DIFFUSIVITIES IN THE ETHANOL-WATER SYSTEM -THE APPLICABILITY OF THE DIAPHRAGM CELL METHOD TO THE CASE OF SYSTEMS WHERE VOLUME CHANGES OCCUR ON MIXING by FRANCIS ANDREW LESLIE DULLIEN Dipl. Chem. Eng., • Technical University of Budapest, 1950 M.AiSc, University of British Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Chemical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, I960 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada. P U B L I C A T I O N S 1. G . Varsanyi, F. Szathmary, and F. Dullien, Photometric Evalua-tion of u.v. Absorption Spectra and Application of Method to Estimation of O-Diethyl-benzene in Benzene, Magyar Kemiai Folyoirat, 58, 13 (1952). 2. Significance of Raman Spectroscopy in the Investigation of Structure of Molecules, Magyar Kemikusok Lapja, 7, 326 (1952). 3. Apparatus for Making Raman Spectra, Magyar Kemiai Folyoirat, 59, 244 (1953). 4. Preliminary Communication on the Problem of Suppressing the Continuous Background of Raman Spectra by the Use of Optical Filters, M . T . A : Kozponti Fizikai Kutato Intezete Kozlemenyei, 223 (1955). 5. Contribution to the Theory of Raman Tubes with Special Con-sideration of the Micro Technique, Acta Phys. Acad. Sci. Hung. 7, 181 (1957). 6. Raman Spectra and Configuration of Some Alpha-Furyl and Alpha-Benzofuryl Ketoximes, Can. J. Chem., 35, 1366 (1957). 3% Pttfogrstig of ^rtitsif Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE F I N A L O R A L E X A M I N A T I O N FOR T H E D E G R E E OF D O C T O R O F P H I L O S O P H Y of FRANCIS A. L. DULLIEN Dipl. Chem. Eng., Technical University of Budapest, 1950 M . A . S c , University of British Columbia, 1958 M O N D A Y , SEPTEMBER 19th, I960 A T 3:30 P.M. IN ROOM 423, CHEMISTRY BUILDING COMMITTEE IN CHARGE F. A . F O R W A R D , Chairman L . W. S H E M I L T D . M . M Y E R S J. S. F O R S Y T H C . A . M c D O W E L L N . E P S T E I N W. M . A R M S T R O N G S. D . C A V E R S A . M . C R O O K E R T . E . H U L L M . D A R R A C H ' S. Z . P E C H External Examiner: A . L . BABB University of Washington DIFFUSIVITIES IN T H E E T H A N O L - W A T E R S Y S T E M — T H E A P P L I C A B I L I T Y O F T H E D I A P H R A G M C E L L M E T H O D T O T H E C A S E O F S Y S T E M S W H E R E V O L U M E C H A N G E S O C C U R O N M I X I N G . A B S T R A C T Smith and Storrow [J. Appl. Chem. (London) 2: 225, (1952)1 and Hammond and Stokes [Trans. Faraday Soc. 49: 890, (1953)3, using their individual modifications of the diaphragm-cell method for measuring liquid diffusivities, reported diffusion coefficients on the ethanol-water system that disagreed by as much as 100 per cent. It was apparently in view of this disagreement that Johnson and Babb gave a very reserved summary opinion on the diaphragm-cell method in a comprehensive review article CChem. Rev. 56: 387, (1956)1. In the present work a diaphragm cell was designed which, unlike prior designs, is suitable for use with practically all organic liquids. A compact machine, accommodating and stirring a battery of six such cells, was designed and used. -The precision of the cell constant determinations, using aqueous KC1 solutions, was ± 0.1%, a scatter completely accounted for by the errors arising from a standard gravimetric analysis of the solu-tions which, in this case, was improved in accuracy by one order of magnitude over hitherto reported analyses. Using this apparatus, the diffusivity results obtained by Ham-mond and Stokes were confirmed within the accuracy of the ethanol-water measurements C ± 2 % ) . Critical experiments on an apparatus similar to that used by Smith and Storrow revealed that there was a possibility of distilla-tion through a wetted ground glass joint from the one solution into the other. The apparent higher diffusivities obtained by these authors were attributed to this phenomenon. Detailed derivation of the general equation of diffusion, using the methods of the thermodynamics of irreversible processes, and discussion of the question of the various frames of references and the various diffusivities defined by some authors is given; Using the general equation of diffusion in a binary system new formulae which apply regardless.of volume changes on mixing were derived for use with the diaphragm cell method. It was shown that, in the case where the density of the solution is linear in the volume concentra-tion, the general equation of diffusion reduces to Fick's first law and the various formulae based upon the general equation reduce to the corresponding simple formulae. The trial-and-error procedure used with electrolytes by R. H . Stokes was generalized and used to compute the true differential diffusivities from the measured integral values and it was shown that, contrary to the contention of Stokes, this type of procedure is applicable in the case of systems where the limiting values of the diffusivity are not known and the diffusivity changes strongly with concentration. Also, an alternate procedure, based on the differen-tiation of the integral diffusivities, is suggested. It was noted that the minimum of the diffusivity-concentration curve and that of the relative volume decrease-concentration curve, and the maximum of the viscosity-concentration curve are at a com-position corresponding to one molecule of ethanol and three mole-cules of water. This coincidence may be evidence for the existence of a molecular complex. It was found that the activity-based diffusivity is practically constant over some 70% of the composition range. G R A D U A T E STUDIES Field of Study: Transport Processes Momentum Heat and Mass Transfer N . Epstein Distillation J. S. Forsyth Solvent Extraction and Gas Absorption S. D . Cavers Fluid and Particle Dynamics N . Epstein Related Studies: Linear Algebra H . Davis Differential Equations C . A . Swanson Complex Variables W. H . Simons Theoretical Mechanics W. Opechowski Statistical Theory of Matter W. Opechowski P U B L I C A T I O N S 1. G . Varsanyi, F. Szathmary, and F. Dullien, Photometric Evalua-tion of u.v. Absorption Spectra and Application of Method to Estimation of O-Diethyl-benzene in Benzene, Magyar Kemiai Folyoirat, 58, 13 (1952). 2. Significance of Raman Spectroscopy in the Investigation of Structure of Molecules, Magyar Kemikusok Lapja, 7, 326 (1952).: l 3. Apparatus for Making Raman Spectra, Magyar Kemiai Folyoirat, 59, 244 (1953). 4. Preliminary Communication on the Problem of Suppressing the Continuous Background of Raman Spectra by the Use of Optical Filters, M . T . A . Kozponti Fizikai Kutato Intezete Kozlemenyei, 223 (1955). 5. Contribution to the Theory of Raman Tubes with Special Con-sideration of the Micro Technique, Acta Phys. Acad. Sci. Hung. 7, 181 (1957). 6. Raman Spectra and Configuration of Some Alpha-Furyl and Alpha-Benzofuryl Ketoximes, Can. J. Chem., 35, 1366 (1957). FACULTY OF GRADUATE STUDIES PROGRAMME OF THE F I N A L O R A L E X A M I N A T I O N FOR T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y of FRANCIS A. L. DULLIEN Dipl. Chem. Eng., Technical University of Budapest, 1950 M . A . S c , University of British Columbia, 1958 M O N D A Y , SEPTEMBER 19th, I960 A T 3:30 P.M. IN ROOM 423, CHEMISTRY BUILDING COMMITTEE IN CHARGE F. A . F O R W A R D , Chairman L. W. S H E M I L T D . M . M Y E R S J. S. F O R S Y T H C . A . M c D O W E L L N . E P S T E I N W. M . A R M S T R O N G S. D . C A V E R S A . M . C R O O K E R T . E . H U L L M . D A R R A C H S. Z . P E C H External Examiner: A . L . B A B B University of Washington DIFFUSIVITIES IN T H E E T H A N O L - W A T E R S Y S T E M — T H E A P P L I C A B I L I T Y O F T H E D I A P H R A G M C E L L M E T H O D T O T H E C A S E O F S Y S T E M S W H E R E V O L U M E C H A N G E S O C C U R O N M I X I N G . A B S T R A C T Smith and Storrow [J. Appl. Chem. (London) 2: 225, (1952)] and Hammond and Stokes [Trans. Faraday Soc. 49: 890, (1953)1, using their individual modifications of the diaphragm-cell method for measuring liquid diffusivities, reported diffusion coefficients on the ethanol-water system that disagreed by as much as 100 per cent. It was apparently in view of this disagreement that Johnson and Babb gave a very reserved summary opinion on the diaphragm-cell method in a comprehensive review article [Chem. Rev. 56: 387, (1956)1. In the present work a diaphragm cell was designed which, unlike prior designs, is suitable for use with practically all organic liquids. A compact machine, accommodating and stirring a battery of six such cells, was designed and used. The precision of the cell constant determinations, using aqueous KC1 solutions, was + 0.1%, a scatter completely accounted for by the errors arising from a standard gravimetric analysis of the solu-tions which, in this case, was improved in accuracy by one order of magnitude over hitherto reported analyses. Using this apparatus, the diffusivity results obtained by Ham-mond and Stokes were confirmed within the accuracy of the ethanol-water measurements C+2%). Critical experiments on an apparatus similar to that used by Smith and Storrow revealed that there was a possibility of distilla-tion through a wetted ground glass joint from the one solution into the other. The apparent higher diffusivities obtained by these authors were attributed to this phenomenon. Detailed derivation of the general equation of diffusion, using the methods of the thermodynamics of irreversible processes, and discussion of the question of the various frames of references and the various diffusivities defined by some authors is given. Using the general equation of diffusion in a binary system new formulae which. apply regardless, of volume changes on mixing were derived for use with the diaphragm cell method. It was shown that, in the case where the density of the solution is linear in the volume concentra-tion, the general equation of diffusion reduces to Fick's first law and the various formulae based upon the general equation reduce to the corresponding simple formulae. The trial-and-error procedure used with electrolytes by R. H . Stokes was generalized and used to compute the true differential diffusivities from the measured integral values and it was shown that, contrary to the contention of Stokes, this type of procedure is applicable in the case of systems where the limiting values of the diffusivity are not known and the diffusivity changes strongly with concentration. Also, an alternate procedure, based on the differen-tiation of the integral diffusivities, is suggested. It was noted that the minimum of the diffusivity-concentration curve and that of the relative volume decrease-concentration curve, and the maximum of the viscosity-concentration curve are at a com-position corresponding to one molecule of ethanol and three mole-cules of water. This coincidence may be evidence for the existence of a molecular complex. It was found that the activity-based diffusivity is practically constant over some 70% of the composition range. G R A D U A T E STUDIES Field of Study: Transport Processes Momentum Heat and Mass Transfer N . Epstein Distillation J. S. Forsyth Solvent Extraction and Gas Absorption S. D . Cavers Fluid and Particle Dynamics . . . . . N . Epstein Related Studies: Linear Algebra , . H . Davis Differential Equations C . A . Swanson Complex Variables W. H . Simons Theoretical Mechanics W. Opechowski Statistical Theory of Matter W. Opechowski i i ABSTRACT Smith and Storrow [ J . Appl. Chem. (London) 2s 225, (1952)] and Hammond and Stokes [Trans. Faraday Soc. 4J?! 890, (1953)], using their individual modifications of the diaphragm-cell method for measuring liquid d i f f u s i v i t i e s , reported diffusion coefficients on the ethanol-water system that disagreed by as much as 100 per cent. It was apparently i n view of this disagreement that Johnson and Babb gave a very reserved summary opinion on the diaphragm-cell method i n a comprehensive review article [Chem. Rev. 5_6s 387, (1956)]. In the present work a diaphragm c e l l was designed which, unlike prior designs, i s suitable for use with practically a l l organic • liquids, A compact machine, accommodating and s t i r r i n g a battery of six such c e l l s , was designed and used. The precision of the c e l l constant determinations, using aqueous KC1 solutions, was ± 0.1$, a scatter completely accounted for by the errors arising from a standard gravimetric analysis of the solutions which, i n this case, was improved i n accuracy by one order of magnitude over hitherto reported analyses.-Using this apparatus, the d i f f u s i v i t y results obtained by Hammond and Stokes were confirmed within the accuracy of the ethanol-water measurements (± 2%), C r i t i c a l experiments on an apparatus similar to that used by Smith and Storrow revealed that there was a possibility of d i s t i l l a t i o n i i i through a wetted ground glass joint from the one solution into the other. The apparent higher d i f f u s i v i t i e s obtained by these authors were attributed to this phenomenon. Detailed derivation of the general equation of diffusion, using the methods of the thermodynamics of irreversible processes, and discussion of the question of the various frames of references and the various diffus-i v i t i e s defined by some authors is given. Using the general equation of diffusion i n a binary system new formulae which apply regardless of volume changes on mixing were derived for use with the diaphragm c e l l method. It was shown that, i n the case where the density of the solution i s linear i n the volume concentration, the general equation of diffusion reduces to Fick's f i r s t law and the various formulae based upon the general equation reduce to the corresponding simple formulae. The trial-and-error procedure used with electrolytes by R. H. Stokes was generalized and used to compute the true differential d i f f u s i v i t i e s from the measured integral values and i t was shown that,, contrary to the contention of Stokes, this type of procedure is applicable in the case of systems where the limiting values of the d i f f u s i v i t y are not known and the d i f f u s i v i t y changes strongly with concentration. Also,, an alternate procedure, based on the differentiation of the integral diffus-i v i t i e s , i s suggested. It was noted that the minimum of the diffusivity-concentration curve and that of the relative volume decrease-concentration curve, and the maximum of the viscosity-concentration curve are at a composition corresponding to one molecule of ethanol and three molecules of water. This coincidence may be evidence for the existence of a molecular complex. It was found that the activity-based d i f f u s i v i t y i s practically constant over some 10% of the composition range. V TABLE OF CONTENTS Page II. Apparatus and Experimental Procedures . 4 a) Preparation of Solutions ....... «•« • 4 b) Apparatus for Diffusivity Measurements 5 1) Design of Cell ' 5 2) Cell Stirring and Support B 3) Constant Temperature Bath 13 c) Diffusivity Experiments 13 1) Degassing and Introduction of the Solutions into the Bottom Parts of the Diffusion Cells.. 13 2) Diffusion 14 3) Sampling 16 d) Experiment on the Effect of the Rate of Stirring ...... 16 e) Determination of Volumetric Data of the Diffusion Cells 19 f) Analysis 21 1) Analysis of the KC1 Solutions 21 2) Analysis of the Ethanol-Water Mixtures 25 g) Investigation of D i s t i l l a t i o n through Ground Glass Joints 31 III. Theory 3.8 a) Derivation of the General Equation of Diffusion for Binary Mxxture . . . . « • . . . . a o . . . . . . . . . . . . . . . . . . . 3B i ) The thermodynamics of irreversible processes ... 42 i i ) The equations of change o . . . . . . . . 45 v i Page III. Theory (continued) i i i ) Derivation of the entropy balance equation .... 51 iv) General case of ordinary diffusion , 57 v) The condition of mechanical equilibrium 63 vi) Ordinary diffusion - binary systems 67 b) Application of the Diffusion Equation to the Diaphragm Cell Method 81 IV. Results and Discussion 107 Results 107 Discussions 118 a) Comparison of Results with Those Obtained by Hammond and .Stokes (3) and Smith and Storrow (2) 118 b) The Relation between the True Differential Diffusivities and the Measured Integral Values 119 c) The Scatter of the Experimental Results ,, 123 d) The Systematic Error Committed by Neglecting Volume Change on Diffusion 125 e) Discussion of the Diffusivity-Composition Relation-ship 130 f) Discussion of the Function: Activity-based Diffusivity, versus Mole Fraction • 133 V. Notation , 136 VI. Bibliography 142 VII. Appendices 144 Appendix I - The Question of Analytical Errors .......... 144 a) General 144 v i i Page VII. Appendices Appendix I (continued) b) Relative Error of the Calculated Value of the Diffusivity Owing to the Error of the Measured Concentrations 145 c) Estimation of fcQ, the Probable Error of the Measured Values of the Concentrations 147 i ) The case of the KC1 solutions 147 i i ) The case of the ethanol-water mixture 152 d) The Error Introduced i n the Calculated Value of the Diffusivity via the Error Committed i n the Determination of the Cell Volumes 157 i ) Wrong values of the compartment volumes are used consistently 157 i i ) Different values for the compartment volumes are used for the calibration run and the actual run. It i s assumed that the effect of different volumes on ^ i s negligible.. 1.60 i i i ) The error in determining volume changes on , diffusion by material balances 162 x D^ ' e) Comparison of Probable Error of as Calculated D from the Analytical Error and as Obtained from the Scatter of the Diffusivity Values 163 i ) The KC1 runs 163 i i ) The alcohol-water runs 164 v i i i Page VII. Appendices Appendix II - Derivation of Some Buoyancy Formulae and Related Expressions 166 a) Derivation of Eq. ( l ) , (Section II, f, l ) .......... 166 b) Derivation of Eq. (2) , (Section II, f, 2) 167 c) Discussion of Eq. (3) , (Section II, f, 2) . . . . . . . . . . 169 Appendix III - Estimation of the Effect which the Existence of Concentration Gradients in the Cell Compartments Has on the Apparent Value of the Cell Constant (or Diffusivity) 171 Appendix IV - Calibration of Thermometers 175 r Appendix V - Numerical Procedure to Check on the Assumption of Uniform Fluxes throughout the Porous Diaphragm 179 Appendix VI - Calculation of the Unknown, Fourth Concentra= tion and the Magnitude of Volume Change in the Case where there i s a Volume Change on Diffusion 186 Appendix VII - Illustration Showing the Use of Some of the Formulae to Calculate Diffusivities 190 ix Table of Contents (Continued) List of Diagrams Page Diagram 1. Diaphram Cell 6 Diagram 2. Cell Support and Stirring Device in Section 10 Diagram 3. The Battery of Six Diaphragm Cells 12 Diagram 4. Sketch of Diaphragm Diffusion Cell used by Smith and Storrow (2) 32 Diagram 5. Apparatus used to Investigate D i s t i l l a t i o n through Ground Glass Joints 33 Diagram 6. Apparatus Used in Pseudo-Diffusion Experiments 35 Diagram 7. Schematic Representation of Concentration and Driving Force Conditions inside the Porous Diaphragm 91 Diagram 8. Integral Diffusivity - Mean Concentration Plot ........ 113 Diagram 9. True Diffusivity - Concentration Plot 114 Diagram 10. True Diffusivity - Mole Fraction Plot 115 Diagram 11. Relative Decrease i n Volume, <|> , on Mixing in the Ethanol-Water System 131 Diagram 12. Activity-Based Diffusivity and „%ln,a. as a Function dlnx of Mole Fraction , 135 Appendix I. Rate of Change of Concentration with Density as a Function of Concentration , 158 Appendix LT. Calibration of Plot Used to Determine Constants i n Equation (3) , 170 X Table of Contents (Continued^ List of Tables Page Table I. Analytical results obtained i n Runs 7 and 8 with the aqueous KC1 solutions . . . . . . . s o o . . . . . . . . . . . . . . . . . 107 Table II. Ethanol concentrations obtained i n the runs with the ethanol-water mixture at 25°C. .................. 108 Table III. Cell-compartmsnt volumes, ml 109 Table IV. Cell constants, ^ > , cm."2 109 2 -5 . Table V. Measured values of d i f f u s i v i t i e s , D, cm. , sec. ' m the ethanol-water system at 25°C 110 Table VI. Diffusivities at 25°C. i n the ethanol-water system as measured by Smith and Storrow (10) ........... 1.16 Table VII. Diffusivity i n the ethanol-water system as a function of the volume concentration C at 25°C 116 Table VIII. Diffusivity and the diffusivity-density product i n the ethanol-wat.er system as a function of weight fraction of ethanol at 25°C. 117 ACKNOWLEDGEMENTS xi The author wishes to express his sincere gratitude for the financial support provided by the National Research Council, by the Eldorado Mining and Refining Company through the Charles G. Williams Fellowship for the academic year 1957-1958, and for a CIL scholarship for the academic year 1959-1960. Thanks are extended to Drs. N. Epstein, D.W. Thompson, and J.S. Forsyth of the.Department of Chemical Engineering for helpful discussions, to members of the departmental workshop for their assist-ance, and i n particular to Mr. E. Rudischer who was always ready to help. The author feels deeply obligated to Dr. L.W. Shemilt for starting him on this project and for his most considerate attitude and help throughout the years the work was i n progress. Contribution to this work by Mr. V.M. Bothner, B.A.Sc, and Mr. R.W. Crossland, B'.A.Sc, i s acknowledged. - 1 -I. INTRODUCTION Babb and Johnson wrote in an excellent review a r t i c l e (l) on liqu i d diffusion of non-electrolytes: "Although the diaphragm-cell technique has been widely used for diffusion measurements, serious d i f f e r -ences exist among data obtained by different investigators using this tech-nique. Recent work on the ethanol water system by Hammond and Stokes [Trans. Far. Soc. 45* 890 (1953)] and Smith and Storrow [ J . Appl. Chem. (London) 2, 225 (1952)]disagree by as much as 100 percent, yet each pair of investigators considered his results accurate to 1 and 3 percent, respectively. One source of error no doubt l i e s i n maintaining a constant cross-section for the calibration runs and the actual experimental runs, since entrapment of air or vapor could seriously alter the effective cross-section for diffusion. Another d i f f i c u l t y i s the requirement that the logarithm of the ratio of concentration differences be used to calculate D. With small differences i n i n i t i a l concentration, extreme precision i s required to reduce the error in D to a tolerable value because of the multiplying effects of the mathematical treatment". Hammond and Stokes wrote i n the introduction to their paper that "correspondence with Dr. Storrow had fa i l e d to reveal any reason for the discrepancy". The fact that two research groups obtained results i n such striking disagreement by apparently identical experimental methods i s certainly bound to arouse suspicion as to the usefulness of this particu-l a r technique of measuring d i f f u s i v i t i e s . - 2 -On the other hand, i t must be borne in mind that the diaphragm c e l l method of measuring d i f f u s i v i t i e s had been applied to a considerable number of systems with satisfactory and often excellent results. The sources of error pointed out by Johnson and Babb had been discussed, along with many other possible sources of error, by numerous investigators using this technique. In the form R. H. Stokes used the diaphragm c e l l method these sources of error had been carefully eliminated or their effect minimized,, ' Nevertheless, by virtue of the very serious disagreement dis-cussed by Johnson and Babb this technique has f a l l e n into some sort of disrepute and the issue certainly deserved attention and investigation. A limited amount of work had been done i n this Department (the Department of Chemical Engineering, University of British Columbia) with the diaphragm c e l l method [Mary McFadden, M. A. thesis (1951) and Carl Shalansky, B. A. Sc. thesis (1957)]but the problems dealt with i n the present work were not broached i n i t . / The author of this thesis set himself the task.of subjecting both the theory and the practice of the diaphragm c e l l method to a certain amount of scrutiny and, i n particular, of establishing the facts on the basis of which the reported discrepancy could be satisfactorily explained. These facts had to include the correct values of the d i f f u s i v i t i e s i n the ethanol-water mixture. Hammond and Stokes (loc. c i t . ) divided the concentration range into three parts in order to minimize the effect of volume changes on mixing, which are appreciable with ethanol and water. This, however, as also pointed out by Johnson and Babb, necessarily resulted in increased error i n the value of the measured d i f f u s i v i t i e s . Hence, i t was f e l t desirable to establish the accurate formulae to calculate d i f f u s i v i t i e s from diaphragm diffusion c e l l experiments regardless of volume changes or diffusion. To be able to do this the physical theory of the so-called "laws" of diffusion had to be examined. Realizing that this f i e l d i s not widely known i t was considered helpful to include a section on this subject i n appreciating the theoretical considerations concerning'the diaphragm c e l l technique. _ - 4 -II» Apparatus and Experimental Procedures. a) Preparation of Solutions "Reagent Special" Code 2059 KC1 was used (Baker and Adamson make) without further purification. Approximately 0.1N stock solution was prepared, 2 liters at a time. The KC1 was weighed on a Mettler "gram-atic" balance with a precision of i 2 x 10"^ g., then introduced into a 2000 ml, volumetric flask to which some distilled water (specific resistance 500,000 S<£ cm.) was added. On dissolution of the crystals the flask was f i l l e d up almost to the mark, shaken, and then adjusted to the mark. Commercial absolute ethanol was treated with activated charcoal and then distilled over in a 3-ft. glass column packed with glass helices at reflux ratio 10:1. The fi r s t tenth of the distillate was discarded and the last tenth was left as residue. After this treatment the density of the alcohol was determined and the corresponding composition looked up in the tables (U.S. Bureau of Standards, reproduced in Handbook of Chemistry and Physics, 37th edition, page 1948). Several dilutions were then prepared using gravimetric techniques and the analytical balance. The density of these dilutions was determined and the corres-ponding compositions found in the tables were compared with the calculated ones. The corresponding values never differed by more than 1 x 10 weight percent and the scatter was random. This was considered satisfactory. The diluted alcohol solutions coming from the diffusion experiments were re-used after being distilled over in the column at - 5 -reflux-ratios varying from l s l to ls3» Whereas the alcohol obtained i n the f i r s t treatment was about 96$ the re-runs produced solutions of about 90-93$ alcohol content. b) Apparatus for Diffusivity Measurements 1, ' Design of Cell The design by Stokes (4) served as a starting point because of i t s apparent great simplicity. There were, however, a few points i n this design that, i n our estimation, could be-improved. One of. these was the use of stoppers to seal the diaphragm c e l l at both ends and the ensuing d i f f i c u l t y of finding a stopper that seals without using stopeock grease and i s not attacked by common organic solvents. Apparently the best answer he found was to grind i n a glass stopper with such precision that, "when the c e l l was f i l l e d with water, no a i r leaked past the unlubricated joint when a filter-pump was attached to the upper end of the c e l l " (5). ,>' jjj the present design the necessity of using rubber stoppers and lubricated or unusually w e l l - f i t t i n g ground glass joints has been eliminated. As shown i n Diagram 1 there i s no stopper used i n the bottom' half cell"B. F i l l i n g and emptying of B i s done via two narrow {>» 0.-5 mm. i.d.) capillaries E and D f i t t e d near the ends with "polytheme-screwclips". These were made from short pieces of poly-thene tubing and commercial screweiips. The piece of polythene tube was f i r s t joined to the capillaries by application of heat. Then, - 6 -- 7 -after heating the polythene tube carefully i t was flattened by squeezing with a pair of p l i e r s 0 The screwclips were equipped with a larger handle made of thick copper wire, and were also reinforced by completing their frame to have four sides. Silver solder was used i n both cases for joining the pieces together. These "'polythene-screwclips" or valves showed no detectable leakage of water under a vacuum of about 20 mm. Hg as observed by the position of the meniscus at the end of the capillary. However, these valves were not gas-tight. Originally Teflon stopcocks were to be used but the polythene valves were found far more convenient. The Teflon stopcocks usually leaked under vacuum unless they were tightened so much as to make turning the plug far too d i f f i c u l t . Also, tightening of the nut often resulted i n breaking the thread thus making replacement of the Teflon-plug necessary. F i s "fine grade" sintered glass diaphragm. G are iron-wire-sealed-in-polythene s t i r r e r s . The question of s t i r r i n g w i l l be dealt with below. A i s the top h a l f - c e l l sealed with ground glass joint C, equipped with a Teflon stopcock. Here the Teflon-stopcock was found satisfactory because the requirement of vacuum-tightness was not essential. The use of unlubricated glass stopcocks was considered objectionable© The two capillary side-arms D and E were situated side-by-side on the same side of the c e l l and were almost touching the outside surface of the c e l l wallst About at the level of "AM i n the diagram the c e l l and the side-arms were reinforced with a few windings of plastic insulating tape. In the course of a l l the experiments described i n this thesis breakage of side-arms occurred only i n two instances and then only * Note that,, for" the sake of c l a r i t y , " i n "Diagram 1 the" two side arms are shown on the two opposite sides of, and set further apart from, the cell-body than i n actuality. ~ 8 -at a time when the tape had been removed for replacement. The length of a c e l l from the very lowest point of D up to the t i p .of E was about 14 1/2 inches and the o 0d. of the cylindrical body was 1 3/8 inches„ 2 0 Cell Stirring and Support To justify the choice of the particular design of the c e l l and the way of s t i r r i n g to be used with i t , a brief survey of the previous methods using this technique seems appropriate„ Roughly speaking, the existing c e l l designs f a l l i n two main categoriess One with the lower compartment consisting of a beaker and the upper one of a bell-shaped vessel with the diaphragm at the bottom and a tube equipped with a stopcock at the top c The other i s a cylindrical vessel divided i n two compartments by a diaphragm and with one or more openings at both ends f i t t e d with a tube or a stopcock,. Regarding the method of s t i r r i n g the solutions inside the c e l l there does not appear to be a complete agreement of views. Earli e r the so-called "density s t i r r i n g " was employed with the denser solution i n the upper compartment (6), Later views were expressed which seemed to justify the necessity of mechanical s t i r r i n g . The f i r s t methods using this latter technique employed glass beads, the c e l l being rotated either round an axis of the plane of the diaphragm (7) or round the longitudinal axis of the c e l l , the latter being i n a t i l t e d position (8)„ S t i l l later doubts were expressed (4) as to the efficiency of this method i n s t i r r i n g = 9 -the adherent layer on the two surfaces of the diaphragm, and simultaneous-l y i t was pointed out that whenever the level of the denser solution comes above the lower level of the less dense solution, there i s a danger of bulk flow from one compartment to the other. It was recommended that the c e l l be i n a vertical position (horizontal diaphragm) with the denser solution i n the lower half and that magnetic s t i r r i n g be employed i n an analogous fashion as i n titrations? a piece of iron wire sealed i n a glass or plastic tube to act as a s t i r r e r i n each compartment.. The density of the capsules i s adjusted so as to have the st i r r e r s press l i g h t l y against the surfaces of the diaphragm. The capsules are made to rotate by a rotating external magnet. One author (9) fearing that direct mechanical contact between the diaphragm and the stirrers w i l l result i n a change of the characteristics of the latter recommended a design where a small paddle mounted on a pivoted shaft acted as a s t i r r e r . In the present work freely rotating iron-sealed-in-polythene stirrers were used. It was found that the stirrers were subject to some wear which, however, did not noticeably change the characteristics of the cells., Considering the long duration of a diffusion experiment i n a diaphragm c e l l , i t was thought advantageous to provide means whereby a large number of experiments could be run simultaneously. Therefore, a machine capable of accommodating a battery of six cells was designed (see Diagram 2). The c e l l s were arranged with their centres on the six corners of a regular hexagon0 - 10 -The body.of each c e l l was surrounded by a brass sleeve D attached to the cells by means of cork wedges. The brass sleeves contained four slots S serviiig to permit the circulation of the bath liquid through the sleeves. The bottoms of the brass sleeves were sealed and the cells inside were sitting on a piece of cork placed i n the bottom of the brass tube 0 On the outside of the bottom of the brass sleeve there were two pegs, the one i n the centre a l i t t l e longer than the other one0 • The c e l l s with the sleeves on them were held i n position by bottom supports C„ The la t t e r consisted of short pieces of brass, rods soldered' on a brass base plate B i n a hexagonal arrangement. There were two holes through each support matching the two pegs on the bottom of the sle eve s holding t he c e l l s , To provide means for s t i r r i n g the contents of the cells a pair of: rotating magnets M was arranged around each c e l l with the poles level with the diaphragms. The magnets were standing upright mounted on pinions E 0 The pinions were driven by a central drive F, They were rotating i n bearings provided by circular holes i n a large disc made of "Tufnol w (linen-laminated phenol-formaldehyde resin) sheet G which was supported centrally by a p i l l a r A consisting of a large brass tube soldered on the base plate B„ In the centre of this p i l l a r there was a foot-bearing i n which the axle H of the centre gear was rotating. The latte r was driven by a variable speed motor accommodated above the bath and held by an angle iron frame. The battery of six cells assembled outside the bath i s shown i n Diagram 3«> - 12 -DIAGRAM 3 The Battery of Six Diaphragm Cells - 13 -3o Constant Temperature Bath A liquid bath was installed on the grounds that i t i s more readily assembled and easier to operate than an air bath 0 The bath, a rectangular aluminum vessel, was of the standard design. It contained a 500 Watt heater and a cooling c o i l as well as a thermoregulator a l l located i n one corner. Stirring was provided most of the time by a 1/15 H . P o variable speed s t i r r e r . The bath has been found to-maintain the temperature constant within +_ COl^G, inside each c e l l . At f i r s t water waa planned to be used as bath liquid but this was replaced by thin paraffin o i l to lubricate the bearings of the pinions. The temperature was measured with the aid of a mercury-in-glass thermometer with a range of 65 = 105 @F o and smallest sub-divisions of 0,05°F o The thermometer was calibrated against a cert i f i e d standard platinum resistance thermometer using a Mueller temperature bridge, (See Appendix IV,.) c) Diffusivity Experiments 1. Degassing- and Introduction of the Solutions into the Bottom Parts of the Diffusion Cells The necessary quantity of the KC1, solution was boiled i n a beaker for about 5 minutes over a Bunsen burner for degassing. Subsequently the solution was cooled i n running water and used immediately. The aqueous alcohol solutions were frozen i n a vacuum flask using cooling mixtures of dry ice-ethanol, Then5 the solution was l e f t to thaw under vacuum produced by water pump. The criterion for satisfactory - 14 -t degassing was complete absence of bubble formation on mixing the two solutions or, on mixing them separately with degassed distilled water0 The solutions were introduced into the diffusion cell as follows (see Diagram l ) . C was connected to the water pump via poly-thene tubing. The lower half of the cell was wound with 100-200 Watt heating tape. The solution to be introduced in B was poured into a small, short beaker. The end of D was submerged in the solution. The valve at E was closed. Opening the stopcock at C the solution was sucked into the cell until i t covered F. Then the valve at D was closed. Heat was turned on and the solution in B was boiled under vacuum for some 5-10 minutes. The purpose of this operation was to remove a l l air from F by means of the vapors passing through the sintered glass disc. Then by opening the valve at D, more solution was sucked into the c e l l to f i l l about one third of A. Boiling was continued for a minute or so. Finally heating was turned off and air was let into the ce l l through C. During a l l this operation capsule G was removed from A. The cell was then inserted in the brass sleeve and placed i n the bath. 2. Diffusion After about 15 minutes the small amount of solution left in A was pipetted out, A was f i l l e d with the other solution, capstiie was re-introduced, the ground glass plug C was placed in position with the stopcock open and the solutions were left to diffuse for 2-3 hours for - 15 -. pseudo-steady state conditions to be established i n F„ The reason for temporarily removing G from A was that with A only partially f i l l e d , under the action of s t i r r i n g , the solution there picked up a i r at a high rate. After 2-3 hours the solution i n A was pipetted out, A was rinsed 2=3 times with small amounts of solution taken from the same stock and then f i n a l l y f i l l e d with this solution,. In the case of the calibration runs with the KC1 solutions, the upper li q u i d was always d i s t i l l e d water at the beginning. In this la t t e r case C was placed i n position only after the elapse of some 10-15 minutes, with the stopcock closedo When in position, the stopcock was opened.* The water displaced i n the seal around the joint served to prevent evaporation of the liq u i d i n A through the ground glass joint. Collecting the sealing liquid from a l l six cells at the end of a diffusion experiment did not give any measurable residue on evaporation, indicating the absence of significant diffusion through the joint. The same method was not considered safe or convenient i n the case of the ethanol-runs. It was not safe because, owing to evaporation,, the composition of the sealing liquid changed rapidly which, combined with the fact that ground glass joints leak under very small hydrostatic heads, could have led to contamination of the solution i n A 0 It was not convenient because the sealing liquid had to be replaced frequently,. Here C was placed in position, with the stopcock open, immediately after f i l l i n g A. After 10=15 minutes the sealing liquid was blotted out, the - 16 -stopcock was closed, and C removed carefully. Rubber bungs, carrying about 3 inch pieces of 0,1 mm, i 0 d , capillary were used to seal .A,, There was no measurable evaporation through the capillaries, There was an a i r -pocket of a few mis. under the rubber bungs0 In case of v o l a t i l e solutions that attack rubber a second set of shorter type of ground glass stoppers should be used instead of the rubber bungs. 3, Sampling After several days of diffusion the stopper was removed from A and the contents were pipetted into a 100 ml. erlenmeyer flask, E was connected to the compressed a i r line, the valve at E was opened f i r s t , then the valve at D. Discarding the f i r s t few mis,, the contents of B were emptied into a second 100 ml,, erlenmeyer flask. The flasks were marked and stoppered, d) Experiment on the Effect of Rate of Stirring It has been realized for a long time that the rate of s t i r r i n g has an effect on the apparent rate of diffusion as measured i n a diaphragm diffusion c e l l . It has been found invariably that, starting at zero s t i r r i n g , the apparent rate of diffusion increases with the s t i r r i n g rate up to a certain point where i t becomes constant. This phenomenon was explained by assuming, at low rates of s t i r r i n g , a stagnant film on the two sides of the porous diaphragm which i s then gradually eliminated as the rate of s t i r r i n g i s increased. This certainly seems a plausible explanation. It i s somewhat peculiar, however, that there i s a general disagreement as to the minimum value of the s t i r r i n g rate at which the - 17 -rate of diffusion becomes independent of the rate of s t i r r i n g . According to Stokes (4), this rate i s around 25 r.p.m., while Lewis (9) found i t to vary from c e l l to c e l l up to about 80 r.p.m. Smith and Stqrrow (?) on the other hand, reported this minimum rate at 3 r.p.m. Considering the fact that i n Lewis's case the s t i r r e r did not touch the surface of the diaphragm, the possibility of obtaining higher values seems plausible enough. It i s not impossible that the variation from c e l l to c e l l which he found bears some relation to the small differences i n the geometry of the various ce l l s he investigated such as slight differences i n the distance between the diaphragm and the paddle. The difference between the limits given by Stokes, and by Smith and Storrow, respectively, i s much more d i f f i c u l t to understand, since both investigators used the iron-sealed-in-plastic type of s t i r r e r driven by rotating external magnets (or magnetic f i e l d ) . Since the s t i r r e r s were i n both cases i n direct contact with the diaphragm i t i s not at a l l l i k e l y that the geometry of the c e l l s could have had much to do with the degree of efficiency with which the "stagnant layer" was being stirred. In the course of the present work, in one of the preliminary runs with the KC1 solutions, at low rates of sti r r i n g (about 10 rbp0m.), the existence of a concentration gradient i n the compartments was esta-blished. In this run, instead of pipetting out the entire contents of the top compartment in a beaker and taking aliquots from there, the f i r s t 25 ml. taken directly from the bottom of the top compartment, and the next two 10 ml. aliquots taken also dir e c t l y from the c e l l , were analyzed - 1 8 -separately with the following results*! Cell no 0 Cell no<> Cel l no 0 Cell no 0 Cell no. Bottom 2 5 mlc 1 0 „ 0 2 ? 6 l 3 0 o 0 3 l 6 2 4 0 . 0 3 2 1 1 5 0 . 0 3 3 2 6 6 0 . 0 3 8 6 4 Next 1 0 m l c 0 , 0 2 6 5 6 0 , 0 3 1 0 3 0 . 0 3 1 3 4 0 , 0 3 2 9 0 0 , 0 3 8 0 4 Top 1 0 ml. 0 , 0 2 6 3 2 0 . 0 3 0 4 4 0 . 0 3 0 4 8 0 , 0 3 3 1 7 0 . 0 3 8 2 0 g./ml. g./ml. g./ml. g./ml. g./ml. It should be remarked here that this run was not done with the specific purpose of establishing the existence of a concentration gradient. Howeverj when pipetting out the liquid from the top compartment the t i p of the pipette was, generally, l i g h t l y resting on the diaphragm. It is possible that this was not the case for the four bracketed samples. Nevertheless, the results shown above prove the "existence of a concentra-tion gradient quite conclusively. Postulating the existence 'of a concentration gradient in both compartments i t i s easy to show that the value of the apparent c e l l constant obtained by using the bulk concentrations i s less than the true value 0 Using simplifying assumptions' the error committed was found to be about =lo3% for the case of the concentration gradients i n the above experiments. If the hypothetical concentration gradient was assumed to be five times as great, the error was found to be -6,3$. These computations are contained i n Appendix III. It should be noted that the probable uncertainty of the figures in the table i s + 2 units in the last decimal place. - 19 -It seems, therefore, that with the polythene capsule type of magnetic s t i r r i n g , the stagnant layer is eliminated at considerably lower stir r i n g rates than i s necessary to ensure homogeneity of the. solutions i n the c e l l compartments. This i s quite plausible i f one considers that the capsules have a smooth cylindrical shape, not at a l l efficient for s t i r r i n g the bulk of the l i q u i d . However, in the case of the experiments of Smith and Storrow there were conductivity electrodes i n the upper half, which acted as baffles, that might have increased the efficiency of s t i r r i n g . In the lower compartment the s t i r r e r was at the level of the free surface of the solution where i t probably gave rise to rippling which, i n turn, again must have resulted in increased efficiency of s t i r r i n g . This i s the probable explanation why Smith and' Storrow found the lower limit of the adequate st i r r i n g rates so much lower than Stokes i n whose experi-mental setup, as i n the present experiments, there were no factors present to make stir r i n g more efficient. To insure homogeneity of the bulk of liquids in the c e l l compartments, very high rates of stirring (about 100 r.p.m.) were used i n a l l subsequent experiments. As expected, no concentration gradient could be detected at such high rates of s t i r r i n g . e) Determination of Volumetric Data of the Diffusion Cells For computing the d i f f u s i v i t y the volumes of the two c e l l compartments and, unless these are equal, the volume of the pores in the sintered glass diaphragm, are required. Contrary to the usual practice - 20 -these volumes were here determined volumetrically. For this purpose the cell was: completely f i l l e d with water, and G (see Diagram l ) was placed in position with the stopcock open. The stopcock was closed and C withdrawn. Contents of A were poured in a beaker and the volume was measured using 25 ml., 10 ml., and 5 ml. graduated pipettes. The smallest sub-division of the latter was 0.5 ml., permitting estimation to 0.1 ml. This procedure was repeated several times and the results were found to agree to within +0.1 ml. The volume of B was determined by two slightly different techniques. In both cases A was f u l l , C in position with stopcock closed. In the case of the first method the cell was in normal position and completely f i l l e d with water. Valves at E and D were opened and liquid from E was forced out by carefully blowing in at the top. Then the contents of B were removed in a beaker, using compressed air at E. The last few drops, the contents of D, were left to flow on the bench. With the other method, the cell was clamped upside down and B was f i l l e d through E by applying suction at D. The meniscus in D was carefully adjusted at 1-2 mm. from the bottom of B. Then the cell was turned in a normal position and the contents were removed through D by applying compressed air at E. The two methods gave results that agreed to within +0.1 ml. The volumes of A and B being not very different, very accurate knowledge of the pore-volume was not necessary. The pore-volume was determined by bringing the dry sintered glass disk in contact with water and measuring the volume of water absorbed by the former. Satura-- 21 -tion was indicated by the fact that at this point the surfaces of the diaphragm became moist. The water was added to the diaphragm by a graduated pipette with smallest sub-division of 0.1 ml. A l l the diaphragms investigated had a pore-volume very close to 0.3 ml., and this value was used for a l l the six c e l l s . f) Analysis 1. Analysis of the KC1 Solutions Regular weighing bottles of 50-200 ml. capacity were carefully rinsed with d i s t i l l e d water, then placed on a clean enamelled tray and dried i n an oven at about 280°C. After removal from the oven and cooling down to room temperature, the bottles were weighed on the "gram-atic" analytical balance specified i n Section II, a. To exploit the potential accuracy of the balance required very careful operation. Six of the weighing bottles were placed in the balance case at a time to assume the temperature prevailaing inside the case. After some t h i r t y minutes each bottle was weighed at least three times accord-ing to the following pattern: The zero of the balance was set. Bottle no. 1 was weighed. Bottle no. 2 was shifted in the empty place of bottle no. 1. Bottle no. 1 was placed from the pan to where bottle no. 2 had been. Again the zero was set and bottle no. 2 was weighed, and so forth. Finally a seventh, standard bottle was also weighed. This bottle, heavier than the rest, was kept i n the balance case a l l the time. The purpose of weighing each bottle three times was to find the value of i t s - 22 -weight with greater certainty. The reason for rotating the bottles with respect to their positions^ i n the case was to minimize their heating due to radiation from the body of the operator. When the same procedure was carried out without rotating the bottles with respect to their positions, the two bottles just behind the front window of the balance case changed their apparent weights from one measurement to another much more than the rest of them. With the rotating system introduced, the variations became random and, on the average, smaller. It i s of interest to note that i f the temperature of the bottle was higher than the balance tempera-ture then the apparent weight of the bottle was lower, and vice versa. The "temperature error" could very easily amount to several mgs. The standard bottle was used to take into account changes i n the a i r density within the time interval between weighing the empty bottles and weighing the bottles containing the residues. The sample taken from a c e l l compartment was pipetted into three weighing bottles i n aliquots of 10 ml. each or, sometimes, two 10 ml. and one 20 ml. aliquots. That way the number of bottles was 36 per run. With the bottles arranged on clean enamelled trays, the solutions were evaporated to dryness i n an air oven at about 60°C. as recommended by McBain ( l l ) . Then the temperature was gradually raised up to about 280oC. After removal of the bottles from the oven their l i d s were immediately placed in position, the bottles were l e f t to cool and then were weighed by exactly the same procedure as the empty bottles. The standard bottle was again weighed with each l o t . - 23 = The amount of the residue being very small, the absolute buoyancy correction on i t was negligible„ However, the difference i n buoyancy on the weighing bottles at the time of the two weighings could give rise to significant error i f there was a change i n the a i r density,. The true weight of the potassium chloride residue was computed by the following formulas (1) where i&g+jj = apparent weight of the bottle plus residue, mB = apparent weight of the bottle, M = weight of standard bottle corrected to vacuo, M-^ = apparent weight of standard bottle at time of weighing residues, M = apparent weight of standard bottle at time of weighing empty bottle. The derivation of Eq. ( l ) i s given i n Appendix II... To prevent any change in the weight of the bottles between the time when they were weighed empty and the time when they were subsequent-l y weighed with the KC1 residue, great care had to be taken. The bottles were stored covered with clean plastic sheets. It was established that each time a bottle was opened there was a measurable amount of chipping at the sharp edges of the ground parts. Consequently, the l i d s were not placed i n position between the two consecutive weighings, - 24 -The methods outlined above were evolved gradually. In the last , or 8th run with the KC1 solutions, where a l l these measures were applied, the most probable error of the parallel determinations was estimated to be + 2 x 10°^ g., which is just about an order of magnitude better than the precision claimed by McBain ( l l ) , who could reproduce parallel determinations to within ± 0.2 mg. It i s shown in Appendix I that the calculated error i n the value of |3 , the c e l l constant, originating from the above analytical error, i s about + 0*11%. This i s roughly the same value as the average of the deviations about the means of the c e l l constants found i n run 7 and run 8. As far as the accuracy of the analysis i s concerned, this was checked i n various phases of the preliminary work directed at accurate measurement of the c e l l constant. The procedure consisted of heating up KC1 crystals to as high as 600°C, then weighing them, dissolving them in d i s t i l l e d water, and evaporating the solution i n the way described above. The results of the two weighings always agreed within the experimental error. No effect of adsorbed moisture on the weight of the bottles could be observed. It was definitely established that the KG1 residue inside the sealed bottle did not change i t s weight from one day to another. The effect of the error i n the volume of the aliquots - 25 -analyzed on the value of the c e l l constant i s shown i n Appendix I. It was found that the error of pipetting was negligible as compared with the error of weighing. 2. Analysis of the Ethanol-Water Mixtures The aqueous alcohol solutions were analyzed using commercial Sprengel pycnometers (Fisher Cat. No. 3-290). These were slightly modified by drawing out the t i p of one of the two arms to a narrow capillary of about 0.1 mm. i . d . and part of the other arm to less than 1 mm. i . d . In this section, using a f i l e , a mark was engraved which was subsequently blackened by a graphite pencil. These pycnometers, as supplied, wsre equipped with ground-glass caps at the end of the two arms, thus preventing evaporation of the solutions. However, because the i.d. of the side-arms was 2-3 mm., the volume error due to the uncertain position of the menisci was prohibitive for the present purposes. Besides, the positions of the menisci were very strongly influenced by t i l t i n g the pycnometer. When drawing out the t i p of one arm to a fine capillary, care was exercised to preserve a large enough part of the ground end to f u l f i l l i t s purpose, i.e., to prevent evaporation with the cap i n position. The modified pycnometers, two at a time, were suspended via hooks made of 1 mm. copper wire into a constant temperature water bath of conventional design. The bath consisted of a 25 l i t e r cylindrical glass container, equipped with heating and cooling coils, a thermoregulator and a small st i r r e r , allowing i t s temperature to be controlled to ± 0.01°C. or better. Temperature was measured using a mercury-in-glass - 26 thermometer with smallest sub-division of 0 o l o C o Using a magnifying glass, the temperature could be estimated to 0.01®C. This thermometer was also calibrated against a standard platinum resistance thermometer using a Mueller temperature bridge. (See Appendix IV.) In this second case the calibration was later found to be absolutely unnecessary since, owing to the particular form of the formula i n which the concentration data were substituted, even a difference of 5 degrees centigrade i n the bath temperature would not have resulted i n any significant difference i n the value obtained for the d i f f u s i v i t y . , Two pycnometers were dried ,by rinsing them with acetone and, subsequently, sucking dry a i r through them for some ten minutes. Then the caps were placed i n position, and the pycnometers were cleaned by wiping them with moist chamois leather. One of them was sus-pended by a hook over the pan of the Mgram-atic M balance specified In Section II, a, and the other, standing in a beaker, was also placed i n the balance case. After some 30 minutes they were weighed i n the following manner. The pycnometer plus the hook were weighed. Removing the pycno-meter from the case, the hook alone was weighed. Then the other pycnometer was suspended and weighed i n the same way. Finally a standard object, kept i n the balance case a l l the time, was weighed to determine the density of the a i r 0 This standard object was prepared by sealing off the neck of - 27 -a 100 ml. flask. Over a couple of weeks the a i r density was carefully determined at the time of each weighing, using the formula 1.7013 x 1 0 ° 6 (p-k) a = 1 + 0.00367 t (2) where k = 0.0038 Hp H2° H = relative humidity, percent, t p n = vapor pressure of water at temperature t, 2° p = reduced barometric pressure. This formula i s given by Weissberger (13) and i t s derivation given i n Appendix II. The measured values 0 a and the corresponding weights of the standard bottle, nig, were used to determine the constants Fg and nig i n the equation mg/nig = 1 which, i n turn, was used subsequently to find the unknown ai r density from measured values of m_. The derivation of Eq. (3) i s also given i n Appendix II. The values of the two constants were found to be Fg = 2.17, m° m 29.12203. ti The pycnometers were then f i l l e d with d i s t i l l e d water, the water from the arm with the mark was drained by t i l t i n g the pycnometers and, subsequently, dried over a small flame. The pycnometers were suspended i n the bath at 20eC, for some 15 minutes, then f i l l e d up to somewhat over the mark by introducing water at the capillary end, using a piece of narrow glass tube. The meniscus of one of the pycno-meters was final-ly adjusted to the mark by removing the excess water at the capillary end, using small pieces of paper towel. At the f i n a l adjustment the meniscus was viewed through a magnifying glass clamped over the marked arm of the pycnometer. To enhance contrast, white back-ground for adjusting the meniscus was provided by using a small piece of white plastic sheet inserted i n a short, small test-tube. The test-tube, i n turn, was slipped over the end of the marked arm0 After" adjusting the meniscus, the pycnometer was removed from the bath and the small caps were placed i n position immediately, starting •always with the one on the capillary side 0 Care was taken that the ground glass ends of the two arms were dry„ The pycnometer was again wiped with the moist chamois leather and suspended over the balance pan. The other pycnometer was treated i n the same way and placed i n the balance case standing i n a beakerD Weighing of pycnometers with water was performed i n the same way as that of the empty pycnometers. The standard bottle was again weighed,, The weight'-6f the wire-hook was in each case subtracted from the t o t a l weight, and?the difference was corrected to vacuo using the following formula T 29 • where ~m0 = apparent weight of object i pa = density of ai r , j)Q *» density of object, w = density of weights 0 The density of the weights, as customary, was assumed to be equal to the density of brass, 8„4 go/ml0 The density of the empty pycnometer was taken equal to the density of borosilicate glass, 2„23 g./mlo The density of the pycnometer with water was found by trial-and-error. This had to be done only once» The difference of the corrected weight of the pycnometer plus water and the corrected weight of the empty pycnometer was taken as the true weight of the water D In an attempt to assess the precision of this methods the weight of the water f i l l i n g the pycnometers was determined a considerable number of times. The most probable error of an individual determination was estimated to be ± 00.12 mg0, corresponding to an error of i 1 x 10 go/mlo i n the value of the density of the analyzed, solutions,, The weight of the alcohol-water mixture f i l l i n g the pycnometers was determined i n exactly the same way as the weight of the d i s t i l l e d water. As a rule, two determinations were made on each solution. The density of the pycnometer plus 96$ alcohol was once determined by t r i a l -and-error and for the rest of. the solutions the value of the factor . 1 1 . (— — — w a s interpolated i n an approximate mariner. Po P* To obtain the specific gravity of the solutions referred t& water at 4°Co, the average weight of the water was divided by 0.99823, - 30 -the relative density of water at 20°Co, and the weight of the solution under consideration was divided by this number which, of course, was numerically equal to the volume of the solution. The grand average of the deviations from the mean taken over a l l the 191 individual deterraina-tions was found to be + 1,3 x 10 g./ml. This includes four cases where abnormal scatter was observed. Omitting these four cases, the average deviation i s found to be ± 0,9 x lO"-* g./ml. The bath temperature of 20°C. was selected because this tempera-ture could be controlled accurately and conveniently. It was found that the difference between the values of the d i f f u s i v i t y computed from the concentrations taken at 20eC. and those taken at 25°C. was not significant. Nevertheless, for the sake of rigour, the 20°C. specific gravities were converted to the corresponding values at 25°C, the temperature of. the diffusion experiments, and the concentrations at 25°C. were always used to compute the value of the d i f f u s i v i t y . Since the specific gravity referred to water at 4°C., i s " numerically equal to the density, for the sake of brevity, the term density w i l l be used from now on in place of specific gravity. To make direct conversion between densities and volume concentrations possible, a table was prepared by multiplying the corresponding tabular values of the weight fraction and the density together and thus preparing a third column containing the volume concen-trations. The error i n the volume concentration corresponding to i 1 x 10 g./ml. error i n the density varies with the composition - 31 -between the approximate limits ± 7 x 10"^ g./ml. to + 1 06 x 10"^ go/ml. Maximum error i s around 0.15 g./ml. ethanol concent ration,. The average relative error i n the value of the d i f f u s i v i t y owing to the analytical errors was calculated to be in the range ± 0.2 to +, 0 .6$. (See Appendix I.) g) Investigation of D i s t i l l a t i o n through Ground-Glass Joints On examining the thesis of I. E„ Smith (10) on which the paper by Smith and Storrow (2) was based, i t was noticed that in their apparatus there was a possibility for mass transport via d i s t i l l a t i o n i n addition to the diffusional transport. Diagram 4 i s a sketch of the apparatus used by Smith and Storrow. A i s a glass tube ending i n a sealed male ground glass joint C. D i s the h a l f - c e l l containing the solution richer i n alcohol. E i s the sintered glass diaphragm.. F are magnetic st i r r e r s . G i s a large test tube serving as the other h a l f - c e l l and B i s a vented rubber bung. Since D was reported to be completely f i l l e d with solution, the latte r could be pumped a l l the way up the joint by capillary action, and from the edges, d i s t i l l a -tion of the solution into the other solution of different concentration could take p l a c e This lat t e r solution was contained i n G and reached up to E. Such d i s t i l l a t i o n would have given rise to increased apparent d i f f u s i v i t i e s . To check on this assumption the simple apparatus shown In Diagram 5 was f i r s t used. B i s a U-tube plugged at one end with a ground glass stopper C and f i t t e d with a piece of narrow capillary E at the other end to prevent both evaporation and the development of vacuum. A i s a jacket. D i s a rubber bung. - 3 2 -D I A G R A M 4 SKETCH OF DIAPHRAGM DIFFUSION CELL USED BY SMITH AND STORROW (2) - 33 -D I A G R A M 5 APPARATUS USED TO INVESTIGATE DISTILLATION THROUGH GROUND-GLASS JOINT - 34 - . ' B was f i l l e d with-over 90% alcohol, C was inserted, the edges of C and a l l the interior of A were dried by pieces of paper towel, and small amounts of water were pipetted into the bottom of A. D was placed i n position<> The meniscus i n the right arm of B was observed to move downwards while the meniscus i n A was moving upwards. This process was observed for several days, when the li q u i d i n A, originally pure water, had increased i t s volume by several times the original value,, This experiment was repeated several times. Occasionally the process would not start or i t stopped at a point. The ground glass stopper C was always inserted very firmly so that i t did not seem to make sense to consider any ordinary leakage through the joint. Whenever the d i s t i l l a -tion was i n progress, the edges of the joint, originally dry, were seen to be moist. It was not considered within the scope of the present work to try to elucidate the mechanism of this d i s t i l l a t i o n effect. In an attempt to measure the magnitude of the d i s t i l l a t i o n effect under conditions similar to those used by Smith and Storrow a c e l l was built (see Diagram 6) that differed from the one used by the above authors only i n that the sintered glass disk has been replaced by a short piece of narrow capillary E. The purpose of E was to permit bulk flow from- G into D i n case part of the liq u i d were removed from D via d i s t i l l a t i o n through C. Other things being equal, the diffusional transport through E i s about one millionth of that through a sintered glass disk and hence i s completely negligible. This c e l l was accommodated, along with another reference-cell of the same construction, but not containing any ground glass joint (A, C, and D were a plain glass tube - 35 -D I A G R A M 6 APPARATUS USED IN PSEUDO DIFFUSION EXPERIMENTS - 36 - i • stoppered at the upper end) i n a constant temperature water bath consisting of a 5 l i t e r cylindrical glass container,, The water in the bath was being circulated by means of an "Ultra Thermostat Type K" product of the Colora Messtechnik GMBH, Wurttemberg, Germany,, The temperature of the water i n • the cylindrical vessel was measured with a mercury-in-glass thermometer with smallest sub-division of 0.1°C. The temperature was constant t© within ± 0ol°C o The bath water was pumped through also the appropriate jackets of a Pulfrich refractometer, used for analyzing the solutions i n these experimentsi then i t entered the bath at the bottom; at the top of the bath i t overflowed to the "Ultra Thermostat",container. The Ultra Thermostat thus having no suction head at a l l had to be placed on the floor for smooth operation. The flow of the cooling water to the "Ultra Thermostat" was kept constant by means of an inverted-jar-type constant head tank. The experiments with this apparatus were carried out by R» W. Crossland (12) under the supervision of the present author. First pure water and concentrated ethanol were used. The liquids were immersed i n the bath to bring them to 25°C. When they had reached this temperature the refractive index of the water was measured with a Pulfrich refractometer with a precision of ± 0.0001 units i n the standard way. D was then f i l l e d with alcohol and C inserted. The outside of the c e l l was wiped thoroughly with tissue' paper to prevent alcohol from being trans-ferred to the other h a l f - c e l l v i a the sides of the upper c e l l . About 30 ml. of water was then placed in G, the two compartments joined via B, and the whole unit re-immersed i n the bath. - 3 7 -The other, the "dummy-cell", containing no ground glass joint, was treated in the same way0 Runs were of 14 to 89 hours duration. At the end of the run a sample was taken from G using a hypodermic needle that was inserted through B and served the purpose of venting G throughout the run. The refractive index of the sample was measured with the Pulfrich refractometer. The c e l l containing no ground glass joint exhibited negligible transference, hence a l l the changes measured in the c e l l with the joint were judged to be due to d i s t i l l a t i o n through the joint. Having obtained a positive effect i n a number of runs i t was decided to do some runs at the low concentration differences used by Smith and Storrow (2) i n their experiments. In two runs i t was found that the d i s t i l l a t i o n effect might have accounted for about 10 and 15$, respectively, of the values obtained for the d i f f u s i v i t y by Smith and Storrow ( 2 ) . The possible error of the d i s t i l l a t i o n values, however, i s estimated to be at least 100$. It should be emphasized that the results of the investigations of the d i s t i l l a t i o n effect are, by no means, considered as a proof of the hypothesis that the higher values of the d i f f u s i v i t i e s obtained by these authors are a consequence of mass transfer through the ground glass joint used i n their apparatus. It i s considered sufficient to say that the existence of such a d i s t i l l a t i o n effect i n the experiments described above seems to have been established. = 38 = i n . THEORY a) Derivation of the General Equation of Diffusion for Binary Mixtures Johnson and Babb aptly stated in their quoted review article (l) that "in liquid diffusion the Fick coefficient has been used almost exclusively". This i s certainly the case as far as the experimentalists are concerned. Theoreticians, however, have introduced quite a variety of different diffusion coefficients. This situation i s further complicated by the existence of a large number of different "diffusion equations", i.e., expressions giving the diffusional flow, or flux, in terms of some "driving force". In this connection Bird (14) remarks? "In reading the literature of the f l u i d mechanics of diffusion, one encounters numerous d i f f i c u l t i e s because of the diversity of reference frames and definitions used by authors in various f i e l d s . Frequently more time is spent in 'translating' from one system of notation to another than i s spent in the actual study of the physics of the problem". This diversity i s caused by the following circumstances? the quantity of the diffusing material can be expressed i n terms of some units of mass or in terms of moles; the frame of reference for diffusion can be different| and the "driving force" can be represented i n various ways. In the resulting maze of subtle differences i t i s not easy to - 39 -t e l l the consequences- of some arbitrary formal step from what follows from a fundamental law of nature. Also one may question whether certain arbitrary steps are consistent with the fundamental laws0 Of the treatments available at the present, the one using the methods of the thermodynamics of irreversible processes seems to have the most solid foundations. This treatment was most thoroughly developed by Prigogine (l$) and subsequently somewhat extended by de Groot (16), Bird (14) modified the form of the final expressions by using only one diffusion coefficient. He also considered the case of what he calls stationary axes,. In the present work the diffusion equation given by Bird for the case of stationary axes is used as a starting point to develop an integrated relationship for use. with the diaphragm cell method in the case where there is a volume change on mixing. The purpose of doing this is to increase the potential accuracy of this method of measuring diffusivities by permitting the use of large concentration differences in the diffusion experiments regardless of volume changes occurring on diffusion. It should.be noted that Hartley and Crank (17) dealt with the problem of the question of reference frames in diffusion experiments in the case where the volumes are not additive. Some of their results^, obtained by intuitive reasoning, are at variance with the corresponding results obtained by'the methods of the thermodynamics of irreversible processes. This disagreement will be discussed in more detail at a - 40 -later point. The result obtained by the methods of the thermodynamics of irreversible processes has received considerable attention mainly for the reason that this has been the only approach so far to show from fundamental laws that the true driving force of diffusion, in the absence of external field forces, is the gradient of the chemical potential (18). Apparently no use has been made of the fact that the diffusion equation obtained in this approach gives an unambiguous definition of the diffusion coefficient regardless of the presence of volume changes on diffusion. In addition there does not seem to be any reference in the literature to the existence of a satisfactory description and explana-tion how this more general equation is related to Fick's first law. To quote Byron Bird'on this question.-(14) J " a. Equimolal Counter Diffusion of A and B When A and B are diffusing'counter to one another in such a way that = -Ng, then NA = ~DAB V ' V tun This is the correct expression for use in the analysis of closed diffusion cell experiments for.the measurement of diffusion co-efficients. Equation [48] is known as Fick's First Law of Diffusion. Note that N^ = -Ng corresponds to saying that w = 0 . " (For meaning of symbols in the above, quotation see "List of Symbols".) This statement may create the impression that the validity of Fick's first law hinges on the existence of equimolal counter diffusion. - 41 -Furthermore Fick's f i r s t law i s not necessarily "the correct expression to use i n closed diffusion c e l l experiments". It w i l l be shown later, that in the case of the diaphragm c e l l method when large concentration d i f f e r -ences are used and the system investigated displays serious volume changes on mixing, application of Fick's f i r s t law may result i n significant errors i n the value of the true d i f f u s i v i t y . Before going into the detailed discussions forming the proper subject of this section, the derivation of the fundamental diffusion equation w i l l be given together with a discussion of the question of various frames of references and various d i f f u s i v i t i e s . This has been done because, although the equation from which the present work develops had been stated, there has not been one treatment i n sufficient d e t a i l , . that the val i d i t y of the steps involved can be assessed. In what follows the works by Prigogine, de Groot, and Bird were used as sources and wherever possible, the notation used by Bird was adopted. In many cases, where the proofs of theorems and details of computations were not given i n the quoted sources, these gaps ware f i l l e d in by the present author 0 To begin with, a brief outline of the relevant methods of the thermodynamics of irreversible processes w i l l be given. - 42 -i ) The thermodynamics of irreversible processes. This branch of science deals with such phenomena as heat flow, diffusion flow, e l e c t r i c a l current and chemical reaction rate 0 A l l these come under the names "fluxes" or sometimes "flows" or "currents". Quite in general these w i l l be denoted by «J\ where the subscript i (i* = 1, 2, . o o , n) allows for the simultaneous existence of n different fluxes. It is an experimental fact that such fluxes are caused by a temperature gradient, a concentration gradient, a potential gradient or a chemical a f f i n i t y , respectively. These quantities are usually called "forces" or " a f f i n i t i e s " and w i l l be denoted here i n general by where 1 = 1, 2, n, indicating that there can be n forces acting simultaneously. It i s a further experimental fact that more than one force can (jointly) give rise t o one and the same flux (which i s some= times termed "interference phenomenon"). Thus heat flow can arise under the influence of potential gradient (Peltier effect) or concentration gradient (Dufour effect); and as a result of temperature gradient, e l e c t r i c a l current (thermo-electric effect) or diffusion may arise (thermal diffusion). / Generalization of these observations results i n the statement that any force can-give rise to any "flow"; n = I^k X^ , ( i = 1, 2, n) (5) stating that any flow i s caused by contributions of a l l forces. The coefficients L., ( i , k = 1, 2, n) are called the phenpmenological • - 43 -coefficients. The coefficients are for example the heat conduct-ivity^ the ordinary diffusion coefficient, the electrical conductivity. The coefficients L... with i / k are connected to the "interference ik phenomena". Examples are the thermal diffusion coefficient, the Dufour coefficient and the like. For the sake of completeness Onsa.ger's fundamental theorem [see (16), Ch. II] should be stated at this point. This theorem is a fundamental statement, similar in its character to the fundamental laws of thermodynamics,'and is central to the thermodynamies of irreversible processes. The theorem„is as follows; Provided a proper choice is made for the fluxes > and the forces X^ , the matrix of phenomeno-logical coefficients is symmetric, i.e. L i k = hd.' k " 1»- 2' n) These Identities are called the Onsager reciprocal relations. It will be shown in what follows that in the simple case of ordinary diffusion of two components there is no Onsager reciprocal relation. Nevertheless the vhole development is based on an important consequence of Onsager's theorem. By the "proper" choice for the fluxes and forces is meant that they satisfy the equation (7) - 44. -where § is the entropy produced per unit time during the irreversible process. <S is also called "entropy production" or "entropy source strength". . It is necessary to obtain the correct expressions for X^ in the case of ordinary binary diffusion. To do that an expression for (5 may be der ived using the fundamental macroscopic balance equations in conjunction with Gibbs* statement of the second law of the thermo-dynamics. It is an extra hypothesis that this relation is assumed to be valid outside thermodynamical equilibrium, which means that the entropy may be supposed to depend explicitly on the internal energy, volume and concentrations. It has been shown by several authors, and most thoroughly by Progogine (15) that this condition is met unless the situa-tion considered is 'too fa r 1 from thermodynamical equilibrium. As the fundamental equations used include the equation of motion, many results are simpler in form when intensive quantities are expressed in terms of unit mass. Therefore such quantities will be used in the general de-velopment and transformations to a molar basis will be dealt with separately. The treatment will be kept fairly general up to the point where the desired expressions for the forces X^ are obtained. Then the special case of ordinary binary diffusion will be deduced from the more general expressions. • - 45 -i i ) The equations of change. The four fundamental equations of change to be used are the following? l ) The equation of continuity for component kg where p^ i s the mass of component k per unit volume, the l o c a l time derivative, u^ the velocity of k with respect to some cartesian coordinate system and r ^ the production of k per unit time per unit .volume • Summing Eq„ (8) over a l l components k » 1, 2, n since k-1 and . - y : y . ( P . k u k ) - = y .(jr pktJi - - V - ( P n a n d 4^ rk = °» k-1 the result i s ( P * ) , (9) 46 -where p> i s the tot a l density and v i s the velocity of the center of mass or, the mass average velocity, defined by the equation n the rate of change with time of the quantity considered at fixed values of the space coordinates i n the cartesian coordinate system, i n which the velocity of the center of mass i s "v. However, i t i s convenient to write Eqs 6 (S) and (9) i n an alternative form by introducing the barycentri© substantial (total or material) time derivative defined as which gives the rate of change with time of the quantity considered at fixed values of the space coordinates i n a cartesian coordinate system, with axes parallel to the former coordinate system but moving with respect to i t at the velocity v „ Simultaneously, the flow of substance k i s defined with respect to the latter coordinate system by (U) Jk " h ( t l •k (12) - 4 7 -Since by Eq. (10) n and also n 5 />k~ = />v> there follows that Jk .• °» ( 1 3 ) Using the definitions given by Eqs. (ll) and (12), may be eliminated from Eq. (8) in the following way: Since i t follows from the definitions that by Eq. (ll) we can write T ^ - l l ^ V - ^ - ^ C V - v ) . (15) Combination of Eqs. (15) and (8) gives ^ - - V * ( ^ ) - k + V . < p k v ) (16) But from Eq„ (12). i t follows that (17) Hence, adding Eq„ (16) to Eq„ (17) yields B " " \ - k = | ) k ( ? ^ (Id) k - - \ i . Eq. (18) can be further simplified i f the following identity i s considered Deis") D0 t £ k But, by definition, £1L . i£L + (•• V ) P . ' ' (20) - 49 Adding Eqs. (20) and (9), g£= ~ V • (/> v) + (v . V )|5. (21) Using an expression analogous to Eq„ (14) S Eq« (21) can be transformed to give . ~jD( V • v) . (22) Introducing the definition of mass fraction o o k = - ^ , x (23) combination of Eqs. (19) and (22) gives P ^ W ' ^ B T * f k < V - v ) . (24) Comparison of Eqs. (18) and (24) gives f i n a l l y p _ _ ^ _ ( V o J k ) + r k . ( 2 5 ) Using definition ( l l ) and an expression analogous to Eq.- (14) 5 Eq. (9) i s transformed as follows? - 50 -By Eq. Eq. 9 becomes whence, using the relationship expressed by Eq. (14) p ( V *v) Introducing the specific volume v, defined 1 v = into the left hand side of Eq. ( 2 7 ) is obtained. 2) The equation of motions Where p is pressure, and P. is external (body) force per unit of mass of substance kB Viscous forces have been neglectedo 3) The equation of energy balances where u is the internal energy per unit mass and q is the energy flux vector 0 4) The second law of thermodynamicsi m Ds. Du Dv ^ , B u \ T DT 3 W +..p W " ^ A "W/ (32) where s is the entropy per unit mass and yu^ is the chemical potential per unit mass of component k„ i i i ) Derivation of the entropy balance equation,. As stated earlier, using the four fundamental equations, an expression for G, the entropy production, will be arrived at« It will be recalled that this expression will have to be of the form 6 s £Jk ; (7) - 52 -and that the purpose of the present computations is to obtain the correct expression for X^. - will be elimi: Du — J k The term --JJ- will be eliminated from Eq. (32) using Eq. (25). To eliminate also the term - j ^ - from Eq. (32), an expression for g^ y- that does not contain v.' will be developed f i r s t . The kinetic energy of the center of mass is eliminated from the energy equation. Eq. (30) is multiplied by"v, giving 2 xx | p g - - - ( # - V ) P * £ <F (33) Subtracting (33) from (31) gives (34) • g , Fk«(uk - *) />k. Using an expression analogous to Eq. (14) and also using Eq. (12), Eq. (34) can be simplified to P W a - (*-V)p- (V-q). * ^ F k W k . (35) - 53 -To eliminate v from Eq. (35), Eq. (29) i s used. Eq. (29) can be rewritten Combination of Eqs. (35) and (36) gives DF ~ p W P q) + O "> F » J k k £=1 [37) The entropy balance i s now deduced by introducing, as already indicated, Eqs. (25) and (37) into Eq. (32)% n /*k rk * Transformation of Eq. (38) into the desired form was not given in any of the quoted sources. This can be done as follows. Let us consider the identity - (V -q) + n (V° Jk) '^ -2- /Vk k^l' - 54 -where T has to be determined from Eq„ (39)« Using an expression analogous to Eq„ (14) again, the right hand side of Eq, (39) can be written whence Y i s found to be 7 T - £ ( V V ) <fe . UG) T k=l 1 Using Eqs. (39) and (40), Eq. (38) may be written i n the form - s < V V ) 4 £ * k • \ -• | E A - K. k=l k=l k - l ' that i s readily rearranged to give * - 2, /Vk Ds. _ „ y . E L 7 k K ET0 (41) Eq. (41) has clearly the form of a balance equations the change of the specific entropy i s due to two causes, the negative divergence of the entropy flow J s given by the expression and an entropy production with a source strength q "Xu •+ ZT "*Vxk + J c where and G = k-1 u T # = -£/* k k^ k-1' which i s the chemical a f f i n i t y . The product of J and y, i s defined by the equation where V i s the volume of the system and mk i s the mass of component k 0 • * It can be seen from,:Eq0 (43) that (S has Indeed been found to be the sum'of the products of fluxes q, J v and J and corres-ponding forces .X^, X k and. According to the scheme of the Onsager theory. In a f i r s t approximation, linear relations are assumed between these fluxes and forces, leading to the following so-called phenomenological equations J i = 2-. h.k X k + L i u \ » • - k=l n ^ — k»l J m L c * It must be realized that any chemical reaction formula can be written i n the most general form n 1 m k - 0 or, differentiating with respect to time and dividing by V, k=l Finally, letting dm^ = ^ d ^ there follows - 5 7 -Curie's theorem* was observed when the relationship between the flows of matter and heat on the" one hand and the chemical affinity on the other, as well as the relationship between the chemical reaction rate and the forces and was omitted. Onsager's relations are, for the case considered [cf. Eq. (48)3 Ik ha. 3 1 1 ( 1 L i u = L u i ' ( 5 1 ) iv) General case of ordinary diffusion. Having kept the treatment fairly general up to this point, next the simpler case of ordinary, i.e., isothermal diffusion will be discussed in more detail.** In this case we have from Eqs. (44) and (45) ^ = 0 , (52) xk = V ( ^ M ' > ( 5 3 ) and since absence of chemical reactions is&also assumed J c - 0 . (54) *• According to Curie's theorem i t is impossible for a force of a certain tensorial character to give rise to a flow of a different tensorial character. The fluxes and forces in Eqs. (4$) and (49) being vectors, and those in Eq. (50) being scalars, they are of different tensorial character and hence, incompatible. ** It should be noted that by one of the fundamental postulates of thermo-dynamics of irreversible processes, the existence of V^uk implies the existence of V T . Hence, letting ^ T => 0 i s , strictly speaking, an approximation, though excellently fulfilled in many actual situatzons. Thus the entropy production becomes n . < S d - ! a i — ( 5 5 ) where 6^ is' the entropy source and Is the diffusion flux of component k for ordinary diffusion, and the phenomonological relations are % k x k « <56) This treatment is considerably simplified from this point by virtue of 2 a number of relationships that exist between the n coefficients of Eq. (56). The existence of these relations is a consequence of Eq. (13) Since, by letting X = X = ... = X - X, Eq. (55) takes the form T 6 = x f f = 0 (57) k=l * by virtue of Eq. (13), hence for this special case, (S^ => 0. The argument now is that since we exclude the physical possibility of reversible diffusion, i.e., one that would take place without entropy production, for this case the diffusion fluxes also have to be zeros - 59 -"J' - X = - J = 0 . But then by Eq. (56)? 1 2 n k=l which can be satisfied only i f E L i k = 0 ( i = 1, 2, n.), (58) k=l. since X is not in general zero.* Another set of relationships between L^ k is obtained when Eq. (56) is inserted into Eq. (13), giving * It should be remarked here that the "somewhat more rigorous proof" forwarded by de Groot on nape 102 10c. cit, is apparently based on a misprint. His formula [44] is as follows; 1 T 6 d - | f £ L ^ x , . ( X , - ^ ) . i=l k=l From this equation he obtains his formula [45] by making X^-^ =» 0 for i^js " . -»2 However, the middle member of [45] should read correctly; a^ J + ^ L j k x k ) . ( x r x ; ) , hence the expression in the right hand mpmber of [45] is incorrect. The next step in the proof being based on this expansion, the proof cannot be accepted. - 60 -2 f hk\ ' °' (59) i-1 k-1 Evidently• z i L l kx k = r e L i k x k = c \ Z % k - °/ <60> i=l k=l k=l i=l k=l 1=1 which can be satisfied for arbitrary X k only i f n X T L, k - 0 (k - 1, 2, n). (61) 1=1 Eqs 0 (58) and (6l) form 2n relations which are, however, not a l l independent. De Groot (16) states without proof that the. number of independent coefficients i s |n(n - l ) . It can be shown as follows that this i s true. Considering! the relations (58) and (6l) 2n - 1 of them are seen to be independent. This can be seen by showing that any of these relations can be obtained by linear combination of the rest of the 2n - 1 relations.* * Adding up a l l n relations (58) n n C £ L i k= 0. ( i ) i-1 k-1 Summing over the f i r s t n - 1 of relations (61) n-1 n k-1 i-1 1 K -continued, p.61 - 61 -With 2n - 1 independent relations between the coefficients ^ik* t* i e n u m D e r °? independent coefficients would obviously be 2 2 n - (2n - l ) = (n - l ) . This i s i n perfect harmony with the fact that by Eq. (13) there are (n - l ) independent relations (56). It can be shown readily that each of the (n - l ) relations contains only (n - l ) independent coefficients L ^ . Eq. (56) can be written k - ft Li.k \ + hn V <62> k=l and by Eq. (58) n-1 ' L i n = " £ L i k . (63) k=l Subtracting ( i i ) from (i) n / n n-1 \ £ ( r hk - Z, L i k ) - °-i=l \ k=l k=l ./ Considering the identity there follows from ( i i i ) £ L, - 0, (iv)" i-1 i n which i s the (2n)-th relation sought. Q.E.D. - 62 -Inserting Eq. (63) into Eq. (62) and rearranging the resulting expression h = L i k ( Xk " Xn' ( i = 1, 2, n-l). (64) Q.E.D. To obtain ~ n(n - l ) , the number of coefficients L j ^ that are truly independent, the Onsager relations between the ( n - l ) coeffic-ients i n Eq. (64) have to be considered.* Before considering the more particular case of binary systems, the very important condition of mechanical equilibrium w i l l be dealt with. * The number of these relations can be found,according to the following pattern: There are n-2 Onsager relations L.Q = n-3 Onsager relations a n-4 Onsager relations = l=n-(n-l) Onsager relation L . . L l,n-2 n-2,1 Their total number is given by the sum (n-2)n - £ p = (n-2)n - [2 + (n-l)] i S s l i d L - |(n-#n-2). P=2 Using this result the number of independent coefficients i s found readily ( n - l ) 2 - i ( n - l ) ( n - 2 ) = \ (n-l). Q.E.D. -63-. v) The condition of mechanical equilibrium. Mschanical equilibrium means the absence of acceleration of the center of mass. Hence, i n mechanical equilibrium Eq. (30) takes the form o = = 7 P * E V / V ( 6 5 ) k=l The importance of this condition i s that-it i s , at least very nearly, f u l f i l l e d i n a large number of actual physical situations, Prigogine has shown that for this case some theorems, are valid which simplify the description of diffusion processes. The f i r s t of these theorems i s the following n ± X ± - 0• (66) Progogine's proof is simples inserting Eq. (53) in Eq. (66), n n . n_ (67) Introducing 0Oj_ « - j ^ and considering that (jp-^ ° (-OyPlt (j =1, 2, n), the last term on the right of Eq, (67) can be rewritten so that Rewriting the second term on the right of Eq. (68) i n a somewhat different form » since, according to the Gibbs-Duhem equation the result i s - 65 -Using the thermodynamical relationship V ) . v ' ( 7 1 ) where v^ i s the partial specific volume of component i , the third term on the right of Eq. (68) can be written n / n Using Eqs. (70) and (72) i n Eq. (68) there results £ P i ^ • £ P i f i - V p • Comparing Eq. (73) with Eq. (65)j (72) (73) n X, = 0. (66) i A i Q.E.D. As a corollary of Eq. (66) the other important theorem, also due to Prigogine, i s obtained. This theorem states that for mechanical equilibrium the expression for the source of entropy i s independent of the choice of he an velocity v which i s made in the definition of the - 66 -flux [see Eq. (12)], for ordinary diffusion denoted by j ^ . Hitherto we took for v the velocity of the center of mass. Let us now choose another arbitrary velocity v and define the flow \ - />. - v a ) . (74) Then the formal expresion of the theorem stated above i s T G d = E h - h- t,f± '\ • (75) i=l i=l Prigogine's proof of Eq. (75) i s as follows: Forming the difference between the last two members of Eq. (75) and using Eqs. (12) and (74), n a , n ^ n a , ^ I -> " * , L ^ — * r — V f i * . -* i _ k ' h - ^ k - h - £ <J± - k> • x i 1=1 i=l i=l £ P ± \ *(v - v*) (76) i - 1 1 ,a n (v - v ) • ( £ P, X.) = 03 i-1 ' by virtue of Eq. (66). - 67 -vi) Ordinary diffusion - binary systems. Having developed a l l the necessary expressions and theorems for the general case of ordinary diffusion in a multi-component system, as a next step, the particular case of ordinary diffusion in binary systems will be considered. For this case the entropy production beconBS, by application of Eq. (55), k ' h + J B " XB (77) The phenomenological equations [cf. Eq. (56)] are J A - LAA XA * lAB h i = L . X, + L _ X„ JB BA A BB B (78) The relations (58) and (61) become respectively, LAA + LAB " °» LBA +LBB~°> (79) LAA + LBA = °' LAB + LBB " °' (80) u It is recalled that by virtue of these relations and by Eq. (12), the fluxes of Eq. (78) can be written [cf. Eq.(64)] in the form - 68 -J A = L A A ( X A " V - " LAB ( XA " V j (81) The use of theorem (66) results in further simplification of Eq. (81). For the binary case, this theorem is PA XA + PB XB " °- (82) Using Eq. (82) with Eq. (81) - L AA pB (83) But since = - jg f i t is sufficient to consider either of these-two expressions. Thus, comparing Eq; (53) with the first of the expressions ( 8 3 ) , whence, using the same argument leading up to Eq. (68), (84) B T , C O . 'T,p (85) A. ^ _ - 69 -which, using Eq. (71), further simplifies to 1 -i £ AA pB (86) Eq. (86) can be considered the general equation of diffusion (at constant T and at mechanical equilibrium). According to Eq. (86) the diffusion may be considered to result from three different .factors - from the non-uniformity of pressure, and composition, and from external body forces. Disregarding differences in p and the effect of F^, Eq. (86) reduces to (87) At this point the question of the definition of the diffusion coefficient should be considered.' De Groot (loc. c i t . p. 104) defines a diffusion coefficient by the equation 1 _AA AB = pOOB (88) - 70 -Comparing Eq. (88) with Eq. (87) the diffusion equation may be written*s \ ' - P \ B V ^ - ( 8 9 ) This diffusion coefficient i s identical with the one used by Bird [loc. c i t . p. ;174» cf. i n particular his equation (42)]. Originally, however, Bird introduces another d i f f u s i v i t y . This d i f f u s i v i t y f i r s t appears on p: 168, as his formula (33) (loc. c i t . ) . . Since derivation of this formula i s not given by Bird nor * Eq. (89) should be compared with the one suggested by Hartley and Crank (17) for the case where the frame of reference i s the total mass of the system. In the present notation their equation would read , _ DA ^ A > p ( v ° r 3 x D . where D i s their diffusion coefficient when the frame of reference A o i s the total mass, and v i s "some convenient" specific volume such B as the specific volume of pure component B. The latter was introduced in order to rectify the dimension of the right member of the above equation without having to use nonconventional dimensions for the di f f u s i v i t y . Comparison with Eq. (88) shows that M 2 ^ , _.2 = D D (v°) A / AB B It i s shown in the present work at a later point [cf. Eq. (176)] that whenever the density of the solution i s linear i n the volume concentra-tion, the coefficient D._. i s equal to the coefficient i n Fick's f i r s t • AB law. This certainly w i l l not be true for the coefficient D. . - 71 -is any reference to i t given) there does not seem to be any better way of checking how this diffusivity is related to the one introduced by de Groot than to-compare the corresponding two expressions for the fluxes term-by-term. This is best done for the simple binary case. Bird applies his formula (33) to the binary case on p. 173 loc; c i t , [his formula (41)]: 2 £n \ - - j \ f - T T ^ v x » ' < 9 0 ) I dtnx A where C T <=> * Cg = molar density, (91) = molar concentration of A, (92) A M, A « «, molar concentration of B, '' (93) B with and Mg being the molecular mass of A and B, respectively, x^ is the mole fraction of A and a^ the relative activity of Aj related to the chemical potential per unit mass by the equation Eq. (90) is to be compared with Eq, (89). " Since there follows o x ^ A A B By virtue of Eqs. (91), (92), and (93), whence, on introducing u i ^ and C 2 _ D2 ( + OBg 2 Introducing Eqs. (96) and (98) in Eq.(90) the result i s 1 - - P -*>._ V 03.. A ' AB £ n xa A Comparison of Eqs. (89) and (99) gives D = 3 AB AB Q£nx. Comparing Eqs. ^88) and (100), the relationship between A^A c a n ^ e e stablished also. Thus P^B ^ A -" A B . ^ X A ' P - 73 -But and using Eq. (96) VA RT t in\ I (102) (103) Introducing Eq. (98) i n Eq. (103), and multiplying and dividing the right member by x^, using the definition of x^ given by Eq. (95) VA RT 1 _ £ _ ..... But by Eqs. (95) and (98), Eq. (104) becomes 1 A ^GNAA 2 . _ £ -3o)A M A ^ n x A CO, CM, (105) 'A "T"B Combining Eq. (105) with Eq. (101) the result is L ft °T °°A ^B \ MB ( 1 0 6 ) AA = AB'~ R T > ( 1 0 6 ) i.e., <J3 AB i s the physically meaningful part of L^, - 74 -The question of various d i f f u s i v i t i e s and frames of references may now be pursued. Combining Eqs. (96) and (89), ' 4-- P V s f - ^ " A V ^ A '(W) A B or, using Eq. (98) k " " J MA "B DAB V*A • (108) De Groot found i t convenient to define the d i f f u s i v i t y D^g by the equation DAB " DAB ( j T + M A M B > ( l 0 9> A B which converts Eq. (107) to a form analogous to that of Eq. (85), namely J = - PD" V x . (110) A ' AB A In the foregoing treatment the diffusion flow was defined with respect to the center,of mass movement [cf. Eq. (12)]. That description of the diffusion process i s often referred to as "barycentric diffusion". Another type of description i s the so-called "molecular diffusion". In this the diffusion flows are defined with respect to the molecular center - 75 i movement w defined as PA UA . PB UB M M w = B •CATA + °BUB Pk £B A B (111) 2 1 » The diffusion flux i n g. mole cm. sec. defined with respect to w (also called the molar average velocity) w i l l be denoted by (| , Hence and I>B = CB(*B " (112) Let us introduce the forces and -tm —^ • H = MA XA (113) The entropy production w i l l be the same as i n the barycentric case. Applying Eq. (75) T 6 d = W W •A'?A++B-^ - ^ A ° ( M A X A ) + I B ' (114) - 76 -where evidently >^A = £ A = flux of A (g. cm.**2 sec*"1) _j. — i - 2 - 1 and 6 = 3> M = flux of B (g. cm. sec. )^ I B B B (115) Combining Eqs. ( i l l ) and (112) there follows (116) The phenomenological relations are now ^A AA A AB B J (117) Letting X™ = X^ , i t follows from Eq. (134) and Eq. (116) that T 6 d - 0. (118) By the same argument as the one leading to the relations (58) and (6l), expressions analogous to (79) and (80) are obtained _m Tm AA + AB " °> LBA •* ^ B = °> (119) L + L m 0. AA BA m m L + L = 0. AB BB (120) - 77 -Accordingly, Eq. (ll7) can be written = L m ( X m - X m ) . (121) A AA A B It i s now convenient to write theorem (66) in the following form -»m ~»m , x G X + C X = 0. (122) A A B B From Eqs. (121) and (122), m i A = % - ^ . . ' as) B Also, from Eqs. ( i l l ) and (112), Combination of Eqs. (123) and (124) gives T m C C C B T For the case of the barycentric diffusion, from Eqs. (10) and (12) J „ iLl* . t ) . (126) A p A B - 78 -Combining Eqs. (126) and (83) A and realizing that X = ~ , A MA [. (12?) become s ft A Comparison of Eqs* (125) and (128) gives m LAA 1 P LAA °T B^ MA Pk PB XB CA CB (129) From the definitions of the quantities involved in Eq. (128) i t can be shown readily that L m - L ( ^ ^ r 2 H ; 2 H ; 2 . (130) AA AA M Mn A B A B Substituting Eq. (130) in Eq. (123) and letting - 79 -there follows A B B Whence, considering Eq. (88), t - AB * ^'"2 (M4 V"1 °T V WA , (133) s2 <DA - -D A B P V o>A . ' (134) T A B Both Prigogine and de Groot found i t convenient to introduce the di f f u s i v i t y D m = D (^ + ifB)-2 ( M M )"1 ( 1 3 5 ) AB AB MA Mg ; A B * v K ? J Considering Eq. (135), Eq. (133) can be rewritten $ - -C D m V c o . • •• (136) *A T AB A' Using Eqs. (96) and (98) in Eq. (133) - -CTDABV*A • • ( 1 3 7 ) Eq. (137) i s the analogous expression of Eq. (89) i n molar units and with the molecular center as frame of reference. It i s seen that the two equations contain the identical diffusion coefficient D. . De Groot AB denoted the diffusion coefficient i n Eq. (137) by D^. It follows from Eqs. (115) and (136) that 1* - " CT \ D « V»K • < 1 3 8 ) For the sake of symmetry let us introduce also h = T • ( 1 3 9 ) A From Eqs. (89) and (139), ^ - - S ^ A B ^ A - ( U 0 ) A In diffusion experiments with liquid mixtures the fluxes are usually not measured relative to the center of mass or the molecular center of the solution but with respect to some arbitrary coordinate system, the position of which i s fixed with respect to the apparatus i n which the -2 -1 diffusion proceeds. Denoting the flux of A i n g.cm. sec. with respect to this coordinate system by "n^, we can write _x _» -nA = PA \ ( ( H i ) and n B - pB Ug - 81 -The relationship between n and j . can be established by ft, - - H, considering Eq. (141) i n conjunction with Eqs. (10) and (12) and Eq. ( 8 9 ) . From Eqs. (10) and (12) Comparing now Eqs. ( 8 9 ) and (141) with Eq. (142) there results "A = -/>DAB 7 °°A + H ( \ + *B} ' ( U 3 ) b) Application of the Diffusion Equation to the Diaphragm Cell Method Closed diffusion-cell experiments are usually so arranged that i t i s sufficient to consider only one component of the diffusion flux. Hence, from this point on, the diffusion equation (143) w i l l be used i n i t s one-dimensional form nA = "P DAB " T ^ + ^ A ( nA + nBl> where n^ and n B stand for the x-component of "n^ and ng, respectively. Such diffusion experiments are also characterized by the fact that i t i s usually possible to think of an arbitrary number of fixed (imaginary) planes perpendicular to the x-axis, on one side of which the volume of the solution i s constant throughout the experiment. Under such conditions, - 82 -i f h^ and ng are measured with respect to the apparatus, n £ < Q . (145) Taking absolute values, a new symbol Is introduced by n. r BA . (146) n A l It i s convenient to derive an alternative form of Eq. (144), Rewriting Eq. (126) for the one-dimensional case and using Eq. (141), similarly in the one-dimensional form, the result i s V - . W ^ B " ^ - ( U 8 ) Using the following identity i ^ T W - 0 ' ( 1 4 9 ) and considering Eq. (146), Eq. (148) may be transformed to give JA = n A ( ° ° A r B A + ^ P ) ' ( 1 5 0 ) Comparing Eq. (150) with Eq. (89) in i t s one-dimensional form, we - 83 -have f i n a l l y n. - 0 D (oJ r r AB A BA c o B ) -1 9(X>A 3 x (151) Eq. (151) can also be derived from Eq. (144) by defining the quantity which could be termed "apparatus d i f f u s i v i t y " n A - - (152) Eliminating p — — - between Eqs. (144) and (152) and solving for \ = °AB H RBA + ^ -(153) Combining Eqs. (153) and (152) results i n Eq. (151). It i s implicit in the previous development s that the description of the diffusion process i n terms of either component i s equivalent provided that Eq. (151) i s used. Writing Eq, (151) for both components A and B >DAB W A n. ((^ARBA + U V 3x P BA 9 001 (UJBRAB+ ^ (154) - 84 -Multiplying the f i r s t of these equations by (C(JA r B A + C-Og) on both sides, and the second of these equations by (u\ r + CO ) on both M • J AB A sides, and adding the resulting equations, there follows that DAB " °BA ' <155> Here the identities o u j . = - , r . = =. and r„ = - have A WB* BA -n AB n A. D been used. It i s of some interest to show that Eq. (151) is a more general form of Fick's f i r s t law for the mass diffusion flow. It w i l l be seen presently that the essential difference between Fick's law and Eq. (151) i s that the latter gives an unambiguous description of the diffusion process regardless of volume changes on mixing whereas the former does the same only for solutions where there i s no volume change on diffusion. It can easily be shown that whenever this latter condition i s met, Eq. (151) reduces to the simple Fick's law. For convenience we shal l state the condition of additive volumes by the following expression p = a + b pk , (156) i.e., the overall mass density is linear in the mass volume concentration. To prove that Eq. (156) i s truly equivalent to the case of additive volumes, let us introduce OQ. = in Eq. (156). p Hence p = a + b p GOA. (157) - 85 -Whence, solving for p = ^ , i - n = 1 , (158) so that where v = A + BcO A , (159) A - - and B = - - . a a We proceed to show now that whenever Eq. (159) applies for a range of compositions there i s no volume change on mixing two solutions whose composition i s within that range. Assuming the validity of Eq. (159) we can write for any two solutions i n that range, of composition c o ^ and CO A , respectively v = A + B co , (160) 1 and v 2 = A + B uO ^ . (l6l) 2 Suppose we mix m^ grams of solution 1 and grams of solution 2. Since the composition of the resulting mixture w i l l certainly l i e between OJ. and cO« i t can be written for the specific volume of the mixture A l V v = A + Bu> A. (162). - 86 -Multiplying Eq. (160) by and Eq. (l6l) by m^ , on adding up the resulting equations, \ + V 2 = A (n^ + m2) + B (co A + CO A mj, (163) where = v^ m-^ , and V~2 = VgHig are the t o t a l volumes of the two solutions 1 and 2, respectively, before mixing them together. Multiplying Eq. (162)) by (m^ + n^) on both sides V = A (m1 + m2) + B u). (n^ + m^), (164) where V = v(m^ + m2) is the total volume of the solution after mixing. Since both the + <^ A m ? a n d W A ^ m l + m 2^ c ; ] L e a r l y represent the total mass of A in the f i n a l solution, the right hand sides of Eq. (163) and (164) are seen to be equal. Thus V - V 1 + V 2 , (165) and this concludes the proof. It should be noted that mixing solutions at least one of which i s outside the range where Eq. (159) or (162) applies may well result i n volume changes even though the f i n a l composition may f a l l inside the range. We proceed now to show that whenever Eq. (156) applies, Eq. (151.) reduces to Fick's law. Let us digress first and consider Hartley and Crank's (17) treat-ment of Fick's first law. Fick's law can be written for component A and B, respectively; n. - - D. 2 & A " A f\ o x N B - " D B 'x J (166) It has been shown by Hartley and Crank that for the case where there is no volume change on mixing, = Dg. Their proof is simple. For the case of no volume change on mixing, the net volume transfer on diffusion across any fixed plane is equal to zero. Hence DAVA4^ + DBVB4T^ = 0^ ( 1 6 7 ) d x d x where v^ and Vg are the partial specific volume of A and B, respective-ly. In this special case, of course, v^ and Vg are equal to the specific volumes of the respective pure components. It follows from the definitions of the quantities involved that VAPA + VB PB " X- ( 1 6 8 ) whence on differentiating with respect to x: v A 2£k . „ i £ B . 0. ( 1 6 9 ) 4 X 3 X - 88 -Barring the case where v 4 = 0 or v R = 0 , Eqs. ( 1 6 7 ) and ( l 6 9 ) are consistent only i f D A = DB. (170) On the basis of this result Hartley and Crank came to the reasonable con-clusion that i f there i s no volume change on mixing Fick's law applies. Their treatment, however, does not constitute a proof of the validity of Fick's law which was simply assumed and was not derived by these authors. To derive Fick's f i r s t law we may use the following relation dPB rBA nB (171) which follows from an inspection of the physical situation whenever there i s no volume change on diffusion. In writing down Eq. (171) , Eq. (145) and Eq. (146) were also considered. Using the fact that p A + p% = p > ( 1 7 1 ) may be rewritten BA d p A Since u> A + 00g = 1, on combining Eq. (172) with Eq. (151) P DAB 3C0 A n, = (173) PA Consider now p ^ ^ . p * (A)._£1£A- PKr r A r P p 1 r (174) » dpk (1 " ^ >Adp7 ) • - 89 -Comparing Eqs. (173) and (174), ' V ' - A E - ^ ' . ' ( 1 7 5 ) which i s Fick's f i r s t law. From Eqs. (170), (17$), and (166) • D A = D B - D A B , ' (176) whenever Eq. (156) applies. In what follows a formula based on Eq. (151) i s derived to compute d i f f u s i v i t i e s from diaphragm c e l l experiments. The treatment i s in many ways similar to that given by Gordon (19), who limited himself, however, to the additive volume case. ' A brief description of the physical situation i n the diaphragm diffusion c e l l experiments i s given f i r s t . The schematic model of a diaphragm diffusion c e l l consists of two compartments, each containing, a homogeneous mixture of different compositions, and a small section (the porous diaphragm) separating the two comoartments. It i s assumed that inside this section :?;diffusio:n". occurs by molecular mechanism only. The boundaries between the small diffusion section of continuously vary-ing composition and the bulk of the homogeneous solutions on both sides of i t are assumed to be sharp. Either of the two compartments can be open while the other one is closed. A l l volume changes, i f any, occurring on diffusion w i l l be observable in the volume of the solution i n the open compartment. - 90 -Since the entire treatment depends on the important assumption that, at any given moment, the fluxes are constant a l l the way through the diaphragm, this assumption w i l l now be dealt with at some length. Although Barnes ,(20) has given a rigorous solution of the problem without using this assumption for the special case where volumes of the two c e l l compartments were equal and there was the pure component in one compartment, i t was found interesting and enlightening to investigate this problem i n some de t a i l . Dropping the assumption of constant flux, the procedure adopted the density was as follows'. D was assumed to be constant andNlinear i n the volume concentration, so that Fick's f i r s t and second laws could be used. By planes parallel to the faces of the diaphragm, the latter was divided into four sections. The set of dif f e r e n t i a l equations describing the situation under such conditions was solved by a numarical method, the details of which are contained i n Appendix V . Typical diaphragm-cell values of the constants were used. The familiar S-shaped type concentrations profile was obtained which deviated from linearity only negligibly. This profile i s shown on Diagram 7 in a grossly exaggerated way. In the diagram L i s the thickness of the porous diaphragm. One prime refers to the closed h a l f - c e l l whereas two primes refer to the open one. (in t h i s case i t does not make any difference which compartment i s open since the case of no volume change was assumed. However, the same notation w i l l be used later on for the more general case.) Comparison of the times required to produce a given concentration change as computed by the numerical method and the simple logarithmic formula generally used i n conjunction with the diaphragm c e l l method showed a difference not - 91 -SCHEMATIC REPRESENTATION OF CONCENTRATION AND DRIVING FORCE CONDITIONS INSIDE THE POROUS DIAPHRAGM - 92 -exceeding 0.02$. The bulk of the numerical computations was done by 1 Mr. V. M. Bothner (2l) under the supervision of the present author. The simple logarithmic formula is as follows; (A/oA), uTKpf0"^- ( 1 7 7 ) In this equation Ay0^ = Pk~ Pk " s u b s c r i P t s f a n d ° s t a n d f ° r f i n a l and i n i t i a l conditions, respectively, 9 i s the duration of the experiment and i s the c e l l constant defined by the equation where A„ i s the effective cross sectional area and L the effective e thickness of the diaphragm. V M and V are the volumes of the open and the closed compartments, respectively. Since one of the assumptions underlying the derivation of Eq. (177) i s that of constant flux, the above quoted remarkable agreement between the results of the two methods can be interpreted as a strong support for the assumption of constant flux. It should be remarked here that this assumption i s often called pseudo-steady state or steady state assumption. It i s also clear that this assumption i s equally valid for the case where i s not constant but varies with the composition of the - 93 -solution. The variation i n D. w i l l be compensated for by corresponding ^Pk ' changes i n the value of — , so that the flux remains nearly constant 3 x just as i n the case of constant D^. Naturally, the concentration gradient w i l l deviate correspondingly from linearity. What appears to be the essential condition for the v a l i d i t y of the assumption of constant flux i s that the product ^ be small. Stating the definition of constant flow formally c, n« = n£ - n A , (179) where n^ now stands for the flux anywhere inside the diaphragm. By Eq. (151) the expression 1 9 ULi nA " ? DAB ("A rBA + ( 1 5 1 ) i s then constant throughout the diaphragm. Hence we can replace i t by another expression which i s equally constant but i s far more convenient to handle, i.e., l e t II m i , .„1 ^ CO A ~ .V ^ s "1 A " k , PhB < % r BA * ~B> - o - 1 - ? DAB K r BA * MB> " V ^ (l80> where the symbol has been used to denote the arithmetic mean of the corresponding two values taken i n the two c e l l compartments. For the case where Fick's law applies the significance of this substitution i s shown in the bottom part of the Diagram 7. D^g i s a space average - 94 -d i f f u s i v i t y whose value has to be found yet. This i s done by integrating Eq. (180) between the limits as defined by the conditions at the two faces of the diaphragm: - rBA + <^B AB P °AB rBA + A d6u. . (181) t A Eq. (181) i s then the defining equation of D^ g. As a next step, i n the same way as i n the less sophisticated treatment, material balances are set up on the two compartments. In each case the flux, multiplied by the effective cross sectional area for diffusion*, i s equated to the rate of accumulation of material i n the compartme nt s; n t ^ i " A e ^ A B ^ A rBA + ^ r 1 ^ - ^ - V , ( l 8 2 ) The question may arises what i f A e, the cross sectional area for diffusion, i s not constant? Obviously, the previous arguments for the constancy of the flux w i l l equally hold i f the meaning of flux i s re-interpreted as "flow of mass per unit time", thus including A e. It w i l l have been noticed that for the case of constant DA, constancy of flux meant constancy of ^ / 3 x ° for the case of non-constant DA, the constancy of 'dppj^x ? and for the case of non-constant Ae, constancy of the flux i n the extended sense w i l l mean constancy of Ae • - 95 -In setting up the material balance on the open compartment [Eq.(183)3, allowance has been made for the possibility of changing volume. Expanding the right member of Eq. (183), solving the resulting expression for d p A , also solving Eq. (184) for =dyO^ and adding the two equations thus obtained; i? ? - AeJ* DAB^A RBA + — L (V*. + + Pk W = (184) -"d</°A- Pi) 'd^f>k.•• Using previous definitions, we can write - I ( p " L V A » - p * * 6 0 A + p* G O > A * " p ' U ^ ) = (185) = | A/)A + | ( p ' c O j ( - p"WA) . Combining Eqs. (184) and (185), -DAB A e ( % RBA ^ ^B)"1 ,A - + , w l + _L NH< (186) f A f 1 " d ^ '• - 96 -Dividing both sides of Eq.(l86) by &p k -DAB A e ^ A rBA • ^ B ^ 2L 1 + Al /1 1 \ . ft . PA dv»» . * (187) Introducing ^ the c e l l constant, as defined by Eq. (178), and also (188) Eq. (187) can be rewritten -D AB / ^ V \ - l , v 2 ( % FBA + H} ^ +K) d (189) Integrating Eq. (189) from 9 = 0 to Q =0 ( A rBA + ^ B r l ( 1 + K ) D® ' i B ^ 1 J (190) o ^ - 9 7 -where I = ( 1 9 1 ) R = ( ) A^f ( 1 9 2 ) and I 2 ( 1 9 3 ) To be able to carry out the integration of the l e f t member of Eq. ( 1 9 0 ) , i t i s convenient to define a space-and-time average d i f f u s i v i t y ("integral" diffusion coefficient) D AB AB (^ A RBA +£^B)' J D 1 + K A B UQ r +cD A BA B ( 1 9 4 ) Using Eq. ( 1 9 4 ) , Eq. ( 1 9 0 ) may be written DAB= 2 (3& + if^k rBA + ^ B ) * K i + K ; ( 1 9 5 ) Eq. ( 1 9 5 ) i s the formula for computing B^g from the experimental results. Since the quantities involved in Eq. ( 1 9 4 ) are not known as explicit functions of time, a relationship between D^g and D 4 R that AB 98 does not contain time is-desirable„ This can be obtained i f Eq. (189) i s rearranged ^ d 9 = - ^ e (196) ¥ <*A rBA * " * « On integrating, Eq. (196) gives / 3 0 - - / ~ — Q • (197) - f n rBA * " V (i *« Comparing Eq. (197) with Eq. (195) gives, after rearrangement, the desired relationships \ 1. + K AB (198) dV \ u j A r M + u ^ in I f^> 1 AB A/?A v y ' D a p ( i + K) - 99 -If D^g i s known as a function of composition(the integral d i f f u s i v i t y D^g can be evaluated by combined use of Eqs. (181) and (198). The only additional information necessary i s the knowledge of the density of the solution as a function of the composition. The reverse procedure, when C^g i s known from the experi-mental results, and D^g i s wanted as a function of. concentration, requires some trial-and-error. This question w i l l be discussed i n the next section. In an effort to simplify Eq. (198) this formula has been subjected to detailed analysis, the result of which indicated that the expression Jul. A adopted as an approximation, i s not l i k e l y to be in error exceeding 1% i n what are considered typical diaphragm diffusion c e l l experiments. This simplification hinges on the fact that the expression under the integral sign in Eq. (197), ¥ <^A rBA ^B'" 1 <1 • « • does not vary much and hence, the mean of i t can be factorized out before - 100 -the integral sign without committing any significant error.* Using Eq. (181), Eq. (199) i s then obtained immediately. Eq. (198) can be written i n a somewhat simpler form i f Eq. (181) i s considered. Taking the term i n the denominator of the fraction under the integral sign in Eq. (197) and combining i t with Eq. (181) DAB^ A RBA + ^ B^ (1 + K) - (200) A^ FBA * 4 / TAB d u J I A f A + f'^l - P"^ A JA where Eq. (188) has been used. Introducing the function / / n ,' F(w" aj') » / 7 P ° A B . dcoA , (201) and using Eq. (185), Eq. (200) may be simplified to give; * It should be noted that i n the case where the above term i s linear i n In^pX this step i s rigorously correct. - 101 -IT rBA + ° V (1 + K) (202) Introducing Eq. (202) i n Eq. (198) AB ft dV M p A F(ou" co*) ^ A A (203) which i s the simplified form of Eq. (198) sought. Either Eq. (199) or Eq. (203) may be used to compute D^g as a function of composition from a series of values D^g. As already mentioned, this problem w i l l be discussed i n detail i n the next section. The enormous saving in labor resulting from the use of Ea. (199) i s obvious. Having shown earlier that for the case where ^? = a + b p ^ , Eq. (151) reduces to Fick's f i r s t law [cf. Eq. (175.)], there must follow that i n the same special case, Eq. (195) reduces to the simple logarithmic formula [cf. Eq. (172)], while Eq. (199) reduces to the well known expression for integral diffusivity, based on Fick's law, V Pl- p » A (204) - 102 -To reduce Eq. (195) to the simple logarithmic formula, consider that for the case of no volume change on diffusion [cf. Eq. (167)], \ VA + NB VB " °' ( 2 0 5 ) Hence * ^ , . i . . f § a r ( 2 0 6 ) n A V B PA , where and p^ i s the density of pure component A and B, respectively. Consider now the term (<^ A RBA + B^^ * Using Eq. (206) and definitions this term becomes * H RBA B = 2 (^A + " 2 ( WA + °V + 1 = (207) / " ' \ ^ A „ ( c o . + O J . ) ; — ; — - + 2 _ A A Pi . By virtue of linearity £ ^ . _ L ^ L . (208) Using Eq. (208) we can write - 103 II J , (209) II , . ! I f t II S t CO KP -*LP "AP -WKP where Eq. (188) and the definition of p^ have been used. Comparing Eq. (207) with Eq. (209) ^ A - B A ^ - ^ - <210> Since for the case considered dv" = 0, on comparing Eq. (210) with Eq. (189), -DAB | 3 d & - d£n(dpA). (211) Eq, (211) i s the di f f e r e n t i a l form of the simple logarithmic formula 0 Hence, the proof is concluded with the identification D^ g = . Turning to the case of Eq. (199), i t has already been shown [cf. Eqs. (173) and (174)] that for the case of additive volumes, — 22— dco. = D d p . (212) A rBA + °°B A Consider now the term (CU r + UJ ) A BA B V It t p (co - co ) - 1 0 4 -Since - « < ) . / ' t - W ^ A P ' - W f r A A 9 j ( 1 2 3 ) 2 where the definitions of p and ^ O ^ have been used, and Eq. ( 2 1 0 ) may be rearranged to give ~ II i it _ I , , i n II where the definition of K has been used, there follows ^ A r B A + W B ^ r " ' « II t ( 2 1 5 ) A Introducing Eqs. ( 2 1 2 ) and ( 2 1 5 ) i n Eq. ( 1 9 9 ) , Eq. ( 2 0 4 ) is obtained. Q.E.D. As a concluding part of this section let us re c a l l that in the general case Fick's f i r s t law does not apply. Hence, i f i t is used i n a case where there i s a volume change on diffusion, the coefficients . and Dg cannot be regarded as diffusion coefficients since bulk flow w i l l have con-tributed to their values. Nevertheless in such laboratory diffusion experi-ments on li q u i d solutions as yi e l d practically differential (as opposed to integral) " d i f f u s i v i t i e s " , and Dg w i l l have well-defined values such that DA * DB * DAB« ( 2 1 6 ) It can be shown that there i s the following relation between the true diffusion coefficient D^g, and the apparent values D^ and Dg + i t . ( 2 1 7 ) - 105 -The derivation of Eq. (217) i s straightforward. Comparing Eqs. (152) and (166), A/ 9x A 9x > DA VA AB ~ p Using Eq. (166) r. - £ (1. 1L (218) whence D' °A f£A . (219) A p dco^ Comparing Eq. (219) with Eq. (151), a s ? < H r M •«!>>• ( 2 2 0 ) ( i - - J - ) . (221) Combining Eq. (221) with Eq. (220), Eq. (217) follows. For the case where p i s linear i n p^ the equations follow also from Eq. (217). It has been shown that for this case DA . DB . (170) - 106 -Hence, from Eq. (217), D A B3 ?£A(1 jfi.) . A B P d W A dp A fk (222) °A , dp d u \ dp . — M - O ) , + -—~D -co, - r J ~ ) = D, P v dco A dc<-) / A doo A r A A A Q.E.D. - 107 -IV. RESULTS AND DISCUSSION Results The numerical results obtained in the present work are shown in the following tables and graphs. Table I Analytical results obtained in Runs 7 and 8 with the aqueous KC1 solutions. The concentrations, C, are given in g./lO ml., numerically equal to the weight of the residues in grams. Each concent rati on is the mean of three parallel determinations. C 0 = 0. Run 7 Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 "o 0.07165 0.08249 0.07691 0.08045 0.08350 0.06881 0.05477 0.06283 0.06013 0.06251 0.06554 0.05228 =; 0.01711 0.01995 0.01742 0.01819 0.01807 0.01718 ' 9, sec. 335,080 355,680 356,280 356,880 357,480 357,780 Run 8 Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 0 0.05658 0.07388 0.08077 0.09755 0.07159 0.04301 0.05778 0.06279 0.07664 0.05442 0.01377 0.01672 0.01823 0.02104 0.01784 0, sec. 355,800 356,100 356,700 357,000 357,300 - 108 Table II Ethanol concentrations obtained in the runs with the ethanol-water mixtures at 25°C. Here, for simplicity, C denotes ethanol concentration i n g./ml. The densities used to calculate the concentration values were the mean of at least two parallel determinations (exceptions; Runs 4/1 and 5/1). Cell/ Run It t Co c f cav 6,sec. 4/1 0.7A889 0.72991 0.62380 0.64609 0.68718 678,300 5/1 0.61534 0.63782 0.74889 0.72888 0.68273 676,500 6/2 0.74520 0.72163 0.62996 0.65428 0.69142 756,000 1/4 0.74998 0.73268 0.63482 0.65324 0.69268 513,000 4/4 0.61477 0.65276 0.74998 0.71199 513,000 1/6 0.58288 0.59572 0.66426 0.65265 0.62387 693,900 4/6 0.66426 0.65370 0.58131 0.59299 0.62307 696,600 6/7 0.58519 0.59448 0.65421 0.64563 0.61988 609,000 1/8 0.75533 0.73747 0.65196 0.67098 0.70394 601,800 4/8 0.75533 0.73822 0.64878 0.66708 0.70235 602,400 632,100 3/9 0.64931 0.63864 0.55781 0.56836 0.60353 6/9 0.64931 0.63712 0.55738 0.56941 0.60331 632,100 2/10 0.56528 0.55437 0.47330 0.48423 0.51930 685,800 5/10 0.56528 0.55531 0.46478 0.49504 0.51510 685,800 693,600 1/11 0.47924 0.47263 0.42035 0.42693 0.44979 4/11 0.47924 0.47399 0.43238 0.43756 693,600 1/12 0.40465 0.39620 0.32688 0.33533 0.36577 610,800 4/12 0.40465 0.39680 0.32795 0.33579 0.36630 611,100 2/13 0.34232 0.32830 0.23150 0.24556 0.28692 607,500 5/13 0.34232 0.32873 0.22581 0.23976 0.28416 608,100 3/14 0.25615 0.23798 0.14298 0.16060 0.19943 617,400 6/14 0.25615 0.24317 0.17866 0.19123 0.21730 618,000 1/15 0.17892 0.16273 0.08656 0.10230 0.13263 500,300 4/15 0.17892 0.15979 0.06766 0.08650 0.12322 502,400 2/16 0.06832 0.05443 0.00137 0.01505 0.03479 490,500 5/16 0.06832 0.05565 0.00129 0.01407 0.03483 496,500 3/17 0.76493 0.75940 0.73416 0.73971 0.74955 500,300 6/17 0.76493 0.75778 0.72789 0.73511 0.74643 502,100 1/18 0.75945 0.74993 0.69437 0.70443 0.72705 417,000 4/18 0.75945 0.75043 0.69333 0.70288 0.72652 419,400 2/19 0.73779 0.72520 0.60166 0.61505 0.66993 336,000 5/19 0.73779 0.72717 0.61352 0.62503 0.67588 337,800 683,400 3/20 0.69430 0.70768 0.75880 0.74667 0.72686 6/20 0.69430 0.70904 0.75841 0.74502 0.72669 683,400 1/21 0.58663 0.59394 0.64363 0.63653 0.61518 595,500 596,000 4/21 0.58663 0.59361 0.64453 0.63775 0.61563 2/22 0.50952 0.51681 0.58466 0.57760 0.54715 507,600 Notes In Column "Cell/Run" the f i r s t number refers to the cell., the second to the run. Two experiments with the same number "run represent parallel experiments. - 109 -Table III Cell-compartment volumes, ml. Cell Upper compartment Lower compartment 1 47.8 48.45 2 47.5 48.2 3 49.45 51.35 4 50.0 50.7 5 48.8 48.4 6 48.3 50.2 Table IV Cell constants, ^ , cm,"2 Based on D K P 1 values by Stokes (22) Cell No. 1 2 3 4 5 6 Pain 7 Run 8 Mean 0.0969 0,0969* 0.0986 0.0989 0.09875 0.0884 0.0883 0.08835 0.0896 0.0894 0.0895 0.0847 0.0846 0.0840** 0.1006 0.1006 0.1006 * Cell No. 1 had been in use for a considerable time before the rest of the cells were built. The value O.O969 agreed satis-factorily with the average of earlier determinations. ** Cell No. 5 broke after the above calibration runs. After repair its volume had to be re-determined. The value O .O84O contains the correction necessitated by a small change in the compartment volume. - 110 -Table V 2 =1 Measured values of d i f f u s i v i t i e s , D s cm. sec. , in the ethanol-water system at 256C. The second column contains the value of the integral d i f f u s i v i t y as obtained,by the simple logarithmic formula. The third column contains the mean ethanol concentration (g./ml.) over the run. The fourth column contains the concentration at which the integral d i f f u s i v i t y in the f i r s t column i s equal to the true, differential d i f f u s i v i t y . The f i f t h column contains the mole-fraction of the ethanol corres-ponding to the modified C. The second column plotted vs. the third column is shown in Diagram 8. The second column plotted vs. the fourth and f i f t h columns is shown in Diagrams 9 and 10, respectively. (See also sample calculations i n Appendix VII.) Part A Present Measurement Run DxlO5 c a v C X 5/16 1.145 0.035 0.035 0.014 2/16 1.096 0.035 0.035 0.014 4/15 0.929 0.123 0.123 0.053 1/15 0.875 0.133 o.l33 0.058 3/14 0.697 0.199 0.198 0.092 6/4 0.644 0.217 0.216 0.102 5/13 0.528 0.284 0.280 0.140 2/13 0.487 0.287 0.284 0.143 4/12 0.419. 0,366 0.364 0.200 1/12 0.414 0.366 0.363 0.199 1/11 0.377 0.450 0.450 0.276 5/10 0.390 0.515 0.522 0.357 2/10 0.400 0.519 0.525 0.361 2/22 0.423 0.547 0.550 0.394 3/9 0.472 0.604 0.609 0.485 6/9 0.481 0.603 0.609 0.485 1/21 0.505 0.615 0.617 0.499 4/21 0.509 0.616 0.617 0.499 .. continued - I l l -Table V (continued) Run DxlO5 cav G X 6/7 0.514 0.620 0.622 0.509 4/6 0.523 0.623 0.626 0.516 1/6 0.556 0.624 0.627 0.517 2/19 0.639 0.670 0.677 0.623 5/19 1 0.691 0.676 0.681 0.634 4/1 0.692 0.687 0.693 O.664 5/1 0.693 0.683 0.689 0.653 6/2 0.706 0.691 0.697 0.674 1/4 0.747 0.693 0.700 0.684 4/8 0.749 0.702 0.707 0.703 1/8 0.760 0.704 0.709 0.707 3/20 0.834 0.727 0.730 0.773 6/20 0.840 0.72? 0.730 0.773 4/18 0.878 0.727 0.730 0.773 1/18 0.886 0.727 0.730 0.773 6/17 0.972 0.746 0.746 0.829 3/17 1.010 1.010 0.750 0.842 Part B Data by Hammond and Stokes (3) 1.167 0.023 0.023 0.009 1.165 0.023 0.023 0.009 1.136 0.029 0.029 0.012 1.101 0.046 O.O46 0.019 1.093 0.046 0.0ZV6 0.019 1.091 0.047 0.047 0.019 1.084 0.047 0.047 0.019 1.084 0.053 0.053 0.022 1.079 0.052 0.052 0.021 1.004 0.075 0.075 0.031 0.951 0.097 0.097 0.041 0.950 0.097 0.097 0.041 0.853 0.136 0.132 0.058 0.852 0.142 , 0.137 0.060 0.754 0.181 0.174 0.079 0.752 0.182 0.175 0.079 0.396 0.379 0.369 0.204 0.399 0.379 0.369 0.204 0.368 0.427 0.402 0.231 - 0.362 O.427 0.402 0.231 0.395 0.410 0.375 0.209 - 112 -Table V (continued) Part B (continued) DxlO5 C a v C X 0.393 0.407 0.372 0.206 0.379 0.460 0.503 0.334 0.378 O.46O 0.503 0.334 0.406 0.494 0.543 0.384 0.404 0.494 0.543 0.384 0.589 0.657 0.658 0.580 0.575 0.657 0.658 0.580 0.675 0.679 0.683 0.639 0.684 0.680 0.683 0.639 0.927 0.739 0.740 0.805 0.885 0.728 0.731 0.777 0.980 0.749 0.750 0.842 0.950 0.740 0.740 0.805 0.971 0.749 0.750 0.842 Note that the results of 4/4 and 4/11 have been omitted because of some mistakes made during the experiments resulting in very much higher diffusivities than in the parallel run. X Table VI Diffusivities at 25°C. i n the ethanol-water system as measured by Smith and Storrow (10) X 0 0.275 . 0.390 0 764 0.943 DxlO5 1.35 0.4 1.2 1.5 2.17 Table VII Diffusivity in the ethanol-water system as a function of the volume concentration C at 25°C. Values were interpolated using the combined data by the present author and by Hammond and Stokes [.cf. Diagram 9]. For comparison, the values interpolated by these latter authors, using their own results, are also given. Temperature 25°C. G D C D Present work Hammond and Stokes Present work Hammond and Stokes 0,000 1.220 1.240 0.475 0.373 0.050 1.080 1.079 0.500 •. 0.380 0.372 0.100 0.946 0.935 0.525 0.395 0.150 0.817 0.807 0.550 0.415 0.407 0.200 0.695 0.694 0.575 0.442 0.225 0.637 0.600 0.475 0.46? 0.250 0.584 0.595 0.625 0.517 0.275 0.535 0.650 0.570 0.551 0.300 0.490 0.508 0.675 0.637 0.325 0.451 0.700 0.725 0.766 0.350 0.418 0.426 0.725 0.835 0.375 0.392 0.750 0.975 0.891 0.400 0.373 0.377 0.785 1.220 1.132 0.425 0.365 0.450 0.367 0.362 - 11.7 -Table VIII Diffusivity and the diffusivity-density product i n the ethanol-water system as function of weight fraction of ethanol at25°C. The diffusivity values were interpolated using the data i n Table VII. D Lp CO i D Bp 0.000 1.220 1.216 0.575 0.388 0.346 0.050 1.082 1.069 0.600 0.401 0.356 0.100 0.951 0.932 0.625 0.414 0.365 0.150 0.827 0.805 0.650 0.433 0.379 0.200 0.711 0.687 0.675 0.455 0.396 0.225 0.657 0.633 0.700 0.482 O.416 0.250 0.606 0.581 0.725 0.511 0.438 0.275 0.559 0.534 0.750 0.547 O.466 0.300 0.517 0.492 0.775 0.584 0.494 0.325 0.478 0.452 0.800 0.627 0.526 0.350 0.444 0.418 0,825 0.680 O.566 0.375 0.416 0.390 0.850 0.736 0.608 0.400 0.392 0.365 0.875 0.802 0.658 0.425 0.378 0.350 0.900 0.874 0.711 0.450 0.367 0.338 0.925 0.953 0.769 0.475 0.366 0.335 0.950 1.043 0.834 0.500 0.368 0.335 0.975 1.130 0.896 0.525 0.373 0.337 1.000 1.220 0.958 0.550 0.380 0.341 - 118 -Discussion a) Comparison of results with those obtained by Hammond and Stokes (3) and Smith and Storrow (2). Using the simple logarithmic formula [cf. Eq. (177)], integral d i f f u s i v i t i e s were calculated from the data i n Table II. These integral d i f f u s i v i t i e s are given i n Table V and plotted on Diagram 8 together with the measurements reported by Hammond and Stokes (3). It i s apparent from the graph that the two sets of data are in very close agreement. The data of Smith and Storrow (2) are not shown because some of their points could have been plotted only i f the di f f u s i v i t y scale on the graph had been unduly compressed. Since the very large disagreement between the data by Smith and Storrow, on the one hand, and by Hammond and Stokes, on the other, occurred i n the alcohol-rich part of the concentration range, most of the present measurements were done with solutions containing over f i f t y percent alcohol by weight. This region was covered several times i n independent series of experiments. (Note i n Tables II and V that the second number used to designate a run corresponds to chronological sequence.) From one run to another, occasional small alterations i n the experimental technique were made. In the bulk of the runs the solution richer in alcohol was contained in the upper hal f - c e l l and this compartment was open. In runs 5/1, 4/4p 1/6, and 6/7, however, the lower compartment was open, while i n runs 3/20, 6/20, 1/21, 4/21 and 2/22, the solution richer i n water was in the top compartment which, i n these experiments, was again the open half of the c e l l . A l l these modifications were made to see whether or not they would - 119 -cause any significant difference i n the d i f f u s i v i t y values obtained. No such trend could be detected, however. In view of the close agreement between the present results and the data obtained by Hammond and Stokes, on the one hand, and the very significant disagreement between the above two sources and the measurements by Smith and Storrow, on the other, i t seems most l i k e l y that there was some systematic source of error present i n the experiments of the latter researchers. The fact that they made experiments at various temperatures without any apparent intersection of the resulting lines gives support to the contention that their source of error must have been systematic. One such possible source of error i s that d i s t i l l a t i o n can occur through a wetted ground glass joint. This effect has been discussed at some length in Section II, g, of the present work. b) The relation between the true d i f f e r e n t i a l d i f f u s i v i t i e s and the measured integral values. One way of obtaining the true d i f f e r e n t i a l d i f f u s i v i t i e s from diaphragm diffusion experiments has been discussed by Gordon (19) • In the case discussed by him the value of the d i f f u s i v i t y at i n f i n i t e dilution was assumed to be known. In a f i r s t approximation, the measured integral d i f f u s i v i t i e s are identified with the di f f e r e n t i a l ones for the mean concentrations of the experiments. Then an analytical expression i s derived for the ratio d i f f u s i v i t y at concentration C to d i f f u s i v i t y at in f i n i t e dilution, as a function of C. The form of this function i s assumed to be known. The next step is to compute the integral d i f f u s i v i t i e s for the - 120 -actual experiments using this function and an appropriate formula to calculate integral d i f f u s i v i t i e s from differential values, [cf. Gordon's Eqs. (13), (17), and (20)]. Then these integral values are substituted i n the empirical equation mentioned above and the equation i s solved for the concentrations. The second approximation consists i n identifying the observed integral values with the differential values at these new concentrations. On this basis the coefficients in the empirical equation may be adjusted resulting i n an improved tabulation of differential d i f f u s i v i t y vs. concentration. This procedure is continued un t i l the integral d i f f u s i v i t i e s calculated by the empirical equation agree with the experimental values for a l l the runs considered i n the computations within the accuracy of the measurements. It i s quite obvious that, instead of using an analytical expression the same sequence of approximations may be performed by graphical methods. First the measured integral d i f f u s i v i t i e s are plotted against the corresponding mean concentrationsj then, using the curve obtained, integral d i f f u s i v i t i e s are computed for each actual run| the concentrations corresponding to these integral coefficients are read from the graph and the measured d i f f u s i v i t i e s are replotted against these new concentrations, and so forth. Stokes (23) used the graphical pro-cedure in the case of eight univalent electrolyte solutions, where the "Nernst limiting value" of the coefficients could be assumed to be known. In reference 3, however, they write? "The great change of the diffusion coefficient with concentration, together with the lack of any theoretical limiting values at the end' of the concentration range, precludes the use - 121 -of the method developed for electrolyte solutions." Accordingly, they used the method of least squares to compute the coefficients in those power series which were assumed to give the differential d i f f u s i v i t i e s as a function of concentration. Even though the entire concentration range was subdivided by them into three groups, a cubic expression had to be assumed to obtain what appeared to these authors a satisfactory f i t i n the range of the lowest alcohol concentration. It i s d i f f i c u l t to see the advantage of this procedure over the graphical method of approximation. There is an element of arbitrariness i n assuming any particular form of equation to try to f i t the experimental data i n the range of lowest alcohol concentrations. Also, had the marked variation of the diffusion coefficient with concentration been a serious impediment one guess as to the form of these equations could hardly have been sufficient. Obviously, the same element of arbitrariness has been transferred to the value of the d i f f u s i v i t i e s at i n f i n i t e dilution, obtained as the coefficient of the concentration to the zero-th power in the power series. In the case of the cubic, this means a non-linear extrapolation, which i s rendered even more uncertain by the oscillations which a cubic interpolation formula i s bound to have. Consequently, in the present work, the graphical method of approximation was used. This necessitated some assumptions concerning the values of the d i f f u s i v i t i e s at the two ends of the concentration range. Plotting the integral d i f f u s i v i t i e s against the mole fraction corresponding to the mean concentrations, i t was found that the points were best fit t e d by straight lines for x <^0.03 and x > 0.6. - 122 -Graphical linear extrapolation, checked later by the least-squares method, gave, within experimental error, 1.220 as the value of D at both ends of the concentration range. These additional points were then used to construct a best f i t t i n g line through the integral d i f f u s i v i t i e s (see Diagram 8). It should be emphasized that due reserva-tions are made concerning the correctness of the limiting values obtained by this procedure of linear extrapolation. There i s no reason to believe, however, that this procedure i s i n any way inferior to that used by Hammond and Stokes. It should be realized quite clearly that extra» polated values, barring any theoretical or empirical guidance, are always bound to be uncertain. Fortunately the effect of the error i n the value of the two extrapolated limiting d i f f u s i v i t i e s on the rest of the values as obtained in the graphical procedure dies off rapidly as one moves away from the two limiting compositions. After performing one approximation, using Eq. (204' and tabular integration (see Appendix VI), i t was found that the original curve dra^Ya^most as good as the corrected one (see Diagram 9). Hence, second approximation was not necessary. As may be seen from Diagrams 8 and 9, as well as from Table V, the data obtained by Hammond and Stokes were also used i n the approximation. The overall change i n the concentration values i s very small with the exception of a few points by Hammond and Stokes around the strongly curved minimum of the graph. Thevery fact that the f i r s t curve drawn was practically the correct one refutes the argument by Hammond and Stokes concerning the d i f f i c u l t y of using the graphical procedure because of the very sharp change of d i f f u s i -vity with concentration. In any case, i t i s not the rate of change of the - 123 -d i f f u s i v i t y with concentration, but the degree of curvature of the D vs. C curve that can increase the labour involved i n the approximations. If the curve i s linear, the measured integral d i f f u s i v i t i e s are rigorously equal to the dif f e r e n t i a l values at the mean concentration of the experiments (19). Partly owing to the difference i n the computational methods and part-ly due to the larger number of data considered, the resulting D vs. C relationship i s somewhat • different from the one obtained by Hammond and Stokes. The d i f f u s i v i t y values interpolated on the curve shown in Diagram 9 have been tabulated and are shown with the corresponding figures obtained by Hammond and Stokes in Table VII. The most important deviation was found i n the range of high alcohol concentrations, where the assumption by Hammond and Stokes that the D vs. C relationship was linear could not be confirmed with the additional data on hand. c) The scatter of the experimental results. It is shown i n Appendix I that the most probable error of a single d i f f u s i v i t y value, based on the analytical error, i s about ± 0.2$. It has been found, however, that the average deviation of the experimental points from the best line drawn through them (see Diagram 9) i s ±2%. It i s also shown in Appendix I that, i n the case of the calibration runs, the average deviation from the mean of the individual determinations of the c e l l constant i s equal to the most probable error - 124 -as calculated from the analytical error (± 0.1%), There are several possible sources of error that might account for the increased scatter of the alcohol experiments. One of them undoubted-l y is the fact that the thermal expansion coefficient of the alcohol solutions i s considerably larger than that of pure water. Consequently, equal irregular i t i e s i n both temperatures give rise to greater bulk flow through the porous diaphragm, this way or the other, with the ethanol solutions. Another possible source of error is relative loss of ethanol on sampling owing to i t s higher v o l a t i l i t y . Another possible source of error i s the fact that the. diaphragmswere not i n perfectly horizontal position which, under the effect of larger density difference, could have caused some flow through the diaphragm. A l l these sources of error could be reduced by making refine-ments in the experimental technique. It appears that these sources of error played a less important-' role in the experiments of Hammond and Stokes. The scatter of their data seems to be determined primarily by the analytical errors. This may be concluded from the fact that their parallel determinations agree much better i n the range of low and medium alcohol concentrations, where the experiments were done'under conditions of considerably greater concentra-tion differences than i n the case of the more concentrated solutions. Since, i n the present work, there was no detectable trend i n the results of re-runs, i t i s safe to conclude that the values of the c e l l constants did not decrease significantly i n the course of the experiments. - 125 -Also, Cell No. 1 had been used i n about a dozen runs before completing the calibrations with the KC1 solutions without any noticeable trend i n the values obtained for i t s c e l l constant. d) The systematic error committed by neglecting volume changes on diffusiono In Section III, b, formulae have been derived to compute the true d i f f u s i v i t y from diaphragm diffusion c e l l experiments regardless of the presence of volume changes on diffusion. It has been shown that, s t r i c t l y speaking, under such conditions the simple formulae in common use at present do not yield values for the d i f f u s i v i t y that are consistent with the requirements of theory. These inconsistencies may give rise also to apparent differences between the values obtained for"the true-differential d i f f u s i v i t i e s , depending on the concentration differences used in the diaphragm c e l l experiments, on whether or not the concentra-tions of the one or the other component are used to calculate D, and on whether or not the compartment containing the more or the less concentrated solution i s the one which is open during the diffusion experiment. It has been also been shown in Section III, b, that when the density of the solution i s linear in the volume concentration the more general formulae reduce to the corresponding simple formulae referred to above. Since the smaller the concentration differences are the better the condition of linearity i s met, i f small enough concentration di f f e r -ences are used the simpler formulae may be considered completely satis-factory. Small concentration differences, howeverj greatly increase the error i n D due to the analytical error. If for this or any other \ _ reason i t is found convenient to use large concentration differences, the question of the error introduced by the use of the simple formulae may have to be considered. Tc estimate the correction necessary in the case of a diffusion experiment involving a large concentration difference, some computations were performed, the details of which are shown i n Appendix VII. In a hypothetical diffusion experiment, pure water on one side of the diaphragm was l e f t to diffuse into a solution containing about 0 o 6 go/ml. ethanol u n t i l the concentration difference dropped to about 75$ of the original value. With the ce l l s used i n this work, this case would corres-pond to about one week of diffusion. The following quantities com-puted for this hypothetical diffusion experiment are of special interests DA - using Eq. (177), D A B - using Eq. (195), D A B(appr.) - using Eq, (199) and the values Dp i n Table VIII. D A B(acc.) - using Eqs. (203) and (201) and the values in Table VIII. and a quantity D^ g obtained by applying Eq. (204) to the function DAg instead of DA . The D values interpolated i n the present work and contained i n Table VII were used. The following values were obtained; D A * 8 3 0 . 4 3 5 5 , - 0 . 4 7 7 5 , D A B(appr.) » 0 . 6 3 2 , D A B(acc.) = 0 . 6 3 8 , D*B - 0 . 5 8 8 , and also A V = - 0 . 6 9 ml. 4 Hence = 0 . 9 9 6 5 -D (appr.) r = . „ l e Q 9 2 ( 2 2 3 ) DA D (appr.) q = A % = 1 . 0 7 5 , ( 2 2 4 ) DAB — * and £ = _M » 1 . 0 1 6 q DA It follows from the definitions that [cf. Eq. ( 2 0 4 ) ] It * /,DABd/>A - 2A _ ( 2 2 6 ) DAB d f A - 3 28 -or, considering Eq, (225), 1.016 / DAd p A = / D A B d ^ (227) Differentiating Eq. (227) on both sides with respect to the upper limit of integration, —1» 1.016 D A( p k) = D A B ( ^ A ) . (228) Hence, the correction introduced for this particular experiment i s less than 2% in the value of the d i f f e r e n t i a l d i f f u s i v i t y at p A» It could be further remarked that for the same hypothetical experiment D -2- - 1.023, DA where D°B i s the value obtained for the d i f f u s i v i t y i f the concentrations of water rather than alcohol are used in Eq. (177). It i s perfectly obvious from these results that i n the present experiments*, where the volume change on diffusion was less than 0.1 ml., the simple formulae (177) and (204) were quite satisfactory, since the * i.e., the actual runs with the ethanol-water system which should not be confused with the hypothetical experiment discussed here. error introduced by their use was completely masked by the scatter of the results,, Considering the ratio D (acc.) _ A B _ = l e 0 1 D A B(appr.) i t i s noted that the error committed by the use of the simplified formula (199) instead of the more rigorous Eq. (203) may amount to more than could have been anticipated from the statements made on the accuracy of this type of approximation [cf. reference (19), p. 306)] . Of course, in choosing the particular concentration range for the hypothetical experiment discussed here, care was taken to make the problem as unsymmetrical as seemed possible. On the basis of the above results there does not seem to be any rea-son not to use large concentration differences i n diaphragm diffusion c e l l experiments for fear of the error introduced by volume changes on diffusion [cf. reference (3), p. 891)]. Actually i n many cases i t w i l l probably be possible to conduct the experiments in two groups; i n one of them, i n i t i a l l y pure component A w i l l diffuse into a series of solutions ranging i n composition from pure component B to some intermediate value, while i n the other one, i n i t i a l l y pure component B w i l l diffuse into a series of solutions the compositions of which' cover the other portion of the concentra-tion range. Making small corrections at the "pure component-end" so as to adjust the obtained integral d i f f u s i v i t i e s to the same lower limit of integration, differentiation of the values thus obtained with respect to -130 -the upper limit of integration would y i e l d the di f f e r e n t i a l d i f f u s i v i t i e s i n a f i r s t approximation. These values could be checked by integrating them over the concentration range of the experiments and, i f necessary, corrected accordingly. e) Discussion of the diffusivity-composition relationship. Inspection of the diagrams and/or tables w i l l show that the minimum of the diffusivity-composition function i s at a composition corresponding to three molecules of water and one molecule of ethanol, or very nearly so. The exact alcohol concentration corresponding to this molal ratio i s 0.423 g./ml. at 25°C. It i s of some interest to note i f the ralative decrease i n volume on mixing f " ] LV Z~ x 100, (229) where v* = v tx> + v Ga_', (<30) i s computed from the very accurate density data, i t i s found that this function also has a minimum at the same compositions k"** ethanol 0.44 0.45 0.46 0.47 0.48 t=15°C. 3.8862 3.7825 3.7850 3.7845 3.7822 t=20°C. 3.6611 3.6674 3.6709 3.6724 3.6702 t=25°C 3.5759 3.5811 3.5883 3.5844 3.5831 The plot (b vs. co i s shown at 25°C. i n Diagram 11. - 132 -The analogy between the functions $ (O J) and V(co) i s shown by the following ratios obtained by dividing the height of the curves at C O by the maximum height. CO 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 The viscosity-composition function also has a maximum at this composition [cf. reference (3) , p. 894]. The shape of the curve of reciprocal viscosity, vs. composition is also very similar to the corres-ponding d i f f u s i v i t y curve. A notable difference is that the viscosity of . pure ethanol i s some '25% higher than that of pure water. The fact that the above three functions have a maximum or minimum at a composition corresponding to three molecules of water and one molecule of ethanol suggests that at this composition the ethanol-water system i s i n a state of maximum stability or, minimum energy. Whether or not such complexes also exist at compositions different from x = 0.25 i s a question that should be decided by methods of investigation more suited D p - D ( m ) D - D . o min 0.000 0.315 0.595 0.822 0.968 0.996 0.958 0.863 0.694 0.405 0.000 0.000 0.273 0.601 0.856 0.979 0.994 0.940 0.826 0.651 0.394 0.000 - 133 -to this type of problem than diffusivity measurement s. It may be of some interest to note that when dD/dC i s plotted against C the resulting plot consists of two branches intersecting at the composition corresponding to the hypothetical complex, indicating that the diffusivity i s a function of a different degree of the ethanol concentration on the two sides of the minimum. Furthermore, when the diff u s i v i t y i s replotted against the mole fraction of the hypothetical complex, the plot consists of two almost perfectly symmetrical halves on the two sides of the minimum. Also the di f f u s i v i t y would change only a few percent from pure complex to about 0.5 mole fraction of complex i n both directions,, The limited significance of these observations did not seem to warrant attaching these graphs to this work. In particular, i t must be emphasized that this sort of formal manipulation cannot be considered as a proof of the existence of the hypothetical complex. It seems reasonably certain, however, that when approaching pure water or pure ethanol other modes of association become more stable, resulting i n a rapid change i n the value of dif f u s i v i t y . The number of different sorts of possible associations i n the ethanol-water system being enormous-ly large, no explanation seems possible, at the present time, of the peculiar fact that the values of the d i f f u s i v i t y at the two ends of the concentration range are practically identical. f) Discussion of the function; activity-based d i f f u s i v i t y , oZ) versus mole fraction. The relation between the concentration-based diffusivity, D, and the activity-based d i f f u s i v i t y , ^ , i s given by Eq. (100). The quantity ^fo 5- has been computed by Guggenheim (24) and #lnx i s plotted on Diagram 12 against the mole fraction. It should be borne i n mind that this coefficient was obtained by differentiation of scanty iso-thermal partial vapor pressure data and hence are bound to be rather uncertain. Without making any claims that this i s ju s t i f i e d , a small part of the curve has been tentatively modified so as to make the minimum of this curve coincide with the minimum of the d i f f u s i v i t y curve. The broken line i s the one that would correspond to the original point given by Guggenheim. If this modified value of l n a i s reconverted in terms 3lnx of the slope of the partial pressure-mole fraction curve, i t i s found that the difference as compared with the original line i s barely noticeable. The values obtained for &2) using the ^lna vs. x curve and 9lnx the values of D read from the graph i n Diagram 9 are shown also i n Diagram 12. The effect which the modification of the Zlna curve has ?lnx on the ^) vs. x curve is very considerable. Again, the broken line shows the curve obtained when the point at x = 0.3, given by Guggenheim, is used. The long horizontal section of the <?& vs. x line deserves special attention. The meaning of the small minimum at the pure water end of the range i s uncertain. It i s not at a l l impossible that t h i s minimum i s only the consequence of the inaccuracy of the <^lna data. Relatively 9 line-sman adjustments i n these data would result i n the disappearance of the minimum, leaving a practically constant value o f j ^ ) . The comments above on the shape of the curves for the ethanol-water system are of a purely speculative character, and the only reason for making them at a l l has been to point out that i t would be desirable to know the definite answers to the questions raised. - 136 -V. NOTATION Roman Letters a constant coefficient i n linear equation [Eq. (156)] a activity of ethanol a activity of species A i n binary mixture A constant coefficient in linear equation [Eq. (159)3 chemical a f f i n i t y [Eq. (46)3 A effective cross-sectional area for diffusion e B constant coefficient i n linear equation[Eq. (159)3 b constant coefficient i n linear equation [(Eq. (156)] C mass density (volume concentration) of ethanol i n ethanol-water mixtures CA molar density of A i n a two-component system total molar density DA Fick diffusion coefficient of A DAB or D binary diffusion coefficient \B or 3} activity-based binary diffusion coefficient ^AB diffusion coefficient defined by Eq. (109) °AB diffusion coefficient defined by Eq. (135) FB constant coefficient in Eq. (3) Fk external (body) force per unit mass of k in a component system multi-h external (body) force per unit mass of A i n a system binary H relative humidity I quantity defined by Eq. (191) - 137 -mass diffusion flux of k i n a multicomponent system for ordinary diffusion with respect to center of mass movement mass diffusion flux of A in a binary system for ordinary diffusion with respect to center of mass movement x-component of j ^ mass diffusion flux of i in a multicomponent system for ordinary diffusion with re spect to any arbitrary axes quantity defined by Eq. (47) any flux of i i n a multicomponent system mass flux of k i n general with respect to center of mass mcv ement entropy flux quantity defined by Eq. (188)' effective pore length for diffusion i n porous diaphragm phenomenological^coefficient in multicomponent system (coefficient of X k ) phenomenological coefficient in multicomponent system (coefficient of X u ) phenomenological coefficient i n multicomponent system (coefficient of X u ) phenomenological coefficient in multicomponent system (coefficient o f ) phenomenological coefficient for binary system (coefficient of X g ) phenomenological coefficient defined by Eq. (117) mass (or, weight) mass (or, weight) molecular mass (or, weight) of A - 138 -mass diffusion flux of A with respect to stationary axes x-component of "n^ molar diffusion flux of A with respect to stationary axes hydrostatic pressure quantity defined by Eq. (224) energy flux vector quantity defined by Eq. (223) production of k per unit volume per unit time ratio of the magnitude of fluxes in a binary system [Eq. (146)] quantity defined by Eq. (192) universal gas constant electrical resistance at temperature t°C. electrical resistance at temperature 0°C. entropy per unit mass temperature absolute temperature internal energy per unit mass velocity of k with respect to fixed axes in a multi-component system velocity of A with respect to^fixed axes in a binary system x-component of u^ specific volume of ideal mixture [Eq, (230)] specific volume of i in a multicomponent system partial specific volume of A in a binary system - 139 -v velocity of the center of mass with respect to fixed axes v arbitrary velocity V total volume w ' molar average velocity x x-cordinate in cartesian coordinate system x mole fraction of ethanol in the ethanol- water system x. mole fraction of A. A any fo rce in a multicomponent system X u force defined by Eq. (44) "X^ force defined by Eq. (45) X^ force defined by Eq. (45) in the case of a binary system X™ force defined by Eq. (113) Greek Letters |2> cell constant [Eq. (178)] mean cell constant [Eq. (193)] time li, chemical potential of k per unit mass in a multicomponent system ^u^ chemical potential of A per unit mass in a binary system quantity defined by Eq. (47) total mass density P mass density (volume concentration) of k in a multi-component system mass density (volume concentration) of A in a binary * system - 140 -P A mass density of pure component A C entropy production ^ £ entropy production in ordinary diffusion mass diffusion flux of A with respect to the molar average velocity >^ relative decrease in volume on mixing [Eq. (229)] (h . molar diffusion flux of A with respect to the molar average" velocity CO^ mass fraction of k in a multicomponent system CO^ or to mass fraction of A in a binary system Other Symbols A (subscript) i , k (subscript) 1, 2 (subscripts) o, f (subscripts) " (superscripts) V (above symbol) —(above symbol) — (above symbol) (above symbol) quantity pertaining to component A in a binary mixture quantity pertaining to the i-th, k-th chemical species in a multicomponent system quantities associated with two separate systems quantities evaluated at the i n i t i a l and final conditions, respectively, quantity pertaining to closed and open half-cell, respectively mean of the values of the quantity evaluated in the two cell-halves mean of the values of the quantity evaluated at in i t i a l and final conditions referring to diffusivitys space and time average value of quantity referring to diffusivity: space average value of quant ity (above symbol) vector quantity - 141 -Other Symbols (continued) "Nabla" or "del" operator D/D8 "substantial derivative operator" Note on Vector Operations (v«F) = v F + v F + v F „ v ' x x y y z z V T = i(3T/2>x) + jOT/a y) + £ ( W * z ) (V - v) = (^vx/ax) + H y ^ y ) + ( I v ^ z ) (v. V)T = v (^T/2x) + v ( i T / V ) + v ( W d z ) - 142 -VI. BIBLIOGRAPHY 1. Johnson, P.A. and Babb, A.L., Chem. Rev. 56; 387, (1956). 2. Smith, I.E. and Storrow, J.A., J.Appl. Chem. (London) 2: 225, (1952). 3. Hammond, B.R. and Stokes, R.H., Trans. Faraday Soc. 49) 890, (1953). 4. Stokes, R.H., J. Am. Chem. Soc. 72: 763, (1950). 5. Stokes, R.H.,- Dunlop. P.J. and Hall, J.R., Trans. Faraday Soc. 4£s 886, (1953). 6. Northrop, J.N. and Anson, M.L., J. Gen. Physiol. 12: 543, (1928). 7. Mouquin, H. and Cathcart, W.H., J. Am. Chem. Soc. 57: 1791, (1935). 8. Hartley, G.S. and Runnicles, D.F., Proc. Roy. Soc. (London) A168: 401, (1938). 9. Lewis, J.B., J. Appl. Chem., £s 228, (1955). 10. Smith, I.E., (1950), M. Sc. Tech. thesis, Manchester. 11. McBain, J.VJ. and Dawson, C.R., Proc. Roy. Soc. (London) A148: 32, (1935). 12. Crossland, R.W., (i960) B. A.' Sc. thesis, U.B.C. 13. Bauer, N., i n Physical Methods of Organic Chemistry, A. Weissberger, ed., Part I, p. 273, Interscience, 1949. 14. Bird, R.B., Theory of Diffusion, in Advances in Chemical Engineering, Vol. I, Academic Press, 1956. 15. Prigogine, I., Etude Thermodynamique des Phenomenes Irreversibles, Ch. VII - X, Dunod (Paris) and Desoer (Liege), 1947. 16. De Groot, S.R., Thermodynamics of Irreversible Processes, Ch, I and VII, North Holland and Interscience, 1951. 17. Hartley, G.S. and Crank, J., Trans. Faraday Soc, 4J>: 801, (1949). 18. Babbitt, J.D., J. Chem. Phys. 23: 601, (1955). 19. Gordon, A.R., Ann. N.Y. Acad. Sci. 46: 285, (1945). - 143 -Bibliography (Continued) 20. Barnes, C, Physics 5: 4, (1934). 21. Bothner, V.M., (1959) B. A. Sc. thesis, U.B.C. 22. Stokes, R.H., J. Am. Chem. Soc. 73_; 3527, (1951). 23. Stokes, R.H., J. Am. Chem. Soc. 72; 2243, (1950). 24. Guggenheim, E.A., Thermodjmamics, p. 229, North Holland and Inter-science, 1957. i - 144 -VII. APPENDICES It will be noticed that the notation used in the Appendices is somewhat different from the one used in the mainpart of this thesis. It is believed that this constitutes a well-justified sacrifice of consistency for the sake of greater simplicity. Appendix I - The Question of Analytical Errors a) General The error E of a quantity Q function of the other quantities q., q 0, can be given by the expression The quantities 65, ...,6a are the probable errors of the measured E + • • • + 2 (1/D values of q^, q^, ..., q^ . To compute € , we may use (2/1) or, (3/1) - 145 -or, for weighted observations of weight w f = 0.6745' n - 1 (4/D or, 6 = 0.8453 £(^1PD n(n - 1) (5/D Here n is the number of observations and p *s are the residuals, i.e., the differences of the individual observations and the arithmetic mean. It follows from the previous definitions that the probable error of the arithmetical mean is \f2 £2 £2 C = v l -— + — + • • • -*• —— = — = 0.6745 \/ 2 2 2 n I n n n ^P n(n - 1) (6/1) or, for weighted observations £ m = 0.06745 (7/1) b) Relative Error of the Calculated Value of the Diffusivity Owing to the Error of the Measured Concentrations By (1/1), E£nR~^ 3 £nR\ 2 6 • + 9 InT^ C f € „ " + I A if 1 c „ » ^c" / X (8/D - 146 -where Since II i R = II " C°r . (9/1) f f J_ 1 2 f „ II • i \2 R (c - c ) f f ) (10/1) J R (c (c£ (U/I) R / t i i \ : (c - c .) f f ) (12-1) nR! R2 . 2 f f (13/1) and, assuming that * c i = £ c " ^ c " ^ c ) (14/D I I O / furthermore, since co ' c f " c f - co » ' ( 1 5 / I ) also o ^ (16/1) - 147 -there follows from (8/1) E 1.414 e I n n' R 2 + 2 (17/D Using Eq. (177), and assuming no error i n p or 0; D TnR~ ^e / I o o' R2 + 2 ( W D For the special case i f c Q = 0 (19/D and IFI /2(1 + R 2 ) (20/1) c) Estimation of £ , the Probable Error of the Measured Values of the Concentrations i ) The.case of the KC1 solutions We may write for the concentration m m •» m c = = -§±5 B (2i/i) V V where m^ i s the weight of the KC1 residue, m B + g a n c* m g a r e the weight of the bottle plus residue and of the empty bottle, respectively. V i s the volume of the sample. - 148 -Hence, the probable error of the measured values of the concentra-tions i s where the relations Jro B+R 1 V 5 "3, 'm Is. ( mB +R - V V 1 . (23/D have been used. £ w i s the probable error of a weighing and £y the probable error of a pipetting. £ may be estimated from the scatter of the measured weights of the empty weighing bottles. It i s recalled that [cf. Section II, f , ' l l there were six weighing bottles.per diffusion c e l l and each bottle was weighed at least three times, using the rotating method. The following table contains the deviations from the arithmetical mean of the individual weighings for Run 8 with the KC1 solutions. The deviations _5 are given i n units of 10 g. - 149 -Residuals For Run ,8 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Std. bottle 0 -3 -3 0 -3 0 -1 -1 0 -2 1 1 0 1 0 -2 -2 2 1 2 -1 3 0 G •-. 1 -4 -1 4 1 -2 - -1 0 2 -3 0 2 0 4 -2 -1 -3 0 1 0 0 0 -2 0 . -2 -2 1 4 -1 -3 -3 -1 -2 -3 8 3 -2 0 -1 -3 -8 0 -1 2 3 0 -3 0 1 -2 -1 2 -1 •1 ." 5 -1 1 4 2 0 -1 2 1 -4 • 8 1 -1 -4 -2 -1 0 -1 1 1 0 -1 0 0 0 2 1 0 1 -2 =2 6 0 1 -1 0 , -1 -1 -1 2 1 0 2 1 2 Total 44 22 26 35 19 50 n 22 . 20 . 21 21 18 23 - 150 -Using formula (3/l) c v 6 = + 1.3 x 10 g. w Since V = 10 ml., and mR = 0.013 to 0.055g., easy computation shows that for the volume-error to be comparable with the weight-error, in the worst case, where mR » 0.055? the following condition must hold £ y k 0.003 ml. It was established that the precision of pipetting was £ v 7^L3> x 10" 3ml., where the following apparent weights of water, obtained by using the same pipette 5 times, were used: Res. (mg.) 9.97320 -2 9.97476 0 9.97365 -1 9.97528 0 9.97922 4 Hence from Eq. (22/1) , =%t = ± 1.414 x 1.3 . 1 0 " 5 = ± 1 . 8 x 10" 5 g./lO ml. It i s interesting to compare with this the value obtained for £ c directly from the deviation from the mean of the values obtained i n - 151 -each of the 3 parallel determinations. These values, also for Run 8 with the KC1 solutions are shown in the following tables Residuals 10" 5 g./lO ml. Cell 2 3 4 5 6 Top -1 2 2 -5 1 half 0 -2 1 2 3 1 0 -2 2 -3 Lower 6 -1 0 -5 -1 half 2 -3 2 -4 -1 -9 3 -2 8 3 Total 19 11 8 26 12 n 3+3 3+3 3+3 3+3- 3+3 The result i s 6 = ± 2.2 x 1C~5 g./ml. c -5 Which compares excellently with the "calculated" value of + 1,8 x 10 g. The small difference may be due partly to the neglect of € y, when computing the latter value, partly to other causes. It must be considered very reassuring that the contribution of "the other causes", including accidental contamination of the weighing bottles and loss of material owing to various causes, such as "spitting" during the evaporation i s practically negligible. - 152 -i i ) The case of the ethanol-water mixtures , The error in the density values comes chiefly from the d i f f i c u l t y i n securing the same weight of the pycnometers i n consecutive determinations. The deviations from the mean for a l l the parallel density determinations are,shown in the following table. Under the heading "Run" the f i r s t number refers to the c e l l , the second to the run, the letters U and L stand for the upper and the lower compartment, respectively, and as usual, o stands for the i n i t i a l and f for the f i n a l conditions. •it 1 Run i d 5 g./ml. Run 10 5 g. /ml. Run lO"5.g./ml 6/2 Uo. 0.5 6/7 Uf 0.0 5/10 Lf -1.0 6/2 Uo -0.5 6/7 Lf -1.0 5/10 Uf -0 .5 6/2 Uf 0.5 6/7 Lf 1.0 5/10 Uf 0.5 6/2 Uf -0 .5 (1+4)/8 Uo 2.'5 (1+4V11 Uo -1.5 (1+4)A Uo 1.0 (l+4)/8 Uo -2,5 (l+ 4)/ll Uo 1.5 (l+4)/2 Uo -1.0 1/8 Lf -0.5 ' 1/11 Lf -0.5 1/4 Lf 0.0 1/8 Lf 0.5 l A l Lf 0.5 1/4 Lf 0.0 4/8 Uf 0.5 ,1/11 Uf 1.5 1/4 Uf 1.5 ' 4/8 Uf -0.5 l / l l Uf -1.5 1/4 Uf -1.5 4/8 Lf 0.5 4/11 Lf 1.5 6/2 Lf 1.5 4/8 Lf -0.5 4/H Lf -1.5 6/2 Lf -1.5 1/8 Uf 4.0 4/11 Uf 1.0 4/4 "Jf 0.0 1/8 Uf -4 .0 4/11 Uf -1.0 4/4 Uf 0.0 (3+6)/Q Uo -0 .5 (1+4)/12 Uo 0.5 4/4 Lf 1.0 (3+6)/9 Uo 0.5 (l+4)/12 Uo -0.5 4/4 Lf -1.0 6/9 Lf 0.0 4/12 Uf - 0 . 5 (l+4)/6 Uo - 2 . 0 6/9 Lf 0.0 4/12 Uf 0.5 (1+4)/6 Uo 2.0 6/9 Uf -1.0 4/12 Lf -0.5 - 153 -Run lCr g./ml. -5 Run 10 g./ml. Run 10"5 g./ml. (l+4)/6 Uo l.o .6/9 Uf 1.0 4/12 Lf 0.5 1/6 Uf 0.0 3/9 Lf 0.0 1/12 Lf -0.5 1/6 Uf -0.0 3/9 Lf 0.0 1/12 Lf 0.5 1/6 Lf 0.5 3/9 Uf 0.0 1 A 2 Uf -1.0 1/6 Lf -0.5 3/9 Uf 0.0 1/12 Uf 1.0 4/6 Uf 2.0 (2+5)/lO Uo 0.0 (2+5)/l3 Uo 0.0 4/6 Uf -2.0 (2+5)/10 Uo 0.0 (2+5)/l3 Uo 0.0 4/6 Lf -1.5 2/10 Uf 0.0 2/13 Uf 0.5 4/6 Lf 1.5 2/10 Uf 0.0 2/13 Uf -0.5 6/7 Uo 1.5 2/10 Lf -o.5 2/13 Lf -0.5 6/7 Uo -1.5 2/10 Lf 0.5 2A3 Lf 0.5 6/7 Uf 0.0 5/10 Lf 1.0 5/13 Uf -1.5 5/13 Uf 1.5 5/16 Lf -0.5 1/8 Lf -1.0 5/13 Lf 0.0 5/16 Lf 0.5 (2 +5)/l9 Uo 1.5 5/13 Lf 0.0 - 5/16 Uf -15.0 (2+5)A9 Uo -1.5 (3+6)/14 Uf 1.0 5/16 Uf 1.0 5/19 Uf 4.0 (3+6)/14 Uf -1.0 5/16 Uf 2.0 5/19 Uf ' -2.0 3/14 Lf 1.5 5/16 Uf 12.0 5/19 Uf 8.0 3/14 Lf -1.5 (3+6V17 Uo 3.0 5/19 Uf 3.0 6/14 Lf 4.0 (3+6)/17 Uo -3.0 5/19 Lf -1.5 6/14 Lf -4.0 (3+6)/17 Uo 0.0 5/19 Lf 1.5 6/14 Lf 1.0 6/17 Lf -1.0 2/19 Lf -17.0 6/14 Uf -0.5 6/17 Lf 1.0 2/19 Lf 1.0 6/1A- Uf 0.5 6/17 Uf 6.0 1 2/19 Lf 7.0 (1+4)/15 Uo -0.5 6/17 Uf -4.0 2/19 Lf 9.0 (l+4)/15 Uo 0.5 6/17 Uf 0.0 2/19 Uf 0.5 4/15 Uf -0.5 6/17 Uf 2.0 2/19 Uf ,-0.5 4/15 Uf 0.5 3/17 Lf 1.5 ;3+6)/20 Uo 0.0 4/15 Lf 0.5 3/17 Lf -1.5 ;3+6)/20 Uo 0.0 4A5 Lf -0.5 3/17 Uf 1.5 • 3/20 Lf -0.5 1/15 Uf 1.0. 3/17 Uf -1.5 3/20 Lf 0.5 1 1/15 Uf -1.0 (l+4)A8 Uo 1.0 3/20 Uf -1.0 - 154 -Run 105 g./ml. Run 105 g./ml. Run 105 g./ml. 1/15 Lf -0.5 (l+4)/18 Uo -1.0 3/20 Uf 1.0 1/15 Lf 0.5 4/18 Lf 0.0 6/20 Lf 1.0 (2+5)/l6 Uo 1.0 4/18 Lf 0.0 6/20 Lf -1.0 (2+5)/l6 Uo -1.0 4/18 Uf .1.0 6/20 Uf -1.0 2/16 Uf 0.5 4/18 Uf -1.0 6/20 Uf 1.0 2/16 Uf -0.5 1/18 Uf 0.5 6/20 Uf -1.0 2/16 Lf 1.5 1/18 Uf -.05 (l+4)/21 Uo 1.5 2/16 Lf -1.5 1/18 Lf 1.0 (l+4)/21 Uo -1.5 | 1/21 Lf -0.5 1/21 Lf 0.5 | 1/21 Uf 0.5 1/21 Uf -0.5 4/21 Lf -1.5 4/21 Lf 1.5 L/21 Uf 0.0 L/21 Uf 0.0 I 2/22 Uo -1.5 2/22 Uo 115 2/22 Uf -1.0 2/22 Uf 1.0 2/22 Lf -0.5 2/22 Lf 0.5 Since, in most cases, there were only two parallel determinations, a simple average deviation from the means was calculated where 6 , is the probable error of the density determinations. Omitting - 155 -5/15 Uf, 6/17 Uf, 5/19 Uf, and 2/19 L f Gd » + l 6 2 1 ^ 5 1 0 " 5 = ± 0 . 9 x lO" 5 g./ml. It is interesting to see how these values of the error of the density compare with the one calculated from the errors of the individual weighings. The density can be written as m - m p+a p ( 2 4 / D where is the density of the sample, mp+0 is the weight of the pycnometer plus sample, nip is the weight of-the pycnometer, is the weight of the pycnometer plus water, Thus p+a p a is the density of water. ( 2 5 / D where (m - m )2 ' p+a p' ( m „ m )' I (m - m ) p+a p 2 ' (26/1) (27/1) ( mp +a " m p ) 156 -1 2 (m - m ) p+a p' (28/1) We ,have f- ~ £ „ w ! l , 2 x l C f 4 go, cp+a ~ wp+£ where 6 p + a and £*p+£ a r e ^ n e probable errors of the weight of the pycnometer plus water and the pycnometer plus sample, respectively. Furthermore, the probable error of the weight of the empty pycnometer i s e p ^ ± 1 x 10" 4 g. When computing £ d for the sake of comparison with the values obtained directly from the density determinations, i t must be realized that the empty pycnometers were reweighed before each determination but that for the weight of water f i l l i n g the pycnometer a constant value was used throughout. Hence, the second term on the right of formula (25/1) must be ignored. The value obtained for £^ i s e d = i0.9 x 10" 5 g./ml. which agrees with the second, more r e a l i s t i c figure, obtained from the density ratios, 6r p +g was estimated, using formula ( 3/l), from 12 determinations of the weight of a pycnometer plus water. The residuals were as follows: x . 10 5 g.s 24, -27, 0 - 2 , -18, -8, -&\ -5, - 4 , 0, 26, 31. . - 157 -£ was estimated, using formula (3/1), from 25 consecutive values obtained for the weight of a pycnometer. The residuals were as follows: x • 10 5 g.: 11, 20, -6, -8, 4, 29, 11, 16, -10, 14, -22, -4, -8, -10, -19, -9, -3, 3, -28, 2, -1, 18, 16, -20, -1. It i s worthy of note that the uncertainty of f i l l i n g the pycnometers apparently was not significantly greater than the scatter caused by other factors. The probable error in the concentration values i s e c - ed . ( 2 9 / D The variation of the coefficient -~. with concentration is shown on the attached graph. Accordingly, 6 may have values ranging from about + 1.6 x 10~^ g./ml. to about + 7 x 10~^ g./ml., depending on the concentra-tion. • • d) The Error Introduced in the Calculated Value of the Diffusivity via the Error Committed in the Determination of the Cell Volumes i ) Wrong values of the compartment volumes are used consistently. For the calibration run and for the actual run D = gnR . (31/1) A0 - 159 -Hence A £nR* TnR* (32/1) By differentiating (32/1) 9 R 3R* ( 3 3 / D where 9D K 1 TR " TnR* R > ( 3 4 / D and JLL K gnR 1_ &R* " " ( £nR*) z R* * ( 3 5 / D Considering that, for constant volumes, Co " C f + r(°f " 0»' where r • V (36/1) (37/1) one may write in • j cJ L+r(c£ - C^ ) - Cj I II c f - C f Co Sr, (38/1) where „ „ C, - C LO "4—1 • ( 3 9 / D c f - ° f Similarly - 160 -JR* . co* J r (40/1) with * * ^ • • ^ . - ^ . C a / D (cp - (cp Substituting (38/1) and (39/1), together with (34/1) and (35/1) in (33/1) J P - T ^ " O o S r - ,K/"L o r . (42/1) CnR* R ( £nR*) 2 & Dividing (42/1) with (30/1) JLt - r_5±L_ - £ r . (43/1) D [_RTnR R* nR*J V ' Using typical values for the quantities involved i t was found that -^ 2 is completely negligible (0.01/S or less) even i f i r is assumed to be as large as 0.04. i i ) Different values for the compartment volumes are used for the calibration run and the actual run. It is assumed the effect of different volumes on ^ is negligible. Two important sub-cases will be considered: oC.) V is assumed the same value for both runs. V" is different for the actual run but V" - V" . 0 1 - 161 so that i t Cf-Cf Hence, in this case D = RTnR = c' _ c " v OnR "t f (44/1) If follows from (43/l) immediately that I D _ CO £r D ~ R~TnR » and with typical experimental values, and o~ r = 2 x 10"^ •ip x 100 - 0.1$, which is s t i l l negligible. p . ) Same as under o6j but V£ i VJ . In that case R + U m y T ( C f - C 0 ) + C f - C o + C f y (47/D With J V" = 0.1 ml.- and typical values of the rest of the quantities in (47/1) ••i-P- x 100 & 6%, D ' - 162 -which i s significant. This fact i s important i n those diffusion experiments with the diaphragm c e l l where there i s a volume change on diffusion. It i s quite customary to analyze only three of the four solutions i n an experiment and calculate the fourth concentration by material balance. As the above example indicates, i n such cases i t i s not permissible to use the same value for the volume i n the open h a l f - c e l l at the beginning and at the end of the experiment. i i i ) The error i n determining volume changes on diffusion by material balances. Consider the material balances Vo " A (pf V f + p f V f - pQ V Q) , (48/1) Co = ( C f V f + C f V f - Co V o ) • -to/I) o For a small concentration interval, such as to be considered for an error computation, one may write CQ' = A pi + B. (50/1) Combining Eqs. (48/1), (49/1) and (50/1) and solving for V" o " 1 Vo = I vf' Af + vH( Af - A0)J , (5i/i) where _ At - c^> - A # , A f - Cf - A p f , A o -1 c 0 " A p o > (52/1) (53/1) (54/1) '* The two i n i t i a l and the two f i n a l solutions. - 163 -and i t has been realized that V^ = V^ , From (51/1) we have immediately for the volume range Av" -1 (B - aj) - v 1 ( - A 0)J . (55/1) Standard error computation results in the following formula (56/1) Using typical data there results i{ A v " ) 5* + 0.001 ml., which i s completely satisfactory. Hence, the volume change on diffusion in the open half of the diaphragm c e l l may be determined, from material balances, to about ± 1%, e) Comparison of Probable Error of -~. as Calculated from the Analytical Error and as Obtained from the Scatter of the Diffusivity Values i ) The KC1 runs. Here the value of the d i f f u s i v i t y was assumed to be correct and i t s place i s taken by the c e l l constant p in the error formulae. Thus by Eq. (20/1) i & = 4t~ v Ad • R 2) • (57/D - 164 -Using the values R 2, C Q <x 7.5 x 1 0 " 2 g./ml. f ^ i 2 x 10~5 g./ml. we obtain x 100 » + 0.1$ . Taking the deviations from the means of the values obtained in Run 7 and Run 8 with the KC1 solutions, the following value is obtained for the average relative deviation x 100 s + 0.1$ . It may be concluded from this comparison thatv in the case of the KC1 solutions the analytical error fully accounts for the actual observed error of the cell constant. i i ) The alcohol-water runs. "d Using Eq. (18/1) and £ ?± 1 x 10 ^ g./ml. in conjunction with the Ac chart, the average probable error for the 35 runs is estimated to be x 100 ~ ± 0.4$ . On the other hand, plotting a l l the diffusivities vs. the concentration and drawing what appears to be the best fitting line over the points, the average relative deviation of the points from the line was found to be -^P-xl00^± 2$.. - 165 -Comparison of the two results indicates the presence of additional signi-ficant sources of random error besides the error of analysis in the case of the alcohol-water runs. The error formulae considered here were i n many cases based on the simple logarithmic formula. However, since the l a t t e r i s a good approximation also in those cases where the density of the solution i s not linear in the volume concentrations, i t seems quite certain that the magnitude of the probable errors obtained by the error formulae based on the simple logarithmic formula w i l l apply also to the results obtained by the more general expression derived in Section III, b. - 166 -Appendix II - Derivation of Some Buoyancy Formulae and Related Expressions a) . Derivation of Eq. (l),(Section II, f, l ) The buoyancy formulae are based on the following statement. At equilibrium: true weight of object - weight of a i r displaced by object = true weight of weights - weight of a i r displaced by weights. Hence, in particular o Pa nu. - HIT, - = m. o ttB f>B B " mB (1/H) where nig = true weight of bottle, nig = true weight of weights = >^g = density of bottle, p w = density of weights, D = density of ambient a i r . apparent weight of bottle, Eq. ( l / I l ) may be rearranged to give 1- Pa Pw • /, Pa Jk\ m" = n i — • • — l . — /fem L 1— + —*— = m . ^ i - ^ a B l P, Pb b Pw X 1 + (2/II) We may write analogous expressions for the bottle plus residue JL -J the standard bottle, at the time of weighing the empty bottle, (3/ir) (4/H) - 167 -and the standard bottle, at the time of weighing the bottle plus residue, Dividing (2/II) by (4/II) "2 2 Dividing (3/II) by (5/II) (5/II) (6/II) m. o = mB+R B+R (7/II) Subtracting (6/II) from (7/II) and realizing that = = M , ni n • ni 0 . 0 B+R B mR = V H " "B M (8/II) This concludes the derivation. b) Derivation of Eq. (2) (Section II, f. 2) By the ideal gas law f a - " p f - x 2 9 » (9/II) w RT x 18, (10/11) where - 168 -p & = density of dry air, g./ml., pM = density of water vapor, g./ml., p = partial pressure of dry air , mm. Hg, p^ = partial pressure of water vapor, mm. Hg. Realizing that the density of humid air is Pa= Pi + N ' l l r ( P a + P w - I f ) x 2 9 ' (11/11) and that 1 ft Pa + 2? Pw - P . " ° ' 3 8 Pw> (12/11) where P = Pa + Pw (13/II) and also that H t pw 100 P H 2 Q ' which, when introduced i n (12/II), gives (14/H) 18 Pa + Pw 29 = P ~ 0.0038'H p = P - k, 20 (11/11) may be written (15/11) . _ 29(p-k) 29 P-k f a RT ~ 62.361 x 273 1+0.00367 t (16/11) This concludes the proof. - 169 -c) Discussion of Eq. (3) (Section II, f, 2) Eq, (2/II) may be written for t\vo different weighings of the same standard bottle mB ° \ { 1 + FB > (17/11) B B^ B a 2 Solving (17/H) and (18/11) for (18/11) F_ = m B 1 " mB s B 2 P a 2 ' m B 1 P a 1 (19/II) With a number of experimental pairs of values nig , p& on hand, a best value for F could be determined. Doing this was greatly f a c i l i t a t e d by the fact that, in the range in which the atmospheric density was observed to change, the plot nig vs. pa was linear within the experi-mental accuracy. (See attached diagram.) Using the value obtained for Fg, nig could be calculated from any pair of values nig, p^ read from the graph. Eq. (3) i s merely a rearranged form of (2/II): mB/raB - 1 (20/11) 'B 80 1.1600 20 40 60 80 1.1700 20 40 60 80 1.1800 20 40 60 80 1.1900 20 40 60 - 171 -Appendix III Estimation of the Effect which the Existence of Concentration Gradients i n the Cell Compartments Has on the Apparent Value of the Cell Constant (or Diffusivity) ~" Supposing the absence of adequate s t i r r i n g concentration gradients may be set up i n the cell-compartment. The material balances on the two compartments must be written accordingly;' -de av DA c - c 7 d0 (1/IH) where D C " „DA c_=_S_ d 8 av v» L »top *av v | "~" ' 5ottom (2/III) (3/IH) over the whole compartment. As a f i r s t approximation let us assume that the concentration profile i s linear over the whole compartment, or c - c = x_ c - c t "0 or c-= — c' + c ( l - — ) , x t 1 x t (4/IH) » t where c^ and c are the concentrations at the two extreme ends of the compartment. Also dV « Adx , i t av o (5/IH) (6/III) - 172 -Similarly, for the other compartment II i t c. + c t 'av 0 or, introducing & by the equations i i i t II c t - c - c we may write and Hence c + av t i t i J C a v = c A c a v = A c + £ c' d A c a y = d & c + d£ = -D^ 3 A cdO d f n A c + -j^- = - p,m9 A c o I A c (7/IH) (8/III) (9/III) (10/111) (11/III) (12/111) (13/111) (14/111) Supposing that one works-with a concentration gradient existing i n the compartments, the c e l l factor as computed by the last formula may be considered to be the true value i n the approximation employed here. 173 -If one mixes the entire contents of the compartne nts, and makes analyses on samples taken from the homogeneous liquids and mechanically uses the logarithmic formula defined by n (Ac.) < A co>avl (15/111) If one measures the concentrations at any particular level i n the compartments another apparent c e l l constant defined by the equation below i s obtained n ( A c,) X 0 (16/111) Let us compare the c e l l factors obtained for c e l l no. 1 based on the above results, assuming linear concentration gradient and letting Cf = 0.02762, (cj) o 0.02632, < 0 av c Q = ( c D ) x = 0. ( c j = 0.02697, 1 av = 0.00130, |= 0.00065. In the evaluation of the integral let us assume - 174 -• that 1/fA c) is a linear function of £ . Otherwise straightforward computation gives For the case where the concentration is measured near the two extreme ends of the c e l l the following additional deviation is obtained ••/V - = k m 0 5 % t far Let us investigate the hypothetical case, where with the same (c') and (cl!) , & i s assumed to be 5 times larger than i n 1 av 1 av the case just computed.- The following differences between the various values of the c e l l factor are then obtained: (^av and Q>. fi, — • - 6.33$. f 3 - 175 -Appendix IV - Calibration of the Thermometers The Pt-resistance thermometer used as standard was a Leeds and Northrup, metal enclosed knife-type, serial number 679368. The thermometer used with the diffusion experiments was calibrated much earlier than the other thermometer. This calibration was carried out i n a well-insulated, stirred o i l bath which had been used for such purposes i n the department previously. This contained two heaters, one of which was meant to be- on a l l the time to make up for heat losses, whereas the other was controlled by a relay. The output of both heaters could be regulated by two Variacs. Since, for the purpose of the present work, the low tempera-ture range, round room temperature, was primarily of interest, the'perma-nent heater could be dispensed with and, to compensate for the heat generated by the s t i r r e r (that could actually be observed) a cooling c o i l had to be installed. The temperature was raised in steps of about 2°F. over the entire range of the thermometer. At each measurement point the temperature was constant well within 0.01°F. for several minutes as established by the bridge. Computation of the temperature using the measured resistance may be done using the following formulae where R and N refer to the two resistances measured with the poles commuted and Z stands f o r the resistance of the' leads. R t R +,N NZ + RZ 2 2 (l./iv) t = (2/IV) - 176 -There were four calibrations of the Pt-resistance thermometer available. One i s by the National Bureau of Standards, issued on June 25, 1958. Accord-ing to this source, the values of the constants i n Eq. (2/IV) are as follows a = 0.003932, S = 1-49, \*> 25.558 (absolute.fi). In the case of the first calibration these values were used leading to the correction at 8 almost equidistant points i n the range 19.86° to 27.14°C.: -0.15, -0.06, -0.06, -0.04, -0.04, -0.05, -0.05, -0.06°C. -0.1°F. was accepted as correction. Another certificate was issued on October 6, 1959, by the National Research Council; a t h i r d one does not give the source of i t s origin; the fourth one i s by the National Bureau of Standards (no date) and contains only resistance ratios. Although the resistance values based on the d i f f e r -ent calibrations are somewhat different, i t was realized later on that the ratio \ / R q i s the same for a given t, no matter which certificate was used. Accordingly, i n the second calibration, the value of RQ was also determined at the same ambient temperature as the rest of the measure-ments. A brief description of the second calibration i s given now. A Mueller-bridge, a b a l l i s t i c galvanometer, and 4V dry batteries were used. It was found necessary to clean a l l the contacts with polishing - 177 -paper prior to use lest "negative resistance" of the leads be obtained. The water bath used with the pycnometers was used here, too. The temperature was found unchanged, as indicated by both the mercury and the Pt-resistance thermometer while the following sequence of operations was performed! l ) Bridge was compensatedj 2) Mercury thermometer was read; 3) Bridge was read and reversed; 4) Mercury thermometer was read; 5) Bridge was read. To measure the resistance at the ice point a 1 l i t e r erlenmeyer flask was f i l l e d with crushed ice prepared from d i s t i l l e d water; the resistance thermometer was immersed; d i s t i l l e d water was poured over the ice so that the thermometer was covered with water. Occasional stirring was carried out u n t i l the bridge remained compensated, A further 15 minutes, while the bridge stayed compensated, was allowed. The mercury thermometer was then checked at the ice point and no noticeable deviation from the zero mark found. The following readings were obtained! N ( i l ) R(Sl) t(°C.) 27.4495 . Same 18.90 27.5206 Same 19.60 27.5773 Same 20.16 25.5476 Same ice-point N Z ( f t ) R Z ( # ) 0.0007 Same Deflection when balancing bridge: 20 scale divisions, Deflection per 0.0001 Q : 6 scale divisions. Hence zero correction of bridge 0.0003^6 , and each reading had to be corrected by -0.0010 Q . - 178 -The following results were Thermometer reading 18.90°C. 19.60°C. 20.16°C. Correct temperature 18.75°C. 19.45°C. 20.01°C. Hence the correction is -0.15°C. - 179 -Appendix V - Numerical Procedure to Check on the Assumption of Uniform Fluxes throughout the Porous Diaphragm A numerical, iterative procedure, somewhat similar to that developed by Dusinberre(McAdams, W.H., Heat Transmission, McGraw-Hill, 1954, p. 44.) was worked out. The porous diaphragm was divided into a number of sections by planes parallel to the faces of the diaphragm. To allow for the special importance of the value of the gradient at the two surfaces of the diaphragm, the two sections adjacent to the surfaces were taken half as thick as the rest of the sections. It was a matter of convenience to use two "half-sections" adjacent to and outside of the diaphragm. The concentrations i n the planes outside of the diaphragm were.obtained by extrapolation. These concentrations are of no physical significance whatsoever (see attached sketch). U . ^ |_ > Material balance on the shaded volume element gives i n a zero-th approximation (Cjtf* " <CJ'O • ^ 5 [ «>l>o - * « W o ] . ( V O where D was assumed to be constant. 0 i s time and C i s volume concentra-tion. - 180 -j - - 1/2, 1/2, 3/2, 5/2, 7/2, 9/2 refers to the s e r i a l number of the dividing plane; k = 0, 1, 2, ..,, m denotes a certain time and k+1 denotes the time one time increment later than k. Similarly, a material balance may be written on each compartment. In the same approximation (O k+1 o'l (c )k + PI M 0 0 V« £ x 1/2 0 -1/2 0 and ( V i - ( V o + ( C7/2 }o = ( C9/2 )o 3 (2/V) (3A) where A i s the effective cross sectional axes for diffusion, V and V" are the volumes of the respective cell-compartments. However, the l e f t members of (l/V) (2/V), and (3/V) do not, i n general give the new concentrations i n a satisfactory approximation. The reason for this i s that i n these formulae no allowance has been made for the fact that the bracketed terms on the right hand side of the equations undergo simultaneous change. The true value of the changed concentrations at time k+1 w i l l be intermediate to the values (O)^ and (C)^ +^ as given by the above equations. Better values, (C) k +^" are obtained i f the mean - 181 -c oneentrat ions . .k , .k+1 k (c) + (c) (C) = —-2- J L _ (4/V) 1 2 are substituted in Eqs. (l/V), (2/V) and (3A) i n the place of the concentra-tions (C) o In a second approximation the mean concentrations . ,k , vk+l k ( C ) n + ( C ) 9 w i l l be put i n (l/V), (2/V) and (3/V) i n the place of (C)£ . This procedure may be continued u n t i l the condition ( C ) i + l " ( C ) i = ( C ) n ( i =°> 1 ' 2 ' n > n + 1 ' ( 6 / V ) i s satisfied. Eqs. (4/V) and (5/V) may obviously be generalized to give ' ( o k . ( o k * ; ( O k t l - ° 2 " 1 . . (7/V) The corresponding generalization of Eqs. . (l/V), (-2/V) a n d (3/V) gives - 182 -and (9/V) (10/V) Combinations of Eqs. (8/V), (9/V) and (10/V), on the one hand, and Eq. (7/V), on the other, gives (Cj.!)^ - 2 ( C j ) J • ( C j + 1 ) ^ (HA) ( c j k . - ( c j k + * i+1 o'Q 2 V and <c4>* , " « v / + H l+l H o 2V» [ (c 7 / 2) i - (c9/2) (12A) (13A) Eqs. (llA), (12A) and (13A) roay be used most conveniently to perform the iteration u n t i l Eq. (6/V) i s satisfied for a l l C- It i s realized that then also (O k +^(c) k + 1= (c) k + 1. l+l 1 k (14A) Hence, the true value of the concentrations at time k+1 are obtained from Eqs. (8A), (9A) and (10A) by substituting (C) k throughout the bracketed right members. Then a new timing i s considered wherein the same steps as outlined above are repeated. - 183 -It was found that this iterative procedure was not necessary once the system has reached the so-called pseudo-steady state conditions. Otherwise, the following approximation was always found to be satisfactory. It is important to note that with a given set of values D and cannot be chosen a r b i t r a r i l y lest the numerical procedure d i -verge. Since the condition of convergence is C + C Cj + A c ^ i i i i f ' ^ C j < o , (16/v) of and by (8/V) c j + ^ c j < C , 1 ° 1 I 0 , 1 + 1 l f > 0, (17A) Ac. - (c. - c. + c. ), (is/v) there follows or < 2C,1 - C,M - C.M , ( 1 9 A ) (Ax) 2 C j - V i - V i < 1/2 LLzlL , (20/v) D - 184 -The following values of the Quantities occurring i n the equations were useds V = V" = 100 cm.3 A = .4 cm. L = 0.4 cm. D = 2 x 10" 5 cm.2 sec." 1 x = 0 , 1 cm. $ = 200 sec. The computations were carried out, with the cooperation of Mr. Vym Bothner, to six significant figures. Typical results are shewn i n the following table, A C 0 (at 10,000 sees, from the start of diffusion) - 7.14100 x 10"*3 g./ml. Final time sec. A c f x 10 3 g./ml. Time of diffusion from log formula Time of diffusion from numerical method % difference 10,200 . 7.13530 199.923 sec. 200.000 sec. -0.038 10,400 7.12962 398.695 " 400.000 » • -0.033 10,600 7.12392 598.676 » 600.000 » -0.022 10,800 7.11824 798.081 » 800.000 » -0.024 11,000 7.11256 997.659 " 1,000.000 " -0.023 Here the time of diffusion as obtained from the logarithmic formula was compared with the value obtained from the numerical analysis, the i n i t i a l and the f i n a l concentration difference, A C Q a n c * A Cf, respectively, being the same i n both cases. It may be seen that the difference between the two diffusion times i s small and is getting ever smaller as diffusion proceeds. Owing to the very small time increments used, the numerical method is i n no ) - 185 -way less accurate than an analytical solution as indicated by the fact that i f only one section (extending over the entire thickness of the diaphragm) i s used i n the numerical procedure the results obtained are indistinguishable from those obtained by the logarithmic formula. Hence, the minute differences seen i n the table may be interpreted as being due to the fact that the logarithmic formula f a i l s to take into account the slight curvature of the concentration profile across the diaphragm. In view of the very close agreene nt between the results of the two methods one may conclude that, under typical diaphragm c e l l conditions, the assumption of uniform flux (which, at constant D, i s equivalent to linear concentration gradient) i s an excellent one. - 186 -Appendix VI - Calculation of the Unknown, Fourth Concentration and the Magnitude of the Volume Change in the Case where there i s a Volume Change on Diffusion The unknown fourth concentration and the f i n a l volume in the open compartment may be determined from an overall material balance and a material balance on one of the two components? • « «• (Po + Pa\ • " ' • m /Pf + Pi (1/W1) ir tt i t it / C + C \ II „ f t ni /C „ + C„ \ V C + V C +V -2 £ L v c + v c + v - i L (2/VI) o o o I 2 Y f f f [ 2 J ' where V*" i s the pore volume i n the diaphragm, and p - p>(c). ' ( 3 A i ) it One of the unknowns is V^ . The other unknown may be any one of the four concentrations. In part of the present experiments the unknown was C^ . On the other hand, whenever the time of diffusion i s sought for an experiment where the d i f f u s i v i t y and the i n i t i a l concentrations are given, either or the two f i n a l concentrations may be specified arbitrarily, and the other must be found by using Eqs. (l/Vl), (2/Vl) and (3/Vl). Since the function p = p(C) i s usually given i n tabular form, solution of these equations involves very laborious trial-and-error , computations which usually converge distressingly slowly. - 187 -There i s , however, a simple way to practically eliminate t r i a l ~ and-error. In a f i r s t approximation the unknown concentration i s found from a material balance where constant volumes are assumed V C + V C = K . t ? C. , (4/VI) o o I I in where also V has been neglected. Once the approximate value of the unknown concentration i s known, i n that neighbourhood, a linear relation between the density and the concentration may be established using the tabular values of p and C. With the help of this linear relation, taking the place of (3/Vl), the equations ( l / V l ) , (2/Vl), and (3/Vl) may be solved e x p l i c i t l y for the unknown concentration. Three different cases w i l l be considered. . . i (a) C i s unknown. ' o D = a + b c \ (5/VI) fo o Both Eq. (1/VI) and Eq. (2/Vl) are solved for s J I? i t V c + _ o o # (7AD c f - 188 -Equating the right members of Eqs. (6/VI) and (7Al)» and solving for G Q , B - aC f C_ --Q ii ii > bC f " Pf (8/71) where (5/Vl) has been used and v'o'l + V ^ 2 + -L- ( i 1 + ^ ) B in ( 9 A I ) with and . i n t II K = Pf c f - c f i°f * / it H II H *2 ' f>f Co " C f po ' (10/VI) (11/VI) Once C? has been found v" can be obtained from Eq. (7/Vl). o f (b) C n i s unknown. In that case we write P = a + bC o o ( 1 2 A I ) Proceeding as under (a) but using (12/VT) and solving for C n , the result i s II „ F - aC f Co = •» " bC - D f f f (13 A i ) - 189 -Here , .„ r • _ ( H A D t i t ft TT V + - V_ 0 2 (cj i s unknown. In this case we write f>"f = a + bC^ . (15AD Proceeding as under (a) and (b), using Eqs. (6/Vl), ( ? A D a n c * (15AD» a n d tt solving for , the result i s where tt 1 t i t / • V \ / „ • _ 1 . . t t V . ft - ( T - . ( c ; . c;> • <T; • .E.> < ' - 190 -Appendix VII - Illustration Showing the Use of Some of the Formulae to Calculate D i f f u s i v i t i e s Consider a diaphragm diffusion c e l l experiment characterized by the following conditions: c" = 0.00000 g./ml., v" = 47.80 ml. o * o C^ = 0.10405 g./ml., V1 = 48.45 ml. C^ 0.4803 g./ml. Using Eqs. (8/Vl) and (7/Vl) C' and v" were computed: o I C* = 0.58152 o V^ - 47.11 . By Eq. (177) DA ^ & = 0.4353 . Performing the integration i n Eq. (191) by factorizing out the mean of the function under the integral sign gives I = -0.0016. It may be seen immediately that the ratio of the magnitude of the two fluxes, denoted by rg^ i n Eq. (195), i n particular, may be d ^ , where the primes indicate the derivative must i i C expressed as 1 + be taken at the closed h a l f - c e l l . Using this expression and also Eq. (188) - 191 -gives rBA & B ) 1 + K / 1.1012. Hence, by Eq. (195) DA B P 0.4775. To obtain D A B by using Eq. (199) tabular integration of Eq. (199) was performed between the limits UJA = 0.5985 and HJ^ - 0.05312. The data i n Table VIII were used i n the formula n F(x) dx ~ 1/2 ) + F(x 2)(x 3-x 1) + F(x )(x -x2) + ... + F(x : ..)(x -x n) + F(x )(x -x , ) l n - l / v n n - l 7 v n' N n n-l'f Also It was found that the value of the integral was 0.28101, - 2.25433 gxving DAB = °-°32. - 192 Integration of data i n Table VII between the limits C a y = 0.05203 and C* = 0.53094 gave for the value of the integral 0.28143 and hence AB = —r c - c av av DAB d C = 0.588 Considering Eqs. (203) and (201) the value of the function . 1 1 1 ^ A' ^ A w a s c a l c u ^ - a t e < ^ by tabular integration using data i n Table VIII 11 at a series of values of C . The following results were obtained; c" 0.000 0.032 0.058 0.083 0.104 v" 47.80 47.59 47.41 47.24 47.11 II l \ F ( U V WA> 0.355 0,312 0.270 0.231 0.206 A c 0.581 0.518 0.467 0.416 0.376 It was found that C*, v" and F(CAJ", (/J!) could be interpolated A a between the terminal values with a good degree of accuracy. Using Eq. (203) the accurate value of D f^i was obtained by tabular integration of the two terms i n the denominator, giving DAB = 0 , 6 3 8 * The implications of these results are discussed i n Section IV, d. Taking c" as independent variable.
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Diffusivities in the ethanol-water system : the applicability of the diaphragm cell method to the case… Dullien, Francis Andrew Leslie 1960
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Title | Diffusivities in the ethanol-water system : the applicability of the diaphragm cell method to the case of systems where volume changes occur on mixing |
Creator |
Dullien, Francis Andrew Leslie |
Publisher | University of British Columbia |
Date Issued | 1960 |
Description | Smith and Storrow [J. Appl. Chem. (London) 2: 225, (1952)] and Hammond and Stokes [Trans. Faraday Soc. 40: 890, (1953)], using their individual modifications of the diaphragm-cell method for measuring liquid diffusivities, reported diffusion coefficients on the ethanol-water system that disagreed by as much as 100 per cent. It was apparently in view of this disagreement that Johnson and Babb gave a very reserved summary opinion on the diaphragm-cell method in a comprehensive review article [Chem. Rev. 56: 387, (1956)]. In the present work a diaphragm cell was designed which, unlike prior designs, is suitable for use with practically all organic liquids. A compact machine, accommodating and stirring a battery of six such cells, was designed and used. The precision of the cell constant determinations, using aqueous KC1 solutions, was ± 0.1%, a scatter completely accounted for by the errors arising from a standard gravimetric analysis of the solutions which, in this case, was improved in accuracy by one order of magnitude over hitherto reported analyses. Using this apparatus, the diffusivity results obtained by Hammond and Stokes were confirmed within the accuracy of the ethanol-water measurements (± 2%). Critical experiments on an apparatus similar to that used by Smith and Storrow revealed that there was a possibility of distillation through a wetted ground glass joint from the one solution into the other. The apparent higher diffusivities obtained by these authors were attributed to this phenomenon. Detailed derivation of the general equation of diffusion, using the methods of the thermodynamics of irreversible processes, and discussion of the question of the various frames of references and the various diffusivities defined by some authors is given. Using the general equation of diffusion in a binary system new formulae which apply regardless of volume changes on mixing were derived for use with the diaphragm cell method. It was shown that, in the case where the density of the solution is linear in the volume concentration, the general equation of diffusion reduces to Fick's first law and the various formulae based upon the general equation reduce to the corresponding simple formulae. The trial-and-error procedure used with electrolytes by R. H. Stokes was generalized and used to compute the true differential diffusivities from the measured integral values and it was shown that, contrary to the contention of Stokes, this type of procedure is applicable in the case of systems where the limiting values of the diffusivity are not known and the diffusivity changes strongly with concentration. Also, an alternate procedure, based on the differentiation of the integral diffusivities, is suggested. It was noted that the minimum of the diffusivity-concentration curve and that of the relative volume decrease-concentration curve, and the maximum of the viscosity-concentration curve are at a composition corresponding to one molecule of ethanol and three molecules of water. This coincidence may be evidence for the existence of a molecular complex. It was found that the activity-based diffusivity is practically constant over some 70% of the composition range. |
Subject |
Diffusion |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0059085 |
URI | http://hdl.handle.net/2429/39609 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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