2.9.... zaww bun .mwuw. 2.3... .... ”40.0”: o 1”... . 1")? 3.. . h . .. 2.. .. mafia/2“,! ..... 2. "fig... .1222... 2 r...4. .1... ..2:..2.. . : .....x.... .. .1... .43... M25... . 1:... 2.27.”? .Wwyx/ L'. fl #2 .. . 2 21 .. 23.2. ..2.222..2.2.2. . 2.. 2. 3+: (2. ....._.Z2.../.2. .....r........ .x. .u fur-H.114” J?):). 2.? grchauffl. . .. . .2 ... #2229427. r .f22 22.52.1144 ......2 25... WWW2 2.2.2.2....2... ”Hwy/V .f..mli..a H lufWu. 2.2 . p.244. mm¢fl u .2. 2...... 2...... ...2r...u. 2.2! 2.522.. .2 .. 23 1! ..2 .J. .. 0.1;}. . 2 «I. .2. ../..J. 2; 122122.!!! «2...... at... 6.5.9.213. . . . 2.2.... . . .15-... (a9. 1......u7...u~.u. $521.14.? 4.1.14.1.43925... 31i5%z<.$ 14w44 .HJML. franc. dud. . ”finflnflmfi 72v. \ .hueflfl:~fi 2”,. .2... “1551:: 0-169 LIBRA xv». to" R5! ‘ 1 ~ . . I Micbgm fit??? NWNWWWW “WHIIHLW L/ Umw/erslty This is to certify that the thesis entitled APPLICATIONS OF HYDRODYNAMIC THEORY TO MULTICOMPONENT LIQUID DIFFUSION presented by TERENCE K. KETT has been accepted towards fulfillment of the requirements for Ph.D. degree in Qhemical Engineering @M/M gag/WEI Major professor Date May 9, 1968 ABSTRACT APPLICATIONS OF HYDRODYNAMIC THEORY TO MULTICOMPONENT LIQUID DIFFUSION by Terence K. Kett Hydrodynamic theory is applied to multicomponent diffusion of non—electrolytes. Generalized equations are derived for the flows, diffusion coefficients, and the phenomenological coefficients for non—associating systems. Similar equations are derived for associating systems but only for the simplest ternary cases where either one component associates with itself to form dimers or where two components associate with each other to form dimers. The theory indicates that under the condition of constant partial molar volumes, Onsager's reciprocal relations are valid for non-associating systems and for the associating systems with the additional assumption that the partial molar volume of the associated species is equal to the sum of the partial molar volumes of the species making up the dimer. Four ternary systems are studied in this investigation. Experimental diffusion and phenomenological coefficients are obtained using a Mach—Zehnder interferometric technique for the systems dodecane—hexadecane-hexane and diethyl ether—chloroform— carbon tetrachloride. Experimental data for the systems toluene— chlor L brnmrL " r and acetone—benzene—carbon tetrachloride TERENCE K. KETT were taken from the literature. Diffusion and phenomenological coefficients for all four systems are also calculated from equations derived from hydrodynamic theory. The coefficients obtained by the two methods are compared and for the non-associating systems show excellent agreement. For the acetone-benzene-carbon tetrachloride system in which the actual associated species present is not clearly known, reasonable agreement among the diffusion coefficients was evident. The other associating system could not justifiably be compared since satisfactory activity data was not availableo Experimental evidence from this investigation verified within the limits of experimental accuracy that Onsager's recipro— cal relations are valid for non—associating systems. In addition, it demonstrated the applicability of hydrodynamic theory to multi- component diffusion. For associating systems where the degree of association is not clearly known, it indicated that hydrodynamic theory can be applied as a predictive theory in obtaining reasonable multicomponent diffusion coefficients. Based on this evidence, it can be concluded that hydrodynamic theory should play a major role in describing multicomponent diffusion. APPLICATIONS OF HYDRODYNAMIC THEORY TO MULTICOMPONENT LIQUID DIFFUSION By Terence K. Kett A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1968 6;,5:/€/€?} To my Mother and Father and my wife, Betty ACKNOWLEDGEMENT The author wishes to express his appreciation to Dr. Donald K. Anderson for his 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 University and to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for providing financial support. The author wishes to express his sincere appreciation for the patience and efforts of his wife, Betty, in the typing of this manuscript. iii TABLE OF CONTENTS Page ACKNOWLEDGEMENT............................................ iii LIST OF FIGURES............................................ vi INTRODUCTION......................................o..H...° viii THEORY..................................................... l Hydrodynamic Model...................................... 4 Hydrodynamic Flow Equations............................. 7 Current Flow Equations in Terms of Concentration Gradients........................................... l3 Diffusion and Phenomenological Coefficient Expressions.. l6 Non-associating,Systems............................. l6 Associating Systems................................. 29 EHERIMENTALOOOO0.00..0.0.00.0.00.0000000COOOOOOOOO00.00066 46 Apparatus............................................... 46 Procedure for Experimental Run.......................... 52 Purity of Materials..................................... 54 Calculations............................................ 56 Concentrations and Viscosities..........o........... 56 Reduced Second Moment............................... 58 Reduced Height-Area Ratio........................... 63 Differential Refractive Index Constants............. 63 Diffusion Coefficients.............................. 65 Friction Coefficients............................... 67 Activity Data....................................... 69 Error Analysis................................ ...... ,c.. 71 RESULTS ANDDISCUSSIONQOOOOOOOOOOOOCOOOOOOOOOOOOOQOOJOO00.. 77 TheoreticalOOOOOOOOOOOOOOOOOOOOOOOOOOOOGOOOOOBCOOOOOGOOC 77 mperimentalOO 00000 00.000.000.0000000000000000000000000G 78 CONCLUSIONS................................................ 98 iv Page FUTURE WORK......... ..... ..... ....... ....................... lOO Theoretical...OOOOOOOOOOOOOOOOOOOOOO0.0GOOOOOOOOOOOOOOOOO 100 Experimental-0.0.0.o....000OOOOOCOCOOQOOOOOOOO0.0000050000 101 APPENDIX I - Determination of Friction Coefficients....... 103 APPENDIX II Determination of the Ternary Diffusion coeffiCientSOOOOOOOOOOOOOOOOOOOCOOOOOOIOOOOOO 107 APPENDIX III Determination of Activity Expressions........ 120 APPENDIX IV - Fortran Program to Calculate the Reduced secondMomentOOOOOOOIOOIOOOOOOOOOOOOOOOCOO000 130 APPENDIX V - Fortran Program to Solve for the Determinant of the Experimental Ternary Diffusion coeffiCientS.OOOOOOOOOOOOOOOOOOOOOOOOOO00.000 136 APPENDIX VI - Experimental Data...ooooooo-ooooooooooooooooo 138 NOWNCLATUREO0.000...OOOOOOOOOOOOOO0.0.0..00.000000000000000 157 BIBLIOGRAPHY................................................ l6l LIST OF FIGURES Figure Page 1. Schematic diagram of the interferometer showing position of mirrors................................ 47 2. Photograph of apparatus.............................. 48 3. Photograph of the diffusion cell.. ..... .............. 50 4. Diagram of the diffusion cell........................ 51 5. Typical set of photographs taken during a diffusion run...0.0.0.00IOOOOOOOOOOOIOOOOOOOOOOOOOOOOOOIOOOOO 55 6. Determination of the differential refractive index increments, R1 and R2, for the system dodecane- hexadecane—hexane.................................. 80 7. Determination of the differential refractive index increments,Rl and R2, for the system diethyl ether—chloroform-carbon tetrachloride.............. 81 8. Linear relation of the reduced second moment, D Zm’ versus the refractive index fraction of dodecane, al, for the system dodecane—hexadecane-hexane...... 82 9. Linear relation of the reciprocal square root of the reduced height-area ratio, l/VDA, versus the refractive index fraction of dodecane, a for the 1’ system dOdecane-hexadecane-hexaneo o o o e o o o o o o o o o o o o e 83 10. Linear relation of the reduced second moment, D2m’ versus the refractive index fraction of diethyl ether for the system diethyl ether-chloroform— carbon tetrachloride............................... 90 vi Figure 11. 12. l3. 14. 15. 16. Linear relation of the reciprocal square root of the reduced height-area ratio, l/VDA, versus the refractive index fraction of diethyl ether, al, for the system diethyl ether-chloroform—carbon tetraChlorideOOOO0.00000000IOOOOOOOOOOOOOO0.0.0.00. Second moment, m , versus the measured time for the 2 system dOdecane-hexadecane—hexaneo o o o e o o o o o o o o o o o o 9 Second moment, m2, versus the measured time for the system dodecane-hexadecane-hexane (runs 157 and l62)............................................... Second moment, m , versus the measured time for the 2 system diethyl ether—chloroform-carbon tetrachlo- rideOOOOOOOOOOOOOOO0.00I.OOOOOOOOOOOOOOOOOOOOO0.00° Typical refractive index gradient curve at various times during difoSionIOOOOOOOOOOOOOOOOOOOOOOOOOOOO The product of the absolute temperature and the reciprocal of the self diffusion coefficient versus carbon number for several hydrocarbons............. vii Page 91 147 148 149 153 156 Table 10. 11. LIST OF TABLES Page Comparison of physical constants with previous recorded data.‘.0.......0.....OOOOOOOOOOOOOOOOOOO... 57 Variances and 95% confidence limits of the slapes and intercepts of the D versus a and l/v/DA versus a1 curveSOOOOOOOOOOOOO?IP'OOOOOOOOO}O....OOOOOOOOOOOOOOOO 73 95% confidence limits of D m and l/fiZ; ............... 74 2 Comparison of the predicted and the experimentally determined friction coefficients for the system dodecane—hexadecane—hexane.......................... 85 Comparison of experimental diffusion and phenomeno— logical coefficients with those calculated from friction coefficients for the system dodecane- hexadecane—hexane................................... 86 Comparison of experimental diffusion and phenomeno— logical coefficients with those calculated from friction coefficients for the system toluene- chlorobenzene-bromobenzene.......................... 89 Comparison of experimental diffusion and phenomeno— logical coefficients with those calculated from friction coefficients for the system diethyl ether— chloroform-carbon tetrachloride..................... 92 Comparison of experimental diffusion and phenomeno— logical coefficients with those calculated from friction coefficients for the system acetone- benzene-carbon tetrachloride........................ 96 Physical prOperties of the pure components.... ..... ... 138 Initial concentration differences..................... 139 Binary diffusion data at infinite dilution...........° 140 viii Table 12. 13. 14. 15. l6. l7. 18. 19. Page self difoSiondataOOOOOOO0.00.0.000000000000000.0000. 141 Reduced second moment, D2m’ and reduced height—area ratio, DA, dataOOOOOOOOOOOOOOOOOOOOOOOI’IOOOOOOOOOOOOO 142 Second moment, 1112 data................................ 143 Time correction, At , data........................ 146 corr. Data used to calculate the experimental ternary diffusion coefficients.............................. 150 Quantities involved in the calculations of the diffusion and phenomenological coefficients......... 151 Van der Waals constants for various hydrocarbons...... 154 Activity constants used in the ternary activity equationSOOOOOOO......OOOOO.......OOOOOOOIOOOOO0.0.0 155 ix INTRODUCTION Throughout the last twenty years, interest has gradually increased in the area of multicomponent diffusion. There is voluminous . . . . 21 literature available on binary difquion that has built up Since Fick presented his equation defining a diffusion coefficient. However, it . 37,38 . . . was not until Onsager presented equations describing the flux of each component as the linear sum of every concentration gradient multiplied by a diffusion coefficient that any reasonable attempt to describe multicomponent liquid diffusion was made. Even so, no reasonable experimental work in this area was introduced until Baldwin, . . 3,22 . . . . Dun10p, Fujita and Gosting in 1955 presented equations Similar to Onsager's along with experimental techniques for obtaining the diffusion coefficients. This was certainly a major contribution in this area and furnished the impetus for renewed interest and consequent improve- ments in multicomponent diffusion. ,19’21’23 have periodically Since that time Gosting, et. a1. developed improved experimental techniques with Optical methods utilizing a Gouy interferometer. Burchard and Toor7 have also adapted the diaphragm cell method to obtain the multicomponent diffusion coefficients. With these methods, ternary diffusion data have become increasingly available enabling the study of both electrolytic and non-electrolytic multicomponent systems.7’1?"18’19’22’35’41’44’45’49 3 The equations presented by Baldwin, Dunlop and Costing des- cribing multicomponent liquid diffusion for an N component system are N—l ac. =_v __l °=1 N-l l where Ji is the one—dimensional flux of component i in moles/cm.2/sec., Dij is a diffusion coefficient in cm.2/sec., and BCj/Bx is the concen- tration gradient of component j in the x direction in moles/cm.3/cm. Concentration gradients are used because they render themselves more easily to experimental measurement but in actuality the negative gradient of chemical potential is considered the driving force for the flows, Ji. From irreversible thermOdynamic considerations, the flows Ji are also related by the following expression J = X L, Y. i = 1, ..., N — 1 (2) where the Lij are phenomenological coefficients. If in these expres— sions, both the fluxes Ji and the thermodynamic forces Yj are indepen— dent and the sum of their products appears in the expression for the rate of entropy production then under these conditions,28 the theory of irreversible thermodynamics says that the Lij should satisfy the following relations, known as Onsager's reciprocal relations, namely, Lij = Lji (3) A strong emphasis in most multicomponent diffusion studies has been placed in trying to verify these reciprocal relations. This verification has been conclusively shown in a number of cases such as heat conduction in anisotropic crystals, thermoelectricity, electro- kinetic effects, and e.m.f. and transference in electrolyte solutions.34 However, this verification has not been shown in isothermal diffusion without some doubt because of the limitations in the eXperimental techniques employed. The phenomenological coefficients, Lij’ can be related to the diffusion coefficients, Dij’ and therefore if the Onsager reciprocal relations can be verified, the experimental quantities necessary to describe diffusion will be reduced considerably. This explains the interest in the verification. Up till now, verification has only been possible by experimental means and; consequently, because the experimental techniques have been of limited accuracy, question con— cerning the verification has been justified. In this research, hydrodynamic theory ofHartley and Crank27 is extended to multicomponent diffusion in non—electrolytes. Genera- lized equations for the flows Ji are obtained in terms of concentra— tion gradients, chemical potential gradients, and the forces Yj. In addition, expressions for the phenomenological coefficients are obtained in terms of the diffusion coefficients, Dij’ and in terms of friction coefficients, 01. This theory shows that the reciprocal relations are valid for non-associating liquid systems and this is verified experimentally. In addition, experimental evidence is provided which supports the hydrodynamic theory. THEORY Hydrodynamic Model The hydrodynamic theory of'Hartleyand Crank27 considers molecular diffusion of liquid solutions in a closed container. They proposed that diffusion of a particular species could occur in two ways. First, by random molecular motion and secondly by flow of the entire medium itself. The latter Hartley and Crank called mass flow. Since these two methods of movement of a species can occur at the same time, both effects must be taken into account when experimentally studying the overall diffusion process. The interpretation of molecular motion in a liquid can best be explained by assuming that the liquid molecules oscillate within a cage or hole formed by its neighboring molecules. The molecule per- forming this oscillatory motion, will occasionally acquire sufficient energy to overcome the potential energy barrier of this hole and migrate to another hole. Because of the energy required before a molecule can make this jump, typically less than 1% of the total number of molecules are undergoing this motion at any given time. It can be seen, however, that in this way diffusion of a species is occurring. This type of motion, often referred to as intrinsic diffusion, can occur regardless of whether concentration gradients exist or not. However, if a concentration gradient of a component does exist, there will be a net movement of that component to reduce the gradient. The other means by which a particular species can flow from one point in the container to another is by flow of the medium itself. It is easy to see in the case of a liquid solution flowing continuously through a container with open ends, such as a pipe, how a particular species can move from one point to another. It is simply carried along by the bulk solution itself. It was unfortunate that Hartley and Crank referred to a "mass flow" as contributing to the overall diffusion of a species since this implied the flow just described. Actually this bulk motion referred to by Hartley and Crank can occur in a closed container. In order to study molecular diffusion experi~ mentally, the system is closed to enable free diffusion to take place. To obtain an understanding of how a flow of the medium takes place in a closed container and how this contributes to the diffusion of a species, consider a coordinate fixed reference plane somewhere in the container. Now if there are different components in the solution, the molecules will have different sizes and shapes. Hence when a com- ponent of type ”i” migrates by molecular motion across this referent; plane in the closed volume, an increase in volume necessarily occurs unless there is a compensating flow back across the plane. This com— pensating flow is what Hartley and Crank called the bulk motion and what is referred to here as the flow of the medium. Let us now look at the plane across which no net flow of material is occurring. In the case of a pure liquid where all the species are alike, the velocity of this plane relative to fixed coordi— nates is zero. This follows because there would be just as many molecules diffusing by hole migration (intrinsically) across a coordinate-fixed plane in one direction as there would be in the opposite direction. Thus for a pure liquid, any coordinate-fixed plane satisfies the condition of no net flow across it. The same analysis can be applied to a solution of uniform concentration. The concentration of any component would be the same on either side of a coordinate fixed plane located anywhere in the container. Since the concentration is the same on either side, the number of molecules of component i crossing the plane by hole migration would be the sane as the number of component "1" molecules crossing in the Opposite direction. The same applies to all the other components in the solution. Hence, the plane across which no net flow occurs corres- ponds to any coordinate fixed plane as long as the solution is uniform. It follows that if the solution is not uniform, the velocity of the plane across which no net flow occurs will not be zero relative to fixed coordinates. In this case, the number of molecules of "i" crossing a coordinate—fixed plane by hole migration will not be equal to those migrating across in the opposite direction. The same follows for the other components. Since each component will generally have a different rate of migration, there will tend to be an accumulation on one side of the coordinate fixed plane unless there is a compensating flow back. Even.if the molar volumes of each component are the same, there still will be build up if the rates are different. For nonuniform solutions, only in the case of equal constant molar volumes and equal rates of hole migration will the velocity of the plane across which no net flow occurs be zero. In further discussions, this velocity with respect to fixed coordinates will be referred to as the velocity of the medium, v . m,c Hydrodynamic Flow Equations If a molecule in solution is assumed to diffuse at a constant rate, the sum of the forces acting on it must be zero. The driving force per molecule and the resisting force per molecule must be equal and opposite. The question arises as to what are these forces. It is generally accepted that the driving force for diffusion is the negative gradient of chemical potential, -Vui. Intuition says that a particular component will diffuse in such a direction as to reduce its gradient of concentration. In binary solutions this is the case, but in solutions of 3 or more components this does not always follow even though eventually the solution will become uniform through- out. For example, if the velocity of the medium is greater than the intrinsic rate of a particular component it is possible for the net fIUX'Of that component to be in the opposite direction to its negative gradient of concentration. Eventually the velocity of the medium slows down as the concentration gradients decrease and the intrinsic rate of the component surpasses it. Gibbs24 has shown that at equilibrium the chemical potential should be the same throughout and hence a system not at equilibrium always tends to equalize the chemical potential. This applies whether a particular component is present in more than one phase or whether it is present in various subsystems of one phase. The negative gradient of concentration, introduced by Fick21 in mathematically describing the diffusion process, is generally used in diffusion equations because it lends itself more easily to experimental measurement. The resisting force for diffusion is a more controversial subject; however, certain conclusions regarding its description can 4’25’39’45 indicates be drawn. The overwhelming experimental evidence that it is proportional to the viscosity of the medium. Attempts have also been made to correlate it to the radius or some power of the radius of the diffusing molecule.4’20’31’39’45 Although indeed it has been shown that the size of the molecule is a factor, no universal relation to the size has been found. In addition, Bidlack4 has shown that shape should be an important consideration. Based on these observations let us define the resisting force as follows =— N F“ f1 ”“1111 = -o, u. 4 i,r , in i m ( ) 9 where Fi,r is the resisting force per molecule of component i to intrinsic diffusion, fi is a coefficient which is a function of the size and shape of the diffusing molecule and includes effects of the medium, n is the viscosity of the medium, N is Avogadro's number, ui, is the velocity of species i with respect to the velocity of the medium. The proportionality factor, oi = fiN will be referred to as the friction coefficient of component i. The negative sign is a result of the velocity of i being in the opposite direction to the force. The driving force for diffusion as discussed earlier is given by Fi,d = "V“i (5) where Fi d is the driving force for intrinsic diffusion of i and Vui ! is the gradient of chemical potential Of component i. For one- dimensional diffusion in the x—direction, equation 5 becomes F. = - (6) Since the driving force is equal and Opposite to the resisting force, Sui ——=O.nU. (7) 3x 1 i,m Multiplying both sides by Ci’ the concentration of i, and rearranging gives J. =C.U. =— (8) - m where Ji is the flux of component i with respect to the velocity Of the medium (i.e., the intrinsic flux of i). . . .th . . The chemical potential of the i constituent may be written as o “i - pi + RT ln a1 (9) 0 . . . . where mi 18 a function Of T and P, R is the gas constant, and a1 is the activity of i. At constant T and P, substituting equation 9 into equation 8 yields lO C {8 ln 3, C.RT 3 In a. BC. Jm = _ I i = _ 1 [ i __l i o,n [ ax ] on k ac, ] [3x] 1 1 1 T,P T,P T,P (10) Jm = _ RT [3 1n a. sci, i o.n L ac. 3x J 1 l T,P T,P In order to Obtain an expression for the velocity of the medium with respect to fixed coordinates, vm,c, a volume flux balance across a coordinate-fixed plane is made. If we consider the case where the partial molar volumes Of each component are assumed constant and the system is closed, then the volume flux across a coordinate fixed plane must be zero. In other words every time a certain volume of molecules diffuses across the plane, an equal volume of solution must come back in order to keep the total volume constant. Since the system is closed, the only way this could not happen is if the molar volumes changed. However they are assumed constant, thus the volume flux crossing the plane due to intrinsic diffusion plus the volume flux crossing this plane due to the flow of the medium must be zero. That is for an N—component system IIMZ if? + v = 0 (11) . i i m,c i l where F; is the partial molar volume. Since J? has units Of moles/mn.2/ sec. and V; has units of cm.3/mole, notice that vm c has units of , velocity, cm./sec. Equation 11 applies only for the case of constant partial molar volumes and a closed system. What about the case where the molar 11 volumes are not constant? Again, an expression for the velocity of the medium relative to fixed coordinates, vm, , can be Obtained by looking at a volume flux balance across a coordinate fixed plane. In this case, however, there are three contributions. There is a flux of volume due to the intrinsic motion relative to the medium, there is a volume flux as a result of the flow Of the medium, and there is a volume flux across this coordinate plane because the total volume is changing. Physically what happens is that a certain volume Of material diffuses across this plane due to intrinsic motion. As a result of this, to relieve any hydrostatic pressure which might build up since the system is closed, the medium itself flows back. However, because of this diffusion, the concentrations have changed and thus if the molar volumes are functions of concentrations, the total volume may have changed thus producing a net volume flux across the coordinate fixed plane. The net volume flux relative to a fixed coordinate plane N is given by '21 FgJ? + Vm,c which does not equal zero in this case but 1: rather — m ViJi + Vm,c — VV,C (12) IIMZ i l where vV c is the net volume flux relative to fixed coordinates resulting , from the change in volume. It follows that vV c is the velocity of the 9 plane across which the net volume flux is zero. From equations 11 and 12, expressions for vm c can be obtained. 3 These are 12 —m ViJi (13) <2 II I II M2 1 for the condition Of constant molar volumes and - m ViJi + VV,c (14) < II I II M2 i l for the condition of varying molar volumes. The diffusion process that is studied experimentally consists Of both intrinsic diffusion and the flow of the medium. It is desirable then to Obtain expressions for the overall diffusion flux relative to fixed coordinates of each component. These are obtained by summing both contributions. Hence JC. = J”? + ON (15) 1 1 1m C 9 c where Ji is the overall flux Of component i relative to fixed coordi— nates. Most experimental data reported in the literature are considered using overall fluxes relative to the velocity Of a plane across which the net volume flux is zero, that is relative to vV C. Therefore 9 V m = + — Ji Ji Ci(vm,c VV,c) (16) where J: is the total flux of i with respect to this volume flux plane. From equation 14, it follows that equation 16 becomes W? (17) 1 J J <: S IIMZ j For the case of constant molar volumes, equation 16 becomes JY = J? + C.v (l8) 1 1 1mc 9 and substitution of equation 13 into equation 18 gives the same result as in the case of varying molar volumes. That is, 13 i i i ' ° (17) j=l J J Substituting equation 8 into equation 17 gives J: in the desired form. C. Bu. C. N C.V. Bu. V i 1 1 J J J, = — —— + —— —l (19) i o.n 8x n ._ 0. 8x 1 J-1 J Throughout the rest Of this report, the superscript V will be dropped. All fluxes, unless specifically designated, will refer to the volume flux plane. Current Flow Equations in Terms of Concentration Gradients As mentioned earlier, the flow equations required to interpret diffusion experiments are generally written in terms of concentration gradients. For systems of two components, diffusion of either component . . . . 21 . . . is completely described by Fick s first law. For one-dunenSional flow this law is 8C. 1 Ji=—DAB[—O;]t 1=A, B (20) In this equation, Ji is the flux of component i, and (aCi/ax)t is the concentration gradient, at position x and time t. The same diffusion coefficient DAB applies to both components. In systems with three or more components, the flow Of each component depends not only on its own concentration gradient but also on the concentration gradients of other components present. There is 9 . . 3 . interaction of the flows. As a result, Onsager first proposed a description Of this case by expressing the flow of each component as 14 the sum of every negative concentration gradient multiplied by a dif- fusion coefficient. Thus for a system Of N components, his formulated equations were N . J =— 2 D,,—-1 i=1,...,N (21) 2 with N2 diffusion coefficients, Dij' He then showed that only (N - l) diffusion coefficients are necessary to describe the flows however. This can easily be seen in the case Of a constant volume system. It follows for a closed system that C.V. = 1 (22) 1'1 ll M2 1 l and if in addition the molar volumes are constant that VJ. = 0 (23) l 1 1 II M2 i Equation 22 follows from a material balance on the system with each CyVi factor representing the fraction of the total volume contributed by canponent i. It is not necessary that the partial molar volumes be constant for equation 22 to be valid. Equation 23 results from the assumption of a closed volume system with no volume change on mixing for which the net volume flux must be zero. From equation 22, the number of terms in each flux equation can be reduced from N to N - 1. Similarly with equation 23, one of the fluxes Ji can be elfininated from equation 21. Thus there are N — 1 independent fluxes and N — l indepen— dent concentration gradients for an N component system. Baldwin, Dun10p, and Gosting3 therefore presented a set Of flow equations for an N component system as follows 15 3C.\ N J, =— 2 nil—fl i=1, ...,N (24) i j=1 J x t This set Of flow equations defines (N — l)2 diffusion coefficients, Dij’ for a system Of N components, as required by Onsager's theory; however, these coefficients differ from those defined by Onsager in equation 21. They are related to certain combinations Of Onsager's N2 coefficients. These combinations can easily be found by applying equation 22 tO eliminate the concentration gradient Of the Nth compo- nent in equation 21, and then comparing corresponding coefficients Of like concentration gradients in equation 24. These relationships, first Obtained by Dole,l4 are iN 1 (25) C II C?) | < < 2 IL I G H .1. ll |._.| Z I where the Dij are the Onsager diffusion coefficients. The preferred equations for describing multicomponent diffusion are equations 24, since they are linearly independent and contain gradients of the measurable quantity, concentration, not chemical potential. Baldwin, Dun10p, Fujita, and Gosting3’22’23 in a series of articles have discussed in detail various methods of experimentally determining the diffusion coefficients, Dij’ for ternary systems. Their techniques involve interferometric methods and the utilization of resulting refractive index gradient curves. Some of these techniques are adapted in this laboratory and are discussed in the experimental section. Solution Of the describing equations is 16 given in Appendix II. Burchard and Toor7 modified the diaphragm cell to experimentally measure the ternary diffusion coefficients. Diffusion and Phenomenological Coefficient Expressions Non—Associating Systems It is desirable to obtain expressions for the Dij in terms of the friction coefficients, Oi. Since experimental values of the Dij and the Oi can be Obtained, a check would then be available on the hydro- dynamic theory. These expressions will now be derived here. Diffusion Coefficient Expressions - It is convenient to first derive expressions for the Dij in terms Of the friction coefficients, Oi, for the ternary case since the terms, equations, and algebraic manipulations are less involved. From this, the generalization to N components is easier to follow. The flow equations for the ternary case from equation 24 are J = _ D El _ D 8—.C_2 l 11 8x 12 8x (26) J = — D 3:]; _ D E 2 21 8x 22 3x These equations are written in terms Of concentration gradients and equations 19 in terms of chemical potential gradients. It would be desirable to have both sets of equations in terms Of concentration gradients because then, provided the equations are independent, the coefficients of like terms could be equated. For the ternary case, the flow equations for two of the components from equations 19 are 17 C1 — a“1 C1C2V2 a“2 C1C3V3 a“3 Jl=-—(l-C1Vl) a 3x + 3x Oln X ozn O3n _ _ (27) C1C2Vl 3“1 C2 — 3“2 C203 3 3‘13 J2=—_T-—(1'V2C2)ax+ a Oln x ozn O3n X From the Gibbs—Duhem relation 8p Bu 3p 1 2 3 Cl 8x + C2 3x + C3 8x — O (28) Therefore, 33 = _ 31 iii _ 3; 32 (29, 8x C3 3x C3 3x For the constant volume system, equation 23 gives le1 + J2V2 + J3V3 = O (30) and therefore only two fluxes are independent. Substitution Of equation 29 into equations 27 gives J = - El[(l - V101) + V3C1]3“1 _ C1C2 :1 :2 8“2 l T) 01 03 3X T] 03 02 ax _ _ _ _ (31) J _ _ ClC2 Z§._ I}. aul — EZ.[<1 - V2C2) + V3C2] apz 2 n 03 01 8x n O2 03 3x To convert the gradients Of the chemical potential to gradients of concentration, use will be made Of the identity 8x, if. = 2 (AL) (4) (32, t . 3x. at. J J l where f = f(xl, x2, . , x ) and x1 = xi(tl, t2, , tm) 18 At constant T and P, for the ternary case (33) The chemical potentials are functions of only two concentrations, Cl and C2, because the third, C3, is not independent. This is evident from equation 22 which for the ternary case is ClVl + C2V2 + C3V3 = l (34) Applying the identity equation 32 to equations 33 gives ) Bul = aul 3C1 + Bul 3C2 8x BC 8x 3C 3x 1 c k 210 2 l (35) 8n = auz 8C1 + (3U2 8C2 8x 8C 3x 8C 8x 1 C 2 C 2 l which when substituted into equations 31 and rearranged yields _ ”' " if F‘ " 1r 1 _ cl (1 vlcl) V3Cl Sui] c102 v3 v2 Buz acl Jl _ — h—' 0 + O BC + n 6—._ O_- SC_- “5; 1 3 1] 3 2 1 C2 C2 (36) _ _ r .3; (l VlCl) V3Cl apl ClC2 V3 V2 Buz 8C2 ' n o + 0 ac + n B— _ 23— ac ax l 3 2 3 2 2 C1 C1 19 c102 V3 V1 3111 c2 (1 - V2C2) V3c2 {8112 ac J2 = — O_-- O 5C—- + ——. O + O BC 3x “ 3 1 ,, n 2 3 l 1 C ”2 2 (36) _ clc2 y;- _ :1] aul + c_2 (1 - V2C2) + v3c2 3.2 ac n O3 O1 802 c n O2 O3 3C2 C 8x 1 l The chemical potential of the ith constituent can be written as O Hi - “i + RT ln a1 (37) O . . . where pi is a function of T and P, R is the gas constant, and a1 the activity. Therefore, at constant T and P Bu Bln a BC BC 1 C l C 2 2 8p aln a __1 = RT 1 8C BC 2 C 2 C l l (38) an aln a ___2 = RT __..__2. 3C 3C 1 C l C 2 2 8p aln a 3C BC 2 C 2 C l l Equating coefficients Of the corresponding equations 26 and 36 and sub— stituting equations 38 yields the desired eXpressions for the diffusion coefficients, D. 13 20 RT (1 — VlCl) V3Cl Bln al V3 V2 aln a2 D11 = "' C1 0 + 0 ac + C1C2'_" "5_ ac ” 1 3 1 C 3 2 1 C 2 2 RT (1 VlCl) V3C1 Bln al V3 V2 Bln a2 D12 = —3' Cl 0 + 0 ac + C1C2 E—"'E— ac 1 3 2 c 3 2 2 c 1 1 (39) RT V3 V1 a1n al (1 — VZCZ) V302 {Bln a2 D21 = ‘3' C1C2 E—" 3" ac + C2 0 + 0 ac 3 l 1 c 2 3 l 1 c 2 2 RT 3 1 a1n a1] (1 — V2C2) v3c2 {Bln a2 D22 = —F' Clcz 3—" 3" ac J + C2 0 + 0 ac 3 1 2 2 3 l 2 C1 C1 N Bu. 2 c ———- = 0 (40) . 1 8x i=1 therefore auN N-l Ci Sui 73;; = ‘ X 6—7; (41> i=1 N The constant volume constraint is given by equation 23 N — E J,V, = 0 (23) . 1 1 i=1 Thus, there are only N - 1 independent fluxes and N — 1 independent chemical potential gradients. Performing the same algebraic Operations as was done for the ternary case gives the generalized flux equations 21 c, (1 - V c.) c. an. N—l V‘ VI au. J = — _l.[ 1 1 + 1 N:l 1 - 1- c,c, ——-—-—l ..J. n n Oj 3x 1 (42) The same procedure of converting chemical potential gradients to concentration gradients has to be done so that the corresponding coefficients can be equated. From equation 22 it is obvious that one of the concentrations, CN’ is not independent. Therefore ui = f(Cl, C2, ..., CN—l) i = 1, ..., N (43) and an. N—l 8p. 8C, N—l 8C, __l.= V __l. __l.= E u ._Ll ax 5 ac, ax ,_ 1j ax J-1 J c J-1 k (44) 1 = l, 000, N-l Ck - Cl 0°Cj_lC'+l°°°CN_l Sui where u,, = ——— 13 3C. 3 c k Substituting equation 44 into equation 42, collecting the coefficients Of the BCj/Bx, and equating these with those in equation 24 gives the desired generalized expression for Dij D = —l- + n O C, (l — ViCi) CiVN p N—l Cick XE. XE ij i ON ij (45) 22 Phenomenological Coefficient Expressions - It has been found in almost all non—equilibrium situations where thermodynamic variables have meaning that the thermodynamic theory of irreversible processes ’ 3 can be applied successfully. Briefly this theory states that (l) the rate at which entropy is produced within a system undergoing dissipative processes can be written by Ts = g JiXi (48) where T is the absolute temperature, s is the rate of change Of entrOpy created within the system per unit volume; Ji is the generalized flow -2 —1 . . -2 such as matter (moles x cm. x sec. ),electric1ty (faradays x cm. x —l -2 —l . sec. ) or heat (cal. x cm. x sec. ); and X1 are the generalized thermodynamic forces such as chemical potential gradients, temperature gradients, and e.m.f's. (2) The flows Ji are related linearly to the forces Xi by J. = Z L. X. (49) j where Li' are phenomenological coefficients related to electrical resistances, diffusion coefficients, or heat conductivities, etc. (3) If the Ji and X1 are mutually independent, then the Lij satisfy the 2 following relations, 8 called the Onsager reciprocal relations, that is Lij = Lji (50) More details are given by Miller.33’34 It can be shown that the entrOpy production for ternary dif— . l3 . quion is 23 3p 8p 3p Ts=—J —1-J —2—J ——3 51 lax23x33x () However, only two of the fluxes and two of the chemical potential gradients are independent. This is evident because of equations 30 and 29. If these equations are used to elhninate J and 3u3/8x in equation 3 51, one Obtains Ts = JlYl + J2Y2 (52) where 2 C.Vg an. Y.=—Z <5..+—-1——l i=l,2 (53) i . ij - 3x J=l C3V3 and 6ij is the Kronecker delta. The linear relations according to irreversible thermodynamics are therefore J1 = L1111 + L1212 (54) J2 = L2111 + L2212 and because the Ji as well as the Yi are independent, the Onsager reciprocal relation L = L should be valid. This analysis was first 12 21 proposed by Miller33 who from this derived a sufficient condition for which the reciprocal relations could be experimentally checked. This is presented later. If the Yi expressions are substituted into equation 52 and the resulting equations rearranged, one gets an Buz 1 J1 ‘ "(111“ + 112*) '§§" (1118 + 1126) ‘3E' (56) 24 811 311 1 2 J2 ‘ ' (121“ + 122Y) ‘SE" (1218 + 122Y) “3;" (56) where C'V 0‘? a = 1 + 11:1" B ='“f:— C3V3 C3V3 (57) c V, c'V y=—-:— 6=l+ _ c3v3 03v3 Now by equating like coefficients Of the independent equations 31 and 56, four linear equations are Obtained which can be solved for L 11’ L12’ L21 and L22. These equations are given below L a + L y = El.[£3;:;XlElZ-+ 323;] ll 12 n O1 O3 _ C1C2 V3 V2 L11B + 112‘S '6’ "E— n 3 2 (58) LQ+LY=EILC_Q:3__YA] 21 22 n O3 O1 c (1 - c ) V’c _._g 2 2 3 2 L218 + L226 — n [————;;———-4————e] 03 Since we are interested in checking L12 and L21, only these phenomeno- logical cross coefficients are Obtained. These are cl [(1 — V1C1) V3C1] clc2 V3 v2] n O3 02 O O L = 1 3 (59) yB - a6 25 L = (59) From the definitions Of a, B, y, and 6 and equation 34, 1 Y8 - 05 = - —*:7' (60) C3V3 Using equations 34, 57, and 60 in equations 59 and rearranging the resulting expressions gives —- —- —- - -2 L - L _ - C1C2V1(l — VlCl) C1C2V2(l - V202) + C1C2C3V3 12 ‘ 21 ' O ’ O (61) 1D 2T] 0 3T1 Thus, the hydrodynamic theory predicts that L12 = L21 for non- associating ternary systems. It should be emphasized that in the development presented here the condition Of constant molar volumes was imposed on the system. HOpefully this condition, although appearing necessary now, may be removed in some future work. An encouraging point Of interest here is that no activity data are required in order to cal— culate the phenomenological coefficients. Miller33’34 has Obtained expressions for the ternary phenomeno- logical coefficients, Lij’ in terms of the ternary diffusion coeffi— cients, Dij' The method involves using equations 35 to Obtain equations 56 in terms Of concentration gradients. The resulting equation is then compared to equation 26 giving eXpressions for the Dij in terms Of the Lij' These equations are then solved for the Lij in terms Of the Dij' These are 26 aD12 - ch1 L12 = ad — be (62) dD21 — b1)22 L21 = ad - bc where c V’ an O V' an a- 1+ 1_1][_1_ ...}; __2_ v c3v3 acl c c3 3 acl c 2 2 (63) c V’ an c V" an b = ~1:3- ——1- + 1 + —3:?- -—3- c3v3 acl c3v3 ac1 C2 C2 and c and d are the same respectively as a and b except that (8/3Cl)C 2 is replaced by (a/acz)c . l Substitution of the expressions derived for the diffusion coefficients in terms of the friction coefficients, Oi, (equations 39) into equations 62 yields the same expressions obtained for the Lij in tenms of friction coefficients (equation 61). This should follow since the same principles are used. However, it does serve as a check on the derivations performed in both methods. The usefulness of equation 62 is apparent since it provides a check on the hydrodynamic approach and on Onsager's reciprocal relations. The diffusion coefficients can be Obtained experimentally by the methods 22’23 and Burchard and Toor.3 From them, the of Fujita and Gosting phenomenological coefficients can be determined. Also, from equations 39 and equation 61, the diffusion coefficients and phenomenological 27 coefficients can be Obtained using experflnentally determined friction coefficients. Hence, by comparing the diffusion coefficients and by comparing the phenomenological coefficients, the hydrodynamic approach can be checked. Also from equations 62, the condition from which the Onsager reciprocal relations can be verified is Obtained. Simply by equating L12 and L21, one obtains aD12 + bD22 = CD11 + dD21 ; ad - bc # O (64) This relation has been the basis up to this time for experimentally checking the Onsager reciprocal relations. For the case of N components, expressions for the phenomeno- logical coefficients can be Obtained in a similar procedure to that used for the ternary. Extending the analysis to a system Of N components and using equations 23 and 41 one Obtains N—1 J, = Z L ,Y, i = 1, ..., N-l (65) i i=1 1] j where N—l Vflc an Y, = — 2 a, +-:9—3: ——3- (66) 3 k=l 3k v c 3X N N Substituting equation 66 into equation 65 and collecting the coefficients Of the (Buk/ax)'s, yields N-l N—l V C an Ji = E Z Li, 6'k +_':1—k 71:; i = l, ., N-l (67) k=l j=l J 3 v c X 28 2 Now, equating coefficients of equations 42 and 67 gives (N — 1) linear equations with (N - l)2 unknowns, the Lij' These equations are N-l V,c c cc V V 2 L,,T+—‘T+ .x‘ ‘l_V12C)—3_— Oln x Ozn x O3n Olzn x __ __ (76) _ VlClCB Bu1 C2 —- an2 V3C3CB 3“3 C12, —- 8“12 J - -———-—-—-—-— -——(l — V C )-—-+ - (1 — V C )-—-- B Oln 8x Ozn 2 B 3x O3n 8x Olzn 12 B 8x If the associating reaction can be described by an equilibrium constant involving activities, one gets: a K = 12 (77) a132 This combined with equation 37 can be used to Obtain a relation between the gradients Of the chemical potential. This relation is 8“12 = all1 8“2 8x 8x + 3x (78) Substituting this result into equations 76 gives C __ C __ Bu J =— 0—1(1-vch)+———12(1-v c)]—1 1n 012” (79) _ C12 —- V20ch 8“2 V3C3CA 8“3 12A on On 0 n(1'V C)- 12 32 J = B o n 12CB) ' _.[ €12 Vlcch] aul 12 O (1 -‘V In 8x (79) (l - V C ) +---——(l — v c ) ———- Ozn 2 B 012” 12 B 8x O3n 3x From the Gibbs-Duhem relation we can Obtain an 3p 3p Bu 1 2 3 12 C1 ax + C2 ax + C3 ax + C12 ax = O (80) Using equations 74 and 78, this becomes 3p 3p 8n 1 2 3 CA ax + CB ax + CO ax ‘ O (81) With this relation, 3u3/8x can be eliminated from equations 79 to give 2 c c V'c an - l -' l2 - 3 A l J = - ———(1 — v c ) +--———(1 — v c ) + ]-——— A [Oln 1 A Olzn 12 A O3n ax [L 2C2CA C12, —- V3CACB 8U2 - O + O ‘1 - V12CA) + O 8 L 2n 12” 3n x (82) VlClCB C12, —- V3CACB 3“1 J = — - O + O \l — V12CB) + O 3 B 1n 12” 3 x c c Vc2 an 2 -' 12 -' 3 B 2 - ———(1 - v c ) + (1 - v c ) + ]-——— [Ozn 2 B Olzn 12 B O3n 3x It is desirable at this point to Obtain expressions for the Dij in terms Of the friction coefficients. Therefore, the chemical potential gradients Of the actual species must be converted to gradients Of the 33 stoichiometric concentrations in order to equate corresponding coefficients. From equation 22, we have Clvl + CZVZ + C3V3 + C12V12 = 1 (83) Therefore at constant T and P, (84) U2 = f(cl, C29 C12) However, from the stoichiometric relations, equations 74, equations 84 can be written (85) Hence Bul = aul BCA + aul BCB 8x 8C 8x 8C 8x A C B C B A (86) Buz = apz SCA + Buz 8GB 8x 8C 3x 3C 3x A C B C B A For this system, equation 24 is J = .1. fl - D Be A AA 3x AB 8x (87) 8C BC J - -D A D -——- 34 Substituting equations 86 into equations 82 and equating corresponding coefficients with equations 87 gives the desired results. 2 C C C 3n _ 1 _ - 12 _ - 3 A l DAA — l:— (1 V CA) + O n (1 VlZCA) + -———] [—J C Oln l 12 O3n BCA B V'c c c V'c c an 2 2 A 12 —- 3 A B 2 c 2n 12n A O3n 3CA B c c V'c2 an _ 1 _ —- 12 _ —- 3 A 1 DAB [—0 n (1 VlCA) + O n (1 VlZCA) + O n] ——3C 1 1 12 3 B c A Vzcch C12 V3CACB 3“2 + — O n + O n (l - V12CA) + O BC 2 12 3n B C A (88) V1C1CB C12 V3CACB an1 DBA = ' o + o (1 V12CB) + 6 ac 1n 12 3n A c B c c V'c2 an 2 —- 12 —- 3 B 2 4- -——— (1 - V c ) 4—-———— (1 - v c ) +-——-—-] -——— [Ozn 2 B Olzn 12 B 03n 30A 0 B _ VICICB C12 —- V3CACB an1 DBB ' " + (1 ' VIZCB) 1'—_—___' '7?- Oln Olzn U3” 8 B C A C C V02 3 _. __ u + [——3-(1 - vch) + -—lZ-(1 - vlch) + 3 B ] 352' O2n O12n G3n B C A 35 It may appear that a considerable problem has been introduced here because Of expressions like (Bul/BCA)CB. This would appear to involve Obtaining activity data Of the actual species as a function Of the stoichiometric quantities which could be a considerable problem. However, it has been shown by Nikol'skii36 in general and by Prigogine and Defay40 for a two—component mixture with association that the chemi- cal potentials Of a component and the monomer Of that component are equal. They Observed that this result depends in no way on any assump— tion about the manner in which association occurs. Furthermore, it is valid for both associated solutions and associated gases and only . depends upon the assumption that the complexes are in thermodynamic equilibrium with each other. Nikol'skii further stated that it is valid for any equilibrium gaseous or liquid mixtures with any reactions occurring in them. It follows from this that “1 “A (89) “2 = uB It should be recalled that from the Gibbs—Duhem equation one can Obtain BuA auB Bu CA ax + CB ax + CO ax = 0 (90) In order for both equations 81 and 90 to be valid, the chemical potential Of the species must equal the chemical potential of the component. Activity Expressions Based on the Chemical Model - If activity data are not available we are still faced with the problem Of finding values for expressions Of the type (Bui/BCj)C . For associating i 36 systems, Dolezalekls’l6 first proposed the chemical model which states that non-ideality in associated systems results purely from the com— plexing to form other species. In other words, the actual species present in the solution form an ideal solution. This Of course is only an approximation but a reasonable one if the non-ideality is mostly caused by the association. For the system under consideration here, the monomers l, 2, and 3 and the dimer 12 should therefore form an ideal solution according to the model. Furthermore, if it is assumed that the associated species does not exist in the vapor and the vapor is an ideal gas, then Raoult's law for species 1 says that p:L = XlPI = aAPA (91) where pl is the partial pressure, P the vapor pressure Of the pure 1 species, and aA the activity. Since P1 and PA are identical, aA must equal X1, the mole fraction of monomer of A. This analysis can be extended to the other species, therefore the following set of relations relating stoichiometric and true quantities must hold (92) 312 = X12 By Obtaining expressions for the true mole fractions in tenns Of the stoichiometric quantities, values for (aul/acA)C = (apA/BCA)C etc., B B can be Obtained. Details Of this are given in Appendix III. 37 Since pure A and pure B have no associated species present and are assumed to exist only in the monomer form, reasonable values for Vi and Vé are the respective partial molar volumes Of the pure components A and B. Phenomenological Coefficient Expressions — It is of interest to check on the Onsager reciprocal relations for this sfinplest associative case. The rate Of entropy production is given by 11 = ’11 3§%" J2 3§%" J3 3%3" 12 8::2 (93) Utilizing equations 78 and 81, equation 93 becomes Ts=—(Jl+J12-§‘:J3)%-(J2+J12-:—:J3)% (94) For this system, equation 23 is JiVi + JiVé + JéVé + 112112 = 0 (95) Eliminating J from equation 94 with equation 95 and rearranging gives 3 a " a " a a " a '— . _ “1 CAVl “1 CBVl “2 “2 CBVZ “2 CAV2 a“1 Ts - —J -——-+- + - J ———-+--———--—-+--———--——- 1 3x C VI 3x C V. 3x 2 8x C V1 8x C V' 8x c 3 c 3 c 3 c 3 “1 CAVlZ “1 8“2 CBV12 3“2 - J ——— + + + 3 ax c V' ax ax c V' ax c 3 c 3 From stoichiometry J1 = JA " J12 (97) J = J - J 2 B 12 38 Substituting equations 97 into equation 96 and rearranging yields c V an O V an c V an c V an T6 = —J 1 + A 1 __l.+._§_l.__2 _ J 1 + B 21 2 + A 2 1 A C V 3x C V 3X B C V 3X C V 3X c 3 c 3 c 3 c 3 (98) c an 0 an A - - - 1 B —- —- —- 2 - J12[C V' ('Vi'V2+ 12) ax + C $— ("V1'V2+V12) ax ] C 3 c 3 Now a reasonable assumption in accordance with the previous assumption Of constant volume is that 612 = Vi + Vi. If this is the case, equation 98 becomes c V' an c V' an O V' an O V' an Té=-J 14.—Ll __];+__B_l__2__J #44. 14.1.24 (99) A C V. 8x C V. 3x B C V. 3x C V. 3X C 3 C 3 C 3 C 3 From the postulate of irreversible thermodynamics, the following equations are valid. CAVl 3“1 CBV1 3“2 CAV2 a“1 CBV2 3“2 J =—L l+———————+——-——-L —-——+ 1+ — A AA c V' ax ‘ c V' ax AB c V' ax c V" ax C 3 C 3 C 3 C 3 (100) CAVl Bpl CBVl auz CAVZ aul CBV2 an2 J = —L 1 + ————- ———-+-————-——— - L ————-———-+ 1 + -——— B BA CV 3x C.V- SX BB C.V- BX CV X c 3 C 3 c 3 c 3 and LAB = LBA (101) Carrying out the multiplications in equations 100 and equating corres- ponding coefficients with those in equations 82 and then solving the resulting equations for LAB and LBA gives 39 2 _ c c v c c v ——-l(1-Vch)+ 12(1—V120A)+ 3A]—B_l O1n O12 03n 0 v L = C 3 AB _ _1__ CCV3 V'c c c v c c c V’ [E 2 2 A + 12 (1 _ VlZCA) + 3 A B] 1 + A;l O2n O12n °3” c v _ c 3 _ ._l__ CCV3 (102) Vac C(1-V C) Vcc CV [_ lolnB + 12 O n12 B + 30AnB] 1 + B 2 L _ 1 12 3 ch3 BA 1 CCV3 c c V'02 C'V [-—Z Using equations 104, 105, 108 and 109, the flow equations become c 20 czV an 1 —- 11 —- A 3 1 J=-—1-CV + 2-cv +———-— A [oln ( A 1) olln ( A 11) o3n:] 8x _ [_ CBCAVZ + CACBV3]an 2 ozn 03n 8x (110) J = — [— ClCBVl — 2C11CBVll + CBCAV3] Sul B oln 011” o3n 8x 2 8 -[—C—B—(l-CBV V2V)+—3—CB] “2 o n o n " 2 3 3" For this system, equation 22 is CV +cV +cV +CV =1 (111) 1 1 B 2 C 3 ll 11 and this combined with the stoichiometry relations (equations 106) enables us to write 42 f(C C at constant T and P. Since equations 86 and 87 also apply to this system, (112) equation 86 can be substituted into equation 110 and the coefficients compared to equation 87. This gives - 2 C 20 V C Bu 1 —- ll - 3 A 1 D =—(l—CV)+——(2—CV)+ ]— AA [oln A 1 011” A ll o3n BCA C B — " a +[ CACBVZ + CACBV3] “2 2 03n BCA c B Cl __ 2Cll __ V30: Bul DAB=[on(l—CAV1)+On<2_CAV11)+0n BC 1 11 3 B C A + [- CACBVZ + CACBV3] .3112 Ozn 03n . 3GB c A [ chBVl zcllCBVll CACBV3] [Bull D = — ._ + ____..__ BA n c (113) 43 D = [_ C1C13Vl _ 2C11CBV11 + CACBV3] 8“1 BB oln olln o3n aCB C A (113) 2... C C V Bu -+ [-—4§ (1 C v2) + B 3 ] '55g 2 03n B C A Activities Based on the Chemical Model - If no dhner is assumed to exist in the vapor, then pA = pl. If in addition the four species form an ideal solution, then the following relations must hold 3B = az = X2 = YBXB (114) aC = a3 = X3 = YCXC a11 = X11 It also follows that an Bu 1 _ A 8x - 3x (115) Also the fact that B and C do not associate means that 112 = LIB, 113 = 11C (116) v2 = VB, v3 = vC (117) G2 = CB’ 03 = 0c (118) Phenomenological Coefficient Expressions — The rate of entropy production is given by 44 Bu 8p 8p 3p 1 2 3 11 T8 J1 ax J2 ax ' J3 ax J11 3x (119) If it is assumed that 11 = 2Vl (120) then using equations 108 and 109 and the relation le1 + sz2 + J3V3 + JllVll = O (121) equation 119 reduces to C V' an C V' an Té=_J l+..é_];_l+__B__l__2 A - 3x -' 8x CCV3 CCV3 (122) CAV2 Bul CBV2 BUZ — J ----——-+ l +-——- = — J Y — J Y B C V? 3x C V: 3x A A B B C 3 C 3 From irreversible thermodynamics, JA = LAAYA + LABYB (123) B LBAYA + LBBYB J Now, carrying out the multiplications indicated in equation 123 when the substitutions are made for the Y's and equating coefficients with equations 110, LAB and LBA can be obtained. They are C C V C C v _ 1 B 1 —- A B _ -— LAB ' LBA ‘ ' Cln (l CAvl) 0B (1 CBVB) __ __ (124) 2C C v C C v 11 B 1 —- A B C —- -— ' olln (2 CAV11) + oCn (l ' CBVB ’ CAVl) 45 Thus if the assumption that V11 = ZVi is true, then the hydrodynamic model for systems in which one component dimerizes predicts that Onsager's reciprocal relations should hold. EXPERIMENTAL Apparatus The experimental diffusion coefficients for both binary and ternary systems were obtained with an optical diffusiometer. Diffusion took place in a glass—windowed cell immersed in a constant temperature bath and was followed by measuring the refractive index of the solution with a Mach-Zehnder interferometer.51 Two solutions of slightly different concentrations were carefully flowed one on top of the other into the cell and free diffusion allowed to take place. The concentration was taken as the average of the two solutions. This set— up was similar to the diffusiometer described by Caldwell, Hall, and Babb8 and'is described in detail by Bidlack.4 A diagram and a photograph of the interferaneter system are shown in Figures 1 and 2. The canponents were supported by ordinary laboratory bench carriages located along a continuous rail composed of three optical benches. These in turn were bolted to an I—beam mounted on a concrete block to dampen outside disturbances and vibrations. Monochromatic light from a Cenco quartz mercury arc lamp source, filtered to isolate the 5461 2 green mercury line, was col— limated and then split in amplitude by a half-silvered mirror (mirror 1). Half of the beam was refleCted to a full reflecting mirror (mirror 2) and the other half passed through to a full reflecting 46 47 .muouuwa mo coauflmom wafizoam Hmumaoummumuafl mo Emuwmww owumECSUm .H madman mama maflumeflaaoo H Houufla m Houufie Condom O uafiom mama mumfimo N nouufis q uouuwa madam mwmafl 48 .msumummm< mo ammuwouoam K9522 .mm......_._am .N ouswwm .A. , meme;— Kuhtdnm 49 mirror (mirror 3)° The two beams were then combined at a half- silvered mirror (mirror 4). Constructive interference of the two beams occurred when the path lengths 1—2—4 and 1-3—4 were equal or differed by a whole multiple of the wavelength of the incident light. The mirrors were so adjusted as to give straight, vertical, parallel fringes. The interference beam was arranged so that it could be photographed directly by a camera. The camera consisted of a 3 foot long aluminum tube of 3% inches diameter containing a lens with a 343 mm. focal length set in the end towards the interferometer. The lens was focused on a type M, 3% x 4% inch Kodak plate located at the opposite end. A lever mechanism on the plate holder enabled fourteen successive exposures to be taken per plate. The magnification factor of the camera was found to be 1.923.4 The diffusion cell was fixed in a water bath maintained at 25 i 0.030C by a thermoregulator. The water bath consisted of an 18 x 18 x 18 inch stainless steel tank covered with 3/4 inch plywood and rested on the cement block without touching the interferometer. Two 3% inch diameter optical flat windows were clamped and sealed into the ends of the water bath and aligned to allow passage of the light beams through the bath and the cell windows. Distilled water was preferred over tap water since it did not cloud up as fast. In Figures 3 and 4 are shown a photograph and a diagram of the diffusion cell. The main body of the cell consisted of a B x 3% inch slot cut into a stainless steel plate with two Optically flat Figure 3. Photograph of the diffusion cell. 51 glass solution reservoirs made from 50 cc. syringes filling - A. — - B syringe [- P valve 2 valve 1 cell cell window -""" body ‘- L V q- vflye4 boundary .l/é //2 VMNeB sharpening slits _ siphon g valve 5 Figure 4. Diagram of diffusion cell. 52 windows clamped over the slot to form a sealed channel. The channel was situated to allow both light beams to pass through it; thus, a vertical concentration gradient in the solution across one of the beams resulted in a fringe displacement pattern that 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 to enable the study of most corrosive liquids. A framework was bolted to the cement block and positioned above the bath so that the cell could be hung from the top and immersed in the bath. Two small position pins were placed on the framework to insure that the cell was always placed in the same position. The cell was provided with two inlets, one in the top and one in the bottom, and two outlets directly across from each other about one-third the way up the channel sides. Two solutions of slightly different concentrations were then slowly flowed simul— taneously into the cell, the denser solution through the bottom inlet and the other through the top, and out the two outlets. A sharp boundary was thus formed between the two layered solutions. This boundary was located in the center of the lower beam. All the valves were then closed and the solution allowed to diffuse freely. Procedure for Experimental Run 1) The light source and water bath heater were first turned on. 2) The cell was then placed in a rack away from the rest of the apparatus for convenience in filling. 3) 4) 5) 6) 7) 53 All the cell valves except valve 2 were then closed and approxi- mately 25 cc. of the.denser solution were placed in reservoir B. Some of this solution was then allowed to flow into the cell through valve 5 until the liquid level was % to % inches above the outlets. Valve 5 was then closed. Valve 4 was next opened slightly and liquid was forced into the exit line by means of the filling syringe plunger until the liquid level in the cell was just above the outlets. Valve 4 was then closed and more solution from the reservoir was passed through valve 5 into the cell as in step 3. More liquid was forced into the exit line through valve 4 and the whole procedure repeated until liquid dripped from the outlet line. This was done to insure that liquid had filled the exit line as far as the tee. Valve 4 was then closed. The same procedure of adding liquid to the cell through valve 5 was repeated and exit valve 3 Opened. The filling syringe was then used to force liquid into the exit line until the cell liquid level was just above the outlet. At this point, valve 3 was closed° Step 5 was repeated until the liquid flowed freely from the exit line by means of a siphon. A11 valves were then closed except valve 1 and 25 cc. of the less dense solution was placed in the filling syringe. Valve 2 was then slowly Opened so as to allow the solution to trickle down the side of the cell channel and layer on top of the more dense solution. The solution was allowed to flow this way with valve 54 2 being Opened more and more as the solution built up in the cell. After the solution had overflowed up into reservoir A, both valves 1 and 2 were closed. 8) The two reservoirs were then filled to approximately equal liquid levels with the appropriate solutions, remembering to always place the more dense in reservoir B. 9) At this point the cell was placed in position in the water bath. The reservoir valves, valves 1 and 5, were then opened one full turn followed slowly by valve 3 until the rate of flow from the exit line was one drOp every 8 seconds. The opposite outlet valve, valve 4, was then slowly Opened until the combined exit flow rate was one drop every 4 seconds. It was important to maintain balanced flow rates into both halves of the cell as well as through both outlets. 10) When the boundary had formed satisfactorily, valves 3 and 4 were closed followed as soon as possible by valves 1 and 5. The timer was then started and pictures taken at various times during the run. A series of exposures taken for one run are shown in Figure 5. Purity of Materials The chemicals hexane, dodecane, hexadecane, and carbon tetrachloride were Obtained in the purest forms available from Matheson, Coleman, and Bell, Co. The hexane and carbon tetrachloride were spectroquality and the dodecane and hexadecane were 99+% “Blefin free) quality. The chemicals diethyl ether and chloroform ,1 . AéEE§§§§§§§§ ;:_:—-———-—-/ ' r r /”.—_—d - ~%/¢/fiwafi====fi ,,fl===EE:r~T Typical set of photographs taken during diffusion run. Figure 5. 56 were purchased from the Malinkrodt Chemical Co. and were analytical reagent grade. The chloroform contained a slight impurity of ethanol used as a preservative. This affected the density slightly but showed no noticeable change in the refractive index. The measured values of density and refractive index were’compared with those listed in the literature and were found satisfactory. These values are given in Table l. Calculations Calculation of Concentrations and Viscosities In determining concentrations, various predetermined amounts of each component were weighed out together, and the density of the resulting solutiontneasured with a pycnometer. From the known amounts and the density of each solution, the concentrations were calculated. Volumes calculated on the basis of densities of the pure components agreed within 0.15% of the measured volumes but were lower than the measured values in all cases. The very small differences were an indication of the constancy of the molar volumes. For systems in which the diffusion coefficients had been experimentally determined by others, concentrations were calculated using the given mole fractions according to the following equation X. C = —-————1 i = 1,2,3 (125) XV. J J IMO) j- Viscosities Of the solutions were experflnentally determined using a Cannon-Fenske type viscometer. Kinetic energy effects were taken into account by using an equation of the form Table l. 57 previous recorded data. Chemical Hexane Dodecane Hexadecane Carbon Tetrachloride Chloroform Diethyl Ether Density at 25 0C This Work 0.6550 0.7450 0.7698 1.5850 1.4740 0.7075 d (g/CC) Reference 42 0.6549 0.7451 0.7699 1.5845 1.4795 0.7077 Comparison of physical constants with This Work 1.3727 1.4196 1.4324 1.4570 1.4426 1.3500 “1) Refractive index at 25 OC Reference 42 1.3723 1.4195 1.4325 1.4576 1.4422 1.3499 58 ”1:“-.. (126) p 2 where K1 and K2 are experimental constants determined using liquids of known viscosity and t is the time. The Viscosities for the systems reported by other sources when not given were calculated according to the following relationship 3 1n n = Z x, 1n n, (127) i=1 1 1 where Xi is the mole fraction of i and ”i is the viscosity Of pure i. Reduced Second Moment The fringe pattern obtained by the diffusiometer for each exposure may be considered as a plot of the refractive index versus distance in the cell because the displacement of the fringes is pro— portional to the refractive index difference. It is also true that [__. x n - n = kl(jx - jx ) (128) 2 X1 2 1 y where nx is the refractive index i at xi and jx is the number of i J fringes up to the point x,. 1 fringes From equation 128, k1 = An/J where J is the total number of l A\ fringes and An is the total refractive index difference across the boundary. It is desired to obtain a plot Of 59 (an/ax)t versus x for each eXposure, since it is the second moment of the refractive index gradient curve that is required. If we look at the center regions of the exposure (near x = 0), it is noticed that any curvature in the fringes is small. From equation 128, it follows that the refractive index difference across one fringe is k Hence in the center regions of an exposure a 1. reasonable value for the refractive index gradient can be obtained by measuring the distance between two fringes. Thus k (‘33) 2 35—47;.— (129) 3 t x. +x. j+1 j x = .311_1 2 where xj and xj+1 are the distances to fringe j and j+1 respectively and (Sn/8x)t is the refractive index gradient at x = (xj+l + xj)/2. In the outer regions of the exposure (x >> O, x << 0), the curvature of the fringes is more pronounced and therefore equation 129 is not a good approximation. In these regions, tan— gents to the fringes give more accurate values of the refractive index gradient. The tangent at these points gives ay/ax not an/ax, therefore a relation between By and an is necessary. This can be Obtained by measuring the distance in the y direction between a specific number Of fringes. This provides a prOportionality constant, k2 y ' y 2 1 k2 = (130) 43' where Aj is the number of fringes between y2 and yl. Using equation 128,equation 130 becomes 60 An = -—-Ay (131) 1%). where tan 0.x is the measured tangent of any fringe at point xi on i therefore k1 x, k2 1 tan 0 (132) X. 1 the exposure and where O is the angle measured between the tangent line and the x axis. The values x and y which are measured Off an exposure are not the actual distances but rather magnified distances. The camera which records the fringe pattern at various time intervals has a magnification factor, M = 1.923, hence x = xm/M where xIn is the measured distance and x is the actual distance. On this basis, equations 129 and 132 are Mk 9-3 = M ——3“ = 1 (133) EX t x + x 3xm x’+l m — X‘ m -j+l,m j,m t J 9 J9 2M an — 22;— - ——l-tan 01 (134) 3x 3x k I t x m 2 x _B t .31 M M The definition of the rth moment is given by +m - .1 r ‘22 mr — An {00 x ax t dx (135) where An is the total refractive index difference across the initially sharp boundary. In terms of measured quantities this definition becomes 1 +.. a m = f x (41—) dxm (136) 61 From the expressions for the refractive index gradients used in the numerical integrations, equations 133 and 134, and the defini- _ th . . . tion of the r moment, it can be seen that the proportionality constant kl cancels. Thus in evaluating the moments, the constant k1 is not required. It is Obvious then that any value chosen for kl will not affect the results. If k1 is chosen as unity in evaluating the refractive index gradients, then ma—n +°°a ax dx=f (41—) dxm=An=k1J=J (137) 3 t -m xm t J can easily be evaluated from each exposure within 10.2 fringes. Therefore, since it can also be calculated numerically from equation 137, a check on the numerical calculations is provided. For the case of k1 = 1, calculation Of the value of xm at the initially sharp boundary can be Obtained from +00 _ 1_ an x — J I xm (-——) dxm (138) Notice that the choice of the origin of x used in determining the measured x values will give a different value of x m' It is shown 9 in Appendix II that if xC m is taken as the origin, then m1 = 0. 9 This Of course follows from equation 138. Since the moments and the diffusion equations are derived on the basis of xC = 0, the values of x used in the determination of 1112 must be equal to (xm — x m)/M C: where xm is the measured value of x based on any starting point, xC m 9 62 is the value of xm at the boundary based on the same starting point, and M is the magnification factor. The expression for 1112 is therefore +00 _ l _ 2 8n m2 _ 2 1 (Km Xc,m) (3x ) dxh (139) M J —w m t The reduced second moment D2m is defined by m 2 2m 2t In actuality a perfectly sharp boundary will not be formed so that t will not equal the measured time, tm. Rather t = t + At (141) m corr. Hence equation 140 should be In2 D2m = 2(t + At ) (142) m C0rr. Rearrangement of equation 142 gives = D t + (143) “12 'r— D At 2 2m m 2m corr. Therefore, a plot of m2/2 versus tm should give a straight line of slope, D2m’ and intercept DZmAtcorr. The slope gives the de31red value of the reduced second moment, and the intercept provides the time correction, Atcorr From the latter, the absolute time of diffusion can be determined using equation 141. Plots of the second moment versus the measured time for the systems studied are given in Figures 12, 13, and 14 of Appendix VI. 63 It can be seen that they are linear in accordance with equation 143. Tabulations of the time corrections, zuéorr , obtained from these curves are given in Table 15 also in Appendix VI. For convenience the second moments at the various measured times of each run are also provided in Table 14 of Appendix VI. Reduced Height—Area Ratio The reduced height-area ratio is defined by 2 0A 5 (An) (144) an 2 4nt[ 31:1] t max In terms of measured quantities this becomes 2 (k J) D = 1 (145) A 2 an 2 4TrtM l: (—-——) ] 3x 1“ t Again it can be seen from the expressions for the measured refractive index gradients (equations 133 and 134) that kl cancels and hence k1 can be taken as unity in the calculations. Since there are no minima and only one maximum in the measured refractive index gradient curve, DA can be obtained by simply finding the maximum of the measured refractive index gradient curve and substituting this value into equation 145. This maximum which can be obtained either numerically or graphically, decreases as the absolute time, t, increases. Differential Refractive Index Constants 3,19,22,23 It was assumed by Costing, et. a1 and this author that the total refractive index change across the boundary, An, 64 could be expressed by the equation An = RlACl + RZAC2 (146) where Ci represents the concentration in moles/liter and Ri is the differential refractive index increment of component i. Division of equation 146 by ACi, gives An _ 2 AC1 — R1 + R2 BC: (147) Thus, values of R1 and R2 could be obtained from the intercept and slope of a plot of An/ACl versus ACZ/ACl. An can be determined either by direct refractive index measurements or from equation 128 An = R J (128) Based on several runs performed in this laboratory, k1 was found to be 2.10 x 10.5 refractive index units/fringe. To Obtain this value larger concentration differences, and thus larger An values, were used so as to minimize errors in refractive index measurements. The J values, total fringes, could be obtained directly from the exposures. It should be pointed out that in the determination of the ternary coefficients, only the ratios of R and R are used. There— 1 2 fore a value of k1 is not needed since it cancels in the ratios. The I I following equation can therefore be used to calculate an R1 and R2 from which the correct ratios can be determined. 65 AC AJ ' ' 2 — = R + R — (148) ACl 1 2 A01 where v R 1 R l 2 R =— , R =— (149) 1 k1 2 k1 and l R R _} = E1 (150) R2 2 The latter method is considered more accurate because Of the limita— tions in the refractive index measurements. With small initial concentrations difference chosen to minflnize errors resulting from variation of Dij’ VA, and Ri with concentration, the An values are naturally quite small. In fact, they were in the neighborhood of 0.0005 to 0.0010 indicating that small errors in a refractive index measurement could cause large percentage errors in An. The J value, being a strong indicator of refractive index difference, was considered a more accurate measure of An. Diffusion Coefficients The method used in this laboratory of experimentally determining the diffusion coefficients D,, of a ternary system was 13 very similar to the method outlined by Fujita and Gosting.22 In both methods a refractive index gradient curve at various thmes during free diffusion is obtained. Using these curves, reduced second moments and reduced height—area ratios are calculated. Plots of the reduced second moments and of the reciprocal square root of 66 the reduced height—area ratio are then made against the refractive index fraction of one of the components, al- This refractive index fraction is defined by ‘11 = RlACl: R ACRl-lA-CR AC (151) An 1 l 2 2 The slopes and intercepts at a1 = 0 and a1 = l of these plots are then used to calculate the ternary diffusion coefficients. The linear equations of the reduced second moment, D2m’ and of the reciprocal square root of the reduced height-area ratio, l//§;. with a are 1 D2m = 12m + SZmal (152) 1 -—=I+S /5_ A Aal (153) A where 82m and SA are the slopes and 12m and IA are the intercepts at a1 = 0. The intercepts at a1 = l, L2m slopes and intercepts by the following relations and LA, are related to these L = I + S (154) L = I + S (155) The ternary diffusion coefficients in terms of these slopes and inter— cepts are L I S 2m A 2m IDijl + Lzm/lDijl + ———SA Dll = — (156) S 2m 67 I2 LASZ |D_,|+I /|D,,| +————-—m 1“ 13 2m 13 SA D22 = s (157) 2m Rz D12 = EZ-(Izm — D22) (158) R1 D21 = R—2 (LZm - D11) (159) where lDijl’ the determinant Of the ternary diffusion coefficients, is given by 1Dij| = D111122 ' D121121 (160) The lDijl is determined from the cubic equation 2 y— 3 S2m 2 s2m (I-Dij|) + IZm-IAq (VIDij|) — q =0 (161) The development of these equations and the solution to the describing equations for diffusion are given in Appendix II. A computer program to solve for the roots of equation 161 is given in Appendix V. Friction Coefficients Values of the friction coefficients, Oi, for the systems dodecane—hexadecane—hexane, toluene—chlorObenzene—bromobenzene, and acetone—benzene—carbon tetrachloride were calculated from mutual diffusion of the binaries at infinite dilution and from self diffusion data Of each component. Values of the friction coefficients for the system diethyl ether—chloroform—carbon tetrachloride were determined 68 by Wirth.47 For the hydrocarbon system values were determined more exactly using tracer techniques. The necessary diffusion data are provided in Tables 11 and 12. Since the parameter RT/oi always appeared together in the calculations, this value was determined for each component rather than Oi itself. The relationship used in this calculation was 0 ijnj 1,2,3 (162) 14' II 3 -—=X,D_n, + Z X,D l i i . J J j... 2 where Di is the self diffusion coefficient of canponent i in cm. /sec., Dij is the mutual diffusion coefficient of the i—j binary at infinite dilution of component i in cm.2/sec., ni is the viscosity of pure i 2 . . in dynes x sec./cm. , and X is the mole fraction of component i at i which the diffusion coefficients were measured. It should be pointed out that equation 162 is more applicable to non—associating systems since the friction coefficients correspond to actual species. With associating systems where there may be more than three species present, the friction coefficients Obtained do not correspond to actual species. Rather they are empirical factors for each stoichiometric component. They would therefore give less reliable values for the diffusion and phenomenological coefficients calculated from them. The parameter RT/oi for the hydrocarbon system was obtained from tracer techniques according to the equation -—— = D n (163) 69 where D* is the tracer diffusion coefficient, n is the viscosity of the solution, and O1 is the friction coefficient of the tagged species. The concentrations in the tracer runs were the same as those at which the ternary diffusion coefficients were determined. Details on the derivation of-equation 163 are given in Appendix I. This procedure could be applied to other non-associating systems if the components could be obtained in tagged form. Activity Data Dodecane (l) — Hexadecane (2) — Hexane (3) - Activity data at 200C for the hydrocarbon binaries hexane—dodecane and hexane- hexadecane were Obtained from Bronsted and Koefoed.5 Since they concluded that the system exhibited regular solution behavior, the activity data for the ternary system were obtained by a Van Laar fit.43 The equations used were 2 T In Y _ (03’313 + C2A23'312) 1 2 (C1A13 + c3 + C2A23) (164) --— 2 (ClAl3/B21 + C3'Bz3) )2 1A13 + C3 + C2123 T 1n 7 2 (C where Aij and Bij are constants obtained from the binaries and C1 is the concentration of 1. These constants are tabulated in Table 19 of Appendix VI. Further details on the derivation of required activity expressions are given in Appendix III. Diethyl Ether (A) — Chloroform (B) - Carbon Tetrachloride (C) - Activity data for this system was calculated on the basis of the 70 associated chemical model proposed by Wirth47 and Anderson. ’ According to their model, diethyl ether and chloroform form a onezone complex and this together with the uncomplexed monomers and the inert carbon tetrachloride species form an ideal solution. The vapor is assumed to be ideal with none of the dimer complex present. From this model the activities of the species are given by equations 92 A 1 l A A aB = a2 = X2 = YBXB (92) aC = a3 = X3 = YCXC a12 _ X12 where Y is the activity coefficient. Details on the determination of X X X3, and X and of (api/aci)c are given in Appendix III. J Acetone (l) — Benzene (2) — Carbon Tetrachloride (3) — Activity 1’ 2’ 12 data at 45°C for the acetone—benzene and acetone—carbon tetrachloride binaries are given by Brown and Smith5 and for the benzene—carbon tetrachloride binary by Christian, et. a1.9 These have been fitted to two term Margules equations50 and extended to the ternary by the method of Wohl.47 The equations used were 3 _ 2 2 CT 1n yl — CZI:A12CT + 201(A21 - Alzfl -+ C3 [A13CT + 201(A3l -A13% + c2c3[CT(A21 + Al3 - A32) +2C1(A3l - A13) (165) + 2C3(A32 ' A23) ' C123(CT ’ zcli] 71 3 _ 2 2 _ CT 1“ Y2 ‘ C3 [1123111 1 2C2(‘132 ' A23)] + C1 [A2101 + 2C2(‘112 A213] + C103 [CT(A32 + A21 - A13) + 2c2(A12 - A21) (165) 1' 2C1(1‘13 ’ A31) ' C123(111 ' 2%)] where C = 123 (166) NIH (A21 ' A12 1' A13 ’ A31 + A32 ’ A23) and Ci is the concentration of i and the Ai' are the binary coefficients. These coefficients are listed in Table 19. Details of the develOpment of the expressions for (aui/acj)c are given in Appendix III. The 1 activity constants determined by Yon and ToorSO are given in Table 19 of Appendix VI. Toluene (l) - Chlorobenzene (2) — Bromobenzene (3) - This system was assumed to be ideal so that for all three components a. = X. (167) Error Analysis of Experimental Method The accuracy in determining binary diffusion coefficients with this apparatus was found to be 11% by Bidlack.3 He Obtained diffusion coefficients for seven aqueous sucrose solutions and compared them with those reported by Costing and Morris. Diffusion coefficients for aqueous sucrose solutions Obtained by this author also were within 11% of those listed. It is difficult to specify the accuracy in determining ternary diffusion coefficients. The reason for this is the lack of 72 reliable ternary data to compare to. The accuracy of the ternary data that are available in the literature is not given for this same reason, only the precision is generally discussed and sometimes not even that. In determining experfinental ternary diffusion coefficients in this laboratory, the accuracy is not limited by the apparatus but rather by the readings taken from the photographic plates and in the method of calculations themselves. Because of the former, it is expected that values obtained for the reduced second moment, D , and 2m for the reduced height—area ratio, D can be no better than i1%. A’ The precision of the data Obtained in this laboratory was determined by calculating the variance and confidence limits of the lepes and intercepts of the reduced second moment and reduced height- area ratios. These are listed in Table 2. It was found that the 95% confidence lfinits on the slopes were considerably larger than the intercepts. The limits on the intercepts in all cases amounted to less than 1.95% of the actual value used. The limits on the slopes varied from 11.6% to 16%, with the average deviation being approximately 13%. The 95% confidence envelope for each of the least squares fit can be Obtained from Table 3. The 95% confidence limits of the ordinate values (D2m and l//fi; values) were found to vary from i0.l% to 11% of the least squares line through the points. The variances of the ordinate values were determined according to the equation 73 Table 2. Variances and 95% confidence limits of the vs. a and slopes and intercepts of the D l lf/BX-vs. 2m 0 curves. 1 Dodecane — Hexadecane - Hexane Variable Variance 95% Confidence Value used in Limits calculations 12m 1.39x10"14 11.12x10‘7 1.192x10‘5 32m 4.31x10'15 ~:6.32x10'8 0.1492110"5 IA 1.98 11.34 2.923x102 sA 0.61 :0.75 -0.244x102 Diethyl Ether — Chloroform - Carbon Tetrachloride Variable Variance 95% Confidence Value used in Limits calculations 12m 3.21x10'15 5.351110"8 0.716X10_5 32m 4.89x10’14 ~_~2.10x10'7 1.39.1x10'5 1A 1.42 :1.13 2.89ox102 sA 21.6 :4.42 -0.725x102 A _ 74 Table 3. Confidence limits of D2m and l//§X. Dodecane (l) - Hexadecane (2) — Hexane (3) 01 Dsz105 Confidence limits l//BX’ Confidence limits of 02 x10 of 1/75‘ m A -l.134 0.984 10.938 324.7 $1.12 -0.239 1.203 10.380 300.6 £0.45 0.176 1.197 10.121 283.2 10.14 1.352 1.411 10.612 256.0 -0.73 -2.416 0.835 il.737 350.3 12.07 3.599 1.710 12.012 206.9 12.40 1.257 1.388 10.553 261.9 10.66 Diethyl Ether (l) — Chloroform (2) - Carbon Tetrachloride (3) 5 61 Dan10 Confidence limits 1/75Z' Confidence limits of D2 X10 of l/Vv—- m A 1.002 2.113 :0.044 218.8 10.09 0.623 1.585 :0.840 243.3 11.76 1.317 2.577 10.617 193.9 11.30 1.150 2.303 10.267 203.4 10.56 75 N 2 V _ 2 é (Y1 Y1) se(yi) = (168) N - 2 where s:(yi) is the variance of yi, N is the number of data points used in the least squares fit, Yi is the ordinate value calculated from the constants of the least squares fit, and y1 is the ordinate value used in determining the least squares fit. The N — 2 degrees of freedom result from the use of two quantities, the slope and intercept of the least squares line, which are calculated from the data. The variances of the lepe and intercept were calculated from the following relations: 2 2 Se(yi) s (slope) = (169) e N V (x —§E>2 9 i 1 2 2 Se(yi) s (intercept) = ——————— (170) e N where N 2x _ i 1 x ___ (171) N The 95% confidence limits were calculated according to the relation 95% confidence limits on 2 = £0.95 52(2) (172) 76 where z is a dummy variable. These equations and error analysis are discussed by Mickley, Sherwood, and Reed.32 RESULTS AND DISCUSSION Theoretical Hydrodynamic theory predicts that the Onsager reciprocal relations are valid for non—associating systems in which the molar volumes are constant. This latter constraint was experimentally maintained by choosing the initial concentration differences as small as possible. An attempt was made to apply hydrodynamic theory to multicomponent associating systems by considering the two simplest cases where association results in a dimer. It was found that Onsager's reciprocal relations would again be valid but with the additional assumption that the molar volume of the dimer be equal to the sum of the molar volumes of the species associating to form the dimer. This assumption was reasonable but not rigorously valid. It was also found that Miller's equations for the phenomenological coefficients reduced to those derived from the hydrodynamic approach. This was especially encouraging because Miller's equations contain activity expressions but the hydrody— namic equations do not. Activity data for ternary and higher order systems, like diffusion data for these systems, is difficult to obtain and up till now has been a major deterrent in verifying Onsager's reciprocal relations. It was mentioned previously, that Miller's condition for verifying Onsager's reciprocal relations was 77 78 derived from his expressions for the phenomenological coefficients. Since these contained activity terms, they also appeared in the condition used for testing the reciprocal relations. It should also be pointed out that the expressions for the diffusion coef— ficients obtained by the hydrodynamic approach do contain activity terms. When these expressions are substituted into Miller's equations for the phenomenological coefficients, the activity terms drop out. Experimental The systems dodecane-hexadecane—hexane and diethyl ether- chloroform—carbon tetrachloride were experimentally studied in this laboratory, while the systems toluene-chlorobenzene-bromobenzene and acetone-benzene—carbon tetrachloride were experimentally studied 7’41 The hydrodynamic model was applied to all four elsewhere. systems and the reciprocal relations tested when possible. In the derivation of the equations used to determine experimental diffusion coefficients with optical methods, it was assumed that the dependence of refractive index on the two indepen— dent concentrations could be adequately represented by the first three terms of a Taylor expansion (equation II—l6, Appendix II). This is a critical assumption since it relates refractive index measurements, which are the basis of optical methods, to the concentrations. Thus, it is important before any confidence in the experimental ternary diffusion coefficients is possible, that the reliability of this assumption be checked. Fortunately, the trend has been in all systems studied that this assumption is valid. By 79 subtracting the two Taylor expansions representing the refractive index of the two initial solutions used in a run and then rearranging, one obtains -——- = R + R -—— (147) as was shown in the experimental section. Therefore, provided the R1 and R2 are independent of concentration, a plot of An/ACl versus ACZ/ACl should be linear. Since the initial concentration differences are small, the assumption of constant R1 and R2 is reasonable. For both systems studied experimentally in this laboratory, such plots exhibited linear behavior. This is shown in Figures 6 and 7. Non—Associating Systems Dodecane—Hexadecane-Hexane The basis of the optical method in determining the diffusion coefficients other than that just discussed is that the reduced second moment, D and the reciprocal square root of the reduced 2m’ height-area ratio, l/VDA, be linear with the refractive index fraction, 0 It is from the intercepts and slopes of these curves 10 that the ternary diffusion coefficients are calculated. For the system dodecane (1) — hexadecane (2) — hexane (3), these curves are given in Figures 8 and 9. It can be seen that linear behavior does occur. The lepes and intercepts of these curves for this hydro- carbon system are presented in Table 16 of Appendix VI. The ternary 80 0.014- An AC 0.004. -OeOl ' l f I ' 1 -l.l -0.9 -0.7 -0.5 -0.3 -O ACz/ACl Figure 6. Determination of the differential refractive index increments, R1 and R2, dodecane (l) - hexadecane (2) — hexane (3). for the system 81 -0.008- -0.010.‘ -00012 c -0 0014 ‘ An/ACl —O o 016 d -0.018, _00020 I I ' I Fl I -l.l —O.5 0.1 0.7 1.3 1.9 2.5 ACZ/ACl Figure 7. Determination of the differential refractive index increments, R1 and R2, for the system diethyl ether (1) - chloroform (2) — carbon tetrachloride. 82 1.9 007 - I A llfi -2.5 -1.5 -O.5 0.5 1.5 2.5 3.5 Figure 8. Linear relation of the reduced second moment, D , versus 2m the refractive index fraction of dodecane, a for the 1, system dodecane (1) — hexadecane (2) - hexane (3). 83 355~ 3351 315 .1 295 . 275 .p 255 - 235 3 215 . 195 —l.5 Figure 9. -0.5 0.5 1.5 2.5 Linear relation of the reciprocal square root of the reduced height-area ratio, l/VEZ, versus the refract- ive index fraction of dodecane, a for the system 1, dodecane (1) — hexadecane (2) — hexane (3). 3, 5 84 diffusion coefficients obtained from these slopes and intercepts along with the phenomenological coefficients calculated from these diffusion coefficients are given in Table 5. In determining experimental diffusion coefficients and phenomenological coefficients based on the hydrodynamic model, experimental or predictive friction coefficients were required. For the hydrocarbon system, these were obtained both ways. This fulfilled two purposes: (1) to obtain accurate values of the friction coefficients thus placing more confidence on the diffusion and phenomenological coefficients calculated from them and (2) to check on the predictive method of calculating friction coefficients. Values of the friction coefficients calculated by both methods are given in Table 4. Comparison shows that the predicted friction coefficients compared quite favorably with those obtained experi— mentally by tracer methods. The poorer agreement of the friction factor for hexadecane was probably caused by error in estimating the self diffusion coefficient of the hexadecane. The lack of self diffusion data for hydrocarbons higher than decane is evident from Figure 16. The fact that the molar volumes of the hydro— carbons differed considerably and that friction coefficients for the same component calculated from the infinitely dilute binary diffusion coefficients and self diffusion coefficients differed appreciably indicated that the effects of the other components were strong. Under these conditions, the predictive method for non-associating systems would be the poorest. With this in mind, the agreement between the experimental and predicted friction 85 Table 4. Comparison of the estimated and the experimentally determined friction coefficients for the system (RT/ol)X107 dynes Estimated Tracer (Equation 162) 1.089 1.118 dodecane (1) - hexadecane (2) - hexane (3) X1 = 0.350 X2 = 0.317 (RT/02)X107 (RT/o3)x107 dynes dynes Estimated Tracer Estimated Tracer (Equation 162) (Equation 162) 1.062 0.848 1.899 1.873 86 Table 5. Comparison of experimental diffusion and phenomeno- logical coefficients with those calculated from friction coefficients for the system dodecane (1) - hexadecane (2) - hexane (3) O T = 25 C X1 = 0.350 X2 = 0.317 5 5 5 5 DlleO Dl2X10 D2leO D22X10 (cm2/sec) (cmg/sec) (cm2/sec) (cm2/sec) Expt. Calculated Expt. Calculated Expt. Calculated Expt° Calculated Est. Tracer Est. Tracer Est. Tracer Est. Tracer 0.968 1.082 1.115 0.266 0.270 0.386 0.225 0.209 0.167 1.031 1.123 0.971 5 5 L12 X RT X 10 L21 X RT X 10 Experimental Calculated Experimental Calculated Est. Tracer Est. Tracer -0.453 —0.538 40.465 -0.444 -0.538 =0.465 87 coefficients under the least applicable conditions provides reasonable confidence in the predictive method. The diffusion and phenomenological coefficients calculated from the friction coefficients (equations 39 and 61) are given in Table 5. In comparing the values obtained by Optical methods to those predicted from hydrodynamic equations it can be seen that there is good agreement. In fact, the agreement of L12 and L21 appears to be better than that found by other authors on other systems. This conclusion may not be justified, however, since no comparison between the accuracy of experimental methods can be made. The hydrodynamically obtained values are well within the expected accuracy of the experimental values. The agreement of the experimental L and L and the 12 21 12 = L21 obtained from the friction coefficients indicates that Onsager's reciprocal relations are valid. agreement of these with the L Toluene-Chlorobenzene-Bromobenzene Experimental diffusion coefficients were obtained by Burchard and Toor7 using a modification of the diaphragm cell tech— nique. Based on the fact that this apparatus gives less accurate binary values than the apparatus used in this laboratory, it is reasonable to assume that the ternary diffusion coefficients are less accurate than those obtained in this laboratory. This system was assumed ideal and on this basis phenomeno- logical coefficients were determined from the diffusion coeffi- cients given by Burchard and Toor. Friction coefficients were obtained by the predictive method. The fact that the molar volumes 88 were almost equal and that the friction coefficients calculated from the infinitely dilute binary diffusion coefficients were approximately equal, indicated that reasonable values of the friction coefficients could be expected. Table 6 lists the diffusion and phenomenological coef- ficients. Excellent agreement between the hydrodynamically obtained values and experimental values can be seen from a comparison. Except for very few cases, the values obtained from friction coef- ficients were well within the 95% confidence limits. Reasonable agreement between the experimental L12 and L21 is evident and these agree favorably with those obtained from the hydrodynamic approach. Based on the results, this author feels that Onsager's reciprocal relations are verified for this system. Associative Systems Diethyl Ether—Chloroform-Carbon Tetrachloride For this system plots of the reduced second moment, D2m’ and the reciprocal square root of the reduced height-area ratio, 1//?;, against the refractive index fraction, 01, are presented in Figures 10 and 11. It can be seen that these curves exhibit the linear behavior predicted by the experimental equations, 152 and 153. The $1Opes and intercepts of these curves are given in Table 16 of Appendix VI and the resulting experimental diffusion and phenomenological coefficients are presented in Table 7. 89 111.0- 111.0- 144.1- 101.1- 111.4- 111.4- 114.1- 114.1- 114.0- 114.0- 111.1- 111.1- 0mpmafioamo HmemEHMomxm 104 x 11 x 414 111.4 104.4141.4 110.0- 111.4 104.4011.4 110.0- 011.4 104.4411.4 110.0- 410.1 104.4110.1 110.0- 011.4 104.4101.4 100.0- 111.4 110.4111.4 110.0- .0400 .0011 .0410 11 41 1041 0 104x 114.1100.0 114.4114.0- 11041100- 110.“ 410.0- 11011140.0- 110.4 110.0- E Amv oQoNQoQonHQ - Amv oeoNquOHoano - AHV meosaop 804111 map Mom mpeoflofi%%moo q0490111 801M 00¢1H50110 omonp £413 1pemfloflwwmoo HmOHmOHOQoEoeogm 0mm SOH159110 ampeoaflpomxo %o QOmflpmmaoo 111.0- 411.0- 11.0 144.1- 141.1- 11.0 111.4- 111.4- 01.0 114.1- 110.1- 14.0 114.0- 114.0- 10.0 111.1- 111.1- 01.0 0091150110 ampeoaflnmmxm 104 4 am 1 141 1x 140.0- 11014110.0- 111.4 104.4111.4 1:0.0- 11011010.0- 111.4 10 .n100.1 100.0- 114.4110.0 141.4 104.1111.4 110.0- 114.4410.0 110.1 11 .4114.1 110.0- 00414110.0- 441.4 11014011.4 140.0- 104.4110.0- 111.4 11014111.4 .930 .980 .181 14 44 104x 104x 1 14.0 14.0 11.0 01.0 11.0 11.0 .mofifia 2.8 90 2.6"5 2.4-- 5 , 02m x 10 N 'T’ 2.0“ 1.8“ Reduced second moment 1.61- 1.4 0.5 0.7 0.9 1.1 1.3 1.5 Figure 10. Linear relation of the reduced second moment, D2m’ versus the refractive index fraction of diethyl ether, 01, for the system diethyl ether (1) - chloroform (2) - carbon tetra- chloride (3). 91 250 240 n" 230 " 220 " 17/0: 210 ‘- 200 " 190 " 180 I 0.5 Figure 11. 0.7 0.9 1.1 1.3 0‘1 Linear relation of the reciprocal square root of the reduced height-area ratio, 1//D;; versus the refractive index fraction of diethyl ether, 01, for the system diethyl ether (1) - chloro- form (2) — carbon tetrachloride (3). 92 Table 7. Comparison of experimental diffusion and phenom- enological coefficients with those calculated from friction coefficients for the system diethyl ether(l)- chloroform (2) - carbon tetrachloride (3) T = 25 00 X1 = 0.25 X2 = 0.25 5 5 5 5 DllX10 D12Xlo D21X10 D22X10 Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. 2.06 2.76 —0.31 —0.34 0.23 —0.42 2.11 2.23 L X RT X 105 L X RT X 105 12 21 Experimental Calculated Experimental Calculated -0.709 -0.824 0.209 -0.824 93 The necessity of having reliable activity data is clearly brought out here. No experimentally determined ternary activity data were available for this system. As a result, the associated— model of Wirth47 and Andersonl’ was assumed here. This model says that there are four species in solution, the monomers of all three components and the associated dimer of diethyl ether and chloroform. In addition, the model assumes that these four species together form an ideal solution thus assuming that all non—ideality in this solution is caused by the association. For the diethyl ether— chloroform binary, the above model was found by Andersonl’2 to be quite good. The importance of assuming that the species form an ideal solution is that then activity data for the components can be obtained without experimentally measuring pressures, etc. The required ternary activity expressions were calculated in Appendix III. The experimental diffusion coefficients were assumed to be reliable. However, the phenomenological coefficients, calculated from the diffusion coefficients and the required activity expressions, were far from equal. The friction coefficients used to calculate diffusion coefficients were determined by Wirth47 and were assumed to be reasonable. These diffusion coefficients did not compare favorably with the experimental diffusion coefficients. It can be seen that reasonable agreement exists between the D12 and D22 respectively, but this was considered fortuitous. There were two possible reasons for this disagreement assuming that the experimental work was satisfactory: (1) that the 94 hydrodynamic approach as applied to associating systems was not correct and/or (2) that the activity data obtained assuming the chemical model was incorrect. It is highly likely that only the latter was the cause. This belief is based on the fact that the vapor-liquid equilibrium data and the mutual diffusion data of the carbon tetrachloride—chloroform binary were not ideal. According to the chemical model, this binary should be ideal since no association has been detected. The equilibrium and diffusion data are presented by Wirth.47 It was admitted by Wirth that the diffusion data may not be reliable since difficulties in duplicating it were encountered. In any case, it can easily be seen from his data, that the general trend was far from ideal. If ideal, a plot of DAB” versus mole fraction gives a straight line but this plot was quite curved especially in the region towards pure chloroform. The fact that difficulties of duplication were experienced may also reflect the non—idealities of the system. Since the ternary chemical model, assumed to calculate activities, considered this binary as ideal, it is quite reasonable to assume that the acti— vities calculated from this model are in error. The purpose of studying this system was two-fold, to check the ternary chemical model and the applicability of the hydrodynamic theory as applied to associating systems. It can be concluded from this study that the proposed ternary chemical model is incorrect. Since the validity of the model, or at least partial validity, is necessary in order to experimentally verify the 95 applicability of the_hydrodynamic approach, no justifiable conclu- sions on the latter can be drawn. Acetone-Benzene-Carbon Tetrachloride This system can be classified as an associated system but the actual species present are difficult to predict. However, since experimental data had been collected and activity data for all three binaries were available, it was hoped that some use of the system as a check on the hydrodynamic theory could be made. To do this, the system was treated as a non—associating system and friction coefficients calculated for each component from binary and self diffusion data. These friction coefficients are not coefficients for any particular species but rather overall factors prOposed to give approximate diffusion coefficients for any system. In this sense, friction coefficients calculated in this way serve as a predictive method of obtaining multicomponent diffusion data. In any case, if the hydrodynamic theory could lead to an approximate or predictive method of determining reason— able multicomponent diffusion coefficients, it would certainly provide some faith in the theory. Friction coefficients were obtained according to equation 162 and the values are recorded in Table 17 of Appendix VI. From these, the ternary diffusion and phenomenological coefficients were obtained and are given in Table 8 along with the experimental values obtained by the diaphragm cell technique. It is evident that indeed the values do compare favorably and are certainly of the right order of magnitude. It is quite possible 000000000 96 Table 8. Comparison of experimental diffusion and phenom- enological coefficients with those calculated from friction coefficients for the system acetone (1) - benzene (2) - carbon tetrachloride (3) O T = 25 C 5 5 5 5 X1 X2 DllX10 D12X1O D21X10 D22X10 Expt. Cale° Expt. Calc. Expt. Calc. Expt. Cale. .30 0.35 1.887 1.731 -0.213 -0.274 -0.037 -0.029 2.255 2.226 .15 0.15 1.598 1.532 -0.058 -0.176 -0.083 -0.113 1.812 1.731 .15 0.70 1.961 2.200 0.013 -0.131 -0.149 -0.251 1.929 2.072 .70 0.15 2.330 2.366 -0.432 -0.485 0.132 0.146 2.971 3.236 .09 0.90 3.105 2.499 0.550 -0.071 -0.780 —0.410 1.860 2.065 .24 0.75 3.069 2.268 0.603 -0.180 -0.638 -0.059 1.799 2.352 °49 0.50 2.857 2.263 0.045 —o.365 -0.289 0.253 2.471 2.884 .74 0.25 3.251 2.702 -0.011 -0.563 —0.301 0.252 2.896 3.493 .895 0.095 3.475 3.162 -0.158 -0.691 0.108 0.122 3.737 3.905 X X L X RT X 105 L X RT X 105 1 2 12 21 Experimental Calculated Experimental Calculated 0.30 0.35 -0.283 -0.280 -0.286 -0.280 0.15 0.15 -0.026 —0.041 -0.036 -0.041 0.15 0.70 -0.216 -0.277 -0.243 -0.277 0.70 0.15 -0.461 -0.475 —0.437 -0.475 0.09 0.90 -0.230 -0.230 -0.252 —0.230 0.24 0.75 -0.594 -0.567 -0.604 -0.567 0.49 0.50 -1.068 -0.974 —1.072 —0.974 0.74 0.25 -0.969 -0.943 -O-975 -O~943 0.895 0.095 -0.498 —0.517 -0.496 -0.517 97 that they are within the experimental accuracy of the method. Only in the case where the concentration of one of the three components is almost zero do the predicted values show any appreciable discrep— ancy. Better agreement occurs between the main diffusion coefficients, D and D but the cross diffusion coefficients are in the right 11 22’ order of magnitude, and close in comparison, when one considers that they are much smaller and more subject to experimental error. Sur- prisingly close agreement among the phenomenological coefficients was also obtained. The fact that this predictive method produced reasonable results is encouraging and indicates that hydrodynamic theory can be applied to multicomponent associating systems with some success. It should be cautioned that the associating system studied was not a particularly strong associating system and thus poorer agreement could be expected in more highly associative systems. CONCLUSIONS It can be concluded from this study that hydrodynamic theory should play an important role in describing multicomponent liquid diffusion. It predicts that Onsager's reciprocal relations should be valid for non-associating systems of any number of components. Furthermore, it leads to generalized equations which enable the cal— culation of diffusion coefficients for quaternary and higher order non-associating systems from friction coefficients, which can be accurately obtained directly from tracer measurements, and activity data. The number of tracer runs required is equal to the number of components in the system. Calculation of diffusion coefficients in this way would eliminate the laborious and time-consuming techniques now available and provide data for systems of higher order than ternary. Such data, currently not available, would be extremely useful in considering multicomponent mass flow problems. EXperimental evidence obtained in this study verifies the Onsager reciprocal relations for non-associating systems within the limits of experimental error. In addition, it demonstrates the validity and applicability of hydrodynamic theory to liquid diffusion. These conclusions were based on the fact that good agreement between the diffusion and phenomenological coefficients obtained by optical 98 99 methods and those obtained from friction coefficients was found for the non-associating systems studied. Hydrodynamic theory applied to simple ternary associating systems in which only dimers are formed indicates that Onsager's reciprocal relations are valid provided the molar volume of the dimer can be considered as the sum of the molar volumes of the species associating to form the dimer. An attempt to describe an associating system by the chemical model which states that the actual species in solution form an ideal solution was not satisfactory. Since ternary activity data were not available and since the model would have predicted the missing activity data, the reciprocal relations for associating systems could not be experimentally verified. It was found that for associating systems of unknown association reasonable diffusion data could be obtained by treating the system as non-associating and calculating friction factors for each of the chemicals added to form the system. In this way, the hydrodynamic approach served as a useful predictive method. It should be noted however that the predicted data would probably be less reliable in more highly associated systems. FUTURE WORK Theoretical An important area opened up in this study was the appli— cation of hydrodynamic theory to associating systems. It would be worthwhile to try to obtain generalized expressions for the fluxes, diffusion coefficients, and phenomenological coefficients in terms of friction coefficients for systems in which there might be dimers, trimers, tetramers, etc., in various combinations among the compo- nents. In addition, it is hoped that the hydrodynamic theory can be applied to the simplest associating systems such that the assumption concerning the molar volume of the associating species would not be necessary in order to show that Onsager's reciprocal relations are valid. A further area of develOpment would be with systems in which the kinds of associated species are not known. It was found in this study for the system acetone-benzene-carbon tetrachloride that using overall friction coefficients for a component gave reasonable diffusion coefficients. Maybe refinement in this area could produce an improved predictive method for associating systems by combining hydrodynamic theory and binary diffusion data. A critical factor in using hydrodynamic theory is the calculation of friction coefficients. A better understanding of the 100 lOl effects of composition is certainly needed. Improved methods of determining their values for associated systems at various composi— tions would also be helpful. Experimental There is considerable room for improvement in this area. To obtain ternary diffusion data, time-consuming calculations and lengthy experimental work is required. Considerable error is introduced in the method of calculations which uses the lepes and intercepts from a plot of slopes from other curves and involves differences of small numbers in the same order of magnitude. Improved techniques in taking measurements from the photographic plates are also desirable. The improvement in the experimental area of multicomponent diffusion is not needed so much in the apparatus but rather in the application of the apparatus. It seems almost ludicrous to speak of development in experimental techniques with quaternary systems when the techniques in ternary systems are not yet well developed, but such data could be extremely useful in checking the hydrodynamic predictions presented in this study for higher order systems. Activity data are a very limiting factor in multicomponent diffusion since they are required and yet are not sufficiently available. In the last few years, more and more ternary data have become available and even some quaternary data have been reported. However, most of these are not collected at constant temperature but rather at constant pressure. It is suggested that careful consideration be 102 given to the systems chosen before any experimental diffusion work is performed to see that adequate activity data are available. It would of course be convenient if activity data could be collected in conjunction with the multicomponent diffusion studies. APPENDIX I Determination of Friction Coefficients The crux of the hydrodynamic method is obtaining reliable values of the friction coefficients, oi. For the non—associating case, these can be obtained directly by tracer techniques. In tracer diffusion, the concentration is uniform throughout the system and, as a result, the velocities of all reference planes relative to fixed coordinates are zero. That is v = v = 0 (1—1) * C* 3 * * m i ui Ji - Ji - - * -——- (1—2) oin 3x where the superscript * refers to the tagged species. Substituting for chemical potential from equation 37, equation 1-2 becomes 1 1 C. aln a. 1 RT ——-—l (1—3) 0.0 8x 1 which can be rearranged to give X- 1 RT oln a, BCi <1—4) X-H J..- = - * 1 oin aln C. ax H 103 104 The tracer diffusion coefficient which is measured is defined by 7‘: 7: Jo = - DO (1-5) 1 1 Therefore, by comparing equations I-4 and I—5, the measured diffusion 1 coefficient Di is 3(- RT Bln a, (1-6) 0 H 1 u 114 1 0,0 Bln C. 1 P If we make the reasonable assumption that the physical properties of the labeled species are the same as the unlabeled species, then 1 0, = 0 (I—7) 1 For the case of tracer diffusion, >1- aln a, = 1 (1-8) 1H 81n C. H Therefore, from equations I-6 and I-8 * RT Di = 0.0 (1—9) 1 and hence 1 Dan 1 01. = -- (I-lO) RT Thus, by tagging each component separately and keeping the concentra— tions the same as those at which diffusion coefficients, D. , are 11 105 measured, all of the friction coefficients can be determined. If there are N components, N tracer runs would of course be required to obtain the N friction coefficients. If tagged species are not available, reasonable values of the friction coefficients can be obtained from the infinitely dilute mutual diffusion coefficients of the binaries and the self diffusion coefficients of the binaries and the self diffusion coefficients of the pure components. Hartley and Crank27 and others have shown that D = B: :(A + £1; —_aln aA (I-ll) AB Tl OB GA Bln XA where DAB is the mutual diffusion coefficient.of the A—B binary. At infinite dilution of component A, equation I—ll becomes 0 RT D = (1‘12) AB OAnB and therefore RT 0A = D0 (I-13) ABnB where DZB is the mutual diffusion coefficient of the A—B binary at infinite dilution of A and n is the viscosity of pure B. In a B ternary system, two such values of 0A can be obtained from the A-B and A—C binaries; similarly, for 0B and 0C. Another value of the friction coefficient can be obtained from the self diffusion coefficient, D1’ of the pure component. Again, Hartley and Crank27 and others have shown that 106 (I-l4) In summary, if self diffusion data and binary data at infinite dilution in each component are available, several values for each 01 can be obtained. These values will be reasonably constant if the friction coefficient is independent of the other species present. A reasonable value of Oi is therefore a weighted average based on mole fraction. Thus, RT N 'RT 01. pred. = Dini + j=1 Do 0 X3 1 = 1' °°°’ N (1-15) 13 j 111 where 0, is the predicted friction coefficient. i,pred. Determination of 0i in associated systems has been described by Wirth47. This method also involves tracer techniques. APPENDIX II Solution to the Ternary Diffusion Equations and the . . 2 Experimental Determination of the Diffusion Coeffic1ents 2 Solution to the Diffusion Equations The flow equations for a ternary system can be obtained from equation 26. They are repeated here for convenience. J = ...D BELL—D fl 1 11 0X 12 3X (II-1) J = — D 301 _ D 3C2 2 21 8x 22 8x To obtain the equations for one-dimensional diffusion in a three- component system, it is first assumed that the diffusion coefficients are all independent of concentration and and that no volume change occurs on mixing. These conditions can be approached experimentally by keeping the concentration differences across the initial boundary sufficiently small. By making a material balance on a differential volume element of the diffusion solution, the desired relations are then obtained. These are (II—2) 107 108 (II-2) at 8x 8x in which C1 and 02 are the concentrations of any two components and are functions of position x and time t. D11 and D22 are the main dif- fusion coefficients which are generally on the order of mutual diffusion coefficients in binaries, and D12 and D21 are the cross—term diffusion coefficients which are considerably smaller than the main coefficients. For free diffusion a sharp boundary is formed at t = 0 between solutions A and B which are above and below respectively the position x = 0. As a result, the initial conditions for the two components (i = 1,2) are c. = 01 — AC,/2 x > 0, t = 0 1 i i __ (II—3) C, = C. + AC,/2 x < 0, t = O i i i and the boundary conditions are C. + .C, - AC,/2 x + +1, t > O i i i __ (II-4) C, + C, + ACo/Z x + -w, t > 0 i 1 1 Here Ci is the mean concentration of each component and A01 is the con— centration difference of each component across the diffusing boundary. They are defined as follows __ (C.) + (C.) C, = i A i B (II-5) 1 2 AC = (C ) - (C ) i 1 B 1 A (II-6) It is well known that under these initial and boundary conditions, a new variable, y, may be introduced to reduce equations 109 II—2 to a set of ordinary differential equations. The new variable y is given by x = __ (II-7) (2/t ) and the set of ordinary differential equations obtained is dC d2C d2C ' 2y ——l'= D11 7'2;'+ D12'—2"g dy dy d y (II—8) dC dzC d2C 2 1 2 ’ 2y __—'= D21 2 + D22 "—__' dy dy dy In addition, equations II—4 — II-6 reduce to AC. —- i C. 4" C, -' _ y ++oo 1 1 2 (II-9) AC C. "> C. + '—i y —,‘> -oo 1 i 2 The details in solving equations II—8 and II—9 for the con— centrations of the components are given by Fujita and Costing.22 The desired exact solutions which they obtained are ... + _ c1 - C1 + Kl¢(/0: y) + Kl0(/E: y) (II—10) _ + _. 02 - 02 + K2¢(/0: y) + K29(/3: y) in which 2 q _ 2 <11) = — j e ‘1 dq (II-11> /F 0 110 and where V 1 0=1/2H+E+ [(H-E)Z+4FG]l5 1 1 (II-12) 0=1/2H+E- [(H-I~:)2+4FG]Li 7 1 (9+-E)401-F40 (0_ — E)AC1 - FAC 2 — 2 K = K = 1 2(0+ _ 0_) 1 2(0_ 0+) (II-l3) _ _ —H C —GC K+ = (0+ H)Ac2 GACl K_ = (o_ )4 2 4 1 2 2(0+ — 0_) 2 2(0_ - 0+) and D D D D E = '11 , G = 21 , H = —2‘-2—— , F = 12 (II-l4) lD-l lDll lDi-l lDi-l 1J J J J lDij' ' D11D22 ' D12D21 (11—15) It should be noted that the equations for each component concentration are a linear combination of two probability integrals plus a constant term, with the characteristic of each term depending on.ACl, AC2, and the four diffusion coefficients. Determination of Diffusion Coefficients In order to derive equations applicable to experimental techniques utilizing optical methods for studying free diffusion, it is assumed that the dependence of refractive index, n, on the two independent concentrations can be represented by the first three terms of a Taylor series. That is, n = n_ + R1(C-Cl) + 112(02-0 ) (II-l6) C 2 111 where R1 and R2 are the differential refractive index increments defined by R :4 [an(cl,cz)] 1 8C1 1,9,0 2 _ .— C1 ‘ Cl’ 2 2 (II—l7) R .: [an(cl,cz)] 2 3C2 T,P,C and n;_is the refractive index of a solution in which the solute con- C centrations are Ci and Ci. R1 and R2 are assumed to be independent of 1 _ _' _ C1 ’ C1’ C2 ' 2 0I concentration. Again, selecting small initial concentration differences justified this assumption. It follows from equation II—l6 that the total change in refractive index across the boundary may be written as An = RlACl + RZAC2 (II—18) For convenience component fractions on the basis of refractive index are defined by R,AC, a1 = ___1___}_ (II—l9) Z RiACi 1. It therefore follows from equation II-l8 and II-19 that al 02 = l (II—20) For optical methods, it is desired to obtain the refractive index, n, as a function of x and t. This can be done by substituting 112 equations II—lO into equation II-16. The desired refractive index distribution expression which results is then n = n +-§E-[r (975— y) + r 0(/3—'yfl (II—21) '6 2 + + - - where _ 2 + + F+ ' An (R1K1 + R2K2) (II—22) r = LL.(R K + R K") — An 1 l 2 2 From the defining equations for P+ and F_, it can be shown that F+ + F- = 1 (II-23) The reduced height-area ratio is defined by (An)2 DA = 2 (II—24) 4wt (22) . 3x t max Substitution of the maximum value of the first derivative of equation II-21 into equation II-24 yields = r /3- -+ r_/E' (II-25) + + — 30 11> Substituting equations 11-13 into equations II-22 and making use of the relation 01 + 02 = 1, equation II-13 becomes 1 E- = IA + SAOLl (1.1-26) A 113 where H + (R /R )F — E — (R /R )G S = l -2 2 l (II—27) A 73; -+ VE— Vo + E - (R /R )F 1A = + —.__ 1 2 (II—28) /0+ + VET- Thus a plot of l/Vfi; versus a should produce a straight line with l slope S and intercept IA at a = 0. Another useful relation is the A 1 intercept at a = 1. This intercept, call it L l is given by A, 70 0 + H — (R /R )G + - 2 1 LA = IA + sA = (II-29) 70+ + 70 It should be pointed out that equation II—26 gives two equations involving the four unknowns D , and D since E, 11’ D12’ D21 22 F, G, H, 0+, and 0_ are expressions involving only the four diffusion coefficients. R1 and R2 can be experimentally determined. Therefore the 31Ope, SA’ and intercept, IA, are functions of the diffusion coefficients and R1 and R2. Obviously, two more independent equations involving the diffusion coefficients are necessary. These can be found from the reduced second moment of the refractive index gradient curves. th The r moment of the refractive index gradient curve, (an/3x)t versus x, is defined by 00 l r r 8n _ nr _ An 3 x (3x)t dx r — 0,1,2, .. (11-30) -m 114 and the reduced rth moment, Drm’ is defined by m D 2 —£ (II—31) rm 2t Differentiation of equation II—l6 with respect to x and sub— stitution of the result into equation II—30 yields 2 +00 3C m = QL. 2 R1 f xr (——i) dx (II—32) = ' t . . . . th . To faCilitate the derivation, an r moment of each concentration gradient curve is defined as follows: Ri Tm r 3Ci (m ). E -—- x (-—-J dx (II—33) r 1 An _m 3x t so that mr1 + mr2 = mr (II-34) It should be emphasized that only mr is a measurable quantity however. Now differentiating equation II-33 with respect to time and then inverting the order of differentiation with respect to t and x gives d(m ) R, +00 BC —————r i = -—3‘— [ xr [2— (——i) dx (II-35) dt An —w x t These Operations are permissible whenever the derivatives, . 11,29 E3(8Ci/3t)x/8x] t and E(BCi/Bx)t/3tj x are both continuous. By means of the continuity equation for a constant volume system, which says that 115 BC. SJ. 1 _ _ __l. (II-36) at t 3x t the fluxes Ji can be introduced. Substituting equation II—36 into equation II—35 and integrating once the right hand side by parts, gives (II-37) d rR, +oo SJ, in}: = —-3; f xr—l (4) dx 3x t dt An —m since for free diffusion, (BJi/Bx)t is zero at the limits x = 1m. Remembering that Ji is also zero at x = im, another integration by parts gives d(mr)i r(r — l)Ri +m r ._____— = _ f x J, dx (II-38) dt An -m Recalling the assumption that the diffusion coefficients are indepen- dent of concentration, we now substitute equation II-l into equation II-38 and integrate the resulting equations to obtain expressions for the moments. d(m ). r(r - l)R, 2 +00 SC, r i = i 2 D.. f r—2 (3 3) dx dt An j=l 13 -.. X t (II—39) 2 = r(r - 1) j£1(Ri/Rj) Dij (Mr_2)j This equation is seen to be a recursion formula relating the time derivative of any even moment (r) to a sum of the next lower (r — 2) even moments, and the time derivative of any odd moment to a sum of the next lower odd moments. Obviously its use depends on determining values for (mo)j and (ml)j. 116 Integration of equation Il—33 for r = O and substitution of equation II-l9 gives an expression for (mo)j. (m ) = a. (II—40) Remembering that Jj = 0 at x = im, integration of equation II-38 for r = 1 gives d(m ). ___1_.1 = 0 (II—41) dt indicating that the first moment of each concentration gradient curve is zero. Since it doesn't change with time, the first moment must correspond with the position of the initially sharp boundary. In our experiments the origin (x = O) is chosen as the position of the initially sharp boundary, therefore ). = 0 (II-42) The fact that (ml)j = 0 means that all the odd moments are zero; hence, as long as the diffusion coefficients, Dij’ and the refractive index increments, Ri’ are independent of concentration, the refractive index gradient curve should be symmetrical about the position of the sharp initial boundary. This is indicated for a typical run in Figure 15 of Appendix VI. Expressions for the even moments can be obtained from equation II-39. Substituting equation II—40 and r = 2 gives the second moment expression of component j. (m2), = 2t .2 (Ri/iji a. i = 1,2 (II-43) j J 117 Expanding equation II-43 and substituting equation II—34 gives the relation R2 R1 m2 = 2t [(Dll + E; D21)OL1 + (D22 + E; D12) (12] (II-44) Utilizing equation II-ZO and rearranging, equation II—44 becomes (II-45) where D2m is by definition the reduced second moment and the slope, SZm’ and intercept, IZm’ are R2 R1 S2m = D11 + RIDZl ‘ D22 ’ EZ'Diz (II-46) R1 12m = D22 + ngz (II-47) The intercept at a1 = 1, L2m is given by L2m = I2m + 32m = D11 + (Rz/R1)D21 (II-48) Equation II—45 indicates that a plot of D2m versus d1 should give a straight line with the slope, S , and the intercepts 2m at a = O, I , all being functions of the l , and at a =1,L 2m 1 2m diffusion coefficients. Therefore, if at least two experimental runs are made at the same average concentration but with different initial concentration differences, values of the slopes and intercepts from plots of 1/J5; versus a and D versus a would enable deter- 1 2m 1 mination of the diffusion coefficients. 118 It is desired to obtain expressions for the diffusion coefficients in closed form in terms of the slopes and intercepts. First of all, from equations II-27 and II—48 we can obtain the relation 2 lDijI (D11 + D22 + Mung) — (Szm/SA) — o (II—49) where IDijl = det Dij = D11D22 - D12D21 Similarly, combining equations II-27, II—28, II-47, and II-48 gives D + D = I 11 22 2m - IA(SZm/SA) — /|Dij| (II-50) Elimination of (D11 + D22) from equations II-49 and II-SO leads to a cubic equation in VIDijI from which IDij' can be determined either numerically or graphically. This cubic equation is 3 2 2 _ (/IDij|) + [12m — 1A(32m/sA)] (/|Dij|) — (Szm/SA) — 0 (II-51) Also combination of equations II—47 and II-48 with the definition of lDijl gives I D + L D = |D l + I L (II-52) 2m 11 2m 22 ij 2m 2m The value of lDijl is known from equation II-5l, therefore D11 and D22 can be obtained from the linear equations II-SO and II—52 to yield = _IDij| + LZm Dijl + LZmIASZm/SA (II-53) ll 32m 119 lDijl + I2m IDij' + IZmLASZm/SA D22 = S (II-54) 2m Substitution of these expressions for D11 and D22 into equations II-47 and II-48 permit the evaluation of D12 and D21 respectively. R2 D12 = i (1.2m - D22) (II-55) Rl I)21 = i— (LZm ' D11) (“'56) APPENDIX III Determination of Activity Expressions Sui/BCj It was shown previously that for a ternary system there are four terms of the type Sui/acj that are required to determine the diffusion coefficients Dij and to check the Onsager reciprocal 33,34 relation with Miller's condition. These are 8111 am Y1 1 1 V1 —,—— =RT———”—'- +RT —-—1—— (III-1) ac ac c c — 1 c l c 1 T V3 2 2 8p Bln Y 1 V- --1— = RT ————3L - RT -—- 1 - i (III—2) ac2 ac c — c 2 c T V3 1 1 3p Bln y 1 V. 2 __._._ = RT ————2 — RT —— 1 - A (III-3) ac ac c - 1 c 1 c T v3 2 2 3112 am yz 1 1 V2 —— = RT —— + RT — - — 1 - — (III-4) ac2 ac2 c C — c c 2 T v3 1 1 It is obvious that in order to obtain expressions for Sui/8C,, expres— J sions for Bln yi/SCj are required. Most activity data can be correlated in terms of 1n Yi so that the latter expressions can be obtained. 120 121 Dodecane (l) - Hexadecane (2) — Hexane (3) — This system was fitted to the ternary Van Laar equations using the binary Van Laar constants Aij and Bij° Binary activity data for the systems hexane— dodecane and hexane-hexadecane were available.4 To obtain the constants used in the Van Laar equations, only data on two binaries are needed since the following relations between the constants apply. l Aij = 2:i- 1,] = 1,2,3; 1%] (III-5) Jl Aii = l (III-6) VBij = - VBjiAij i,j = 1,2,3; i#j (III-7) B B B w/Xe +W/fl +‘\/—Ai1- = 0 (In-.. 12 22 32 Aik Aij = A.k i,j,k = 1,2,3; i#j#k (III-9) J From the binaries, values for A and B are obtained. Care must be taken to designate the right subscripts to A and B. As a check, the sub- scripts can be determined in the correct order if the relative magni- tudes of the Van der Waals' constants, a and b, are known. For example, if the relative magnitude of bi/bj and A are greater than unity, then the subscripts are ij. The corresponding B value obtained in binary Van Laar fit is then assigned the same subscripts. The ternary equations used are 2 (03013 + C2A23C12) (ClAl3 + C3 + C2A23) T 1n Y1 = - (III-10) 2 122 2 (C A C + C C ) T ln Y2 = _ l 13 21 3 23 2 (III—11) (ClA13 + 03 + C2A23) where Ci is the concentration of component i, and Aij and Cij are constants. The C . are defined in terms of the binary Bij as follows 13 + 0.. = or V B.. (III—12) 1.] 13 It was found for this case that all the binary B values were negative which when substituted into the ternary Van Laar equations 164 yielded a factor of i2, where i is the imaginary index, in the numerator. This explains the minus sign in equations III-10 and III—llo In the case of the value of B, it should also be noted that it occurs in the multicomponent equation as a square root, and this immediately raises the question of whether the value is positive or negative. This question can be answered on the basis of the relative magnitudes of /a;/bi where a and b are Van der Waals constants. If, for example, {/EEVbi — JEE/bj) is positive, then VB;; is taken as positive and VB}; as negative. The term /a;/bi corresponds to the square root internal pressure of the liquid. Thus polar compounds which have high internal pressures would be expected to have high values of this group, while compounds of low polarity would be expected to have low values. From the tabulation of ai, b1 and /a;/bi listed in Table 18 of Appendix V1 for various hydrocarbons, it can be seen that ai increases, bi increases, and VEEYbi decreases as the carbon number increases. These observations were used in determining 123 the sign of Cij° The values of Aij and Cij are given in Table 19 also in Appendix VI. To obtain equations III—1 — III-4, the partial derivatives of equations 111-10 and III—11 must be taken with respect to Cl at constant C and with respect to C at constant C . These, after sim- 2 2 1 plification and use of equation 34, are T ain Y1 _ 2(c3c13 + C2A23012) c :1— 8C1 —(CA +c+CA)2 13V c2 1 13 3 2 23 3 (III—13) + A _.31 (C3C13 + C2A23C12) ] 13 73 (011113 + c3 + C2A23) 31“ Y1 2(C3013 + C2A23C12) T ac = 2 ‘A23012 2 C (C1A13 + C3 + C2A23) 1 (III-14) + C .Z2,+ A _ Xg_ (C3C13 + 02A23012) 13 V 23 V (01.6.13 + c3 + 02A23) 3 3 Bln y Bln y Expressions for T —Efi?—_— and T -ER?——- can be obtained from 2 Cl 1 02 equations III—12 and III—l3 respectively by interchanging subscripts 1 and 2. Diethyl Ether (1) - Chloroform (2) - Carbon Tetrachloride (3) - For this system, the model of Wirth47 and Andersonl’2 which assumes that ether and chloroform form a 1:1 complex was used. The model says that only four species exist in solution: the monomers (unassociated species) of ether and chloroform, the inert species CC14, and the 124 ether—chloroform dimer. These four form an ideal mixture; thus, according to the model, any non-ideality of this system is attributed to the association of ether and chloroform with each other. It was shown previously that 3A = X1 aB = X2 (92) a0 = X3 312 = X12 where the subscripts A, B, C refer to the stoichiometric quantities of ether, chloroform, and carbon tetrachloride respectively and the sub—scripts l, 2, 3, and 12 refer to the actual species present. Therefore it is desired to obtain expressions for the mole fractions of the actual species in terms of the stoichiometric concentrations. This would enable the determination of the Elm ai/aCj expressions. The equilibrium constant for the association reaction is given by K = -———- (III-15) Wirth47 found K = 2.73. Because of stoichiometry, the following relations exist, CA = C1 + C12 CB = C2 + C12 (III-16) C = C 125 and from these, the mole fractions of the actual species can be found in terms of the stoichiometric quantities. We have x = Cl _ CA ‘ C12 1 C1 + C2 + C3 + C12 CA + CB + CC - C12 x _ C2 _ CB ‘ C12 2 C1 + C2 + C3 + C12 CA + CB + CC — C12 (III-17) x _ C3 _ Cc 3 C1 + C2 + C3 + 012 CA + CB + CC - C12 X = C12 12 CA + CB + CC - C12 Defining X12 as follows C o 12 X = (III-l8) 12 CA+CB+CC Equations III-l7 become 0 x _ XA—XIZ 1 _ o l X12 0 KB X12 X2 = o 1 X12 (III-l9) XC X3 = o 1 — X12 0 X12 X12 _ o 1 - X 126 With these substitutions, equation III-15 becomes 0 O x (l-X) K = 12 12 (III-20) O O (X. - X1209. - X12 which can be used to eliminate X0 in equations III—l9 to give 12 x = (2XA - l) + K(2XA + XC - l) + ® 1 1 + K(XC + 1) + o x = (2XB - l) + K(2XB + XC — l) + ® 2 1 + K(XC + 1) + o (III—21) 2X (K + l) X = C 3 l + K(XC + l) +-¢ l + K(1 - X ) — ¢ X = C ' 12 l + K(XC + l) + ¢ where 1/ 2 2 Q = fi§(l - XC) + f) — 4XAXBK(K + li] (III—22) Notice that if K = O (i.e., no association), 6 1 and equations III-21 reduce to X1 = XA’ X2 = XB, X3 = XC’ and X12 = 0. In terms of stoichiometric concentrations, X and X become 1 2 X = (20A - CT) + K(CA - CB) + w 1 CT + K(CT + CC) + w (III—23) X _ (20B - CT) + K(CB — CA) + w 2 CT + K(CT + CC) + W 127 1 /2 2 where ‘1’ — [@(CT - CC) + CT) - 4CACBK(K + 1)] Therefore aina 1 ax V . A =——-—i = (l+—‘f\‘-+K+‘¥)X 3C X 3C. —' A B l A . V CB CB C (III-24) VA VA ' 2 2CA-‘-CT+K(CA-CB)+‘P l-_—+Kl-2_—_—+‘PA /(Xl>\) V V C C alna 1 ax V . A =————]; = (-1+—B-—K+\y)). BC X 3C - B B 1 B V CA CA C (III—25) VB VB ' 2 - 2CA—CT+K(CA-CB)+‘¥ l-%—+K1-2-V—+‘PB /(Xl)\) C C where A = CT + K(CT + CC) + W (III-26) , aw V w - —— =—l.— 2|:K(C -C)+C]x K+1-—-4‘- —4c1<(1+1<) A BCA 2‘? T C T V- B cB C (III-27) , aw 1 VB C C (III—28) A Bln aB Sln aB The --—- and -—-—- are the same as equations III-24 and 3C BC B CA A CB III—25 respectively except that every A is replaced by B and vice versa. 128 Acetone (l) - Benzene (2) - Carbon Tetrachloride (3) — This system was fitted to Wohl's ternary equations using binary constants Aij and a ternary constant C obtained from the binaries.50 The ijk ternary expressions for ln Y1 and 1n Y2 are given by equations 165 and 166. The respective partial derivatives of 1n 1 are Bln Y 1 V.‘ C V. 1 _ __. 2 _._1 _ _ .4111 acl ] ” ‘C3] C2 [A12 1 + 2("“21 AlZi] 2 V' [213CT + C T 3' 3 2 + 2C (A — A ) + C2 A l - Vi + 2(A A ) l 31 13 3 13 V. 31 - 13 3 V1 + C2C3 [(A21 + Al3 - A32) (1 — fif) + 2(A31 - A13) (III-29) v V" c'V ._1 _ .41 _._Z_l _ 2( ) (A32 A23) + 0123(1-+__ 1] __ [CT(A21 + A13 A32) 3 V3 V3 < + 2C1(A31 ’ A13) + 2C3(A32 ' A23) ' C123(CT _ zclfl " 4 (1 _VV) 31m y. 1 2 v2 ac _ ‘3' 2C2 [éizci + 2C1(A21 ' A12} + C2A12(1 ’1:— 1 3 (III-30) V” 2c v 2 2 3 2 +C —— — - 3A13 G V') —- [213CT + 2C1(A31 A13)] 129 <1 |N<1| )1 U.) U») U.) V 2 + [c3 - c2 V] [013le + A13 — A32) + 2C1(A31 - A13) (III-30) 3 3 1n Y1 V; + 2C3(15‘32 A23) C123(‘31 201) " C4 1 ’ V 3C BC aln y2 Bln Y2 Similar expressions can be obtained for -—————— and -—-——- 2 Cl 1 C2 by taking the apprOpriate partial derivatives of equation 165. 130 APPENDIX IV Fortran Program to Calculate the Reduced Second Moment, D2m This program obtains the refractive index gradient curve for each exposure and from this calculates the number of fringes, J, the centroid, x , and the second moment, m Using these values c,m 2' of m and the measured time for each exposure, the reduced second 2 moment, D2m’ and time correction, t , are calculated. The tan— corr. gent method is used at the ends of each exposure to obtain the refrac— tive index gradients while the difference between fringes is employed in the center regions. The gradients in the center regions are fitted to a quadratic, AX2 + BX + C, in an attempt to obtain the best maximum. By obtaining a smooth curve by the least squares technique, it is thought that a more reliable maximum can be obtained. It was found that extensive scatter would give unreliable maxima by this tech- nique but would not affect the values of x calculated at which the maxima actually occur. Areas under curves were obtained by summing the areas calculated under various regions of the curves. At each end of the curve, the points were fitted to a quadratic and the area obtained by integrating the resulting quadratic equation. In the center 131 region, the area was obtained by using the trapezoidal rule between each adjacent point. This latter technique was very reasonable since the points were so close together in this region. Calculated areas agreed very well with those obtained graphically. Loading, fortran evaluation, and set-up on the Control Data Corporationfls 3600 computer require approximately 45 seconds for this program. Execution time for one run with an average of 5 ' exposures per run is approximately 3 seconds. Thus in one minute, the computer is capable of calculating 5 runs. Fortran Program for CDC 3600 PROGRAM CALCD2M DIMENSION RUN(20),DN(lOO),X(100),Y(100),T(15),SECMOM(15),D2M(15), lXR(lOO),XL(lOO),THETA(100),RAD(100),BETA(100),XN2(100),XM(lOO) CONST = THE MAGNIFICATION FACTOR. FOR THIS CASE IT IS = 1.923 NRUNS = THE NUMBER OF PLATES EVALUATED J = THE NUMBER OF EXPOSURES EVALUATED T = THE EXPERIMENTALLY MEASURED TIME IN SECONDS WHICH HAS PASSED UP TO THE PARTICULAR EXPOSURE BEING EVALUATED M = THE NUMBER OF DATA POINTS FOR THE PARTICULAR EXPOSURE EVALUATED DN = THE DERIVATIVE OF THE REFRACTIVE INDEX WITH RESPECT TO THE MEASURED DISTANCE. DELN = THE TOTAL REFRACTIVE INDEX CHANGE, CALCULATED NUMERICALLY, ACROSS THE BOUNDARY XCENT = THE CENTROID OF THE REFRACTIVE INDEX GRADIENT CURVE. IT IS = THE FIRST MOMENT OF THE CURVE SECMOM = THE SECOND MOMENT OF THE REFRACTIVE INDEX GRADIENT CURVE 1 FORMAT (8F10.5) FORMAT (I3) 3 FORMAT (*-*,*THE NUMERICALLY INTEGRATED VALUE FOR THE TOTAL REFRAC 1TIVE INDEX CHANGE ACROSS THE BOUNDARY IS -*) FORMAT (*0*,30x, E13.7) FORMAT (F10.4) FORMAT (*0*,*THE CALCULATED CENTROID OF THE GRADIENT OF THE REFRAC 1TIVE INDEX CURVE IS —*) 9 FORMAT (*0*, *SECOND MOMENT*, 5X, *TIME IN SECONDS*,10X,*D2M*) 10 FORMAT (*0*, El3.7, 6X, F10.4, 10X, E13.7) 11 FORMAT (*O*,*THE RESULTS OF THE LEAST SQUARES ANALYSIS ARE -*) 12 FORMAT (*O*,5X,*D2M*,17X,*INTERCEPT*,lOX,*ACTUAL INITIAL TIME*) 0000000000000 N O\U1-L\ 13 14 15 16 17 18 19 101 102 103 1000 1001 35 36 37 70 132 FORMAT (*0*,El6.9,5X,E16.9,15X,F10.4) FORMAT (*1*,*HERE ARE THE RESULTS FOR RUN NO. *14) FORMAT (10F5.3) FORMAT (* *,E13.6,2X,El3.6,8X,E13.6) FORMAT (* *,5X,*X*,l4X,*DN/DX*,6X,*((X — XCENT)XX2 X DN/DX)*) FORMAT (*0*,5X, *XMAX*, 9X, *MAX DN/DX*) FORMAT (*0*,2x, E13.6, 2X, El3.6) FORMAT (.1615) FORMAT (314) FORMAT (2F10.7) FORMAT (*0*,5x,* A *,10X,* B *,10X,* C *) FORMAT (*0*,1X,El3.6,2X,E13.6,2X,E13.6) READ 5, CONST READ 2, NRUNs READ 101, (RUN(I), I = 1,NRUNS) DO 40 N0 = 1, NRUNS PRINT 14, RUN(NO) READ 2, READ 2, J NSQ READ 103, CONVl, CONV2 DO 20 L = 1,J READ 5, T(L) READ 102, N1, N2, N3 N1 = M M: Ml READ 1, READ 1, READ l, READ l, READ 1, DO 35 II = N1 + N3 - 1 + II THETA(III) RAD(III) III DN(III) X(III) DO 36 I6 = = N1 DN(N) X(N) = DO 37 111 = RAD(III) DN(IIl) DO 70 I Y(I7)= M2= M 0 0 =0 + N2 + N3 - l - 1 (X(I): I = (THETA(I), (XN2(I), I (BETA(I), I (XM(I), I 1,N2 ,Nl) 2) ,N2) 1,N3) ) 1 N ll Ill-IF ,N1 1, 1 = BETA(II) = THETA(III) * 3.1416/180 = TANF(RAD(III))/CONV2 = XN2(II) 2,N3 + I6 - l l.0/(XM(I6) - XM(16 — 1)) (XM(16) + XM(I6 - 1))/2.0 1,N1 = THETA(III) * 3.1416/180 = TANF(RAD(IIl))/CONV1 = 1,M DN(I7) 2 * (NSQ - 1) * (NSQ - 1) -4*(NSQ-l)+l 7 <3<3t§rol 00 8O 31 32 60 20 133 SX2 = 0.0 SX3 = 0.0 SX4 = 0.0 SYX2 = 0.0 DO 80 J4 = NSQ2, M2 SX = SX + X(J4) SY = SY + Y(J4) SXY = SXY + X(J4) * Y(J4) SX2 = SX2 + X(J4)**2 SX3 = SX3 + X(J4)**3 8X4 = 8X4 + X(J4)**4 SYX2 = SYX2 + Y(J4) * X(J4)**2 DET =NSQ3*SX2*SX4 — SX**2*SX4 -NSQB*SX3**2 — SX2**3 + 2.0*SX3*SX2* lSX A = (SYX2*(NSQ3*SX2- SX**2) - SX3*(NSQ3*SXY- SX*SY) + SX2*(SXY*SX 1—SX2*SY))/DET B = (SX4*(NSQ3*SXY— SX*SY) — SYX2*(NSQ3*SX3- SX2*SX) + SX2*(SY*SX3 1 - SX2*SXY))/DET c = (SX4*(SX2*SY — SX*SXY) - SX3*(SX3*SY— SXY*SX2) + SYX2*(SX3*SX 1 - SX2**2))/DET XMAX = -B/(2 * A) YMAX = A * XMAX**2 + B * XMAX + C PRINT 1000 PRINT 1001, A, B, C PRINT 18 PRINT 19, XMAX, YMAX CALL AREANUM (X,XR,XL,Y,AREA,NSQ,M,M6,A,SX,SY,SX2,SX3,SX4, lSYX2,DET,B,C,J1,M1,J3,NSQl) DELN = AREA PRINT 3 PRINT 4, DELN DO 31 12 = 1,M Y(12) = X(I2) * DN(12) CALL AREANUM (X,XR,XL,Y,AREA,NSQ,M,M6,A,SX,SY,SX2,SX3,SX4, lSYX2,DET,B,C,J1,M1,J3,NSQ1) XCENT = AREA/DELN PRINT 6 PRINT 4, XCENT DO 32 I3 = 1,M Y(13) = (X(I3) — XCENT)**2 * DN(I3) PRINT 17 DO 60 I9 = 1,M PRINT l6, X(I9), DN(I9), Y(I9) CALL AREANUM (X,XR,XL,Y,AREA,NSQ,M,M6,A,SX,SY,SX2,sx3,SX4, lSYX2,DET,B,C,J1,M1,J3,NSQ1) SECMOM(L) = AREA/(CONST**2 * DELN) D2M(L) = SECMOM(L)/(2.0 * T(L)) CONTINUE PRINT 9 DO 33 14 = l,J 134 33 PRINT 10, SECMOM(I4), T(I4), D2M(I4) = 0. 0 = 0. 0 = 0 0 SXY= 0.0 D0 34 15 = 1,1 SX = SX + T(15) SY = SY + SECMOM(I5)/2.0 52 = $2 + T(IS)**2 34 SXY = SXY + T(15) * SECMOM(IS)/2.0 SLOPE = (SX *SY - J *SXY)/(SX**2 - J * sz) CEPT = (SY - SLOPE * SX)/J TINIT = -CEPT/SL0PE PRINT 11 PRINT 12 PRINT 13, SLOPE, CEPT, TINIT 40 CONTINUE END SUBROUTINE AREANUM (X,XR,XL,Y,AREA,NSQ,M,M6,A,SX,SY,SX2,SX3,SX4, lSYX2,DET,B,C,J1,Ml,J3,NSQl) DIMENSION X(100), XR(100), XL(100), Y(100) AREA = 0.0 NSQl NSQ - 1 D0 100 I = 1,NSQ 100 XL(I) = X(I) - X(l) M6 = M - NSQ DO 110 I = M6,M 110 XR(I) = X(M) — X(I) = Y(2)/XL(2)**2 AREA = AREA + (A/3.0)*XL(2)**3 =Y(M-1) /XR(M-l)**2 AREA = AREA + (A/3.0)*XR(M—l)**3 SX 0 0 SY 0 0 SXY - SX2 SX3 SX4 . SYX2 = 0.0 DO 200 J2 = 2 ,NSQ SX = SX X+ X(J2) SY = SY + Y(J2) "ll OCDCDCD0 OOOO SXY = SXY + X(J2) * Y(J2) SX2 = SX2 + X(J2)**2 SX3 = SX3 + X(J2)**3 SX4 = SX4 + X(J2)**4 200 SYX2 = SYX2 + Y(JZ) * X(J2)**2 DET = NSQ1*SX2*SX4 - SX**2*SX4 -NSQ1*SX3**2 - SX2**3 + 2.0*SX3*SX2* 18X 135 A = (SYX2*(NSQ1*SX2— SX**2) - SX3*(NSQ1*SXY- SX*SY) + SX2*(SXY*SX 1—SX2*SY))/DET B = (SX4*(NSQl*SXY— SX*SY) - SYX2*(NSQI*SX3— SX2*SX) + SX2*(SY*SX3 1 - SX2*SXY))/DET C = (SX4*(SX2*SY — SX*SXY) - SX3*(SX3*SY- SXY*SX2) + SYX2*(SX3*SX 1 - SX2**2))/DET 300 AREA = AREA + (A/3.0)*(X(NSQ - 1)**3 - X(2)**3) + (B/2.0)*(X(NSQ - 1 1)**2 - X(2)**2) + C*(X(NSQ - 1) - X(2)) SX = 0.0 SY = 0.0 SXY = SX2 SX3 SX4 SYX2 = 0.0 DO 700 12 = 1,NSQI O 0 O. 0 0 GOOD J2 = M - NSQ + 12 SX = SX + XR(J2) SY = SY + Y(J2) SXY = SXY + XR(J2) * Y(J2) SX2 = SX2 + XR(J2)**2 SX3 = SX3 + XR(J2)**3 SX4 = SX4 + XR(J2)**4 700 SYX2 = SYX2 + Y(J2) * XR(J2)**2 DET =NSQ1*SX2*SX4 - SX**2*SX4 - NSQ1*SX3**2 — SX2**3 + 2.0*SX3*SX2 1*SX A = (SYX2*(NSQI*SX2- SX**2) - SX3*(NSQ1*SXY- SX*SY) + SX2*(SXY*SX l-SX2*SY))/DET B = (SX4*(NSQ1*SXY- SX*SY) - SYX2*(NSQ1*SX3- SX2*SX) + SX2*(SY*SX3 1 - SX2*SXY))/DET C = (SX4*(SX2*SY - SX*SXY) — SX3*(SX3*SY- SXY*SX2) + SYX2*(SX3*SX 1 - SX2**2))/DET J1 = M - NSQ + 2 M1 = M - 1 AREA = AREA + (A/3.0)*(XR(Jl)**3-XR(M1)**3) + (B/2.0)*(XR(J1)**2 1 — XR(M1)**2) + C*(XR(J1) - XR(M1)) J3 = J1 - 1 DO 500 J = NSQI,J3 500 AREA = AREA + (l.0/2.0)*(Y(J) + Y(J+1))*(X(J+1) - X(J)) RETURN END 'RUN,1,1200 APPENDIX V FORTRAN PROGRAM TO SOLVE FOR /|Dij| This program utilizes the Newton—Raphson method to Obtain a root to a polynomial and then factors this root by synthetic division in a subroutine. A root to the resulting polynomial (which is Of one less degree than the previous) is again obtained by the Newton—Raphson method and is factored out by synthetic division. This is repeated until only a linear equation remains. This program obtains only real roots. If both imaginary and real roots are present, the real roots are determined first and then the computer cycles until the designated time limit is reached. This program can be generalized to an Nth order polynomial by simply reading in all the coefficients of the polynomial rather than calculating them, as was done here for convenience. Fortran Program for IBM 1800 // JOB // FOR PRTS *IOCS(CARD, 1443 PRINTER) *NONPROCESS PROGRAM DIMENSION A(50), B(50), C(50) 1 FORMAT (12) 2 FORMAT (4E13.6) 3 FORMAT (' COEFFICIENTS OF POLYNOMIAL OF DEGREE '12,2X,'(STARTING IWITH THE HIGHEST POWER OF X) ARE') 136 O\U'l 14 10 11 13 12 137 FORMAT (5X, E13.6) FORMAT ('0', ' ROOT NUMBER ' 12,1X, ' IS ' E13.6) FORMAT (' NEW COEFFICIENTS AFTER DIVIDING OUT ABOVE ROOT ARE') READ (2,1) N N1 = N + 1 READ (2,2) SA, CTA, 82M, CT2M A(l) 1.0 A(2) CT2M - CTA * szM/SA A(3) 0.0 A(4) -(82M/SA)**2 WRITE (3,3) N DO 14 I = 1,N1 WRITE (3,4) A(I) N IS DEGREE 0F POLYNOMIAL P = POLYNOMIAL EVALUATED AT X DP = DERIVATIVE OF P EVALUATED AT X READ (2,1) NRTS DO 12 J = 1, NRTS X = 0.000001 CALL SYND (A,B,C,X,P,DP,N,N1) XPRE = X X = XPRE - P/DP IF (1.0E-11 - ABS (X— XPRE)) 10, 11, 11 WRITE (3,5) J,X WRITE (3,6) DO 13 I = 1,N WRITE (3,4) B(I) A(I) = B(I) N = N - 1 N1 = N + 1 CALL EXIT END // FOR SYND *NONPROCESS PROGRAM SUBROUTINE SYND (A,B,C,X,P,DP,N,N1) DIMENSION A(50), B(50), C(50) B(l) A(l) DO 8 = 2, N1 B(I) A(I) + X * B(I — 1) C(1) B(l) DO 9 = 2, N C(I) B(I) + X * C(I — l) P = B(Nl) DP = C(N) RETURN END H II H II II // XEQ PRTS *CCEND APPENDIX VI EXPERIMENTAL DATA Tflfle9. Component Hexane Dodecane Hexadecane Toluene Chlorobenzene Bromobenzene Diethyl Ether Chloroform Carbon Tetra- chloride Acetone Benzene Molecular weight 86.17 170.33 226.44 92.13 112.56 157.02 74.12 119-39 153.84 78.11 * Reference 50 Reference 42 138 Physical properties of the pure components. Temper- Measured Molar ature Viscosity Volume OC cp cc/mole 25 0.296 131.6 25 1.338 228.6 25 3.031% 294.1 30 0.515* 107.4 30 0.713* 106.0 30 0.985 102.8 25 0.217 104.7 25 0.542 80.7 25 0.890 97.12 25 0.308 73.99 25 0.597 89.40 Measured Refractive Index 1.3727 1.4196 1.4324# 1.492407% 1.5221# 1.5576 1.3500 1.4422 1.4570 1.3566 1.4981 139 Table 10. Initial concentration differences. Dodecane (1) - Hexadecane (2) - Hexane (3) T = 25 00 ‘61 = 1.615 '02 = 1.464 ‘03 = 1.533 Run NO. a1 (mOTSE/l.) 154 -1.134 -0.120 155 -0.239 -0.0185 157 0.176 0.0168 158 1.352 0.0788 160 -2.416 0.156 161 3.599 0.128 162 1.257 0.0716 Diethyl Ether (A) - O T = 25 0 CA = 2.646 CB = 2.653 CC = 5.309 Run NO. 0 AC 1 (moleé/l.) 175 1.002 -O.0864 176 0.623 -0.0481 178 1.317 -0.0629 179 1.150 -0.0568 moles/liter AC2 (moles/1.) 0.134 0.0571 0.0467 -O.0122 -O.131 -O.OS48 ”O o 0087 Chloroform (B) - Carbon Tetrachloride (C) moles/liter ACB (moles/1.) 0.0006 -O.l315 0.0683 0.0334 Table 11. Binary system (i dilute in j) Hexane in Dodecane Hexane in Hexadecane Dodecane in Hexane Dodecane in Hexadecane Hexadecane in Hexane Hexadecane in Dodecane Chlorobenzene in Bromobenzene Chlorobenzene in Toluene Bromobenzene in Chlorobenzene Bromobenzene in Toluene Toluene Toluene Acetone Acetone Benzene Benzene in in in in in in Chlorobenzene Bromobenzene Carbon Tetrachloride Benzene Carbon Tetrachloride Acetone Carbon Tetrachloride in Benzene Carbon Tetrachloride in Acetone * Reference 7 # Reference 12 140 Temperature Binary diffusion data at infinite dilution. * «Joaoaowu #*4(D-F * #fiéfiéh-QFJWD$T\HRD FJCD—q * B>OJ~JLJPFGHO HBEBHE I414LMFDCh\flLUn)FJRDRDUJK>O\$TODUHO C)U)C>#‘¢WD\OCD—QO\N)¥TC>Ovfl|4(DF4 F’F’F’FJFJFJFJFJFJFJFJFJC>C>FJC>RJFJ O 141 Table 12. Self diffusion data. Component Temperature D93X1O5 DIJX ni X107 OC cm /sec dynes Hexane 25 4021 1025 Dodecane 25 0.87: 1.20 Hexadecane 25 0.51b 1.67 Toluene 30 2.60 1.34 Chlorobenzene 30 1.79 1.28 Bromobenzene 3O 1.23b 1.21 Acetone 25 4.77b 1-57 Benzene 25 2.16 1.29 Carbon Tetrachloride 25 1.32 1.18 Chloroform 25 2.44 1.32 Diethyl Ether 25 8.73 1.96 a Obtained from a plot of self diffusion coefficient versus carbon number for various hydrocarbons (Figure 16, Appendix VI) reference 30 Table 13. ratio, 0 Reduced second moment, D A, 142 data. 2m Dodecane (l) - Hexadecane (2) — Hexane (3) Run No. 154 155 157 158 160 161 162 —1.134 -0.239 0.17 6 1.352 -2.416 3.59 9 1.257 2 0.984 1.203 1.197 1.411 0.835 1.710 1.388 Diethyl Ether (A) - Chloroform (B) Run No. 175 176 178 179 2.646 1.002 0.623 1.317 1.150 T 25 2 °C .653 2 2.113 1.585 2.557 2.303 D X10 m D X10 m C 5 = 1.533 DAXIO 0.947 1.107 1.247 1.526 0.815 2.335 1.467 5 and reduced height-area moles/liter 1ND; 324.7 300.6 283.2 256.0 350.3 206.9 261.9 - Carbon Tetrachloride (C) E = 5.309 C 5 DAX1O 2.184 1.765 2.711 2.509 5 moles/liter 1M: 218.8 243.3 193.9 203.4 143 Table 14. Second moment, m2, data. Dodecane - Hexadecane - Hexane Second Moment Measured Time m X 102 t 2 2 m (cm ) (secs) Run No. 154 00832 305 1.163 485 1.434 605 1.887 845 Run No. 155 0.636 125 0.917 245 1.209 365 1.356 425 Run NO. 157 0.955 245 1.388 425 1.959 665 2.536 905 Run No. 158 0.820 125 1.032 185 1°370 305 1.546 365 1.825 485 2.377 665 Run NO. 160 0.631 125 1.010 365 1.326 545 1.517 665 144 Table 14. (continued) Second Moment Measured Time m X 102 t 2 2 m (cm ) (secs) Run No. 161 1.350 125 1.776 245 2.198 365 2.571 485 3.414 725 Run No. 162 1.026 185 1.213 245 1.497 365 1.867 485 2.363 665 Diethyl Ether - Chloroform - Carbon Tetrachloride Run No. 175 0.901 125 1.427 245 l~972 365 2.459 485 2.948 605 3.441 725 Run No. 176 1.272 305 1.660 425 2.080 545 2.662 725 3.224 905 3.725 1085 Run No. 178 0.801 65 1.123 125 1.509 185 1.786 245 2.735 425 3.278 545 3.577 605 145 Table 14. (continued) Run No. 179 Second M0ment Measured Time m X 102 t 2 2 m (cm ) (secs) 0.926 125 1.200 185 2.010 365 2.260 425 2.925 545 3.380 665 146 Table 15. Time correction, At r’ data. COT Dodecane - Hexadecane - Hexane Run No. Am corr (sec) 154 115.3 155 137.6 157 154.0 158 174.6 160 254.4 161 272.4 162 185.0 Diethyl Ether - Chloroform - Carbon Tetrachloride Run No. Am corr (sec) 175 93.6 176 103.8 178 100.1 179 74.6 3.5 3.0 2. 5 2.0 1.5 1.0 0.5 0.0 147 1' 160 ._ 158 'A 4_ 2—155 N _- <3 H e x I N a /54 J , ..- c: ‘ ..L. m g c a " / '8 o . 160 u m 41- U) '1 A .— fl ’4 #1. db I 441 T I 1* -300 —100 100 300 500 700 900 measured time, tm, in seconds Figure 12. Second moment versus the measured time for the system dodecane (1) — hexadecane (2) - hexane (3). 148 3.5 3.0 '- __ 2.5 " N O L l 2 ...—I U1 1 r Second moment, m2 X 10 1.0 44 0.5 ‘r 0.0 I l T I -200 O 200 400 600 800 1000 Measured time, t , in seconds m Figure 13. Second moment versus the measured time for the system dodecane (1) - hexadecane (2) - hexane (3). 149 179 3.5+ ' -- 178 3.0+- ,, 1. A 2.5“ ' 175 .- N c: ‘lly ’ :1. 2.0" ; ; 176 .. N E 1? C1 0) -. 5' -_ 1.5 a ’ " / G O U , (D U) 1.0" .. 0.5" 1- 0.0 4 : 5 : + 4 : 12 : I. -100 100 300 500 700 900 1100 Figure 14. Measured time, tm, in seconds Second moment versus the measured time fo system diethyl ether (1) - chloroform (2) carbon tetrachloride (3). r the (3| (3| in 150 Table 16. Data used to calculate the experimental ternary diffusion coefficients 0 T = 25 C Dodecane(l)- Hexadecane(2)- Hexane(3) 292.30 -24.4 5 267.8 5 1.192 0.149 1.341 X10 0.939 0.01091 0.01803 1.615 1.464 1.533 Diethyl Ether(l)- Chloroform(2)- Carbon Tetrachloride(3) 289.05 -72.53 216.52 0.716 10391 2.107 4.401 -0.01143 -0.00253 2.646 2.653 5.309 151 004.4 000.4 404.4 000.0 004.0 004.0 000.4 400.0 000.0 000.0 004.4 000.4 440.4 040.0 000.0 000.0 404.4 000.0 00.0 40.0 044.4 000.4 000.4 040.0 040.0 040.0 000.4 004.0 00.0 04.0 004.4 000.4 000.4 000.0 040.0 040.0 004.0 440.0 00.0 40.0 404.4 000.4 400.4 000.0 404.0 404.0 000.0 000.0 00.0 00.0 044.4 000.4 000.4 000.0 000.0 000.0 000.0 000.0 04.0 00.0 004.4 400.4 000.4 000.0 000.0 000.0 000.0 400.0 00.0 04.0 004.4 000.4 000.4 040.0 040.0 040.0 000.0 040.0 04.0 04.0 044.4 000.4 040.4 004.0 044.0 044.0 000.0 400.0 00.0 00.0 00 mm B Amv mcflMOHgowhme moppwo n Amv mummmwm u AHV 0GO400¢ 440.4 000.4 000.4 000.0 000.0 000.0 000.0 400.0 00.0 04.0 000.4 000.4 040.4 000.0 000.0 000.0 040.0 000.0 00.0 04.0 000.4 000.4 000.4 000.0 000.0 000.0 404.4 040.0 00.0 04.0 004.4 040.4 000.4 004.4 000.0 000.0 000.0 000.0 04.0 00.0 004.4 000.4 000.4 000.0 004.0 004.0 000.0 400.0 00.0 00.0 000.4 000.4 000.4 000.0 004.0 004.0 000.0 040.0 00.0 00.0 00 om n B Amv 0Q00Q0po804m . Amv 0:00Q090400flo n AHV 0Q05008 000.4 000.4 000.4 400.0 044.0 044.0 040.0 000.4 040.0 000.0 00 0 mmsmw mmmzw 00404 0\000 00404 0\000 004x440\emv 00\0 00\0 am\4 em\0 4000 004 4x 00 mm H B Amv 0Q0xmm u Amv 0Q000©0x0m 1 Adv mudomwom 044 000 040 40 00440440400 044 04 00>40>04 0044440000 .04 04400 152 Table 17 (continued) Diethyl Ether (A) - Chloroform (B) - Carbon Tetrachloride (C) T = 25 0c 6A 2.646 moles/liter EB 2.653 moles/liter EC 5.309 moles/liter _12 0.880 moles/liter 0 0.573 cp. a/RT 0.631 b/RT 0.025 c/RT 0.025 d/RT 0.554 (RT/01)X107 1.956 dynes (RT/02)X107 1.322 dynes (RT/03)X107 1.174 dynes (RT/012)X107 0.896 dynes K 2.73 153 100 90 ‘” 8O 1' 70 ‘4 6O du- 10 1‘ 1.5 1.9 2.3 2.7 3.1 Figure 15. Typical refractive index gradient curve (run no. 160) at various times during diffusion. Hydrocarbon ethane propane butane pentane hexane heptane octane 154 Table 18. Van der waals constants, a and b, for various hydrocarbons a . 2 liter atm mole2 5.489 8.664 14.47 19.01 24.39 31.51 37.32 b liter/mole 0.06380 0.08445 0.1226 0.1460 0.1735 0.2107 0.2368 a/b (atm) 36.7 35-0 31.0 29.9 28.4 26.6 25.8 n\"' Table 19. Dodecane (l) - Hexadecane (2) - Hexane (3) i3 12 21 13 31 23 32 155 activity equations. A.. 1J 0.9815 1.0188 1.0521 0.9505 1.0717 0-9331 Activity constants used in the ternary 2-0937 -2.1136 -3.7004 3.6066 -5.8491 5.6502 Acetone (1) - Benzene (2) - Carbon Tetrachloride (3) T 13 12 21 13 31 23 32 C 123 : 0010 13 0.49 0.39 0.98 0.69 0.10 0.11 25 70 60 50 40 30 20 10 156 I l (T/D.) x 106 1 ’l '4 1 I I l 1 T l l I 6 7 8 9 10 11 12 13 14 15 16 17 carbon number Figure 16. The product of absolute temperature and the reciprocal of the self diffusion coefficient versus carbon number for several hydrocarbons. 18 NOMENCLATURE constant used in Van Laar and Wohl equations Van der Waals constant, literzatm/mole2 activity of component i constant used in Van Laar binary equations Van der Waals constant, liter/mole concentration of component i, moles/cm3 total concentration, moles/cm3 constant used in Van Laar ternary equation constant used in Wohl ternary equation reduced height-area ratio self diffusion coefficient of i, cmz/sec tracer diffusion coefficient, cmz/sec reduced second moment mutual diffusion coefficient, cmz/sec binary diffusion coefficient at infinite dilution, cmz/sec multicomponent diffusion coefficient, cmZ/sec Onsager diffusion coefficient, cmz/sec driving force for diffusion of species i frictional resisting force to diffusion of species i friction coefficient of species i 157 158 intercept (at a» = 0) of the reciprocal square root of 1 reduced height—area ratio curve intercept (at al =0) of the reduced second moment curve total number of fringes on an exposure flux of 1 relative to the coordinate of the plane across which the net volume flux is zero, moles of i/cm.2/sec. flux of 1 relative to a coordinate-fixed plane flux of i relative to the medium flux of 1 relative to the coordinate of the plane across which the net volume flux is zero tracer diffusion flux fringe number association equilibrium constant viscosity measurement constant proportionality constant between refractive index difference and fringe number proportionality constant between an increment of measured distance and fringe number intercept (at a = 1) of the reciprocal square root of l the reduced height—area ratio curve intercept (at a1 = l) of the reduced second moment curve phenomenological coefficient magnification factor, 1.923 rth moment of the refractive index gradient curve th . . r moment of component 1 of the refractive index gradient curve Avogadro's number refractive index 159 gas constant differential refractive index constant of component i differential refractive index constant of component i based on total fringes lepe of the reciprocal square root of reduced height- area ratio curve slope of the reduced second moment curve rate of internal entropy production per unit volume variance of z absolute temperature actual time time correction velocity of species 1 relative to the velocity of the medium, cm./sec. partial molar volume of i, cm.3/mole velocity of the medium relative to fixed coordinates velocity of the plane across which the net volume flux is zero relative to fixed coordinates mole fraction of 1 distance along direction of diffusion centroid of the refractive index gradient curve independent force for diffusion in constant volume system distance perpendicular to the direction of diffusion defined variable = x/(Z/E) refractive index fraction of i activity coefficient of i Kronecker delta Subscripts 11 12 viscosity 160 angle formed by the tangent to the refractive index curve, degrees chemical potential of i partial derivative of “i with respect to Cj kinematic viscosity friction coefficient of i (01 = Nfi) refers refers refers refers refers refers refers t0 t0 to to t0 to to stoichiometric quantity of A arbitrary component or species measured values true quantity of A true quantity of B true quantity of AA complexes true quantity of AB complexes 10. 11. 12. 13. 14. BIBLIOGRAPHY Anderson, D. K., Ph. D. Thesis, University of Washington, (1958). Anderson, D. K., and Babb, A. L., J. of Phy. Chem., 65, 1281 (1961). Baldwin, R. L., Dun10p, P. J., and Gosting, L. J., J. of Am. Chem. Soc., 11, 5235 (1955). Bidlack, D. L., Ph. D. Thesis, Michigan State University, (1964) Bronsted, J. N. and Koefoed, J., Kgl. Danske Videnskab. Selskab., Mat. — Fis. - Med., 22, N0. 17, l (1946). Brown, I. and Smith, F., Aus. J. Chem., 10, 423 (1957). Burchard, J. K. and Toor, H. L., J. of Phy. Chem., 66, 2015 (1962). Caldwell, C. S., Hall, J. R., and Babb, A. L., Rev. Sci. Instr., 28, 816 (1957). Callen, H. B., Thermodynamics, John Wiley and Sons, Inc., New York and London, 1960, Part III. Christian, S. D., Naparko, E., and Affsprung, H. E., J. of Phy. Chem., 64, 442 (1960). Courant, R., Differential and Integral Calculus, Volume II, Blackie and Son, Ltd., London and GlangW, 1936, p. 55. Cullian, Jrs., H. T. and Toor, H. L., J. of Phy. Chem., 62, 3941 (1965). De Groot, S. R., Thermodynamics of Irreversible Processes, Interscience Publishers, Inc., New York, New York, 1951. Dole, M., J. of Chem. Phy., 25, 1082 (1956). 161 162 15. Dolezalek, F., Z. Physik. Chem., 64, 727 (1908). 16. Dolezalek, F. and Schulze, A., Z., Physik. Chem., 66, 45 (1913). 17. Douglass, D. C. and McCall, D. W., J. of Phy. Chem., 62, 1102 (1958). 18. Dun10p, P. J., J. of Phy. Chem., 63, 612 (1959). 19. Dun10p, P. J. and Costing, L. J., J. of Am. Chem. Soc., 21, 5238 (19559. 20. Einstein, A., Ann. Physik, Series 4, 41, 549 (1905). 21. Fick, A., Ann. Physik, 24, 59 (1855). 22. Fujita, H. and Costing, L. J., J. of Am. Chem. Soc., 64, 1256 (1960). 23. Fujita, H. and Costing, L. J., J. of Phy. Chem., 64, 1256 (1960). 24. Gibbs, J. W., "Collected Works", Volume 1, Thermodynamics, Yale Press, New Haven (1948). 25. Glastone, S., Laidler, K. J., and Eyring, H., The Theory of Rate Processes, McGraw—Hill Book Co., Inc., New York, 1941. 26. Costing, L. and Morris, M. S., J. of Am. Chem. Soc., 1;, 1998 (1949). 27. Hartley, G. S. and Crank, J., Trans. Far. Soc., 46, 801 (1949). 28. Hooyman, G. J. and De Groot, S. R., Physica, 2;, 73 (1955). 29. Kaplan, W., Advanced Calculus, Addison — Wesley Publishing Co., Inc., Cambridge, Mass., 1953, p. 218. 30. King, C. J., Hsueh, L., and Mao, K., J. of Chem. Eng. Data, 46, No. 4, 348 (1965). 31. Kirkwood, J. G., J. of Polymer Sci., 42, 1 (1954). 32. Mickley, H. S., Sherwood, T. K., and Reed, C. B., Applied Mathematics in Chemical Engineering, McGraw-Hill Book Co., Inc., New York, Toronto, and London, 1957. 33, Miller, D. 0., J. of Phy. Chem., 23.: 570 (1959). 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 163 Miller, D. G., Chem. Rev., 66, 15 (1960). Miller, D. G., J. of Phy. Chem., 62, 3374 (1965). Nikol'skii, Teoreticheskaya i Eksperimental'naya Khimiya, 2, No. 3, 343 (1966). Onsager, L., Phy. Rev., 61, 405 (1931); 66, 2265 (1931). Onsager, L., Ann. N. Y. Acad. of Sci., 46, 241 (1945). 0thmer, D. F. and Thakar, M. S., Ind. Eng. Chem., 46, 589 (1953). Prigogine, I. and Defay, P., Chemical Thermodynamics, Longmans, 1954, p. 409. Shuck, F. 0. and Toor, H. L., J. of Phy. Chem., 61, 540 (1963). Timmermans, J.,Physico-Chemical Constants of Pure Organic Compounds, Elsevier Publishing Co., Inc., New York, N. Y., (1950). Van Laar, Z. Physik. Chem., 13, 723 (1910); 66, 599 (1913). Weir, F. E. and Dole, M., J. of Am. Chem. Soc1,_§Q, 302 (1957). Wendt, R. P., J. of Phy. Chem., 66, 1279 (1962). Wilke, C. R. and Chang, P., A. I. Ch. E. Jour., l, 264 (1955). Wirth, G. B., Ph. D. Thesis, Michigan State University, 1968. Wohl, K., Trans. A. I. Ch. E., 42, 215 (1946). Woolf, L. A., Miller, D. G., and Costing, L. J., J. of Am. Chem. Soc., 64, 317 (1962). Yon, C. M. and Toor, H. L., Private Communication. Zehnder, L., Z. Instrumentenk., ll, 275 (1891). "I111111114111111113