a mirror and observed by eye through a telescope or by swinging the mirror out of the way, be photographed directly by the camera. The 34 camera was essentially a 3 foot long by 3—1/2 inch diameter aluminum tube with the lens set in the end facing the inter- ferometer focused on the photographic plate at the other end. The achromatic lens of the camera had a 343 mm. focal length. Directly behind the lens was an Ilex No. 5 Universal shutter. The photographic plates were 3-1/4 x 4-1/4 inch, Type M, Kodak plates which were exposed through a 1/4 inch aperture. After each exposure, the plate was translated to the position for the next exposure by a lever mechanism attached to the plate holder. Fourteen successive exposures were thus possible on a single plate. As mentioned above, the camera shutter caused vibration within the interferometer; therefore, the shutter could not be used directly. The interferometer beam was hidden from the camera by a small piece of cardboard before the shutter was opened. After the opening of the shutter, the resulting vibrations were allowed to dampen for a few seconds before the cardboard was removed to expose the photographic plate. One of the most important requirements for correct results was for the camera to be exactly focused on the cell; therefore, much care was taken to locate the camera. Caldwelllls) shows that the correct focal plane within the cell is 2/3 of the cell width measured from the inside of the window nearest the camera. 35 The camera location can be estimated from the focal length of the camera lens but this was not exact enough because of the series of materials all with different refractive indices between the camera and the cell. The procedure chosen for locating the camera was as follows: 1) 2) 3) 4) A glass gauge plate with a scale of 0.005 inch intervals was placed over the side of the cell nearest the camera. The camera was focused on this plate by taking a series of exposures moving the camera along the optical bench at one millimeter intervals in the vicinity of the estimated focus distance. The exact focus distance could easily be seen from these exposures. The gauge plate was placed on the side of the cell away from the camera and step 1 was repeated. From the exact focus distances Obtained in steps 1 and 2 for the front and back side of the cell and the refractive indices of the cell components, the correct position for the camera was calculated. A bonus of this procedure was the determination of the factor by which the image is magnified by the camera. This factor is found to be important for the experimental calculations. At the correct focus distances the magnification M, is equal to the apparent gauge interval 36 on the photograph divided by the actual interval on the gauge plate. Because of the interferometer beam is collimated, M was found to be constant at both the front and rear of the cell and thus for any position within the cell. M was found to equal 1. 923. The Waterbath - The diffusion cell hung in a constant temperature waterbath, an 18 x 18 x 18 inch stainless steel tank supported by 3/4 inch plywood. The plywood support rested on the cement pedestal but did not touch the interferometer at any point. The wood also served as insulation for the bath. Two 3-1/2 inch diameter optical flat windows were clamped and sealed into the ends of the waterbath to allow passage of the interferometer beams. Distilled water was used in the bath because tap water was found to be too cloudy for light passage. The temperature of the bath was controlled by a mercury thermoregulator, sensitive to i0. 03°C, which was coupled to an electronic relay circuit. Within the circuit was a 5-1/2 foot COpper sheathed immersion heater. The data were all taken at 25°C (77°F), a temperature usually above room temperature sufficiently enough to allow use of only the heating circuit except in extremely warm weather. When the weather did become too warm for self-cooling, a lOOp of copper tubing carrying cold tap water had to be placed on the floor of the bath. The bath was 37 stirred continuously except during the time a photograph was being taken. 38 Diffusion Cell Figure 3 presents aphotograph, and Figures 4 and 5 diagrams of the diffusion cell. The cell consisted essentially of a slot, 1/4 x 3-1/4 inches, cut into a stainless steel plate, with two optical flat windows clamped over the slot to form a sealed channel. If the cell, containing a liquid solution was placed in one of the interferometer beams, the vertical fringes would be displaced horizontally a distance proportional to the refractive index of the solution. Actually the channel extended into both beams to assure Optical paths of approximately equal length. Thus, a vertical concentration gradient in the solution across one of the beams resulted in a fringe displacement pattern which was a direct plot of refractive index versus distance. All parts of the cell which would be in contact with the liquid solutions were stainless steel or glass thus making a rather versatile set-up which could handle almost any corrosive liquid. The glass windows were clamped to the stainless steel body by four brass clamps screwed directly onto the body. Care had to be taken to tighten all the screws the same amount in order not to strain the windows and cause distorted fringes. Teflon gaskets of O. 005 inch thickness were placed between the glass and the body of the cell to prevent leakage. ‘4 1,, “£411! v-. Figure 3_ Photograph of Diffusion Cell for Measurement of Diffusion Coefficients 39 glass solution reservoirs made from 50 cc. syringes 23:21:. (TAT 1371 valve 2 valve 1 cell cell window-e / bOdY A_ r valve 4 / valve 3 boundary / sharpening J slits siphon ® valve 5 Figure 4. Diagram of Diffusion Cell. 40 Section A-A Showing Full Assembly BRASS \\\\\ .\\\\‘(| ‘| V \\\\\\\\\\ \\ \\\ \l\\\\\ \ \\\\\\ I \ Boundary Sharpening Slit I 7 l 4 I G) G) Pfi rd fife-.5“ C 5:31;:— O O me e “I“ O r 'l I). . I . Stainless Steel Body Only Section Through Stainless Steel Body Figure 5. Detailed Drawing of Diffusion Cell. Only Major Dimensions are Shown. 41 42 The cell hung from a framework which was bolted to the cement pedestal and rose above but did not touch the bath (see Figure 2). For easy cleaning, the cell and its hanger were made detachable from the framework. To assure the cell being replaced in the same location each time, the hanger slipped onto two small position pins on the framework. The cell was provided with two inlets , one in the top and one in the bottom, and two outlets directly across from each other in the channel sides. To form the concentration gradient, two solutions of slightly different concentration were allowed to flow simultaneously into the cell--the denser solution through the bottom inlet, the other in the top--and out the outlets. The two solutions were thus layered one atop the other with a fairly sharp boundary between. The height of the cell was adjusted so that this boundary was in the center of the lower interferometer beam. At time zero, all the valves are closed and the solution was allowed to diffuse. Stainless steel needle valves for the inlets and outlets were fitted directly to the cell or as near as possible to the cell and fed by 1/8 inch stainless steel tubing. The solution reservoirs were made from two 50cc. syringes with all but the top of the plungers cut off to form a tOp covering. A small vent hole was put in the t0p of both plungers to allow the liquid to flow out freely. For the filling operation, an additional line was attached to the cell with inlet at the 43 tOp corner of the channel. An inlet here permits a slow flow of liquid down the side of the channel. 44 Adjustment of Equipment The final fine adjustments of the interferometer mirrors were made with the aid of a set-up whereby the combined inter- ference beam was reflected by a full mirror into a telescope where the beam could be seen by eye. The rough positioning of the mirrors was mentioned previously but in order to obtain interference, the paths of the two interferometer beams still had to be equalized. The method of path equalization is known as the method of "far and near cross hairs”. The near cross hair was a pointed wire protruding into the beam at point A (see Figure 1) and the far cross hair was the point source. Method of Near and Far Cross Hairs 1) The telesc0pe was first focused on the far cross hair which appeared as two separate images. 2) Mirror 2 was adjusted until the two far cross hair images coincided. 3) The telescope was next focused on the near cross hair which also appeared as two images and mirror 3 was adjusted until these two images coincided. 4) The telescope was refocused on the far cross hair and step 1 was repeated. 5) This process was repeated until both images coincided. At this point fringes were usually seen at the near cross hair. 45 If none were apparent, mirror 3 had to be rotated slightly on its vertical axis. Further details of this procedure are provided by Caldwellus). If the paths lengths were not exactly equal, the vertical fringes would appear slightly curved. An illustration of the fringe pattern as a function of path difference is shown in Figure 6. To correct this curvature, the paths were equalized by the fine longitudinal adjustment of mirror 3 until the fringes were straight and vertical. The attainment of straight, vertical fringes was aided by first adjusting the hairline of the telescope until it was exactly vertical. The hairline was adjusted by aligning it with a plumb line of fine piano wire hung from a wall bracket. .4 l m I Mm' Path Length Difference Figure 6. Appearance of Vertical Fringe Pattern as a Function of Path Difference. 46 At this point, the fringes still had to be focused on the diffusion cell. It was easier to focus the mirrors without the cell in place; therefore, the adjustment of the telescope was marked where it was focused on the cell. This way, the cell could be removed and the fringes could be brought into focus with the telesc0pe . 47 Procedure for Experimental Run 1) 2) 3) 4) The cell with all valves closed, except valve 1, was first clamped in a rack outside the rest of the apparatus. A few milliliters of the denser solution was placed in reservoir B and allowed to flow into the cell through valve 3 until the liquid level in the cell was about 1/4 inch above the outlet. Valves 1 and 3 were then closed. Valves 4 and 2 were opened slightly and the exit line as far as the tee junction was filled with liquid by forcing the liquid out the exit with the syringe connected to valve 2. The liquid level was forced down with the syringe to just slightly above the level of the outlets. A few more milliliters of solution was allowed to flow into the cell as in step 3. This time the liquid was forced into the exit line through valve 3. Again the liquid level was forced down until it was just slightly above the outlets. Step 4 was repeated until liquid completely filled the exit line and started to drip from the end. Then valve 1 was Opened and the liquid was allowed to flow through the exit by means of the resulting siphon for a few seconds. This removed any small bubbles still present in the exit line. 5) All valves were again closed except valve 1 and the plunger 6) 7) 48 was removed from the syringe. About 25 cc. of the less dense solution was put in the syringe. Valve 2 was opened slightly until the solution started to trickle slowly down the side of the cell channel and layer on tOp of the other solution. The solution was allowed to flow this way until the cell was filled and had just started to overflow up into reservoir A. Then valves 2 and 1 were closed. The two reservoirs were filled--A with the less dense solution and B with the denser solution--making sure that their final liquid levels were approximately equal. The cell was placed in position on the rest of the apparatus. Valves 1 and 5 were first Opened about one full turn and then valve 3 very slowly until the rate of flow from the exit was 5-6 dr0ps per minute. Then the Opposite outlet valve was slowly Opened until the combined exit flow rate was 9-12 drops per minute. Care had to be taken to get the flow rate of solution into the top of the cell exactly balanced with the rate into the bottom. The formation of the sharp boundary was watched through the teleSCOpe. The boundary formation was aided by the boundary sharpening slits in the two outlets. These slits allowed the liquid to flow evenly out the entire width of the cell. 8) 9) 49 When the boundary was formed satisfactorily, valves 3 and 4 were closed followed by valves 1 and 5. The solutions were allowed to diffuse for a few minutes until the fringes could be seen distinctly completely across the diffusion zone. Then the mirror reflecting the image into the telescope was swung away from the beam so that the beam was in view of the camera. The interference fringe patterns caused by the diffusion were photographed at predetermined time intervals. The series of exposures taken for one run is shown in Figure 7. After the run was completed, the cell was again clamped in the rack and allowed to drain. It was then rinsed twice with toluene and then once with acetone. Finally, the cell was thoroughly dried with air. Avg 62 sum Runofiwuoaxmv cam guano acid :33. Easemouoem no 3m ASE: .N. ousmwh 50 51 Calculations Consider a differential volume element of the diffusing solution and set up a material balance equation which describes the diffusion in and out of the element. The net result of such a procedure is known as Fick's Second Law , 320 _13c 8x2 — .15 OT (59) and will have the boundary conditions: Case 1(x > 0) i) x-) 00 ii)t=0 iii)x=0 -Case II (x (O) vi)x—>- 00 v)t=0 vi)x=0 x>0 x<0 dx x\\\\\\\\ \X‘ - ---q \\\\\\\\\\\‘ cell body The assumptions are: (l) the concentration dependence of D is negligible over the small concentration differences involved, and (2) the diffusion gradient has the prOperties of normal distribution CLII‘VOS . Equation 59 may be solved with Laplace transforms or by some other meanswo) to give the following identical solution for both Case I and Case II. 52 c - c 1 ...__.._° = — erf. ( x ) , (60) c2 - cl 2 4Dt where co is the concentration at the zero position in the cell and as a result of assumption (2) above is equal to 1/2(c1 + Ca). The refractive index, n, may be assumed prOportional to the concentration<6o), so that n "" no 1 x —-:—-—- : '2 erf. ( ) . (61) Essentially the fringe pattern is a plot of the refractive index versus distance in the cell so that the refractive index difference may be represented by the number of fringes dis- placed. For the relationship between the method develop- ment and the fringe pattern. // l/ / refer to Figure 8, where J is the total number of fringes from top to bottom; k is the local T7713) \\ fringe number in the top half I of the cell andj is the local : j R J fringe number in the bottom 1‘ l l 1 half. Let x1 and xk be the Figure 8. Fringe Pattern. measured distance corresponding to fringes j and k respectively. 53 Thus, where x > 0 n'no _ k-2J = Zk-J n-n — J J and equation 61 becomes xk - .. ~/—4_D—t = erfl (Elf—Ti) . (62) Similarly, where x < 0 n-n o J - 2j n2 " n1 .1 and X. . fin— = erf-1( iLJ—zl) . (63) The exact midpoint of the diffusion zone is difficult to determine; however, the distance, xk + xj, is easily determined by difference measurements. Therefore, éjD—t- +4317: =erf-1(J32j)+erf-l(2kjj). (64) The cell distance is not equal to the distance measured on the photographic plate because the camera magnified the image by the factor, M. Therefore, I I xj-I-xk xj +xk = —— (65) N} 4Dt MV4Dt where x-‘ and xk‘ are distances on the photographic plate. Hence I + I 2 1 Xj "k D12: j . . (66) -I J - 2] -T 2k - J 4M erf ( J )+ erf (__J__) 54 For each exposure , values of the function +xk' Zj) + erf-T(2k-J x.' 1(JJ erf J J were determined for several j's and k's and averaged. The averages for several exposures were plotted versus exposure time and the slope of the resulting line was determined. Thus, D : £312%e_ . (67) 4M See Appendix A for details of a sample experimental run. This diffusion coefficient is assumed equal to the mutual diffusivity at the average concentration, C() = 1/2(c1 + Ca). As mentioned previously, M = 1. 923, so that the factor 4M2 equals 14. 792. The distances on the photographic plate were measured with an optical comparator made from a Gaertner microsc0pe fitted with a travelling eyepiece. The travelling eyepiece could scan a total distance of 5 centimeters by turning a crank and the distance travelled was indicated on a vernier scale accurate to 0 . 0001 centimeters . 55 Accuracy of Experimental Method and Purity of Components Accuracy 9_f Experimental Method - The accuracy of the method was tested by comparing diffusion coefficients at 25°C for seven aqueous sucrose solutions with those reported by Gosting and Morris<29). Gosting and Morris fitted their data to the following empirical relationship using the method of least squares: D = 5.226(1- 0. 01480?) x 10'65: 0.002 (68) where O is the concentration, gm. sucrose/100cc. solution. The results of the comparison are summarized in Table I. The diffusivities deviated by less than 1% from equation 68 and had an average deviation of 0.5%. Purity o_f Components - The chemicals were obtained in the purest forms available. Hexane, heptane, and dodecane and hexadecane were purchased from Matheson, Coleman and Bell, Co. The hexane was spectroquality, the heptane chromatoquality, and the dodecane and hexadecane were 99+% (olefin free) pure. Spectro grade carbon tetrachloride was purchased from Eastman Organic Chemical Company. The purity of the chemicals were further confirmed by comparing their densities and refractive indices with values given by Timmermans. See Table II. .wm Kg :2 mOOCOHOmOH OmHm mom m 56 mm .0. N: .m mm: .m bmom .o #0 mm .01 Q: .m NmH .m wmom .o is ow .o+ omm .m mm: .m wmom .0 me. 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Refractive Index 0 Density at 250C 2'5 C nD Reference Reference This work 58 This Work 58 Hexane 0. 6550 0. 6549aL 1. 3720 1. 37.23a Heptane 0. 6796 0. 6795aL 1. 3855 1. 3852a Dodecane 0. 7450 0.7451 1.4193 1.4195 Hexadecane 0. 7698 0.7699 1. 4319 1. 4325 Carbon a a Tetrachloride 1. 5842 l. 5845 l. 4570 1. 4576 3'Average of several recorded data. 57 58 Experimental Results The experimental diffusion coefficients, viscosities, and densities for the full concentration range for the four binary systems, hexane -hexadecane, heptane -hexadecane, hexane- dodecane, and hexane-carbon tetrachloride, are given in Table III. Viscosities were obtained with an Ostwald-Fenske typeiviscometer and densities were determined with a 10 ml. glass specific gravity bottle. 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For a solution in equilibrium with its vapor and whose vapor follows the ideal gas laws, the fact that the attractive forces are identical results in the number of molecules of a component escaping into the vapor being proportional to its mole fraction. This latter statement is the well-known Raoult's Law: pi = Nip: (69) where pi and Ni are the vapor pressure and mole fraction, respectively of component i. If Ni = 1, it will be noticed that the prOportionality constant pi. , is equal to the vapor pressure of pure i. Thermodynamically, it is preferable to use the generalized form of Raoult's Law, fi = Nifi. (70) where fi is the fugacity of component i, either in the liquid or in the vapor and fi. is the fugacity of pure i. Remembering also the definition of the activity, f. a, = _1_ (71) 62 63 it is seen, that ai/Ni = l for an ideal solution. Other properties of a binary ideal solution are: M AG 2 RT(NA1n NA + NBln NB) (72a) ASM .-. -R(N In N + N In N ) (72b) A A B B AHM = 0 (72c) AVM = 0 (72d) M M M where AGM, AS , AH , and Av are the Gibbs free energy, entrOpy, enthalpy, and molar volume of mixing per mole of solution, respectively. Most solutions, however, are non-ideal and the fugacity is not proportional to the mole fraction except at the extremes of the concentration range. When Ni approaches unity, equation '70 is still valid. When Ni approaches zero, fi = kNi (73) an equation known as Henry's Law and k is the proportionality constant. As a consequence of the Gibbs-Duhem equation, for a binary solution, in the range that the solute obeys Henry‘s Law, the solvent must obey Raoult's Law. Over the rest of the concentration range, equations 70 and 73 do not hold true. Of course, here ai/N.1 7! 1, so that an additional quantity is defined, 64 ai Y. : -I\I—' . (74) 1 i This term, the activity coefficient, is a measure of the non-ideality of the solution at concentration Ni‘ When Yi > 1, the solution is said to be positively deviating from Raoult's Law and when Yi < 1 the solution is referred to as negatively deviating. In many of these non-ideal solutions, the non-ideality is caused by the attractive forces being so strong that the molecules are forced together into clusters of two, three, or more molecules and move about in the liquid as a single molecule. This phenomenon is called "association". These clusters may be formed of molecules of one component only or may be complexes of molecules of more than one species. The effect of these clusters became negligible only in dilute solution where the clusters are widely separated from each other. Also in dilute solution, in the case where there is no interaction with the solvent, the solute clusters will dissociate into single molecules. Thus, the ideal solution laws, equations 70 and 73, are applicable only for dilute solutions. (35) Regular Solutions - Hildebrand called attention to another class of non-ideal solutions which he called regular. Regular solutions are defined as solutions whose molecules are more or less randomly mixed just as the ideal solution and therefore has approximately the same entropy of mixing as an ideal solution. 65 Unlike ideal solutions, the intermolecular forces between molecules of regular solutions are not longer equal and the differences in size and shape may be greater. These charac- terizations of regularity are somewhat vague and thus the boundaries of this classification are rather indistinct. There is, as one would suspect, no association between the molecules of regular solutions so that, in general, the deviations from ideality are fairly small. Hildebrand(36’ 37) found that for binary regular solutions, _ K 2 1” YA " if NB (753) K 2 = — 7 1n YB RT NA ( 5b) where K is a constant that is not only independent of composition but temperature as well. Combining equation 74 with the definition of the regular solution results in the thermodynamic properties: AGM = RT(NAln NA + NBln NB) + KNANB , (76a) ASM = -R(NAln NA + NBln NB) , (76b) AHM = KNA B , (76c) Av = 0 . (76d) 66 The similarity of equations 76 and equations 72 may be noted for the ideal solutions. It may also be noted that the maxima for AGM and AHM are predicted to occur at NA = NB = O. 5 by equations 76a and c. These relationships were derived by Hildebrand<37) by means of some basic assumptions implied by the definition of regularity, that is, the distribution functions (2.0) _ (2.0)_ 2.0) are approximately equal (gAB — gAA — g(BB ) and independent of mole fraction and the molar volumes are approximately the same. The distribution functions as will be recalled, is defined: the probability of finding a type B molecule around an arbitrary 2, 0) central molecule of type A is proportional to (NBN/v)gAB , and the probability of finding a type A molecule around the central . . (2. 0) type A molecule 18 pr0portional to (NAN/v)gAA , etc. Guggenheim(30) derived these same relationships using the so- called "lattice theory" of liquids. 67 Discussion of Previous Experimental Work As mentioned earlier, Caldwell and Babbué) studied several binary ideal systems over the complete concentration range. They found that just as predicted by theory (See Previous Theoretical Work), the function DU/(d 1n ai/d 1n Ni) is linear with mole fraction. For ideal systems, of course, d lnai/d lnNi is unity so that Dr) is linear with mole fraction. However , it has been found that associative non-ideal solutions do not follow theory. A cluster of molecules moving through the liquid medium as a single molecule complicates any diffusion study. The activity correction term, overcorrects the diffusivity-viscosity product, Dn , by a large factor, sometimes by as much as several hundred percent. Anderson(2) had some success by considering the associated molecules as a separate entity whose concentration is a function of the concentration of the nonassociated molecules. As a whole, though, the study of associating systems to prove the applicability of the theory is inconclusive. Therefore, for the present study, systems were carefully chosen which are non-ideal but which give no indication of any association. Acfivfi not as inthe the tl afac 0f the comp its ml Predh assun “the . to this ‘ acmmg 68 Activity Data and Selection of Systems There are probably many non-ideal systems which are not associative but few whose activity data have been reported in the literature. Therefore, since activities are required for the theory comparison, the availability of activity data became a factor for choosing the systems to be studied. The properties of the systems chosen are shown in Table IV. Hexane -hexadecane - This system was first studied by Bronsted and Koefoed(l3) using a highly accurate method for determining the vapor pressures which were then corrected for vapor phase non-ideality to obtain the activities. As indicated by Table III, the data follows equations 75 where K/RT equals -0. 111 at 20°C. Although the molar volumes of the pure components differ by a factor of Z. 2, the enthalpy of mixing has its maximum at NA = 0.48, very near to NA = NB = 0.5 predicted by equation 76c for regular solutions. Thus, it is assumed that this system has approximately the prOperties of a regular solution. The diffusivity and viscosity coefficients were all obtained in this study at 25°C; therefore, the factor K/RT had to be converted to this temperature. This may be done quite readily, since K/R is a constant, independent of temperature. However, this was not m .mm- 23 .o- om 2 as N2 Am: 38¢on .. 33 23on S. .o om 3. o .3. 33 .o- 8 2 gm :1 Am: oncogene: .. :3 ocmumom om 3.. 03m we .o om :12. e .3- A: .o- om 8m 2 gm N2 EV ocmuowmxmm n :«v ocmxom Amy mm .0 mm mm m .mnw m3 .0 om mm no 02 opAuosomfloB sonumO u :3 ocmxosvofnu w .2: om: .o om S 3 N2 :5 mucozumtmh conumO :33 23:3 . . 0 com .83? PM. 0 sum Sum? xme <2 .QEWH. Amman VA VA damp... lawman m> <> 00mm on mafia: .Ao >mAd5sm masvoAmmooO 33304 mo§A0> nonSA E396 .>»Em3mou .365 @5986ro 3 maoumfam mo mofinomoum mo COmAummEoO .>H 3nt 69 70 necessary for this case because, in a later study, McGlashan and Williamson148) obtained activities at several different temperatures and correlated them by the following equation: _ 2 2 1n YA — aNB +pNB (3-4NB). T- 313.15 T (1- -0.1173-3.798( T )+5.960 (m) - 0.00722 (T - 313.15) (77) p = - 0.0080 - 0.118 (T "313' 15) T where subscript A refers to hexane and B refers to hexadecane. (13) and the The agreement between the Bronsted—Koefoed McGlashan-Williamson(48) data were quite good. Ln a differed by only 0. 15% at 0. 5 mole fraction. The constants of equation 76 at 25°C are: 0. =-0. 1105 and B = -0. 00736. Substituting these two values into equation 77, the activity correction term of equations 20 and 57 may be obtained: C] 1n a (1 1n a (1 1n y A ___ B = l + A dInNA dInNB dInNA = 1 + 0. ZZIONANB + 0. 0442NANB(1-2NB) . (78) Heptane-Hexadecane and Hexane-Dodecane — The activities of these two systems were reported in the same paper (13), cited previously by Bronsted and Koefoed As can be seen in Table III, the enthalpy of mixing for heptane -hexadecane showed 71 a maximum at close to the expected equimolar mixture predicted for regular solutions. The thermodynamic properties of hexane- dodecane are not reported in the literature; however, because of the similarities of the two systems , they will both be assumed approximately regular. The activity correction for regular solutions may be calculated from equations 75, dlnaA dlnyA 2K —— = -— o 7 d In NA 1 + a—fi—ln A 1 RT NANB ( 9) It will be noticed that all three of these systems mentioned thus far are negatively deviating from Raoult's Law. Cyclohexane-Carbon Tetrachloride - The diffusion behavior (31) of this system was studied by Hammond and Stokes who used the activity data of Scatchard, Wood and Mochel152). Scatchard, and co-workers used a fairly accurate method for determining vapor pressures and compositions of the vapor and liquid phases at several temperatures. To obtain the activities, the vapor phase was corrected for non-idealities. This data showed good agreement with a later study by Brown and Ewaldué‘) for the same system at 700C. Values for activity coefficients from the two sets of data differed by only 0. 5% at NA = 0. 5. Table III shows that this system has pr0perties very close to those predicted for regular solutions--fairly close molar volumes 72 and the maximum enthalpy of mixing near the equimolar mixture at 25°C. Hexane-Carbon Tetrachloride - The activities of this system were obtained by Christian, Naparko, and Affsprung(zo) at 20°C. In this same paper, activities for the system benzene- carbon tetrachloride were also reported and were found to agree within 0.4% with data for this system by Scatchard, Wood and Mochel(53). Christian, Naparko, and Affsprung1zo) assumed the correction for non-ideality of the vapor phase was negligible. Since the data are so close for benzene-carbon tetrachloride which was corrected by Scatchard, Wood, and Mochel and the correction was generally small for the other systems, this seems to be a rather valid assumption. The other thermodynamic properties of this system are not reported in the literature but since the molar volumes are fairly close, the activity coefficient data follow equations 75, and the similarity with the previous systems, this system too will be assumed approximately regular. These latter two systems will be noticed to deviate positively from Raoult's Law. Thus, this study has three negative and two positive deviating systems. Also, it appears that most nonassociative systems generally have regular properties. This would, of course, be exclusive of high polyme r s olut ions . 73 Discus sion of Results The experimental results are plotted in Figures 9 to 17 (See Appendix B for listing of experimental data). Viscosity and density data are plotted versus mole fraction in Figures 9- 12 for the systems hexane-hexadecane, heptane -hexadecane, hexane-dodecane , and hexane-carbon tetrachloride. Kinematic Viscosities are obtained directly with the Ostwald-Fenske viscometer so that densities were required to convert to absolute viscosity, 71- A smooth curve was drawn through these data so that the viscosity or density of any of the systems at any mole fraction may be taken directly from the graph. In Figures 13-17 are plotted the factors D11 and Dn/(d 1n a/d In N) as a function of mole fraction. Also plotted on these figures is the linear behavior expected of these systems according to the present theories (See equations 20, 57, and 58). The Viscosities for these plots were, of course, taken from Figures 9-12. Four of these graphs show that the activity overcorrects the diffusivity- viscosity product. In one case, hexane-dodecane, D0 was nearly linear with mole fraction but was overcorrected by as much as 4%. The other three overcorrected systems deviated from theory by 2. 5 to 4% from a deviation from Raoult's Law of 2 to 9%. In only one system, hexane -hexadecane, Figure 13, did the diffusion behavior seem to follow theoryul); however, in view of the results from the other systems, this apparent agreement is probably fortuitious. vis cosity, centipoise s 2. l. 01.. . -o.750 . 0- o E ' — 0.700 50 g H '5 C Q) "U C O.— A 0.650 _ l 1 J 1 0 0.2 0.4 0.6 0.8 1.0 mole fraction, hexadecane Figure 9. Experimental Viscosity and Density versus Mole Fraction for Hexane - Hexadecane. 0 Experimental Density; 0 Experimental Viscosities. 74 ....§.4 4.1L .1.) «(fly .u. . . . I. .v .131 .uilii 'J .I . . l.- . centipoises viscosity, 3. 2. l. 0. 800 0 r- O - 0. 750 O 0 ~— O \E 8 00 O a >~ .p '8 C! -« 0. 700 .8 0 _ 1 I I I 0. 650 0 0. Z 0. 4 0.6 0. 8 l. 0 mole fraction, hexadecane Figure 10. Experimental Viscosity and Density versus Mole Fraction for Heptane - Hexadecane. 0 Experimental Density; 0 Experimental Viscosities. 75 vis cos ity, centipoise s density, gm. /m1. 1. 5 0. 800 1. 0. 750 0. 0. 700 l l L l 0. 650 0 0.2 0. 4 0.6 0. 8 1. 0 mole fraction, dodecane Figure 11. Experimental Viscosity and Density versus Mole Fraction for Hexane - Dodecane. 0 Experimental Density; 0 Experimental Viscosities. 76 centipoises viscosity, O O. 2. 0 8 l. 5 6 E E 00 >1 *4 '53 r: a) "o 4 1. 0 Z L l l I 0 0. 2 0.4 0. 6 0. 8 1. 0 mole fraction, carbon tetrachloride Figure 12. Experimental Viscosity and Density versus Mole Fraction for Hexane - Carbon Tetrachloride. 0 Experimental Density; 0 Experimental Viscosities. 77 3.0 2.0 (D Q) G > 'U h. 53 X «SN 5510 “(O C a 'U C ('0 C o O l I 1 I 0 0.2 0.4 0.6 0.8 1.0 mole fraction hexade cane Figure 13. Dr; and Dn/(d 1n a/d 1n N) as a function of mole fraction for the system hexane - hexadecane. 0, Dn; and o, Dn/(d 1n a/d In N); , linear or ideal behavior. 78 2.0- 1. D77 and Dn/(d 1n a/d 1n N) x 107, dynes 1. I l 1 l 0 0.2 0.4 0.6 0.8 1.0 mole fraction dode cane Figure 15. Dn and Dn/(d 1n a/d 1n N) as a function of mole fraction for the system hexane-dodecane. 0, D77; 0 , Dn/(d 1n a/d 1n N); —, linear or ideal behavior. 80 1.6- 1.4- 1. Dn and Dn/(d 1n a/d 1n N) x 107, dynes 1.0— I I I I 0 0.2 0.4 0.6 0.8 1.0 mole fraction carbon tetrachloride Figure 16. Dn and Dn/(d ln a/d 1n N) as a function of mole fraction for the system cyclohexane-carbon tetrachloride. , linear or ideal behavior; 0 , Dn; o, Dn/(d 1n a/d ln N). 81 83 A clue to these deviations from theory may be found by examining the assumptions that were made in deriving the various theories. First of all, for their hydrodynamic theory, Hartley and Crank<33) assumed that the proportionality constant, (1'i , of equation 12, depends only on molecular shape and size at constant temperature and pressure and is independent of solution viscosity and composition. It is generally maintained that this prOportionality constant, also known as a friction coefficient, is independent of viscosity and, indeed, is related to viscosity and the self-diffusion coefficient by equation 58. The statistical mechanical theory shows the self- diffusion coefficient-~viscosity product to be independent of composition but only for "regular" solutions. Thus assuming the validity of equation 58, the statistical mechanical theory suggests that the friction coefficient is independent of composition only for regular solutions. The question is, then, how good is the regular solution assumption and for what systems does it apply? It has been seen that all five of the systems studied have properties that would indicate that they are regular. Even in cases where the difference in size of the molecules is fairly large, the solutions seem to be regular. (49) Miller and Carman studied the self—diffusion behavior of one of the systems listed in the present study, heptane -hexadecane. * They found that neither the function, D T? , nor the function DB*TI, A a“; , . 84 where the subscript A, in this case, refers to hexadecane and B to heptane, were independent of composition as expected by theory. Some insight into causes for this behavior may be gained by examining equation 50 of Statistical Mechanical Theory. Here, it is found that the self-diffusion coefficient of "A", DA*’ is a function of the influence between each molecule of species A and a neighboring B molecule (A-B interactions) and between each A molecule and another neighboring A molecule (A-A interactions). 7% Simiarly, the value of D depends on A—B interactions and B-B B interactions. Since the mutual. diffusion coefficient is a function of the self-diffusion coefficients , as may be suspected, DABn/d 1n ai/d 1n Ni) is dependent on all three types of these interactions. However, it is noted that only one of these interactions, A-B interactions, are common to both DA* and DB):< . Since both DA* and DB* are dependent on composition and only A-B interactions influence the behavior of both self-diffusion coefficients , it would seem that this could be the determining factor of whether or not DA*n and DB*n are independent of composition and DABn/(d 1n ai/d 1n Ni) is linear with mole fraction. One of the stipulations of the regular solution is that the radial distribution should be approximately independent of concentration. Probably, a small deviation from this rule will produce only small departures from other expected properties of regular solutions such as the enthalpy of mixing but 85 may cause large departures from such non-equilibrium processes as diffusion. Thus, the unequal size of the molecules which is the probable cause of the departure from Raoult's Law for these solutions causes the interactions between A and B molecules to be dependent on concentration and the deviation from theoretical behavior of the diffusion process. In fact, it would be difficult to imagine a system which did approximate regular properties any better than the systems of this study. Therefore, except for a few cases such as hexane -hexadecane, where the agreement with theory appears to be fortuitous, the regular solution stipulation does not appear to be stringent enough to make a generalized statement that regular solutions as a group will have the function, DABn/(d 1n ai/d 1n Ni) linear with mole fraction. In fact, the stipulation apparently should be that only ideal solutions follow this behavior for diffusion. Other clues may be found by a look at two assumptions used in deriving the basic non—equilibrium. statistical mechanics for liquid tran5port processes. First of all, it was assumed that the molecules have only the three degrees of freedom attributed to translational motion. All other degrees of freedom are assumed negligible. Eyring, et.al. (54), in their derivations of diffusion equations using absolute rate theory, made the same assumption but also suggested that perhaps rotational motion should also be 86 considered. Statistical mechanics also assumes that the inter- molecular forces between molecules are two-body forces only and that forces involving three or more molecules are negligible. This assumption is usually considered quite adequate for gaseous phases but for condensed phases where the molecules are packed much closer, the validity of the assumption may be doubtful. Part II Diffusion of Solutes at Infinite Dilution 87 911‘ it... I .I..I. WW0 nil :. Afl- Jiur! .I- I rim-52.3». .nlv] u m xi I... INTRODUCTION AND BACKGROUND From the results of the hydrodynamic theory (equation 20) or the statistical mechanical theory (equation 57 along with the remarks of Discussion of Previous Theoretical Work of Part I) it is found for diffusion at infinite dilution of component A, _ RT DAB .. (TAT) . (79) The viscosity, 77 , is the viscosity of the diffusion medium, but, because the concentration of A is infinitely small, 77 will be the viscosity of pure component B, i.e. the solvent. At constant temperature and pressure, the friction coefficient, GA , intro- duced in the Hydrodynamic Theory of Part I, is independent of the viscosity and depends on the size and shape of molecule A. The absolute rate theory of Eyring<27) predicts a similar result: R 1 RT : ._ , 8 DAB 17; n ‘ °’ The parameters, Al. A2. and K3 are the distances between adjacent molecules in the liquid. The parameter X1 is the perpendicular distance between two layers of molecules sliding past each other in viscous flow; A is the distance between neighboring molecules 2 in the direction of flow; and X3 is the distance between two molecules in the plane normal to the direction of flow. It is evident from. equations 79 and 80 that o-A = XZX3/X1. 88 89 Other investigators have derived relationships similar to equations 79 and 80. The original derivations of such equations were by Einstein124) and Sutherland(55). All the parameters of these equations are well-defined, measureable properties of the system except the friction coefficient. The purpose of this study is to give some light to the dependence of the friction coefficient on molecular size and shape at infinite dilution. , i 111...... I] 90 Previous Theoretical Studies of Liquid Diffusion at Infinite Dilution Stokes-Einstein — The Einstein(24) and Sutherland(55) models for diffusion of a solute at infinite dilution was a spherical solute molecule moving at a very slow rate through the solvent considered as a continuous medium. This is just the problem solved by Stokes who considered a sphere moving through a fluid at a Reynolds' Number less than one, where the Reynold's Number is based on the solute radius. The result of Einstein and Sutherland therefore gives the Stokes value for the friction coefficient, that is, “A = 6171'. where r is the radius of the solute molecule. (54) showed a similar result by Stearn, Irish and Eyring considering a large solute molecule moving through a solvent composed of small molecules. According to this model, the diffusion coefficient should be inversely preportional to the one- third power of the molar volume. They attempted to prove their model by comparing the observed with calculated diffusion coefficients for forty organic solutes in benzene as solvent. If they had displayed their observed data on log-log plot of diffusivity versus molar volume, they would have found that not only did their data show a large degree of scatter but also that the trend of the data showed a sloPe greater than the theoretical one -third. Anderson(2) observed that the slope was closer to two-thirds and therefore suggested that possibly the diffusion 91 coefficient should be inversely prOportional to the square of the radius. Kirkwood Equation - Kirkwood(41) pr0posed a model in which a solute molecule such as a normal paraffin moves through a continuous solvent medium as a chain, each -CHZ- group being a segment of the chain. This model may be likened to a string of beads falling through afluid. . The motion of each segment, of course, is dependent on the motion of every other segment in the chain. As a result of this study, . __ -1 a=n§1+5r21:(—1) (81) A 6"“ i j=l Rij 150' where n is the number of monomer units in the chain, (l/Rij) the average reciprocal of the distance between the segments i and j, and g is a friction factor for each segment. To test the Kirk- wood theory experimentally, Dewan and VanHolde(22) compared equation 81 with diffusivity data for ten normal paraffin solutes , ranging in size from pentane (CSH12) to octacosane (C28H58), at near infinite dilution, in carbon tetrachloride as solvent. In order to obtain a workable theoretical model, Dewan and Van- Holde assumed the following: 1) The monomer units are Spheres of radius b/2, where b o is the C-C bond distance, that is, 1. 54A. The friction coefficient was then represented by the Stokes friction coefficient, 5 = (Nb/2- 92 2) Long chain molecules assume numerous and continuously changing configurations as a result of internal rotations about the individual bonds of the chain. However, Taylor(57) calculated the most probably configuration of successive carbon bonds and found that the planar trans-configuration is the most probable at room (22) temperature. Therefore, Dewan and VanHolde assumed a rigid, all-trans model for the calculation of (l/Rij)' Thus, averages were not required and the values of (l/Rij) where simply calculated from the bond length and angle. With this model, the calculated diffusivities deviated an average of 6% below the observed value for seven normal paraffins between pentane and hexadecane. For the longer chain paraffins, the model becomes more unrealistic so that the deviations tend to become larger. A highly complicated model in which the internal rotation was taken into account statistically was also considered by Dewan and VanHolde. This model, however, gave only slightly better results than the rigid, all-trans model. 93 Engineering Correlations Wilke16l) drew attention to the fact that according to the absolute rate theory of Eyring (equation 80) and the Stokes- Einstein equation, the factor T/Dr) should be independent of temperature for a given solute-solvent system. He correlated diffusion data from 178 experiments of various solutes at infinite dilution in water, methyl alcohol, benzene and fourteen miscella- neous solvents. The observed and calculated data had an average deviation of 10%. Later, Wilke and Chang<62) correlated data for 285 experiments for 251 solute-solvent systems at infinite dilution and from this prOposed the following emperical equation: 1/2 D = 7.4x10'8 (XM) T AB nvA 0. 6 (82) where M is the molecular weight of the solvent; T is the absolute temperature, 0K; r) is the solvent viscosity, centiposes; and VA is the molar volume, cc. /gm. mole. The parameter x is intro- duced to correct for association in the solvent. In a nonassociated solvent, x is unity. For associated solvents, x > 1; for instance, for water, x = 2. 6. Wilke and Chang claimed an average accuracy of 10% for equation 82. Another important correlation of experimental data, is one by Othmer and Thakar(51), who derived the following equation based 94 on the properties of water as a reference solvent: D _ 14.0 x 10’5 _ AB " o. 6 1.1 L n80 VA ”WI s/LW) (83) where 7780 is the viscosity of the solvent at 20°C, centipoises; nw is the viscosity of water, centipoises; LS the latent heat of vaporization of the solvent; and LW is the latent heat of vapor- ization of water. Although equations 82 and 83 vary rather widely in most aSpects, they do have one major point in common, that is, the diffusion coefficient is inversely proportional to the molar volume to the 0.6 power. This is in direct disagreement with most existing theories (See Previous Theoretical Studies of Liquid Diffusion at Infinite Dilution). PRESENTATION AND DISCUSSION OF RESULTS The purpose of this study was to learn more about diffusion behavior of solutes at infinite dilution. As discussed in the previous section, the theoretical studies agree in general with experiment that the coefficient is proportional with the absolute temperature and inversely proportional with the viscosity coefficient. (24.55) (27) Both the Stokes-Einstein and the Eyring derivations also predict that the diffusion coefficient is inversely proportional to the radius (of the diffusing molecule. However, as pointed out in Introduction and Background, the experimental evidence of Eyring(54), Wilke and Chang(..61), and Othmer and Thakar(5l) show that the diffusion coefficient is inversely proportional to the solute radius to some higher power. Most of these experimental data were with solvents that were highly associative, such as water, methyl alcohol, etc. or were obtained with experimental methods of questionable accuracy. It was hoped, by using nonassociating solvents and data from improved experimental methods, to clarify some of the apparent discrepancies between theory and experiment. The solvents chosen for this study were hexane and carbon tetrachloride. These two solvents give no evidence of association either by forming dimers or complexes with solutes. These two solvents did not prove to be as heavily investigated as some other solvents, and therefore the literature did not produce as much data 95 96 from previous studies as it would have for, say, water and benzene. Water, however , is highly polar and therefore highly associative. Benzene, on the other hand, is not usually considered associative but in a few casesuo) has been found to show basic and thus associative properties even in extremely dilute solutions. For this reason, it was considered too unreliable for the present study. To supplement the experimental data found in the literature, additional data were obtained using the same apparatus and method of calculations described in Experimental Method of Part 1. One of the two solutions in the diffusion cell was pure solvent. The other solution contained a small concentration of solute. The diffusion coefficient calculated in this case was considered as the diffusion coefficient at infinite dilution of the solute instead of at the average concentration of the two solutions as in Part I. This is in agreement with the assumption made for the calculations that the diffusion coefficient is approximately constant over the concentration range between the two solutions. Additional assurance of the validity of this assumption is shown in Table V using data obtained from Part I. From this table, it is seen that the differences caused by this assumption are within the experimental accuracy of the apparatus. Solutes - In surveying the literature for experimental data that has already been reported in the literature for solutes diffusing in hexane and carbon tetrachloride, it was found that much of the opAHoAnumuuoH «v.0 wnvA thuA mvod o mANod 93on conan 0630309308 mo .0 com .m wmw .m Nwoo .o Mvwoo .o o ConHmO 02,833 A4 .o amp .N one .N wmoo .o 020 .o o osmoopofl 05333 N. 6 wow .N T: .N Nwoo .o wwoo .o o osmompmxmvm ocmxom oom\NEo coAu—mhucoo 00m\NEo 1:00 Ohms op sofiwuucoocoo 69395 :00 no 200 we oosouomma pmuonmmh—Xo omenocwm Ao>oA Ao>oA owmucoouonA CA on CA x .3253 nomad 3.30m «Cozom usonAmwooo an Aoflmooo Goflomnm voE coAmsmfip 208.336 .oudAom mo cowpmuucoocou .GoAu—mhucoocoo Open 0» nouonmmnfino was Zoo ”83.936 6:» CA mcoAfiAOm 03¢ 0:» mo comumuucoocoo owmum>m um «88.3.5380 mucvoAmwooo £39336 mo GOmAHMQEoO .> 3an 97 98 data is for rather small spherical solutes and normal paraffins. Very little data were for larger unsymmetrical solutes such as side-chained paraffins. Thus, additional experiments were made concentrating on these unsymmetrical solutes with a few experiments with more or less symmetrical smaller solutes to increase the range in molecular size. The experimental results are shown in Table VI. Again these results were all obtained at a temperature of 25. 1 i 0.03°c. The materials used were all in the purest forms available. Hexane, pentane, 2,2,4-trimethylpentane, Z-methylbutane, and toluene were obtained from Matheson, Coleman, and Bell Company and the carbon tetrachloride from Eastman Organic Chemical Company. These chemicals were all spectro- or chromatoquality and were used without further purification. 2,2-dimethylbutane (guaranteed normal boiling point, 48-490C), tetralin (b.p. é0-920C at 17 mm Hg pressure), phenanthrene (normal melting point, 99- lOOOC), and Z-methylpropene, trimer (normal b.p. 174-1780C) were also obtained from Matheson, Coleman, and Bell and used without further purification except for Z-methylpropene, trimer which was distilled twice. Before and after each distillation of Z-methylpropene, trimer, the refractive index was determined. No refractive index for this chemical is reported in the literature for comparison but since the reading was the same previous to and after the last Table VI. Experimental data of solutes diffusing in infinite dilution in carbon tetrachloride and hexane as solvents. Solvent: Carbon Tetrachloride Concentration of Solute, mole fraction Experi- Diffusion Solute upper lower mental coefficignt level level run x 10 of cell of cell number cmZ/sec 2, 2, 4- Trimethyl- pentane .0049 103 l. 126 2, 2, -Dimethy1- butane 0. 0045 104 1. 247 Phenanthrene 0. 0049 106 1. 031 2-Methy1butane . 0038 121 1. 487 2-Methy1propene Trimer 0.0140 120 0.8835 Tetralin 0.0059 105 1. 099 Solvent: Hexane Pentane 0. 0478 107 4. 588 Toluene 0. 0030 108 4. 134 Acetone 0. 0265 109 5. 257 2, 2, 4-Trimethyl- pentane 0.0253 110 3. 375 2, 2, -Dimethy1butane 0. 0588 112 3. 630 Tetralin 0.0037 113 3. 270 2-Methy1propene Trimer 0.0083 119 2. 679 2-Methy1butane 0. 0185 122 4. 404 99 .441?! 1‘ if... I ._r.. .1 - 2...“, . 100 distillation, it was assumed that the chemical was pure enough for this study. A fairly pure grade of acetone was used and was distilled several times until its refractive index was constant before and after the last distillation and compared well with the value reported in the literature. As further assurance of the purity of the materials used, the refractive indices of the chemicals were compared with values reported by Timmermansws) (See Table VII). Table VII. Chemical Carbon tetrachloride n-Hexane 2, 2, 4- Trimethylpentane 2, 2- Dimethylbutane Tetralin n- Pentane 2-Methy1propene, trimera 2-Methy1butane Toluene Acetone Comparison of refractive indices of chemicals used with values in the literature. Refractive Index at 25°C with Sodium D Line This Study Ref. (58) 1.4570 1.4576b 1.3720 1.3723b 1.3888 1.3891 1.3662 1.3660 1.5388 1.5392 1.3548 1.3548b 1.4285 --- 1.3508 1.3507 1.4940 1.4941b 1.3562 1.3566b a. apparent normal boiling temperature, 174°C b. average of several recorded data 101 102 Discussion of Results Correlation - Diffusion coefficients at infinite dilution for 24 solutes diffusing in carbon tetrachloride are plotted against the molar volume on log-log scale in Figure 18. Figure 19 shows a similar plot for 12 solutes diffusing in hexane. The molar volumes for the solutes are usually at 25°C, except for octadecane, eicosane, octacosane, and phenanthrene. These four compounds are solid at 25°C and therefore the molar volumes at their normal melting temperatures had to be used. Observing these data, it becomes apparent that the solutes fall into three rather distinct categories for each solvent: (1) symmdrical long-chain molecules with no side chains, i. e. normal paraffins, (2) large irregularly shaped molecules with the same range of molar volumes as the normal paraffins but with lower diffusion coefficients, and (3) smaller molecules of both symmetrical and unsymmetrical shapes. Each of these three categories may be correlated on a straight line, and it is found that in all three cases for both solvents that these lines have identical slopes equal to -0. 77. Thus the data can be correlated by the following equation: D = —7—° (84) AB 0. 7 ’ where Go is a constant which depends on solute size and shape for each solvent at constant temperature and pressure. The values for .vnH 3an 5 moudfiom on. nomad mnongz .> E @593; Q 5 .GOBSZQ BEG“; um 0030300308 :03900 E moudfiom .wa onswrm b :4 o .o o .m o 0v / _ _ / / vu / / I / I I / 1 mm .wm .Gofimfionnoo o¥§>>l\v // .w ,, _. 22.. ,.,. . moflJOm Hofimfim Sm III . / O m h /m / / I l/ .N > . N . Ed- 2% Q // /u o woudHOm awownuoggmcd l l l A3 // MI; I £65888 no // my mouSHOm Hwomnuogm I A3 / Clu'I 103 H0m0h mu0n§2 .> CH mzmn0> Q CH .N 0308 5 0330 0» 53355 0323“"; «0 05003 cm m0ufifiom .3 0H5mrm >G1— o.m w.v 0A4 0.0 N > an . H tho- mu: 0 m3 0 mOHUHOm HHN > ofixmdbdufl NF .0. m 1 m0fi30m Hmownu0§§mcd I l > N . H 2.61 muoH m S; Q - 1 - cm 000.300 303098.50 I .U . I 3 mm m 280.2028 853 /m/ 3v o .H (Iu'T 104 105 for Go for each of the three categories are shown in Table VIII for the two solvents , carbon tetrachloride and hexane. Tables IX and X show comparisons of the observed values for the diffusion coefficient with values calculated by equation 84 and the data from Table VIII. The correlation was very good for category 1 where the average deviation was less than the experimental accuracy for diffusion in hexane and only 1.9% for carbon tetrachloride. In the other two categories where the bounds are not quite so distinct, the accuracy of the correlation is not as good but even here the average deviation was less than 4% for all cases. Thus, as found previously by other investigators, the experimental diffusion coefficients of solutes diffusing at infinite dilution are not just simply inversely pr0portional to the radius of the solute but rather are inversely proportional to the radius to some higher power. Never before, however, has anyone shown that diffusion is so dependent on molecular shape. By having accurate enough data to distinguish the three categories, it is also found that the dependence of the diffusion coefficient on the radius of the solute is even higher than that found by Wilke and Chang, and Othmer and Thakar. Both of these previous correlations show that the slope of the plot of ln DAB versus ln v is -O. 6 instead of -1/3 as predicted by the Stokes-Einstein and Eyring theories. However, the present correlation with more accurate data and therefore less scatter of Table VIII. Tabulation of the values for the constant Go of ‘ equation 84 calculated from the intercepts of plots of Figure 18 and 19. Category 1. Symmetrical long-chain molecules with no side chains (normal paraffins). Category 2. Large irregularly shaped molecules with same range of molar volume as category 1. Category 3. Smaller molecules of both symmetrical and unsymmetrical shapes 106 Solvent CCl4 Hexane 62. 9 179. 8 57. 1 172. 2 47. 0 145. 0 Table IX. Tabulation of experimental and calculated diffusion coefficients for solutes diffusing in carbon tetrachloride. Calculated values are from equation 84 using Table VIII. Numbers refer to points on Figure 18. . . . . 5 Category 1. Diffusmg Coeff1c1ent x 10 Molar cm faec. volume Calc- Obser- Ref. of Percentage Number Solute cc/mole ulated ved observer Deviation 9 ’ Pentane 116.1 1.61 1. 57 22 -2. 5 11 n-Hexane 131.6 1.47 1. 50 22 +2.0 11a n-Hexane 131.6 1.47 1.47 0.0 (this work) 14 n-C7H16 147.4 1.35 1.34 22 -0.7 15 n-C8H18 163.5 1.24 1.26 22 +1.6 18 n-CIOH22 195.9 1.08 1.09 22 +0.9 20 n-ClZH26 228.6 0.960 0.964 22 +0.4 21 n--C16H34 294.1 0. 790 0.765 22 -3. 3 22 n-C18H38 328 0. 728 0. 690 22 - 5. 5 23 n-CZOH42 363 0.673 0. 667 22 -0. 9 24 n-CZ8H58 507 0. 519 0. 534 ._ 22 , +2. 8 Average Deviation 1. 9% Category 2: 8 .7 Cyclohexane 108. 8 1. 55 1. 49 31 -4. 0 10 2-Methylbutane 117.4 1. 45 1. 49 +2. 7 12 2, 2-Dimethy1- ' butane ' 133.8 1.32 1.25 -5.6 16 2, 2,_4-Trimethy1- . ’ pentane 166.1 1.11 1.13 +0.2 17 Phenanthrene 167. 7 1.11 1.03 -7. 2 19 2-Methy1- 221.8 0.894 0.884 -1.1 propene, trimer Average Deviation 3. 5% 107 Table IX. Nurnbe r Cate gory 3: U15§UJNH 0‘ 13 (continued) Solute Methanol Nitromethane Ethanol Acetone Methyl Ethyl Ketone Benzene CCl (self diffusion) Tetralin Molar cmz/§ec volume Calcr Obser- Ref. of cc/mole ulated ved observer 40. 73 2. 71 2. 68 2 53. 97 2.18 2. 00 18 58. 68 2. 04 2. 02 2 73. 99 1. 71 1. 70 90. 13 l. 47 1. 55 2 89.40 1.48 1.38 16 97.12 1. 38 1. 37 32 136. 9 l. 07 1.10 Diffusion Coefficient x 105 108 Average Deviation -1. -1. -0. +5. Pe rcentage Deviation O‘OOr—I N -7.3 .0, +2. . 5% Pe rcenta ge Deviation -0.4 +0. -2. . 7% +0. +0. 2. 0% +3. 0‘ Table X. Tabulation of experimental and calculated diffusion coefficients for solutes diffusing in hexane. Calculated values are from equation 84 using Table VII. Numbers refer to points on Figure 19. DiffusionzCoefficient x 105 Molar cm ngec volume Calc- Obser- Ref. of Number Solute cc/mole ulated ved observer Category 1: 9 Pentane 116. 1 4. 61 4. 59 11 Hexane (self) 131. 6 4. 20 4. 21 23 20 Dodecane 228. 6 2. 74 2. 74 21 Hexadecane 294. 1 2. 26 2. 21 Average Deviation Category 2: 10 2-Methy1- butane 117. 4 4. 39 4. 40 12 2, 2-Dimethy1- butane 133. 8 3. 98 3. 63 16 2, 2, 4- Tri- methylpentane 166. 1 3. 37 3. 38 19 Z-Methylpropene, trimer 221. 8 2. 70 2. 68 Average Deviation Category 3: 4 Acetone 73.99 5. 26 5. 26 7 Carbon tetra- chloride 97.12 4. 28 3. 86 25 Toluene 106.9 3. 98 4. 13 13 Tetralin 136. 9 3. 29 3. 27 109 Ave rage Deviation -o, 3. 8% 110 the data shows that the s10pes of the lines are -0.77. As a comparison, the 'Wilke-Chang correlation, equation 82, is also plotted on Figures 18 and 19. Another property made evident by the correlations shown in Figure 18 and 19, is the remarkable similarity in pattern of the data for the two solvents. This is rather surprising since the molecular shapes of the two solvents are so dissimilar, one being a linear paraffin and the other approximately spherical. Evidently, the shape of the solvent molecule has no effect on the diffusion for non -interacting solvents . Possible Model—Turbulent Solvent Model - The Stokes- Einsteinl24 ’ 55) equation was derived using as a model an anology to a sphere moving through a fluid at some constant, extremely slow rate. The sphere moving at this extremely slow rate will cause no turbulence, i. e. eddies, in the fluid by its motion. The result of such a model, as has already been‘ discussed, is that the friction coefficient is prOportional to the solute radius. Because the solutes, have approximately equal size as the solvent molecules , a more realistic viewspoint may be to consider the solute forcing its way through the solvent molecules and causing a great deal of disturbance among the solute molecules in the immediate region of the Solute. The motion of the solute might be simulated by a sphere moving through a fluid at a constant rate causing turbulence 111 in the fluid such as is the case of higher Reynolds' Numbers. Several studies have been made of spheres moving through fluids and have been summarized by Bird, Stewart, and Lightfootllz), who based their discussion largely on a paper by Lapple and Shepherd(43). It has been found that when complete turbulence exists around the sphere, the kinetic force on the sphere caused by the flow of fluid past the sphere, sometimes called the ”drag", is proportional to the radius of the sphere squared. With this model, the friction coefficient of equation 79, should be, 0A = constant ° r2 . (85) If this is the case, the exponent of the molar volume in equation 84 would be 0.67. Although the exponent has actually been found to be 0. 77 by this study, it is a closer approximation to fact than the non-turbulent model proposed by the Stokes-Einstein theory. Kirkwood Equation - The other concept tested here is the (22). The one by Kirkwood(4l) as modified by Dewan and VanHolde Kirkwood equation, equation 81, along with a discussion of the assumptions and the derivation of this equation are given above in Introduction and Background. Only the rigid all-trans model will be tested here. The statistically oriented model requires such laborious calculations while giving only meager improvement over the rigid all-trans model that it was not considered. As stated 112 previously, Dewan and VanHolde tested their model for several normal paraffins diffusing in carbon tetrachloride and found that they obtained fairly reasonable results. However, it is interesting to see how this same set-up would fare for a different solvent. Figure 20 shows a plot of the friction coefficient versus the number of segments in the solute chain for four normal paraffins diffusing in hexane compared with equation 81. The experimental friction coefficients were calculated by a rearrangement of equation 79, RT “A D (86) .413n where R, the gas constant, equals 8. 317 x 10'7 erg mole'ldeg. -1 in order to have consistant units. It is seen that the comparison is even better than for carbon tetrachloride as solvent. The average deviation from theory for these data was within the experimental accuracy. Thus , it may be concluded that the Kirkwood theory describes the diffusion behavior of normal paraffin solutes in nonassociating solvents fairly accurately. Dewan and VanHolde(Zz) suggested that equation 81 should apply to any solute structure. It was therefore applied to a few of the irregularly shaped solutes. Those solutes were chosen which have no freedom of motion about the chemical bond leading from one segment to the next. No assumptions or laborious statistical calculations had to be made in order to determine the values for r! (a .. .. r .w mm“ friction coefficient, O'A x 10_l7, cm. Us) assumption 0 observed data Kirkwood Eq., Eq. 81, with rigid all-trans number of segments in the n-paraffin chain diffusing Figure 20. in hexane 15 Comparison of Kirkwood Equation with Expe rimental Data. 113 114 Rij since each of the molecules is made rigid by its own structure. The values of Rij were then simply calculated from the bond length and angle as before. Table XI shows the values for the observed and calculated diffusion coefficients. Obviously, the theory breaks down for these structures. Instead of having a higher resistance to diffusion and thus a decrease in diffusion, the theory shows just the Opposite effect. The ring structure of these compounds brings the individual segments in close proximity of each other thereby decreasing the values of Rij which decreases the interaction term of equation 81. This tends to decrease the theoretical friction coefficient. Evidently, the solvent flowing past the close structure of these solutes causes a great deal of drag on the molecule. In order to correct equation 81, another term would have to be added which accurately describes the friction drag on the molecule as a whole by the solvent. This term, of course, is seen to be negligible for the Open structures of the normal paraffins. Viscosity - Another concept that may be investigated by the experimental data is whether or not the solvent viscosity is inversely proportional to the diffusion coefficient as predicted by the Stokes- Einstein, Eyring, and Kirkwood formulas. If this is true, the ratio of the Viscosities of hexane and carbon tetrachloride should equal the reciprocal of the ratio of the values for Goof equation 84 given by Table VII for the three categories mention under Correlation. The Table XI. Comparison of observed diffusion coefficients with values calculated by means of equation 79 combined with equation 81, the Kirkwood equation for irregularly shaped solutes. Diffusion Coefficient x 105 , cm /sec Solute Solvent Ob s e rve (1 Calculated Tetralin Carbon tetrachloride 1. 099 1. 256 Benzene Carbon tetrachloride 1. 38 1. 661 Phenanthrene Carbon tetrachloride 1. 031 0. 998 Tetralin Hexane 3. 270 3. 806 Toluene Hexane 4. 134 4. 560 115 116 ratio of the Viscosities is "(C(34) _ 0.7835 Cp. _ 3 03 n (C6iil4) - Go 2586 CP. _ . This is compared with the reciprocal ratio of the values for Go as follows: GO(C6H14)/GO(CC14) Category 1 179.8/62. 9 = 2.86 Category 2 172. 2/57. 1 = 3.02 Category 3 145. 0/47. 0 = 3.09 The values compare fairly well, so that it may be concluded that the diffusion coefficient is inversely proportional to viscosity. From their own data, Wilke and Chang(62) came to this same conclusion. Therefore, equation 84 may be written, G I _ o DAB ‘ W ‘87) where n is the solvent viscosity, and Go' is another constant that depends on the size and shape of the diffusing solute but is independent of the solvent except that it must be non-interacting. SUMMARY OF CONCLUSIONS Part 1. Mutual Diffusion of Nonassociating Binary Liquid Solutions. 1) The function Dn/(d. 1n ai/d 1n Ni) does not in general follow the linear behavior predicted by the hydrodynamic and statistical mechanical theories except for solutions which are ideal over the entire concentration range. Non-ideal solutions, both associative and nonassociative, both negatively and positively deviating from Raoult's Law, show incongruity with theory. 2) The Eyring and hydrodynamic theories both predict the linearity of the function Dry/(d 1n ai/d 1n Ni) with mole fraction for all non-ideal systems. However, the statistical mechanical theory shows this prediction only for regular solutions. This study shows experimentally that even the restriction to regular solutions is not stringent enough. Apparently, theory is followed only in the limiting case, that is, for ideal solutions. 3) For negatively deviating, n-paraffin systems, the activity correction, d In ai/d ln Ni, in most cases, overcorrected the diffusivity-viscosity product by more than its original deviation from linearity. Therefore, for this category of solutions where the deviation from Raoult‘s Law is small, a better, empirical correlation is obtained by plotting DTl against mole fraction. 117 118 Part II. Diffusion of Solutes at Infinite Dilution. 4) The diffusion coefficient for solutes diffusing at infinite dilution depends not only on the size of the solute but also rather heavily on its shape. For inert solvents, the diffusivity is found to be inversely proportional to the solvent viscosity but other than that, the diffusivity seems to be independent of solvent properties such as size and shape of the solvent‘molecules. 5) The diffusion coefficient is inversely proportional to the solute molar volume to the 0. 77 power. Previous correlations of data showed this exponent to be approximately 0. 6, but the data of these correlations showed a great deal of scatter in most cases so that it was difficult to give any value to the exponent. On the other hand, the maximum deviation of the data used in the present investigation from the value of 0. 77 was only eleven percent and for regularly shaped, long chain molecules, such as the normal paraffins, the average deviation was less than 2%. 6) The theoretical model proposed by Einstein(24) and Sutherland(55) which predicts the diffusivity coefficient to be inversely pr0portional to the solute radius“ is found not to coincide with experimental results. This theory does predict correctly the dependence of the diffusivity on solvent viscosity. Actually, a 119 model which predicts that the radius should be squared may be a closer approximation to experiment. 7) An equation proposed by Kirkwoodl‘“), equation 81, as modified by Dewan and VanHoldelzz), was tested for accuracy for small n-paraffin solutes diffusing in n—hexane. The theory proved to agree even closer to experiment in this study than in a previous (22). 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APPENDICES 124 APPENDIX A SAMPLE CALCULATION Experimental run number: 104 Data: April 17, 1964 Type of diffusion: 2, 2-Dimethy1butane diffusing in CCl4 Solution A (For upper level of diffusion cell) Weight of bottle + CCl 173. 8569 Weight of bottle only 125. 5737 Weight of CCl4 48. 2832 Weight of bottle + CCl + 2, 2-Dimethy1butane 173. 9780 Weight of bottle + CCl only 173. 8569 Weight of 2, 2-Dimethy butane 0. 1211 Mole fraction of 2, 2-Dimethy1- butane in solution A 0. 0045 Solution B (For lower level of diffusion cell) Pure carbon tetrachloride Photographic plate See Figure 7 for actual plate Exposure number Time, seconds Exposure nmnber Time, seconds 1 0 8 840 2 120 9 960 3 240 10 1080 4 360 11 1200 5 480 12 1320 6 600 13 1440 7 720 14 1560 J = 30. 4 Exposure 1 t = 0 seconds (For definition of measurements see Figure 8) 125 126 . 1 1 1 1 1 1 I 1 J (x() - xj 1 cm. k (x0 + xk), cm. [(x0 + xk)- (xo - x‘j )], cm. 4 2.552 16 2.663 0.111 6 2.576 18 2.679 0.103 8 2.596 20 2.697 0.101 10 2.614 22 2.717 0.103 12 2.630 24 2.739 0.109 14 2.646 26 2.766 0.120 Exposure 3 t = 240 seconds . 1 1 1 1 1 1 1 I J (xo - xj), cm. k (x0 + xk), cm. [(x0 + xk)-(xo - xj)], cm. 4 2.460 16 2.666 0.206 6 2. 505 18 2. 697 0.192 8 2.541 20 2.729 0.188 10 2.575 22 2.763 0.188 12 2.606 24 2.803 0.197 14 2.636 26 2.852 0.216 Exposure 6 t = 600 seconds , 1 1 1 1 1 I 1 I J (xo - xj), cm. k (x0 + xk), cm. [(x0 + icky-(xo - xj)], cm. 4 2.361 16 2.658 0.297 6 2.427 18 2.702 0.275 8 2.479 20 2.748 0.269 10 2.529 22 2.797 0.268 12 2.573 24 2.853 0.280 14 2.615 26 2.921 0.306 Exposure 9 t = 960 seconds 1 I 1 1 l 1 1 1 j (x0 - xj), cm. k (x0 + xk), cm. [(x0 + xk)- (xo - xj)], cm. 4 2.281 16 2 650 0.369 6 2.363 18 2.702 0.339 8 2.430 20 2.760 0.330 10 2.489 22 2 819 0.330 12 2.544 24 2 887 0.343 14 2.596 26 2 971 0.375 Exposure 12 t = 1320 seconds I l 1 I j (xo - xj), cm. k (x0 + xk), cm. [(x0 + xk)- (x; - x3”, cm. 4 2. 247 16 2. 671 0. 424 6 2. 339 18 2. 731 0. 392 8 2. 415 20 2. 795 0. 380 10 2. 484 22 2. 865 0. 381 12 2. 547 24 2. 946 0. 399 14 2. 609 26 3. 045 0. 436 127 Exposure 14 t = 1560 seconds 1 1 I 1 1 j (x; - x3), cm. k (x0 + xk), cm. [(x0 + xk)- (x; - xj)], cm. 4 2.220 16 2.679 0.459 6 2.319 18 2.745 0.426 8 2.406 20 2.815 0.409 10 2.478 22 2.890 0.412 12 2.547 24 2.976 0.429 14 2.613 26 3.084 0.471 J-Zj .J..-_ZJ. erf-1(fl)§A J J 22.4 0.74421 0.80356 18.4 0.61132 0.60954 14.4 0.47842 0.45319 10.4 0.34553 0.31648 6.4 0.21263 0.19073 2.4 0.07974 0.07078 (For inverse error functions, see reference (56) Zk-J 35315 erf-1(Zk5J)5B 1.6 0.05316 0.04715 5.6 0.18605 0.16641 9.6 0.31895 0.29064 13.6 0.45184 0.42463 17.6 0.58474 0.57606 21.6 0.71763 0.75859 (A.+:B) l A.+:B 0.85071 1.1755 0.77595 1.2887 0.74383 1.3444 0.74101 1.3495 0.76679 1.3041 0.82937 1.2057 128 Exposure 1 Exposure 3 1 1 1 1 1 ' xk1-x. xk4-x. (x’k+xj)' cm. ITBL , cm. (xi{+x3), cm. IT)- , cm. 0.111 0.130 0.206 0.242 0.103 0.133 0.192 0.247 0.101 0.136 0.188 0.253 0.103 0.139 0.188 0.254 0.109 0.142 0.197 0.257 0.120 0.145 0.216 0.260 average 0.1375 average 0.2522 1 1 2 1 l 2 -+x. ~1x. (%—BJ-) = 0. 018906 cmz {—fii-T-é) = 0. 063605 cmz avg. avg. Exposure 6 Exposure 9 1 1 1 1 4—x. i-x. ( 1 + 1 (xk J 1 + 1) ( J) cm "1. ’3" 6““ “ATE— ' cm' (*1. xj “m m ' ° 0.297 0.349 0.369 0.434 0.275 0.354 0.339 0.437 0.269 0.362 0.330 0.444 0.268 0.362 0.330 0.445 0.280 0.365 0.343 0.447 0.306 0.369 0.375 0.452 average 0.3602 average 0.4432 .xL1nx: 2 2 xii-x: 2 17171 = 0.129744 cm (TTBJ) = 0.196426 cm avg. avg. Exposure 12 Exposure 14 1 1 1 1 ' ' (xki-x.) ' ' (xk1-x.) (xk + xj), cm. fi—B-J— , cm. (xk + xj), cm. _KT-Bg- , cm. 0.424 0.498 0.459 0.540 07392 0.505 0.426 0.549 0.380 0.511 0.409 0.550 0.381 0.514 0.412 0.556 0.399 0.520 0.429 0.559 0.436 0.526 0.471 0.568 average 0.5123 average 0.5537 1 1 2 1 1 2 xki-xj) - 2 xk+xj 2 (m - 0. 262451 cm W — 0. 306584 cm avg. avg. 1.1“!1II. 00“: 03; oom~ .3: 5m 40 8363360 .3 .4 «Sufi m 950 on . 08$ H u. mama“; ooo~ .Ihl h-w~ cow coo cow com 5 u q d 666$ .86 33885 n 64on h. H0 guano + Awllhuh VFW mo 60E 9 ._.. . .. cm 0% 1‘ - 8 ”1211-“? +(Fz-111-I'I °uxo ‘ X Z 201 I Z 130 1 1 2 xk + x. Slope of the plot of (fi-fil versus time, t. avg. is 0. 00018441 cm. 2/sec. See Figure A. 1 for plot. Slope _ 0.00018441_ _. -5 2 DAB- .2322— - 14. 792 — 1.247X 10 cm. /sec. 3‘1 b Ci C0 2' C(l)(? ) 6131132) C9213)(?1’?) D DAB D.* .01 15311971) F311, 1)*(-1-:)1) A PPENDIX B NOMENCLATURE activity of component i radius diffusing segment in Kirkwood equation, cm. concentration of component i, moles/cm. 3 concentration at zero position in cell, moles/cm. 3 average cell concentration of sucrose, gm. sucrose/ 100 cc. solution concentration of molecules of species 0. at point $31 in singlet space, molecules/cm. concentration of molecules of species [3 at point 1’2 in singlet space, molecules /cm. average concentration of molecules of species 0. at point ‘f’l , and of type 8 at a distance '1’ relative to ‘11 in pair space, molecules/cm. Z/sec. diffusion coefficient of molecule i, cm. binary mutual diffusion coefficient of molecules A and B, cm.2/sec. self-diffusion coefficient of molecule 1, cm.Z/sec. 2/sec. intrinsic diffusion coefficient, cm. mean frictional force exerted on a molecule of species 0. at location 7’1 equilibrium frictional forces exerted on a molecule of species a. at location T71 perturbation frictional forces exerted on a molecule of species G at location I" 131 AGM f . 1 f C i G O gffp’fi’l .E’) sf“; (”(71 .7) gffé ”(71.7) AHM J 132 Gibbs free energy of mixing, cal. /mole fugacity of i in solution, atm. fugacity of pure i, atm. constant depending on molecular shape and size of empirical equation 84 pair correlation function equilibrium pair correlation function or radial distribution function perturbation correlation function enthalpy of mixing, cal. /mole diffusional flux, moles i/cm.2‘/sec. total number of interference fringes intrinsic diffusional flux, moles i/cm.2‘/sec. overall diffusional flux, moles i/cm.Z/sec. self-diffusional flux, moles i/cm.2/sec. fringe number in lower level of cell regular solution constant fringe number in upper level of cell Boltzman's constant, 1. 380 x 10'16 erg/deg. magnification factor , l. 923 number of 0. molecules in a system number of [3 molecules in a system mole fraction of component i refractive index "1~L AS 133 number of elementary diffusing segments in solute for Kirkwood equation vapor pressure of pure 1, atm. vapor pressure of i in solution, atm. distance between segments i and j in Kirkwood equation, cm. gas constant radius of diffusing molecule, cm. representative point for (1 molecules in phase space representative point for [3 molecules in phase space distance between 131 and 1’2 enthropy of mixing, cal. /deg. /mole absolute temperature, 0K time, sec. overall rate of motion due to diffusion of molecule i, cm. /sec . intrinsic diffusional rate of i, cm. /sec. mutual potential energy between a and {3 molecules molar volume of diffusing solute at infinite dilution, cm. 3/mole solution molar volume, cm. 3/mole molar volume of molecule 1, cm. 3/mole change of molar volume due to mixing, cm. 3’/ mole (l) 11105 4’06 Subscripts A,B 134 potential of mean force between molecule 0. and [3, erg/molecule distance, cm. activity coefficient statistical mechanics friction coefficient for diffusion solution viscosity , centipoises distances between neighboring molecules in absolute rate theory chemical potential segmental friction factor for Kirkwood equation friction factor of component i parameter of equation 34 *) 41a gum: + D parameter of equation 51, equals 13 refers to components A and B respectively of binary solution non labelled species of component A for self-diffusion labelled species of component A for self-diffusion arbitrary component in solution refers to measurement in lower level of cell refers to measurement in upper level of cell refers to components a and [3 in a V-component system number of components in a multi-component system 135 1 refers to upper level of cell 2 refers to lower level of cell 0 refers to center of diffusion interface in cell Superscripts ' refers to measurements on photographic plate 563 Illilllllllflllfllll 3 03057 12 Ill " u N“ Am mll Hl