1. THE ELECTROPHORETIC CONTRIBUTION TO EQUIVALENT CONDUCTANCE USING THE COMPLETE EXPONENTIAL DISTRIBUTION FUNCTION: 1-1 SALTS IN DIOXANE-WATER MIXTURES 11. TRANSFERENCE NUMBERS AND ACTIVITY COEFFICIENTS OF AQUEOUS SOLUTIONS OF TRIS-(ETHYLENEDIAMINE) COBALT (III) CHLORIDE AT TWENTY-FIVE DEGREES CENTIGRADE Thesis Ior IIw Dog". oI DILED. MICHIGAN STATE UNIVERSITY David J. Karl 31960 a. (2/ .19.. ' ‘J LIBRARY“ ' Michigan Stan: University - 9.?" ii ’3”? U3f‘“,“ "\‘T‘T' I-.H-, «Ixxo J‘IC‘. .‘.' " I WE £13; 5 .QwIY rr 1:; - ‘ " Dr Iig-‘.l-u-l- ‘~_ ‘A.’ ‘1.) ..' _“\;.'.IGE EAST LANSING MICHIGAN I. THE ELECTROPHORETIC CONTRIBUTION TO EQUIVALENT CONDUCTANCE USING THE COMPLETE EXPONENTIAL DISTRIBUTION FUNCTION: 1-1 SALTS IN DIOXANE—WATER MIXTURES II. TRANSFERENCE NUMBERS AND ACTIVITY COEFFICIENTS OF AQUEOUS SOLUTIONS OF TRIS- (ETHYLENEDIAMINE) COBALT (III) CHLORIDE AT TWENTY—FIVE DEGREES CENTIGRADE BY David J. Karl AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1960 Approved ABSTRACT An outline of the Onsager theory of conductance is presented as an introduction to the study of this difficult treatment. Higher order concentration dependent terms of the electrophoretic effect, which are neglected in the usual treatment, have been evaluated, using a digital computer, for 1-1 salts in water and in 10% to 70% dioxane-water mixtures. Significant differences are found to occur between the Onsager and the extended electrophoretic expressions. These differences increase rapidly with decreasing ion size and with decreasing solvent dielectric constant. The electrophoretic calculations are applied to equivalent con- ductance data in several dioxane-water mixtures for tetraisoamyl- ammonium nitrate and tetra-E-butylammoniurn bromide taken from the literature. For these data it is found that deviations from the Onsager- Fuoss conductance equations, which previously have been attributed to ion-pair formation, can be interpreted instead using two constant dis- tance parameters; the minimum distance of approach, and the cation hydrodynamic radius. It is concluded that much of the deviation from theory, heretofore ascribed to electrostatic aggregation of ions, arises from an incomplete treatment of the‘model used rather than from physical phenomena which cause the model to be inaccurate. The pro- gram developed to compute the electrophoretic higher terms is also applicable to other charge types. In addition to these theoretical considerations, experimental data are presented for aqueous solutions of tris-(ethylenediamine) cobalt (III) chloride: transference number values obtained by the moving boundary method and activity coefficients determined from the electro- -motive force of concentration cells with transference. Deviations from the predictions of theory occur for both ion mobilities and activity co- efficients . ii I. THE ELECTROPHORETIC CONTRIBUTION TO EQUIVALENT CONDUCTANCE USING THE COMPLETE EXPONENTIAL DISTRIBUTION FUNCTION: 1-1 SALTS IN DIOXANE-WATER MIXTURES II. TRANSFERENCE NUMBERS AND ACTIVITY COEFFICIENTS OF AQUEOUS SOLUTIONS OF TRIS-(ETHYLENEDIAMINE) COBALT (III) CHLORIDE AT TWENTY-FIVE DEGREES CENTIGRADE By David J. Karl A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree Of DOC T OR OF PHILOSOPHY Department Of Chemistry 1960 ‘U .o f } (\s 7/2.. ",1 f <1» 5 “/ 3- ml". 1‘ . v. r J ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. James L. Dye for his counsel, direction, and patience throughout the tenure of his graduate studies and particularly during the course Of the present work. Appreciation is also extended to the National Science Foundation for financial assistance provided. *********** iv TABLE OF CONTENTS Page PART I THE ELECTROPHORETIC CONTRIBUTION TO EQUIVALENT CONDUCTANCE USING THE COMPLETE EXPONENTIAL DISTRIBUTION FUNCTION: 1-1 SALTS IN DIOXANE-WATER MIXTURES I. Introduction. . ..... . . . . . .. . .. .. . .. . . . . . 1 II. The Interionic Attraction Theory Of Conductance 2 A. Introduction . . . . . . . . . . . . . . . . . 2 B. The Equilibrium Distribution Flinction and Potential. 3 C. The Form of the Conductance Equation . . . . . . . . . 8 D. The Onsager Conductance Equation . . . . . . . . . . . 10 1. General Approach. . . . . . . . . . . . . . . . 10 2. The Boundary Conditions . . . . 14 3. Order Of Terms and Method Of Solution 00f the Continuity Equation . . . . . . . 16 4. First Order Approximation to the Distribution Function and Relaxation Field . . . . . . . . . . l8 5. The Second Order Approximation to the Relaxation Field. . . . . . . . . . . . 21 6. Solvent Velocities and the Electrophoretic Effect . . . . . . . . . Z3 7. Higher Terms in the Relaxation Field. . . . . . 28 8. The Electrophoretic Effect . . . . . . . . . . . 31 E. Recent Modifications of the Onsager Equation . . . . . 33 l. The Kinetic Effect . . . . . . . . . . . . . . . . 33 Z. Einstein Viscosity Correction. . . . . . . 34 3. The Effect of the Relaxation Field on the Electrophoretic Term . . . . 35 F. The Complete Onsager Conductance EquatiOn and Its Limitations . . . . . . . . . . . . . . . . . . 36 G. Ion Association and Conductance . . . . . . . . . . . . 37 III. The Effect Of Higher Terms in the Distribution Function on the ElectrOphoretic Effect for 1—1 Salts in Dioxane~Water Mixtures............................ 39 TABLE OF CONTENTS — Continued Page A. Introduction . . . ........ . . . . . . . . . 39 B. The Extended ElectrOphoretic Effect ..... . . . . . . 40 l. The Exponential Distribution Function . . . . . . 40 Z. Formulation of the Electrophoretic Term . . . . 42 3. The Evaluation of Extended Electrophoretic Effect for Univalent Electrolytes in Dioxane- Water Mixtures. . . . . . . . . . . . . . . . . . 45 C. Applications and Conclusions . . . . . . . ...... . . 59 IV.Summary.................... ....... .. 76 PART II TRANSFERENCE NUMBERS AND ACTIVITY COEFFICIENTS OF AQUEOUS SOLUTIONS OF TRIS-.(ETHYLENEDIAMINE) COBALT (III) CHLORIDE AT TWENTY-FIVE DEGREES CENTICRADE I. Introduction 0 o I 0 O U 0 0 0 0 0 0 0 0 0 0 O O 0 0 0 0 0 0 9 0 0 o 78 II. Transference Numbers . . . . . . . . . . . . . . . . . . . . . . 78 A. Introduction . . . . . . . . . . . . . . . . . 78 B. The Moving Boundary Method. . . . . . . . . . . . . . . 79 C. Experimental. . . . . . . . . . . . . . . . . . . . . . . . 83 1. Materials . . . . . . . . . . . . . . . . . . . . . . 83 2. Apparatus . . . . . . . . . . . . . . . . . . . . . . 84 3. Procedure . . . . . . . . . . . . . . . . . . . . . . 87 4. Results . . . ...... . . . . . ..... . . . . 89 III. Activity Coefficients . . . . . . . ..... . . . . . . . . . . . 91 A. Definitions . . . . . 91 B. Activity Coefficients from the E M. F. Ofo Cmells with Transference. . . . . . . . . . . . . . . . . . . . . . . . 98 C. Experimental . . . . . . . . . . . . . . . . . . . . . . . . 101 1. Materials . . . . . . . . . . . . . . . . . . . . . . 101 2. Apparatus. . . . . . . . . ..... .. . . . . . . 101 3. Procedure. . . . . . . . . . . . . . . . . . . . . . 103 4. Results . . . . . . . . . . . . . . . . . . . . . . . 104 IV. DiscussionofResults. . ..... . . . . . . . . . . . . . . . .109 vi TABLE OF CONTENTS -- Continued Page V. Summary . . . . . . . . . . . . .............. 112 BIBLIOGRAPHY. . . .......... . . . . .. ..... . 113 APPENDIX I . . . ....................... 117 APPENDIX 11 . . . . . . . ..... . . . . . . . ..... . 120 vii LIST OF TA BLES TABLE 1. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.:Water . . . . . ........ 2. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 10% Dioxane—90% Water . 3. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 30% Dioxane-70% Water . 4. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 45% Dioxane-55% Water. . 5. The Extended ElectrOphoretic Conductance Term for 1-1 Electrolytes at 25°C.: 50% Dioxane—50% Water. . 6. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 55% Dioxane-45% Water . 7. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 60% Dioxane-40% Water 8. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 65% Dioxane-35% Water . . . 9. The Extended Electrophoretic Conductance Term for 1-1 Electrolytes at 25°C.: 70% Dioxane-30% Water ..... 10. Contribution of Various Partial Conductance Functions to a Calculated Phoreogram for a 1-1 Salt in 60% Dioxane—40% Water Solvent . 11. Comparison of Calculated and Experimental Values of the Conductance of Solutions of i—Am4NNO3 In Dioxane- Water Mixtures at 25°C.: _a_L_= 4.50 A ; Ti: 7.50 A . 12. Comparison of Calculated and Experimental Values Of the Conductance of Solutions of Bu4NBr in Dioxane-Water Mixtures at 25°C.: a = 5.00 X; R = 7.00 X. 13. Cation TranSference Numbers and Equivalent Conduct - . ancies Of Tris-(Ethylenediamine) Cobalt (III) Chloride Solutions at 25°C . . 14. A Typical Set Of Data for a Concentration Cell with Transference............ ....... . 15. E.M. F. Data and Activity Coefficients of Tris-(Ethylene- diamine)COba1t (III) Chloride at 25°C. . viii Page 47 48 49 5O 51 52 53 54 55 61 64 69 92 105 110 LIST OF FIGURES FIGURE 1. 10. 11. 12. Comparison of several radial distribution functions for 0. 01 normal solution of a 1-1 electrolyte in water at 25°C.: a = 4.0 X. . The ratio of the extended and Onsager electrophoretic terms versus the square root of concentration at several values Of a: 60% dioxane-40% water solution Of a 1-1 salt at 25°C. . The ratio Of the extended and Onsager electrophoretic terms versus a at several values Of the square root of concentration: 60% dioxane -- 40% water solution Of a 1-1 salt at 25°C . . The ratio of the extended and Onsager electrophoretic terms versus dielectric constant for 1-1 salts in dioxane- wateé mixtures at 25°C.: V c = 0. 07; a = 4. O X, 5. 0 X, . The contribution of several terms to the calculated con- ductance Of a 1-l salt in 60% dioxane—40% water solution versus the square root of concentration:a __~, 5. 0 . . Phoreograms for i-Am4NNO3 solutions at 25°C.: 0% and 10% dioxane. . Phoreograms for i-Am4NNO3 solutions at 25°C.: 30% and 50% dioxane. . Phoreograms for Bu4NBr solutions at 25°C.: 0% and 10% dioxane . . Phoreograms for Bu4NBr solutions at 25°C.: 30% and 50% dioxane . Phoreograms for Bu4NBr solutions at 25°C.: 55% and 60% dioxane. Phoreograms for Bu4NBr solutions at 25°C.: 65% and 70% dioxane. Various theoretical phoreograms for a 1-1 salt in 60% dioxane-40% water solution at 25°C.: A0 = 39. 75, a = 5. 0 ix Page 43 56 57 58 62 65 66 7O 71 72 73 74 LIST OF FIGURES - Continued FIGURE 13. 14. 15. 16. 17. Moving boundary transference cell ............. Circuit diagram of current controlling apparatus ..... Cation transference number Of tris-(ethylenediamine) cobalt (III) chloride versus the square root of normality in aqueous solution at 25°C. . . . . ............ Concentration cell with transference ........... Debye-Hiickel plot Of activity coefficient data for aqueous tris-(ethylenediamine) cobalt (III) chloride solution at 25°C ................ . Page 85 88 PART I THE ELECTROPHORETIC CONTRIBUTION TO EQUIVALENT CONDUCTANCE USING THE COMPLETE EXPONENTIAL DISTRIBUTION FUNCTION: 1-1 SALTS IN DIOXANE-WATER MIXTURES I. INTRODUCTION Solutions of ionOphores (1) or ”strong" electrolytes consist Of ions dissolved in dielectric media. The electrical conductivity Of such solutions clearly indicates that the ions are free to move more or less independently. For completely independent ions, the equiva- lent conductance should be independent Of concentration for a given electrolyte and solvent. Actually this quantity varies markedly with concentration. This behavior has been the subject of intense experi- mental and theoretical study for over sixty years. The modern theory Of electrolytic solutions is based on the interionic attraction theory Of Milner (2) as formulated by Debye and Hiickel (3). According to this View, coulombic forces between ions cause any chosen reference ion to be surrounded by an excess of ions Of Opposite charge, and this ion excess can be treated as a uniform charge cloud or ”atmOSphere. " Using the Debye-Hii.cl>kT (14) is met, the terms in equation (12) other than the first may be neglected and one may write vzwj°= K2 W (15> where [41T Z n eff;- DkT i 1 1 K (16) HI It should be noted that K has the units of reciprocal length and is pro- portional to the square root of the concentration through nj. Because of the symmetrical nature of the equilibrium distribution, on depends only on}; = I 321]. The general solution of equation (15) is then Kr Be 2 + I‘ I“ -Kr Wjo A8 (17) . . O . The boundary condition W- ——-> O as r ——->oorequires that B = 0. To maintain charge balance, the total charge outside the central j-ion must be equal and opposite to its own charge. This statement may be formulated as 00 uejzaf 41Trzpdr (18) where a is the distance of closest approach of the ions. A comparison of equations (9) and (15) gives p : ... DKZ ij/41T (19) Combination of (18) and (19) with equation (15) yields °° JCr ej = ADKZ’I e r dr (20) a Integration of this expression serves to evaluate the constant 13: as A = ej e [ca/DU + Ha) “=5 Ej/Dp, (21) The final expression for the potential then becomes 0 . e'e Ka e “ xr Wj 2' l _.l_____.__ 1 “7“" (22) D(1 + Ma) The physical significance of l_(_ can best be illustrated for the simple case in which the ions are considered as point changes (a z 0). Equation (22) becomes, upon expansion of the exponent W-0 5:: e-/Dr .1 e- jC/D J J J This is the simple expression for the potential due to two point charges fj and :ej, at distances 3. and (_1_[_)_(,_) from origin. Accordingly, (1_/_I_£._) has been called the "radius" of the atmosphere. A more refined analysis shows that the charge density is a maximum at this distance. A useful form of the distribution functionfjio can be obtained by expansion of the exponential equation (8) using on from equation (22) )(a - Kr 2 aka -_2xr _ egeie . e .+ eiei e . o e DkT(1 + Ka) r Zfik2T2(1+ Kay! r2 O fji ‘-‘-‘ njni [ 1 (23) This expression for the equilibrium distribution. function is the one used by Onsager in his treatment of the conductance problem. With these background considerations completed, a discussion of conductance theory itself will now be presented. C. The Form of the Conductance Equation From a qualitative discussion of the two major effects considered by Onsager, the conductance equation can be written in symbolic form. This result helps to clarify the logic of the detailed development. If an unsymmetrical force, such as an electric field, is applied to a solution of an electrolyte, the average velocity of all ions of a given type becomes non—zero. An ion, which is "wet” by the solvent, moving through a solution will drag solvent with it. The ions of opposite charge in its atmosphere “£3.11 be moving in the opposite direction and will, in effect, be moving against a local solvent flow. The effect is reciprocal and the net result is a lowering of the average speed of all ion types. This is known as the electrophoretic effect... A second effect is produced when the external force is an electric field. The tendency of a given central ion and its oppositely charged atmosphere to move in opposite directions leads to an asymmetric distribution about the central ion. A finite time (the ”relaxation time”) is required for the atmOSphere to build up and decay about the moving ion. The net result may be pictured as an excess of oppositely charged ions behind a given ion. This effect can be treated as a small restoring force opposite in direction to the applied force. In the case of conductance, the applied force on an ion is just the product of the charge of the ion and the potential gradient 3: The small restoring force is described in terms of a correction to the field, AX, called the relaxation field. A symbolic conductance equation may now be formulated. If a force .ISj were applied to an isolated .j-ion in a solvent, the ion would . .5 assume a veloc1ty v-. Here (ii is the reciprocal of the coefficient of friction of the ion. In the case of conductance the force is given by equation (24) Kj = eJX (25) In a solution of many ions, the average force felt by the jnions is (X + AX)eJ- and the average velocity is VJ 7:: VjS + (X + AX)€jwj (26) where the term ijs is the retarding velocity of the solvent in the neighborhood of the juions which results from the electrophoretic solvent drag by atmospheric i-—ions. The above velocity expression is easily converted to an equation for equivalent conductance through equation (3) to give _ 96, 500 AX 96, 500 j“ 77c?“ 1” 32') 'ej'wj ‘ m “(M (37’ )\ Since the solvent velocity (til-S) depends upon the velocity of ions of type _i_, which in turn depends upon the field (X + AX), the last term on the right may be written as AX 96,500 IVis' _) X 300 X' (28) = AXje (1 + where Akje is the electrophoretic contribution to the conductance. If only nonwinteracting j—ions were present in the solution (an infinitely dilute solution) the equivalent conductance would be 0 96,500 *3 " 300 IBJW (29) o . . . . where Xj is called the equivalent conductance at infinite dilution. Equation (27) may now be written as AX N = (X "5(— ) (30) 0 8 J '“AXJ)(1+ J 10 or, for all types of ions in the solution /\ 2 (A0 w A/\e) (1+- 9515-) (30a) The explicit evaluation of the electrophoretic term .A_/§e, different from that of Onsager, is the major concern of this work. The derivation of an expression for the relaxation term (AX/X) is both difficult and lengthy. Accordingly, only an outline of the Onsager pro--- cedure will be presented here. For anyone seriously interested in the details it is recommended that the following presentation be regarded as an introduction to be read prior to the study of the original papers of Onsager and Fuoss and their amplification by Fuoss and Accascina (19). D. The Onsager Conductance Equation 1. General Approach A In order to find the ionic. velocity Yd" and from. this the equivalent .m-b conductance, it is necessary to find Vi the local solvent velocity, ”‘11 A and the relaxation field AX. The first of these quantities, v55, 8’ involves the solution of a hydrodynamic problem and may he found either by use of Stokes Law (an integration treatment) or through the general hydro~ dynamic equations of motion (a differential treatment). The relaxation field AX is more difficult to evaluate. This field is obtained from the negative gradient of the asymmetric potential W)- evaluated at the surface of the ion, or more properly, at the distance of closest approach of the ions AX = — (VW)aL (31) It is assumed that the potential function Wj can be written in terms of an asymmetric distribution function iii through a Poisson equation analogous to equation (9, 10). In order to find fji? a general expression A for 1,31 the velocity of an imion relative to that of a neighboring ,jwion is 11 written. This expression contains fji and Ejs (which can be evaluated by hydrodynamic equations). The final link in the chain is provided by the equation of continuity which relates E31 andiji. Solution of the resulting differential equation with suitable boundary conditions givesiji. Integration of the Poisson equation leads to the evaluation of the potential \f- from which the relaxation field AX is easily found. The general expression for the relative velocity v~ may be written. J1 as _: .5 A .5. .4. mm. r...) = mar.) + «(K31 - kTV. 1n fji) (32) where ._S Vjs : the solvent velocity in the neighborhood “’i : reciprocal of the coefficient of friction of ions of type i Kji = total force on an inion in the neighborhood of a jmion The last term on the right of equation (32) arises from the Brownian motion of the ions which tends to restore symmetry to the ionic distriu bution. It was this term which was overlooked in an early attempt by Debye and Hiickel (20) to solve the conductance problem. The subscript on the gradient operator is due to the use of two volume elements with different sets of coordinates in defining the distribution function. A com- .3 pletely analagous expression for Vij, the average velocity of a j-ion near an i—ion may be written, Vij(rzs r1.) = Vis(r1) + wj(Kij - M V. In is) (32a) It is assumed that the distribution functions and potentials may be treated as the sum of a symmetric part (denoted by superscript zero) and a perturbation term (denoted by primes) due to an external force. ' I O fji :: fji + fji Wj ‘3 Wjo + Wj' (34) (33) 12 The asymmetric (primed) quantities are then related through PoiSSOn equations , for example 8 qu/j. = - (4 Tr/D) 2 ii; ei/nj (35) i=1 The total force Rji can be specified in terms of external and internal forces. 7“ Aji : ei X 1A - ei vzwi' (a) «=- ei v2 W)“ (:1, 1:2!) (36) The first term on the right is the external force and is chosen to have only an 3: component (i is the unit vector in the x direction). The second term represents the force on an imion due to its own atmosphere and third term is the force on the iuion due to the neighboring j-ion and the atmosphere of this j-ion. The symmetric part of V2 Wfla) cancels out since it represents a balanced force and can produce no net motion. Combination of equations (32) and (36) gives A.) Vji(rz, r12) : VjS (1.2) + wi[eix L. "° ei quji' (a) *- ei V2Wj(rlr21) wkT V, In Fji] (37) _>. A .5 A companion expression for Vji (r1 r21) may also be written. To relate the relative velocities and distribution functions, the hydrodynamic equation of continuity for stationary states is used in the form Vz - (f When integrated, this is merely a statement of charge conservation: __\ v _\ the net flux of charge through the system is zero for a system in a steady state. The complicated expression which results upon combination of equations (37) and (38) may be greatly simplified by using the Onsager symmetry relations. If the solution as a whole is fixed in space, the potentials and distribution functions can be described by the relative .3 distance between any chosen pair of ions. Accordingly r1 is chosen as a 13 A new origin and the system is described by the vector r. .3 r1 3 O A A .3 1‘ 3* r12. ~ a 1"2.1 _\ A r2 '5 1° (39) Since the applied field is in the x—direction, the perturbed potentials and distribution functions will be symmetric about the x-axis and can A be written in terms of the variable r without complication: ‘10 (-3?) 2 $11,; (3‘?) ..'b .. A iii. (‘1’): “‘ iij’ (17°) fji' (r) 2 fij' (wr) .3 .3 fj,’ (r) - if (r) (40) The operators can also be simplified: V: V,: - V1 V.~ V: V... V. V2 (41) The array of terms which arises from the combined equations (37) and (38) are then taken pairs-wise (one from each of the two terms in equation 38 and simplified by the symmetry relations (40) with the aid of equations (39) and (41). Further, all terms which are quadratic in the field are neglected; for example, terms such as :7; . Vf’ and X ( gi— Terms which are completely symmetric such as fovwme no effect on the asymmetry properties and therefore vanish. Assembling all the terms, the continuity equation may be written in the form of equation (42). 14 X (eiwi - ejooj) ( bfjiO/a x) Tx .. kT(wi + wj)v2 fji' Tk - fjio(eiwivz\i/j' - ejijZq/i' Tg : (eiinWj' _, ejwjv Vi.) . ijio Tg + fji'(eiwi' V2 Wjo + eij-Vz Wjol Tg + (eiwiva‘o + ejwjvwio) ' iji' T " [eiwiW'(a) - eij-VLVj'M) ] °ijio Ta 0 + (vis - Vjs)°iji TV (42) The column of symbols on the right indicates the origin of the terms in the nomenclature of the original paper (9). The terms 2X are called the field terms; Tk, the Boltzmann terms; '_I‘_ v, the velocity field terms; I and la, the terms containing VW (a) . The remaining terms are designated by Eg° The quantity which is ultimately sought, the relaxation field, will be given by . a AX=-v\y(a)=—< ax) (43) a Dependence upon x only arises for the conductance case since the dis- tribution function and potential are axially symmetric with reSpect to the direction of the field. 2. The Boundary Conditions Inspection shows that equation (42) is a fourth order non-linear, non-homogeneous differential equation in the asymmetry potentials it, since the quantity 23' appears and_f' itself is proportional to V235. . The solution of the differential equation therefore requires four boundary conditions. Three of the conditions are simple electrostatic requirements: 15 1. The field of the central ion must vanish at infinity 5‘?» _. ‘7?“ *0 00 2. The potential must be continuous across the boundary at r = a ' \Pj' (a-o) :Wj' (a +"l/o) 3. The-field strength-must be continuous at _r_ = _a_ (M) = (a?) Bra-o a a+o Thetfourth conditionis hydrodynamic in nature and while simply stated is very. complicated when put into usefulmathematical form: 4. The radial components of the relative velocities of any two ions must vanish on contact A A .s .3 _s [(fijVij-fjivji).r]a (Y. r)a:0 I“ By further defining the function Z(r) by A .A .A Z(r) = (Y- r) /x = IYI /cos-9 (44) This condition may be stated as a scalar equation. Z(a) = O (45) Tobe of use, equation (45) must be expanded in the same manner as was the equation of continuity. A complicated eight-term. expression ..\ v. __is and . . 0 . . containing f , _f_1 and 11: as well as the local solvent veloc1ties A lie-results. The second and third (electrostatic) boundary conditions may be conveniently combined. .If the ions are considered as conducting (spheres, thenfor r_< a the asymmetric part of the solution of the LaPlace equation (V2 Wj ‘= 0) gives Wj'(a-o)=|Brcos-9|a_o=[r 35““!111'3'] o (46) a- 16 Rearrangement with use of the second and third conditions [Wj' (a — O) = W,” (a + 0)] gives equation (47) - D‘Vfi [r a?) _ W3" ] a + o I: O (47) The four boundary conditions may be summarized by the three equations (48) (0Wj' / '01“) = 0 [1' (AW/JV 31") a Wj' la '~‘ 0 Z(a) :: O (48) 3. Order of Terms and Method of Solution of the Continuity Equation The differential equation (42) has been arranged so that the terms of higher order in the charges are collected on the right. The reason for this division can be clearly shown only after an approximate solu» tion is obtained. Initially it assumed that the primed potentials are proportional to the first power of the corresponding charges. Since ijio starts proportional to eiej the terms on the right are all of ~ order ale-3 e.e,3 e-3e. , The last term on the left is of order as 1 J a l j 7 1 l 1 J and is assumed to give the leading term of the solution. The problem is now specified to a solution of a single electrolyte giving only two kinds of ions. Accordingly let .i_:=_1; j -_: _2_ Equation (42) can then be written as X(eiwl ‘ 62.902) (ale/ 0X) " kTWi + Q32)V2 £21t ‘ f210(61‘91Vz Wz' " ezwzvzWU) '3 2 Ti (49) 1 where 2 T1. denotes the higher order terms. An approximate solution _.L___‘ for the asymmetric distribution function is then found ignoring the higher order terms. Let 17 W2' 2 $2 + P2 (50) and f 21' 2 F21 + 821 (51) where (\Ipand E are the first order approximate asymmetric potential and distribution function obtained by solution of equation (49) without the terms 23 Ti‘ The contributions of these terms are denoted byp and _g_ respectiv'éTyT—The potentials and distribution functions are assumed to be related by Poisson equations: V2 q}, : —4 n Fu/D n1 (52) and v2 P2 = " 4 1T gal/D n, (53) From equation (49) making use of equation (52) and rearranging terms, the differential expression to be solved for F21, the first order approxi- mation to the distribution functions, becomes 4n f 0 e 2w e Zoo 2 21 1 1 2 z o X(€1w1 '“ ezwz) bfu ( ) (54) kT (031 + (02) 3x Having obtained 521, the potential W2 is found by solution of the Poisson equation (52). Differentiation of _g gives the first order approxiu mation to the relaxation field, Al“ . Then, the expression for {‘21 is substituted, with equation (51), into the differential equation (49) keeping the higher terms '_1_‘_i. This gives a differential equation, the solution of which is g2]. The potential B; is then found from the Poisson equation (53). A correction to the first order relaxation field is then obtained from 22- A glance at the form of the higher terms (shown in equation 42), clearly indicates that the second order solution is not easy to find. The method actually used by Onsager consists of splitting_g21 into four 18 parts and treating each part as a separate problem. Further comments about the higher-order terms will be reserved until after the first order solution is set down. 4. First Order Approximation to the Distribution Function and Relaxation Field To obtain the first order solution 15:21 from equation (54) it is necessary to know £210, the equilibrium distribution function. The com- plete function was derived above and is given by equation (8). Onsager and Fuoss, in stressing consistency of the order of terms, used the expanded forms of £330 given by equation (23) to various degrees of accuracy. For example, for the derivative term of equation (54), two terms of_f_210 are used: .. X. £210:- n1n3(1 _ eleze I'/,U.DkTr) (55) where (a a (1+ Xa)/e 34a 3 (56) To this approximation ( bfmo/ ax) ~ el and the term of the right of equation (54) begins prOportional to 312. The second term of the left is already proportional to e12 so that in this term, the approximation 0 £21 :1. nlnz (57) is used. Making these substitutions and the electrical neutrality requirement (equation 13), equation (42) to first order reduces to XI 2 2 2 _, 111317 X a e“ VF21‘7F21"‘["T41THT][—ax( r )] (58) where 4 'Y2 = C12 X2 =[ Tr ] [nieizwi + nzezzwz ] (59) DkT( 0.)} + (Dz) 19 Subject to the first boundary condition (fig/Vb r) :2 0, this equation has the s olution: __ nlelqzx A eixr Ae" yr F21 _ [41Tka(l_qZ)] [ 6X ( I" . I" )] (60) Substitution of this expression into the Poisson equation (52) and sub- sequent integration gives the potential W elezng a [ e“ xr Ae—yr + B ( l 1 )1 : ———-—-———-2— -. , — ——2- - ——z- uDkT(1-q) 3x x; r 'y\2r r y X (61) where A and B are constants of integration. Since this solution contains the charges only as the productm the solutions for “If! and E are identical and are denoted by W. The combined form of the second and third boundary conditions serves to evaluate A_ in terms of B. The constant B is obtained from the fourth boundary condition in which higher-order terms are neglected. The relaxation field is found from the relation AX = «- VX'Qfla) = ~ (3‘17 3 x), (62) For the simple case of point charges (a 2: 2) A and B are both unity. The field for point charges, _A_Xo, is given by AX = [ elezgzx ] Lim [ em Kr'(1 + Kr) _ e ”719(1 + yr) 0 DkT(l - q?) r——) o 34,2 r3 72 r2 + 1 1 ] TS"— "' T? V r x r (63) . " I“ '“ I' ' . ExpanSion of the factors e X and e 7 to order re, and taking the indicated limit (triple differentiation of numerator and denominator) leads to Axe __ (318qu X. x ‘ 3DkT(1+q) (64) This is the classic result obtained by Onsager in 1927 and is usually 20 written as 3:: 1 AYXO = - o. C T (64a) where c,“ is the normality of the solution and a is a constant. For the case in which the ions are represented by hard Spheres with minimum distance of approach a, the constants A and B of equation (61) are complicated functions of )ca . From equations (61) and (62) an expression for AX for this model is obtained. The result may be expressed as a correction, -AX0A1, tolpthe solution for point charges. Accordingly Al is defined by the r elation AX =.- AX0 (1 — A1) (65) So that A1 = 1 — (AX/AXO) (66) The ratio (AX/AXO) is found to be l AX 1+(E)(1+q)[1_ +xa] AX 3 2 b 0 (1 + Na) P3 (67) where b : lelegl /aDkT (68) and P3=1+qxa+ qZK/zaZ/3 (69) The quantity 9.1 may be written xa(1+q-) xa(1+q) )6?‘ a2 2 A : .___..._____. _.__._.__.... r—-—---—-- + 3 70 1 p3(1 + K8.) pr3 P3(1+){a) [C1 C1/ ] ( ) At this point Onsager and Fuoss Specialize to the case of symmetrical electrolytes, for whichq_‘2 : 1/2, and to the approximation that (1 + 6g)/3 (1 + g) 251 , The expression for Al can be written in the much simpler form of equation (71). A1: Ka(1+q) (1+b) /2 bp3 (71) 21 It should be pointed out that with equations (65) and (70), the relaxation field to first approximation may‘ be calculated for any electrolyte charge type. The result is consistent to terms of order E. The quantity of interest (AX_/)_() can now be written to first approximation as AX] €16qu (1 '- A1) * 212‘ -——- : ‘ :2 - l - X 3DkT (1 + q) a C ( A1) (72) 5. The Second Order Approximation to the Relaxation Field The second order approximation to_fji' and AX1X is now found by returning to the continuity equation (42), (49) including higher terms. Substitutions of equation (51) f21‘ = F21 4' 821 (51) into equation. (49) gives a differential equation in gm. Expansion of the higher terms, _T_i leads to a multiplicity of inhomogeneous terms of unknown order insofar as they will effect the solution of. the equation. An ingenious method of classifying these terms was devised by Onsager and Fuoss. It consists of writing the unknown solution of the differential equation in gal as a product of an unknown dimensionless function E and a determinable power of ii. The part of the complete solution which will result from each n . term can then be expressed as K U where n is the lowest power of )6 which will appear in the solution from the term in question. When the method is applied to the single inhomogeneous term retained in the first order solu- tion, it is found that Ellvfl. For the second order solution all terms which give iii are retained and those giving .5152 and higher are neglected. The terms Ii, when examined in this manner, are all of order l_6_:. This justifies the initial separation of terms according to equation (42). It should be noted that while this procedure separates terms of higher order, no information is obtained concerning the relative magnitude of the terms. When the distribution function in the TL terms is expressed as (£21 +g21), all the terms in gal are, by definition, of higher order in M than those in F31 22 and may be neglected. This tremendous simplification has the effect of changing the form of the equation to be solved from V2321 - 72821 = § 61 (1”,COS 9) (73) in which the C'i are unknown functions, to the form V2821 “‘ 72821 3 213 'F‘i (1‘, COS 9) (74) in which the TH involve only known functions of the variables evaluated through the use of E21 and?. The equilibrum distribution function is again used in various approximate forms, the number of terms used always being one more than was used in the corresponding location in the first order solution. It is important to note that this requires retension, in one instance, of the quadratic term in the distribution function. In all cases, the equilibrum potentials are represented by the Debye-Hiickel expression. If the radial functions 1121(r) and E (r) are defined by F21(r,0) = h21(r) cos 8 ”FR, 9-) = E (r) cos 9 (75) The expanded equation in 821 is vngI " 72821 : 2 2 ~2 Kr 1" n,e,e, 7 X —-"> (——2——e > (T ) 8nDuzsz2 ax r x: gB -Xr .. )cr _ elezyze F21 _ elezxze F21 (T ) uDkTr uDkTr g, g23 2 - K r d + nzez.’Y [ ( e )] [Q—i] (T , gas) 4nukT dr r C) x g ._ Kr ._..._._..elez [.9— (8 )][__a_.}}.é_1. (T a) uDkT dr r O x g: g2.) 2 .1413 _ nzez’Y d 8 BE (41"ka ) [ dr ( r )] [ bx ]a (T3,, g8.) .. )(r nlnzelez (V13 " V35) d e . + [ )LDkTTZ(wl + 0.2) 1 [ dr .( m1. ) 1 (TV, gv) (76) 23 The boundary conditions applied to this equation becomes VPj(°°) -‘-‘ 0 [r bpj/ 6r - Pj]a: 0 Z'(a) = O (77) The function girl) is obtained from the expression for Z(r) in the same manner in which the differential equation for gal was obtained from the continuity equation. Rather than attempt to solve equation (76) directly for gm, Onsager and Fuoss have devised a method whereby the result can be obtained from the particular solutions found for certain groups of inhomogeneous terms. They therefore divide gal into the sum of four terms. g21=gB+gz,3+ga+gv £233 gj (78) The first symbol to right of the inhomogeneous terms of equation (76) indicates the origin of the term in the continuity equation (’_I‘_i) and the second indicates the part of __g_21 to which each term gives rise. The problem then is divided into four simpler ones. Rather than finding the potentials £1 from Poissan equations in_g_j and differentiating to get the second order parts of the relaxation field (ABS), the latter are found directly from the boundary conditions. In order to find the solution gv, it is necessary to first find the radial components of the local solvent velocities in. and X21” These quantities also appear in the boundary condition [Z' (1')]a . 6. Solvent Velocities and the Electrophoretic Effect It may be well, at this point, to recall that Vis is the velocity produced in the solvent at a distance _i; from an i-ion by a j-ion, and that these veloci— ties have come into the equation from the velocity field terms. (Vis ' V fij - Vjs fji) 24 A When reduced to the single variable r, and the problem is Specified to i = 1, j = 2, these terms are denoted by (v... - 1?...) Vs. These solvent velocities describe the very important electrophoretic effect as well as being necessary for the evaluation of part of the relaxa— tion field. Insofar as electrophoresis is concerned, a rather Simple method based on the integration of Stokes Law in differential form is sufficient. This method is presented later. In order to find suitable expressions forglr and 37:21. for use in finding the potential gv and the part of the relaxation field _A_)_(V, it is necessary to start with the general hydrodynamic equation of motion .A nvzv :VP‘F (79) where n = Viscosity R7 2‘ solvent velocity at distance}: from a selected ion p '3 pressure F = volume force (force per unit volume of solution) The ionic subscripts are dropped for convenience. A scheme for separat- ing the pressure and force terms of equation (79) is then devised. Since the solvent velocity must satisfy the continuity equation, V°V= 0 (80) equation (79) may be written as .3 .5 nVXva=F-Vp (81) By defining the axial vector if by 7):: V(V u ) ~Vzu J (82) .3 .4 and constructing Vx VX v in terms of u, equation (81) becomes 25 —-\ .A vzvz u -- V(V°VZ u):F-z Vp (83) The form of equation (83) allows the identifications .3 __s VZVZ u = F (84) V(V"Vz :3) = Vp (85) The problem is now reduced to findingii from a knowledge of F . The ..5 _> desired quantity, v, is then constructed from 11 according to equation (82). Since the force may be expressed as A .5 szpi where E , the charge density, is approximately given by the Poisson (86) equation 9 - (D/4w1VZ‘V" (87) The differential equation infi- becomes _.5 -A . VZVZ u = ~-V‘\(J° (88) A part of W0 is due to the central ion and part is due to its atmosphere. It is expected that the resulting velocity, 3, will contain a term describ» ing the local disturbance of the solvent due to the central ion as well as the comparatively long range effect of its atmosphere. In relation to the relaxation field, the use of the equilibrium potential W0 is justified since the solvent velocities enter as a part of the second order solution (gv), and ‘1' would lead to an even higher-order term. The effect on the elect-irophoretic term is negligible by hypothesis since this leads to a term which is quadratic in the field. Integration of equation (66) gives .s .3 .5 V2“ 2 ~(DX 1/4n W0 +172 w (89) A A where v2 w is the general solution of the homogeneous equationV‘ZV2 u z: 0. 26 Through rather subtle reasoning concerning symmetry and boundary .3 4 conditions, it is concluded that the vector land V2 w may be disregarded. The potential W0 is then Split into two terms representing the contribution from the atmosphere ., ._, KI' WA ~ (e/D) (e /}.1r =- 1/r) (90) and one representing the central ion WC 2 e/Dr (91) _> A 1 The solutions EA andmuC are found from equation (89), and the velocity .3 .3 —\ vector X 2: v9 + vc is constructed according to equation (82).. The .5 integration constants that appear from the solution for 11A are found by A — requiring that uA and its first. derivative be continuous at r : a d" dA uA UIA (‘6‘?)aw o _ (‘3'?)a+o (92) 2 A. f A ne w c n n 1 " e r u on i- eren if}... ‘ .' :15: u v , O of the t o o sta ts d sapp a s p d ff t a 1on or cor ‘1‘ ct A A and its evaluation is not necessary, The solution uc is found by requirr» .3. ing the disappearance of the radial component of uc at. I. '3 IS, where .— is the hydrodynamic radius of the ion, The radial component of the solvent velocity in the neighborhood of an ion is simply found from the 4.3 A relation VI. = r1 . v (where r1 is the unit vector in the r direction) and is __Xecose ~ 4177; VF [Z(1+ )ca +X2aZ/Z + )(3 a3/6] 2e x,(a ' Iu)(1 ~ )Cr) RZ no IK3 rT(l +~ Ka) 3r” (93) In the 1957 treatment of Onsager and Fuoss (8), the hydrodynamic radius Ii is, very reasonably, set equal to _a_/_E , thus eliminating an added dis- tance parameter. A more recent modification by Fuoss and Accascina (19) uses Ii 2: _a_._. This seemingly contradictory choice comes about through .3 a consideration of the expression for v evaluated at r :2 a, Z7 Xe 5. 3 1 274, —> . 1 R2 .u _— .. _——_._———- 4: : -—— .u —-—-,, 4m; [ 6a? + 2a 3(1+ )4 a) ] ' r1 [cm a ( 2a 2519)] v(a): (94) A Since one of the boundary conditions used to obtain v requires the radial component to vanish at}: :2 a and cos 9 not. always zero, it. must be concluded that I: r: a in order that the radial term of equation (94) vanish unequivocallyo This same requirement can be shown to be mathe- matically necessary in several similar ways, While this result is operationally satisfying, physical justification necessitates a modification of the model“ If it is assumed that the kinetic entity called a "free ion" is really the ion plus a solvation sheath (22), the dilemma can be rationale ized by allowing that the solvent. molecules are "squeezed out” upon contact of the ions so that two solvated ions of radii I33 and R1, upon contact, have a minimum distance of approach a-L+ ai r: a .. If ai .-_-. ai 2 a/Z and if the solvation sheath is assigned a thickness 3.12, the desired result R 7: a is obtained, This explanation is not altogether satisfactory since it requires a "thicker" solvation sheath for larger ionsuma condition which is hardly a general truthi It should be pointed out that this difficulty arises from the boundary condition which requires the vector E to have a continuous derivative at I; 2: 3 (equation 71),, The only alternative is to require continuity at _r_ = I: and to proceed under the aesthetically distress- ing circumstance of requiring two distance parameters, Fortunately it turns out that the value of I: used in the relaxation term affects the final calculated conductances only slightly. If it is conceded that I: r: a, the xucomponent of}; (a) (equation 94) becomes Xe Z(eJC 6Nn(l V(a) : 61Tna _ + Ka) (95) This expression gives the velocity of the solvent at the surface of the ion, which must also be the velocity of the ion itselfo The first term is 28 just the velocity of an isolated ion of radius I: 2 a as given by Stokes Law ; ._E_ 61m R (96) The second, concentration dependent, term must then be a slowing effect produced by other ions in the vicinity, This is exactly the description of the electrophoretic effect given previously; hence the second term of equation (74) is identified with the electrophoretic velocity correction _. .. X e}(, VS ‘ 6m (1 +){a) (97) The contribution to the equivalent conductance becomes ij: —96500le-I . H. 1800 n n 1 +Ka (98) For the simple case a 2 0 (point charges), the correction may be written -%500 le-IK _ 96% .2 ———————4— : _ . “J 1800 n n 53 C (99) This is the limiting form obtained in the original Onsager treatment (4) while the result expressed by equation (98) is used in the new treatment. A different derivation of the electrophoretic correction, based on Stokes Law is presented below in section (8) of this chaptera Having obtained the expression for the radial component of the local solution velocity (equation 93), the part of the relaxation field due to these and other terms of higher order in the continuity equation are found. 7. Higher Terms in the Relaxation Field The final step in the evaluation of the relaxation field is the solution of equation (76) for second order correction to the asymmetric distribution function g“. Integration of the Poisson equation (53) would give the corresponding potential p2, from which the higher order correction to the relaxation field could be found. As described above $21 is divided into four parts 29 according to equation (78), and each part is found separatelyo Rather than integrating the Poisson equation gj in Ej and to expressions for pi, and then differentiating to find 438, a method was devised whereby the parts of the relaxation field were expressed directly in terms of the boundary conditions and particular integrals of equation (76), The details of the method have been presented by Fuoss and Accascina (23). The results are summarized by the following formulae: 9353 : -(ele2;c )[ cha 1 1+K,a T2 X 3DkT .p2(1+xa)z][ 8' ‘ T + 7r] (100) T2=Tr[(2+q)Ka] x 00 —t X Tr(x) '2 e f e /t r: e Ei(x) x AX — e e g, b J-(a g __A : _J_.§___ 3--— 1 l x [3DkT :png, (1 +J£a73 ] [q 3 ] ( O ) 1 gxa 87‘ch aZ 2714a ’02 32 :: -—-—— “—— : 1 1 __ ZTMa x2 a": pa -- 1+ "-7—— + 4 AX; 3 : - [6132.“ [bjta _§5(}(a) ] (102) x 3DkT Paps“ Heal)Z ' 1 _ 22- 9-9 3P3T2 p’l?p3T1 - - - T0 55. 16 +T+T - Pl-Pz-P3‘4— 2 2 pl : 1+ 16a + K: To =- Tr(xa) = eKa Ei (JCa) T1 2 Tr[(l + q));< l A—X : (1C T(I+A1—A3+ BA3'/C1/\o) (110) Further discussion of this expression will be reserved until after an alternate treatment of electrophoresis is presented. 8. The Electrophoretic Effect The electrophoretic effect is usually treated by the following method based on the integration of Stokes Law. Consider a volume element, (ill, near a central j-ion. The number of ions in excess of the stoichiometric average is s 23 (nji — ni) dV i: 1 and the net force on the volume element is s i r: 1 Similarly, the force on a spherical shell at a distance _r_ from the central ion, and of thickness _d_r_, is S dF : 417 r2 X 12; 1(njiei - niei) dr (111) This force is in the direction of the field and is distributed evenly over the surface of the shell. According to Stokes Law, the force d_F_‘will cause the spherical shell to move with a velocity _d_\_r, Opposed to the directionoz‘notion of the central ion, dF 691an (112) dv= The solution inside the shell will then move with this velocity also, 32 imparting it to the central ion so that dv : (.11.) . To obtain the total decrease in velocity 933- equation (92) is integrated from the distance of closest approach a to infinity. Combination of equation (111) and (112) gives 2X 00 AVj : 3? 31“ r ? ei (nji «- ni) dr (113) or, withpli obtained from equation (23) . . .3 K8” 00 Av. = 2X63 ( T n1e1)e f e - Kr dr J 3n DkT (1 +x.a) a Xejz (? niei3)eZKa ‘3? e-axr 3n [.DkT (1 +Ka)]2’ a “—17“- ' (114) The integration yields: Avj : _ 2)“? niei2)ej + X i? nieirs) [ e-e Ka E" 2 3nDkT (1+)(a) 3,, . DkT(1+Ka)] 1( Ma) (115) In the cases treated by Onsager, the presence of the second term is purely formal; for point charges it was ignored as being small and for symmetrical salts it vanishes because ei : "ej and Z; nieiz‘ = O. - - 1 Using the definition of 5— given by equation (16), the first term gives Avr-EL J 61177 (1+Ka) (116) which is seen to be identical with the relation obtained above (equation 97) in the discussion of the velocity field. The correction to the equivalent conductance is given by equation (98) and is repeated here for clarity. AX-e 2» -96, 500 lejIK, (1 +Jca) (98) J 33 For two kinds of ions, the expression is "96,500 (1311 + 1821 )K ,,1 “‘7.- 1+.)Ca -8c /(l+,l(a) (117) lll AAe= It will be noted that in equation (111) the force on each i-ion was taken to be 313;. Stokes and Robinson (24) have pointed out that it should be more correct to use ei(X +. AX) since the force actually felt by the ion is the external force 8.135. diminished by an amount fl because of the relaxation effect. This leads to the conclusion that m~(x + AX) which was assumed in obtaining the symbolic conductance equation (30). This is equivalent to introducing the cross-term, ije(AX/X), into c onductanc e equation. The terms of the symbolic conductance equation (30a) are now known, and the equation in the form presented by Onsager and Fuoss in 1957 can be written. Two additional terms have since been recognized and evaluated, however, and the writing of the explicit conductance equation will be postponed until after their discussion. E. Recent Modifications of the Onsager Equation l. The Kinetic Effect As discussed above, an applied electric field produces an asyrn— metric distribution of ions of opposite charge about a chosen central ion. There is, in essence, a larger number of atmosphere ions "behind" the central ion so that thermal motions will cause the central ion to be struck from behind, more often than from the front, by these ions (9). The result is an increase in the velocity of the central ion. This effect is described by a small virtual force in the direction of the field or as an osmotic pressure on the reference ion which moves it with the field. The osmotic pressure TI, due to the field, is given by equation (118). 34 O 0 , TT : (n12 ‘- n12 ) kT : (£21 " £21 / n1) kT 2,118) N _ If the approximation f2, ._ £210 + F21 is used, the pressure is 77' = F21 kT/nl (119) The force on the central ion, due to this pressure is found by integrating Foyer the surface of a sphere of radius a corresponding to a region into which no ions can penetrate. The resulting force, AP, in the direction of the field is 2a2 (b - 1) APZXV‘ 12b ] (120) The conductance equation (30a), with the inclusion of this effect, becomes /\= (N - A/\e) (1+ AX/X + AP/X ) (121) The AP/X term is clearly linear in concentration through K232. 2. Einstein Viscosity Correction It has been shown above that the electrophoretic term AAe is inversely proportional to the viscosity. Similarly, if an isolated ion obeys Stokes Law, A0 has the same dependence. It is concluded then that /\~1/n The original model used in conductance theory considers the solvent as a continuum. At finite concentrations, an ion moving with the field through the (assumed) structureless solvent will "see” ions of opposite charge as obstacles to be passed if it is to continue its course. This effect can be treated as a correction to the viscosity (11). The Einstein viscosity expression (25,26), serves to evaluate this correction. n = no (1 + 5 SV/Z) (122) 35 where r) "—' "solution" viscosity solvent vis c osity l. T70 30 2 ion volume fraction The volume faction is given by ,_ 411113 NC 2 99--'—3—(m) " Fe “23’ where 1_\I_ is Avogadro's number and 2 is the molarity of the ions which contribute to this effect. Since, in reality, many ions are not much larger than the solvent molecules, Fuoss recommends that only "bulky" ions (such as quaternary ammonium cations) be considered as ”contributing" ions. The problem of distance parameters is again introduced. Consistent with the chpice made in the velocity field terms, the value Ii: 3 is taken. The effect of this term on the calculated conductance is, in some cases, . much larger than is generally supposed. The effectpof the value of R on the course and shape of some phoreograms (1) (plots of Avs. 613') will be shown in Chapter 4. 3. The Effect of the Relaxation Field on the Electrophoretic Term In the discussion of the electrophoretic effect, it was noted that the assumption that the force per atmosphere ion in a spherical shell about a central ion is given by ei(X + AX) leads to the cross-term (Akie)(AX/X) in the conductance equation. If, however, the atmosphere is represented as a charge cloud, as is the case, Dye (27) has taken the view that the correction to the force e-IAX should not be treated as constant but as a function of 3; and Q. The term éXis regarded as shielding of an ion by its atmosPhere and changes as r and 2 are varied. This effect can be expressed as a correction to the cross-term and is, of course, very small. Using the asymmetric potential for point charges (28) to obtain AX(r, 0 ), the cross-term correction is shown to be 36 - 1 (A+) (Axe) (Ax/X) 1 (124) where e K8. A+= 1 E1[(1+<21) wal-Eflzxa] ‘q - ' (125) F. The Complete Onsager Conductance Equation and Its Limitations The complete conductance equation may be written in the form A = ( /\0 - AAe) (1+ AX/X + AP/X) ( 1+ F c) (126) Values of A may be calculated using the various expressions for the terms of (126) given above. A more convenient expression is obtained by expansion of this equation. The relaxation field through the terms A; and A3' contains several transcendental functions all of which are related to the function 00 Ei(x) = f (e’t/t)dt (127) . x For small values of 5, this function may be approximated by Ei(x) 3:" r-lnx+ x . . . (128) in which F: 0.5772. Using this approximation and (1 + Fe)‘1 3 F c (129) the conductance equation may be written *1 a}: :k 3:: 0 /\ =/\°- Sc T+Ec logc + Jc -F/\ c (130) where all terms of orderij and higher have been dropped. The expressions for the constants .8. E, g and E are summarized by Fuoss and Accascina (29). The first two terms of equation (130) give the Onsager limiting law . The higher terms give theoretical justification for the long- standing practice (30, 31) of fitting conductance data with empirical terms of the form Dclogc and E2. 37 The conductance equation (126, 130) is completely consistent to terms linear in the concentration. Higher order concentration dependence is ignored. The results are limited to solutions of completely dissociated symmetrical salts. Mathematical approximations limit the range of applicability to values of xa < O. 2 or approximately 0. 1 normal for 1-1 electrolytes in water. This limit is also physically reasonable since the representation of the ionic atmosphere as a charge cloud certainly becomes invalid at higher concentrations. The equation contains at least two para- meters, _/_\_O and the distance of closest approach a. The relationship between the hydrodynamic radius 1:, and _a_., should be clarified by extensive studies of viscosity effects as suggested by Stokes (32). G. Ion Association and Conductance The idea of ion-association was first suggested by Bjerrum (10) as a means of explaining observed deviations from the Debye-Hfickel expression for activity coefficients. The postulation of an equilibrium caused solely by electrostatic interactions between "free" ions and neutral ion—pairs according to the scheme c+ + A” 33-9 (cJr A')° V‘ leads to an expression for the association constant K K: (1- v) /c 72,; (131) where l is the fraction of ions which are free and _f_ is the ionic activity coefficient given by the Debye-Hfickel expression. The activity coefficient of the neutral species is assumed to be unity. Bjerrum obtained a theo- retical value for If. by considering the probability of finding an anion in a spherical shell of thickness _<_i_r_ and radius 3 around a reference cation. The resulting expression for the association constant is 4TrN K 2 1000 (ab)3 Q(b) ' (132) 38 where Q(b) = 21)— {e2 - 131(2) + Ei(b) — (eb /b)(l + 1/b + 2/b2)} (133) andb has been previously defined by equation (68) as 18132.,1 /aDkT . This expression suffers from the mathematical necessity of considering ions which are not in physical contact , (less than the distance ab/2 apart), as pairs. Furthermore it predicts abrupt cessation of ion pairing at a critical value of the dielectric constant. For large values of b b 3 v 4w a N . e V 1000 b (134) Using a thermodynamic approach, Denison and Ramsey (33) and Gilkerson b (34) conclude that E is proportional to e_ . The Gilkerson expression is = [G exp (~-ApS/kT)]eb (135) where G and A are constants and P3 is the dipole moment of a solvent molecule. The quantity Aps represents the difference in solvation energies of the ions and the ion pair. In a mechanistic treatment, Fuoss (35) has also reached the conclusion that K ~ e . The application of the concept of ion pairing to conductance is obvious. The concentration termsin the Onsager expression refer to ion concentrations. The presence of a pairing equilibrium means that the average concentration of ions is less than the stoichiometric amount and >:< that c everywhere, except in the viscosity correction, in the conductance equation should be replaced by the ion concentration ci ='.,*yc The observed conductance for 1-1 electrolytes where c = _c_ , is given by A: y A calc or in the expanded form of equation (130) A“ Ao-Scj—+Ecilogci+Jci-KcifzA(1+Fc) l+Fc (136) 39 Unfortunately, the theoretical expressions for I_{_ prove to be unreliable. For solutions of low dielectric constant, the best numerical values are given by the Bjerrum equation (132). However, recent careful exami- nation (12) has shown that IE is more nearly proportional to e: than‘eljlb . As a result the ion pairing constant is now treated as a parameter chosen to give the best fit to the data. Fuoss (36) has outlined several graphical methods for the selection of the parameters _/_\ 0, a, and IE of equation (136). It should be noted, that through the linear superposition of fields approximation, (equation 7), only approximate expressions for the ionic potential and distribution function are obtainable. If exact expressions were known, and the model of nonmpolarizable spherical ions represented real ions accurately, there would be no need for consideration of ion pairing. Since, in reality, ions _a_5_e; polarizable, the introduction of ion- pairing equibria must be considered to represent in part real phenomena and in part a device to hide inherent mathematical inadequacies. In the hope of bringing to light the effect of some of the mathematical approximations, the following study of the higher terms of the electro- phoretic effect is presented. III. The Effect of Higher Terms in the Distribution Function on the Electrophoretic Effect for 1—1 Salts in Dioxane-Water Mixtures A . Introduction When the Onsager equation is applied to aqueous solutions of multiple- charged ions or to nonaqueous solutions of univalent ions, large discrepa- ancies between observed and calculated values of conductances often occur. Much better agreement was obtained for aqueous solutions of certain multivalent ions, whose activity coefficients obey the Debye-Hiickel equation, 40 when the complete electrophoretic effect was employed by Dye and Spedding (13). This treatment of electrophoresis uses an exponential distribution function. Since the extended Onsager expression for the time of relaxation effect is specialized to symmetrically charged ions, it seemed desirable to study the complete electrophoretic effect for symmetrical salts for which a suitable account of the relaxation contribution to con- ductance may be taken. Accordingly, an examination of the magnitude of the higher terms of the electrophoretic effect of "uni-univalent" salts in dioxane-water mixtures has been made and is presented after a brief rationale of the Dye-Spedding treatment. B. The Extended Electrophoretic Effect 1. The Exponential Distribution Function Onsager (37) has shown that the statistically correct expression for the equilibrium distribution function fjjo is fjio 22 njni exp [ .. UJ'iO/kT] (137) where Ujio is the timewaverage energy required to charge an iwion at a distance r from a j wion, less the time average energy required to charge the i-ion at an infinite distance from the j—ion (but still in the solution). The linear superposition of fields approximation Ujio = 61 Wjo (7) gives £310 = ninj [eXpt-eiWJ-c/k'l“) ] (138) It is not until the Poisson equation (11) is used to obtain Wjo that it be- comes mathematically necessary to expand the exponent in equation (138). Consistency requires the retension of the first two terms only. fjio = ninj [142:in 0/1 eiej/Dr as r—> 0 a a f n Combinationyequation (140) and (137) give the "complete” distribution function a .. eje;elc e Kr "Dkfu+xa)° "“r ] f~0 = n-n- exp [ 31 13 u4n An alternative argument for the use of this expression has been made by Kirkwood (38) who pointed out that, while the Poisson equation may be an incorrect ex ression for the second derivative of the otential 2 -°, p p V'Wfi the resulting value of the potential itself may be only slightly in error. 42 It is interesting to compare the exponential distribution function (equation 138) and the linearized form (equation 139) with the function fjio Z? ninj exp [ - eiej/DrkT] which was used by Bjerrum (19) in his treatment of ion pairs. Figure l is a sketch of the radial function (fjiO/ni) 4 1T r2 for various distribution functions. The plot refers to a 1-1 electrolyte at _c_ = O. 01 with the contact distance a z 4. 0 I? . It is clear from the figure that the linearized function ignores a large interaction force between oppositely charged ions at closed distances. The Bjerrum function accounts for these forces at small distances but diverges at large values of _r_. The exponential function, however, gives the correct limiting behavior at both large and small dis— tances. It is this distribution which is used in the Dye—Spedding treatment of electrophoretic theory. 2. Formulation of the Electrophoretic Term In the preceding treatment of the electrophoretic effect, the expression for the ionic velocity correction due to this effect is given by equation (113) 2x:°° Av- 2 —— r .... . . 11 J 3% [§<> AX). 1" z " P 103.450nlvtZ xfp {ex}? [ "2+ P e ‘/‘O 1. + exp[ 1 Z+Zw I Fe. 0/ p] dp (149) 45 The definition of M is given by equation (16) and may be written as 411,62 n-Z-Z" 4TT€ZNC a- , .. , 2. 2. M " DkT § 11 “ 109DkT [141+ IV-Z-] (150) Combination of equations (149) and (150), making use of (148), gives <><> AX+ :: M fpiexp [ - Z+2Pe p/ p ] — exp [ Z+Z_Pe p/ M} dp x (151) where 96,500 DkT MS . . 1800 “IT 1’) 6 (12+) + I Z_I) (152) A similar expression for._A)\_ is 00 A),_ :: M fp{exp [ - Z_2Pe p/p] -. exp [I Z+Z_I Pe p/p]} dp x (153) Equations (151) and (153) are the final expressions obtained by Dye and Spedding. These integrals are functions of the charge type, dielectric constant, viscosity, temperature, concentration and, the minimum distance of approach _a_. It should be noted that. i appears alone through the exponential coefficient E (equation 146) so that several values of an integral can occur for a given xa . 3. The Evaluation of Extended Electrophoretic Effect for Univalent Electrolytes in Dioxane-Water Mixtures In order to evaluate the electrophoretic integrals (equations 151, 153), a program was written for the Michigan State University MISTIC high speed digital computer. A description of this program is given in Appendix I. Tables 1 through 9 give the electrOphoretic correction 9A.? to the equivalent conductance of 1-1 salts in water and dioxane-water mixtures from 10 to 70 weight percent dioxane. The dielectric constants and viscosities are taken from several sources and have been summarized 46 by Fuoss and Accascina (19)° The computational accuracy of the AAe values is O. 004 conductance units. For 1—1 salts _A_x_+:- A__X- = AAe/Z. Because of the method used in computing these values, the single ion conductances are accurate to i 0. 002 units. (See Appendix I.) The extended term is compared with the Onsager electrophoretic term 1 .1. ((3 cT/1+)(a) and the ratio [AAeU +Ka)/(3 c2 ] is given for a series of concentrations at each of several values of the distance parameter a. Figure 2 shows the change in the ratio of the extended term AAe to the 1 Onsager term 8 c 271 + Ma with the square root of the normality for various values of a in 60% dioxane (D :2 27. 21). Plots for the other solvent mixtures exhibit the same features. In all cases for which values were obtained, the extended term is larger than the Onsager term. The l T ratio of the terms rapidly increases from unity at c =2 0 and passes through a maximum in the neighborhood of )ca a: 0. 3_5. 1 Figure 3 is a plot of AAeU +}(,a)/(3 c 7 v_s_._ a_ for several values 1 of cTin 60% dioxane -- 40% water. The extended term becomes larger ‘ as a decreases. Evidently the ratio approaches unity as a approaches _ 1 _ infinity. As a —>- 0, the Onsager term approaches (3 7, while the extended term and ratio become infinite. For the model of hard spheres used in the theory, however, a can never be zero. Figure 4 shows the increasing importance of the higher terms of the electrOphoretic effect as the solvent dielectric constant is decreased. 1 1 The ratio AAeU +K,a)/fi c T ‘E’ dielectric constant is plotted at _c_ 7' : 0. 07 for water (D = 78. 54) and dioxane—water mixtures from 10% (D = 70. 33) to 70% dioxane (D 2: 19. 07). Curves at_a_ values of 4. 0, 5.0 o and 6. 0 A are shown. Below D r: 70 (10% Dioxane), the curves increase smoothly with decreasing dielectric constant. In the region 80 >D > 40 , (3 decreases since the drop in dielectric constant is overpowered by an 1 increase in the viscosity. As a result both [3 cT/(1+J(a) and AAe decrease with decreasing D. The extended term decreases more slowly, 47 TABLE 1 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 250C. :WATER D = 78.54 0 2 0.8937 8 = 60.19 IO‘BJCN c = 0.3286 0 BR] C AA (1 +}(,a) a N} C AA ___..__ e e 1+)Ca. (3V c .0 0.07 3.678 3.629 1.014 0.04 2.228 2 205 1.010 0.02 1.160 1 151 1.008 0.01 0 592 0.588 1 007 .0 0.07 3.764 3.702 1 017 0.04 2_258 2.232 1.012 0.02 1.168 1.158 1.009 0.01 0.594 0.590 1 007 .0 0.07 3.862 3.778 1.022 0.04 2.294 2.260 1.015 0.02 1.178 1.166 1.010 0.01 0.598 0.592 1.010 .5 0.07 3.914 3.818 1.025 0.04 2.313 2.274 1.017 0.02 1.182 1.169 1.011 0.01 0.598 0.593 1.008 .0 0.07 3.974 3.858 1.030 0.04 2.335 2.288 1.021 0.02 1.191 1.173 1.015 0.01 0.600 0.594 1.010 .5 0.07 4.040 3.899 1.036 0.04 2 361 2.302 1.026 0.02 1.197 1.177 1 017 0.01 0.602 0.595 1 012 48 TABLEZ THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 10% DIOXANE-90% WATER :D = 70.33 71: 1.073 8 = 53.08 10-8)(fiJ c = 0.3275 g \f—C— AA (3N1 C AAeUJ-Xa) e 1+)(a. 5N] c 7.0 0.07 3.224 3.202 1.006 0.04 1.958 1.944 1.007. 0.02 1 021 1 015 1.005‘ 0.01 0 522 0.518 1.003 6.0 0.07 3.308 3.280 1.072 0.04 1.989 1.968 1.010 0.02 1.031 1.022 1.008 0.01 0 522 0.520 1 003 5.0 0.07 3 400 3.333 1.020 0.04 2.023 1.992 1.015 0.02 1.038 1.028 1.009“ 0.01 0.525 0 522 1.005 4.5 0.07 3.452 3.368 1 024 0.04 2.043 2 004 1.019 0.02 1.046 1.032 1.013 0.01 0.527 0.522 1.009 4.0 0.07 3.512 3.403 1 032 0.04 2.063 2.017 1.022. 0.02 1.050 1.035 1.014 0.01 0 529 0.523 1 011 3.5 0.07 3 581 3.439 1.041 0.04 2.089 2.030 1.029 0.02 1.057 1.038 1.018. 0.01 0.529 0.524 1 009 49 TABLE3 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 30% DIOXANE~70% WATER I): 53.28 0 = 1.505 8 2 44.69 lo-tx/\/<: = 0.3286 6 1'1 KB. [3N] C 7.0 0.07 2.613 2.548 1.026 0.04 1.598 1.565 1.021 0.02 0.835 0.824 1.013 0.01 0.427 0 423 1 009 6.0 0.07 2.699 2.608 1.035 0.04 1.632 1.588 1.028 0.02 0.845 0.830 1.018 0.01 0.429 0 425 1 009 5.0 0.07 2.800 2 672 1.048 0.04 1.669 1.611 1 036 0.02 0.856 0 837 1.023 0.01 0.431 0.426 1.012 4.5 0.07 2.861 2.705 1.058 0.04 1.692 1.623 1.043 0.02 0.862 0.840 1 026 0.01 0 433 0.427 1.014 4.0 0.07 2.934 2.739 1.071 0.04 1.721 1.636 1.052 0.02 0.868 0.851 1.020 0.01 0.436 0.428 1.019 3.5 0.07 3.025 2.774 1.090 0.04 1.755 1.648 1.065 0.02 0.879 0.846 1.039 0.01 0.438 0.429 1 021 50 TABLE 4 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 45% DIOXANE-55% WATER I): 40.20 0 = 1.830 8 = 41.17 10~fixyd c 2 0.4593 0 ___ a N] c AAe (ix/c AAe(1+Ka) 1+xa F3N/c 6.0 0.10 3.458 3.227 1 071 0.07 2.578 2.415 1.067 0.04 1.562 1.483 1 053 0.02 0.808 0.780 1.035 0.01 0.410 0.400 1.025 5.0 0.10 3.700 3.348 1.105 0.07 2.720 2.482 1.095 0.04 1.620 1.508 1.074 0.02 0.824 0 780 1.056 0.01 0.414 0.402 1 029 4.5 0.10 3.858 3.411 1.131 0.07 2.812 2.518 1 117 0.04 1.656 1.521 1.088 0.02 0.836 0.791 1.057 0.01 0.416 0.403 1 032 4.0 0.10 4.064 3.478 1.168 0.07 2.932 2.553 1.148 0.04 1.072 1.534 1.109 0.02 0.842 0.794 1.068 0.01 0.420 0.404 1 039 51 TABLES THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 50% DIOXANE-50% WATER 1): 35.85 17: 1.913 8: 41.700 10-91/0 c = 0.48635 3. N] c AAe (Six/c AAe(1+)€a) 1+Xa 5J7: 7.0 0.07 2.510 2.356 1 065 0.04 1.548 1.467 1.055 0.02 0.810 0.780 1 038 0.01 0.413 0 403 1.025 6.0 0.07 2.638 2.423 1.089 0.04 1.600 1.492 1 072 0.02 0.826 0.788 1.048 0.01 0.417 0.407 1 025 5.0 0.07 2.813 2 494 1.128 0.04 1.672 1.520 1.100 0.02 0.847 0.795 1 065 0.01 0.421 0 407 1.034 4.5 0.07 2.931 2.531 1.158 0.04 1.720 1.533 1.122 0.02 0.861 0 799 1.078 0.01 0 426 0.408 1.044 4.0 0.07 3.092 2.569 1.204 0.04 1.786 1.547 1.154 0.02 0.880 0.803 1.096 0.01 0.432 0.408 1.059 3.5 0.07 3.326 2.666 1.248 0.04 1.880 1.582 1.188 0.02 0.902 0.811 1.112 0.01 0.441 0.411 1.073 52 TABLE 6 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 55% DIOXANE-45% WATER D: 31.53 77: 1.964 8: 43.37 1078;6/9 c = .51861 3 9c AA. wa?“ AAfiu+)m) li-Ka. fiN/C 7.0 0.10 3.457 3.177 1.088 0.07 2.626 2.417 1.086 0.04 1.625 1.513 1.074 0.02 0.852 0.807 1 056 0.01 0 432 0.418 1.034 6.0 0.10 3.714 3.303 1 124 0.07 2.787 2.489 1.120 0.04 1.693 1.540 1 099 0.02 0.869 “0.815 1 065 0.01 0.436 0.420 1 038 5.0 0.10 4.086 3 439 1.188 0.07 3.018 2 566 1.176 0.04 1.791 1.569 1.142 0.02 0.899 0.823 1.092 0.01 0.444 0.422 1 052 4.5 0.10 4.360 3.511 1.242 0.07 3.187 2.606 1.223 0.04 1.862 1.584 1.175 0.02 0.919 0.827 1 111 0.01 0.450 0.423 1.064 4.0 0.10 4.744 3.596 1.323 0.07 3.423 2.647 1.293 0.04 1.958 1.599 1.224 0.02 0.949 0.831 1.141 0.01 0.458 0.424 1.081 3.5 0.10 5.341 3.665 1.457 0.07 3.791 2.689 1.409 0.04 2.113 1 614 1.309 0.02 0.997 0.835 1.194 0.01 0.472 0.425 1 111 53 TABLE7 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTESAT 25°C.: 60% DIOXANE-40% WATER D = 27.21 0 = 1.980 8: 46.24 1042097: .55827 g. '\/C AAe (EM/c AAe(1+Ka) 1-+)Gi 8 0‘; 7.0 0.10 3.727 3.324 1.121 0.07 2.852 2.541 1.123 0.04 1.770 1.600 1.106 0.02 0.923 0.858 1.076 0.01 0.465 0.445 1.048 6.0 0.10 4.068 3.463 1.175 0.07 3.070 2.622 1.171 0.04 1.868 1.631 1.146 0.02 0.953 0.867 1.100 0.01 0.473 0.448 1.059 5.0 0.10 4.600 3.615 1.272 0.07 3.413 2.707 1.261 0.04 2.018 1.664 1.213 0.02 1.001 0.875 1.144 0.01 0.487 0.449 1.085 4 5 0.10 5.015 3.695 1.357 0.07 3.681 2.752 1.337 0.04 2.136 1.680 1.271 0.02 1.038 0.880 1.180 0.01 0.497 0.451 1.103 4.0 0.10 5.640 3.780 1.492 0.07 4.082 2.799 1.458 0.04 2.314 1.697 1.364 0.02 1.090 0.885 1.232 0.01 0.512 0.452 1.130 3.5 0.10 6.703 3.784 1.771 0.07 4.767 2.847 1.674 0.04 2.614 1.715 1.524 0.02 1.185 0.890 1.331 0.01 0.538 0.453 1.187 TABLE 8 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 65% DIOXANE—35% WATER D = 23.14 17: 1.962 8 = 50.60 lo-BM/«l—c: .60537 g. '\/—c_ AAe 80c AAe(1+Ka) 1+Ka W C 7.0 0.10 4.168 3.554 1.172 0.07 3.219 2.732 1.178 0.04 2.011 1.730 1.162 0.02 1.042 0.933 1.117 0.01 0.521 0.486 1.072 6.0 0.10 4.662 3.712 1.256 0.07 3.552 2.824 1.255 0.04 2.169 1.768 1.227 0.02 1.094 0.943 1.160 0.01 0.537 0.489 1.099 5.0 0.10 5.506 3.885 1.417 0.07 4.121 2.923 1.410 0.04 2.435 1.805 1.349 0.02 1.183 0.950 1.245 0.01 0.561 0.491 1.144 4.5 0.10 6.225 3.977 1.565 0.07 4.610 2.974 1.550 0.04 2.663 1.825 1.459 0.02 1.258 0.959 1.312 0.01 0.584 0.493 1.184 4.0 0.10 7.425 4.074 1.822 0.07 5.411 3.028 1.787 0.04 3.041 1.846 1.648 0.02 1.379 0.965 1.363 0.01 0.618 0.494 1.251 55 TABLE 9 THE EXTENDED ELECTROPHORETIC CONDUCTANCE TERM FOR 1-1 ELECTROLYTES AT 25°C.: 70% DIOXANE-30% WATER I): 19.07 77: 1.914 (3: 57.11 10-02/J<:= .66660 3 0c: AAe 806 AAeU+KM 1+X.a (3(5— 7.0 0.10 4.934 3.896 1.266 0.07 3.874 3 014 1.285 0.04 2.447 1.925 1 271 0.02 1.261 1.045 1.206 0.01 0.617 0.546 1.130 6.0 0.10 5.756 4.079 1.411 0.07 4.469 3.123 1.431 0.04 2.751 1.970 1.397 0.02 1.367 1.058 1.291 0.01 0.649 0.549 1 182 5.0 0.10 7.365 4.284 1.719 0.07 5.641 3.241 1.740 0.04 3.355 2.016 1 664 0.02 1.579 1 071 1.474 0.01 0.713 0.553 1.290 4.5 0.10 8.916 4.394 2.029 0.07 6.782 3.329 2.037 0.04 3.948 2.040 1.935 0.02 1.788 1.077 1.660 0.01 0.776 0.544 1.400 A /\ e<1+xa>/rs 0 c 56 CD! 3.53. {9-5.03. :\\\® 1. Figure 2. The ratio of the extended and Onsager electrophoretic terms versus the square root of concentration at several values of a: 60% dioxane-40% water solution of a 1-1 salt at 250C. AAe( 1+xa1/ w c 57 \ 0.10 O 0. 04 0. 02 g:§:\o _ \O ‘O 1 I ! 1 I I I I I n 3.5 4.0 4.5 o 5.0 5.5 6.0 6.5 7.0 a (Angstroms) Figure 3. The ratio of the extended and Onsager electrophoretic terms versus a at several values of the square root of concen- tration: {60% dioxane- 40% water solution Of a 1-1 salt at 250C. A/\e(1 +J'éa)/£3 ~/ c 58 I I 1 4.03. o O 5.0 o ‘0 O G\ .. \o o \o\ 0 \g\n 0 \\ :" I- ‘8 8: 1 J I l I l - 4 20 30 40 50 6O 70 80 Dielectric Constant Figure 4. The ratio of the extended and Onsager electrophoretic terms versus dielectric constant for 1-1 salts in dioxane-water mixtures at 250C.: '\/ c = 0.07; a = 4.0 21,, 5.0 31,6.0 A. 59 however, giving a slight increase in the ratio. Below D2.“ 40 (45% dioxane), the viscosity increase is less important than the decrease in dielectric constant, and both terms become larger. The extended term increases very rapidly in the region 40 > D > 20 (45 to 70% dioxane), giving rise to the large slopes seen in Figure 4. 1 For a = 4. 0 X and CT: 0. 07 in 70% dioxane, the extended term is more than twice the Onsager term-~a difference of 4. 5 conductance units between the two values. This difference in electrOphoretic terms repre- sents about 50% of the total change in conductance with concentration which is usually observed with this solvent! The differences in the magnitudes of the Onsager and extended electrophoretic terms having been established, some comparisons of calculated conductances with experiment follow. C. Applications and Conclusions In order to test the utility of the extended electrophoretic correction, equivalent conductances were computed with the aid of MISTIC according to equation (154). /\ 2 (/\° - AAe) (1 + AX/X + AP/X) - IAtAAejAX/X + AP/XH (1 + (10/3) 71 N Rfc) (155) using AAe values from the above tables. The relaxation expression of Fuoss and Onsager, AX _- i- t 0 -)-C --~ ac (1-A1+A2) —A'3 (Sc/A (156) was employed. The expressions for _/___\.1 A} and 9'3 are given by equations (71), (104), and (109). The kinetic term 913132 has been defined by equation (120). The denominator of equation (155) represents the Einstein viscosity term which corrects for the presence of ”bulky" ions of hydrodynamic radius 12: and concentration _c_. The term IAfAAGJAX/X + AP/XH is the correction to the cross—term discussed 60 above. In the calculations discussed in this thesis, this term was never larger than 0. 01 conductance units and could well have been dropped. A description of the MISTIC program for the computation of conductances is given in Appendix 11. Before presenting a comparison of conductance values calculated by equation (1.55) with some Observed values, it is of interest to briefly note the relative importance of some of the terms in the theoretical expression. For illustrative purposes, partial conductance functions have been calcu~ lated for 60% dioxane-40% water as the solvent ('D =: 27. 21) withl_\O 2 39. 75, _a_ = 5. 0 A) at 3%: 0. 07 and 0. 04. Table 10 gives a summary of these calculations and the various functions are sketched in Figure 5. A com- parison of the simple limiting law relaxation correction (Curve B), A°(1 - a. CT), with the extended relaxation and kinetic effect function A°(1 + AX/X + AP/X), (Curve E), shows that the simple function accounts for nearly 75% of the change in conductance due to these effects. For 1 2fl: 0. 07 it is found that - (AX/X + AP/X) = 0. 08685 The velocity field contribution to AXV/X is AXV/X = 0. 00524 or approximately 6% of the total. Under the conditions of this example, the velocity field term increases the calculated conductance by about 0. 028 units. The change from R_ = flto R represents a change in conductance of only +0. 004 units, so that no serious Operational difficulties are introduced by this modification. The difference between the Onsager and the extended electrophoretic contributions (Curves g and 2), while not extremely large, is certainly significant. Curves I_)_ and 1:: show that the extended electron phoretic term and relaxation-kinetic term contribute approximately equally to the total calculated change in conductance. Curve A shows the viscosity correction term for R r: 7. 5 X. This contribution increases rapidly above 61 TABLE 10 CONTRIBUTION OF VARIOUS PARTIAL CONDUCTANCE FUNCTIONS TO A CALCULATED PHOREOGRAM FOR A 1«-1 SALT IN 60% DIOXANE~40% WATER SOLVENT D=27.21 /\°=39.75 8.35.028 R:7.5 Function AA N} c 13A (A0 - A/V Description /\°/(1+ 2. 5 9)) .07 39.. 33 0.42 Viscosity cor-- rection (R=7.5X) .04 39.61 0.14 0(1 - (1 N/ c ) .07 37.26 2.49 Limitin relaxa— /\ g . 04 38. 33 1. 43 tion correction A0 — IN c /l+)£a .07 37.04 2.71 Onsager electro- . 04 38. 09 1. 66 phore tic cor— rection A0 - AAe . 07 36.43 3. 32 Extended electro- . 04 37. 73 2. 02 phoretic cor- rection A°(1 + 9% + 921—: . 07 36. 30 3. 45 Relaxation and . 04 37. 49 2. 26 kinetic terms 62 40 A 39 .. l. 38 ._ B A) /\°/1 + 2.51;) l B)N(1- a c?) 1 ~ C)/\°-Bcz_/l+)éa D) A0 — AAe E) A0 (1 + ax/x + AP/X) 37 _ L. 36 1 L i 1 J L 1 0 1 2 3 l4 5 6 7 10’- 62“ Figure 5. The contribution of several terms to the calculated conductance of a l--1 salt in 60% dioxane~~~40% water solution versus the square root. of concentration: a 2 5.. O 63 1 a radius value of Rt: 6. 0 18. At c T: O. 07 the term amounts to about 0. 4 conductance units and cannot be dismissed as a small correction. Conductance data for salts in dioxane-"water mixtures are available in the literature. While limited in quantity, these data are of high quality and, in general, are accurate to about 0. 01 conductance units. Martel and Kraus (39) have reported values of the equivalent conductance of tetraisoamylammoniuni nitrate (iwAm4NNOg) in water and in 10% to 50% dioxane-water mixtures. These data have been treated by Fuoss (12) using the expanded Onsager equation (with R : a/Z) and including a con--- sideration of ion pairing (equation 1.36). The data were fit with one con» stant parameter (a 2 5. 83 A) and three variable parameters: R values of 7. 3 X in water to 8. 4 X in 50% dioxane (D 2 35. 85); association constants from Lito 21.2.; and the best value of A0 for each solution. Conductance values for this salt have been calculated for water, 10%, 30% and 50% dioxane—water mixtures using the complete equation (155). All four sets of data can be reproduced using a variable A0 for each solvent mixture and two constant distance parameters: a '1 4.. 50 X and I: 2 7. 50 28. Table 11 gives observed and calculated values of /_\ for these cases. The "observed" values were obtained from large plots of the original data. A parenthesis around the "observed” value at £217: 0. 07 indicates that a rather long extrapolation was necessary. The average deviation for the twelve compared points of water, 30% and 50%, is d: 0. O3 conductance units. The deviation for 10% data is +0. 06 units, the calculated curve lying slightly above the experimental data. The Fuoss treatment experiences the same difficulty in that a larger association constant was required for the 10% dioxane data than for the data in water. Figure 6 shows the phoreograms for water and 1.0% dioxane. Similar plots for 30% and 50% dioxane are given by Figure 7. The broken. curve above the 30% data represents the calculated conductance values with R :.- 0 . This curve —_ lies above ,the limiting law slope while the data are below. The importance 64 TABLEll COMPARISON or CALCULATED AND EXPERIMENTAL VALUES OF THE CONDUCTANCE or SOLUTJIZONS OF i--Am4NNO3 1N DIOXANE-WATER MIXTURES AT 250C.:a = 4.50 X; R r. 7.50 X _.‘ Water 10% Dioxane D = 78.54 A0 = 89.30 D 70.33 A0 74. 55 " C Aobs. Acalc. "j C Aobs. Acalc. 0.01 88.45 88. 47 0.01 74.12 74.. 10 0.02 87.60 87.62 0.02 73.31 73.32 0.04 85.79 85.85 0.04 71.69 71.73 0.07 82.95 83.06 0.07 69.09 69.25 30% Dioxane 50% Dioxane D: 53.28 A0: 74.85 D2: 333.85 A0: 4255 4 C AObS. A calc. N/ C A obs. ACaIC. 0.01 53.34 53.33 0.01. 41.77 41.77 0.02 52.65 52.64 0.02 40.96 40.94 0.04 51.24. .51., 25 0.04 39.30 39.27 0.07 49.07 49.11 0.07 36.94 36.91 A (Water) 65 89 1.. O 88 - \ 0 Water 9.\ 4 75 C 87 — \ \ ®\. \ .1 74 o\ \ \ 86 l- l \ \ \ a1 73 \ A 85 r ' \ a n3 \ \ >4 0 , \ 2 \ 4 72 Q \ \ as o e \ \ H 84 I. \ \ 2 Experiment \ ' (Martel and Kraus) a 71 9 Calculated 83 1- —— —Limiting Law - 70 82 t; 1 . J 1 I J 1 0. l 2 1 4 5 6 7 102 c7 Figure 6. Phoreograms for i-Am4NNO3 solutions at 250C.: 0% and 10% Dioxane. A(3 0% Dioxane) 54 53 52 51 50 49 48 66 O ' \o\ 30% Dioxane O\® \\ .\ \ r— \ - 9. , \ . 50% Dioxane \\ _ \ Experimental 0 (Martel and Kraus) _ 9 Calculated 1 "' — “" Limiting Law .1 i l "I i J I 0 1 2 3 4 5 102 CT Figure 7. Phoreograms for imAm4NNO3 solutions at 250C.: and 5 0% Dioxane . 42 41 40 39 38 37 A (50% Dioxane) 67 of the viscosity term is evident. At first glance, the distance para- meters R = 7. 5 .8 and a = 4. 50 .8 seem rather contradictory. Molecular models (Stuart-Briegleb type) show, however, that the outer radius of the cation is about 7. 3 18. Further, the ion is not completely spherical but a shorter anion-cation contact distance of about 3. 6 28 is possible. If all anion-cation collisions are random, an average a of 4. 50 A is only slightly smaller than one would expect. The "planer" nature of the nitrate ion may cause some unusual effects. A more critical test of the conductance equation (155), using the complete electrophoretic term, is provided by the data of Mercier and Kraus (40) for tetraun-butylammonium bromide (BuéNBr) in water and 10% to 70% dioxane-water mixtures. The original analysis of these data by Fuoss (12) was performed in two parts. The data for 0% to 45% dioxane were fit with a constant _a_ of 5. 5 A; variable association constants from 0.6 to 3.16; and variable values of R from 6. 0 A to 7. 2 A. The data for the same salt in 50% to 70% dioxane were fit with 13 values from 4. 6 to 103; Rwas held constant at 6. 0 A and‘_a_ varied from 5. 22 A to 8.12 X. In a more recent analysis (41), using 3 = a in the velocity field term of 3 the relaxation effect and ignoring all terms of higher order than c Tin the expanded equation, the data from 0% to 55% dioxane were fit with a = R 2 4.8 X and I_{_ varies from 1. 3 to 6.9. In the 60%, 65% and 70% solutions, a = R values are 5.4, 4. 88 and 5. 00 respectively, while the K's are 12. 7, _2_4, and 82. By allowing a = I: to vary in the latter solvents, the values of I_(_ for 15% to 70% dioxane can be expressed as K = Koeb. The water and 10% data require larger 15 values than expected. In all cases A0 is treated as an adjustable parameter for each solvent. Conduc-t-ances for Bu4NBr have been calculated for water and seven dioxane-water mixtures from 10% to 70% dioxane using the com- plete electrophoretic effect. With a ”best" value for each solvent all eight sets of data can be reproduced with two constant distance parameters: 68 O a = 5. 00 A and R = 7. 00 .8. Table 12 shows "observed" and calculated 1 values of A. The observed values at round c "s were obtained “" 1 graphically as described above. Excluding the point at CT: 0. 07 for 60% dioxane, the average deviation between the observed and calculated values for thirtymone points in eight different solvents is only :1: 0. 023 conductance units. Models of the ambutylammonium ion give R = 6. 7 R and a shortest ion contact distance of about 3. 7 18. An average a value of 5. 0 28 seems quite reasonable. Figures 8 through 11 show the phoreow grams for these solutions. Figure 12, pertinent to the data in 60% 1 dioxane, is a plot of several conductance functions vs. c: ‘5 for A0 = 39. 75. The broken line is the limiting Onsager tangent: the upper curve (A) represents the calculated conductances using the Onsager electrophoretic term [ B C %-(1+ )fiafl witha 2’ 5. 00 X and Ii :: 0; “ the lower curve (B), with the same parameters, uses the extended electrophoretic term. The effect of the viscosity correction with R 1: 7. 0 X is inferred by the experimental points which are reproduced by the complete conductance expression including the Einstein term. While these applications are by no means comprehensive, they serve to illustrate that it is unnecessary to invoke ion pairing to explain all deviations from the Onsager theory in its present form. The advan- tages of using constant parameters directly related to the dimensions of the ions rather than variable ”association constants" is obvious. It is not implied that ion association never occurs, but that properly "pairing" should describe only non-«coulombic interactions which are not considered by the hard sphere model. Data for Bu4NI (39) in dioxane— water mixtures cannot be fit by the above treatment using reasonable and constant values of _a_ and Ii. Undoubtedly the highly polarizable iodide ion (42) gives rise to ion-induced dipole interactions which cause some "pairing. " It is expected that sodium 'bromate (39) solutions would 69 TABLE12 COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF THE CONDUCTANCE OF SOLUTIONS OF Bu4NBr 1N DIOXANE-WATER MIXTURES AT 25°C.:a. = 5.00 X, R = 7.00 A Water 10% Dioxane D: 78.54 A": 97.45 D: 70.33 /\°= 80.85 V C Aobs. Acalc. ‘1 C Aobs. Acalc. 0.01 96.64 96.61 0.01 80.09 80.08 0.02 95.75 95.75 0.02 79.29 79.31 0.04 93.97 94.00 0.04 77.68 77.74 0.07 91.3 91.31 0.07 75.4 75.35 30% Dioxane 501% Dioxane [)2 53 28 /\°: 57.25 I): 35.85 /\0: 43.70 ‘1 C Aobs. Acalc. “ C Aobs. Acalc. 0.01 56.57 56.57 0.01 42.90 42.92 0.02 55.88 55.88 0.02 42.10 42.10 0.04 54.51 54.50 0.04 40.48 40.50 0.07 52.50 52.49 0.07 38.30 38.31 5 5 % Dioxane 60% Dioxane I): 31.53 fif:=41.56 [)2 27 21 /\o: 39.75 “ C Aobs. Acalc. " C Aobs. Acalc. 0.01 40.71 40.69 0.01 38.75 38.75 0.02 39.77 39.78 0.02 37 65 37.66 0.04 37.98 37 98 0.04 35.54 35 54 0.07 35.60 35.59 0.07 32.65 32.82 6 5% Dioxane 70% Dioxane I): 23.14 [0): 38.30 I): 19 07 /V’“ 37.05 “I C Aobs. Acalc. “ C Aobs. Acalc. 0.01 37.05 37.07 0.01 35.43 35.43 0.02 35.69 35.64 0.02 33.48 33.55 0.04 33.03 33.08 0.04 29.97 29.97 0.07 29.70 29.74 0.07 25.90 25.94 A(Water) 97 96 95 94 93- 92 91 70 0 ~. '\ \10% Dioxane \\ Experimental 0 (Mercier and Kraus) - 9 Calculated ... _. — Limiting Law \ e \ i- 1 . L 0 1 Z 3 1 5 6 7 102 c 7‘— Figure 8. Phoreograms for Bu4NBr solutions at 25°C.: 0% and 10% dioxane. ' 81 80 79 78 77 76 A(10% Dioxane) /\ (30% Dioxane) 57 56 55 54 53 52 71 F’ G O\ ‘ \ \ - \ Experimental \ ' (Mercier and Kraus) . \ O Calculated \ —- -- - Limiting Law \ \ _ O \ 1 L L l 1 I l 0 1 2 3 4 1 5 6 7 10?- c?" Figure 9. Phoreograms for Bu4NBr solutions at 25°C.:30‘7o and 50% dioxane . 44 43 42 41 40 39 38 A (5 0% Dioxane) /\ (55% Dioxane) 42 41 40 39 38 37 36 35 72 6)\ C L- 0\..O\\ 55% Dioxane 1.. .\®..\ \\ \\ 1.. 60% Dioxane \\::\:\\\ ®\\ \ 4 5 Experimental (Mercier and Kraus) O Calculated —" '“ "'" Limiting Law "' r. fl 1 1 l g 1 l 1 0 Z 3 5 6 7 102C Figure 10. Phoreograms for Bu4NBr solutions at 25 O.C 155% and 60% d1 oxane 40 39 38 37 3.6 35 34 33 A (60% Dioxane) 39 37 35 33 31 29 27 25 73 O .7 \O \ 65% Dioxane b _. G)\\ 0Q \ " o \ . \ '1 - \. \\\ . O . \ _ \ \ O \ _ \ \ 1— 70% Dioxane \ \ . \ \ 7 l. \ \ q O L \0 \ _ \ O\ t \ ‘ Experimental ' ‘- ° (Mercier and Kraus) \ 7 0 Calculated - — — ‘— Limiting Law T _ O .(l \ 1 1 1 1 1 1 J 0 1 2 3 4 1 5 6 7 107‘ c7 Figure 11. Phoreograms for Bu4NBr solutions at 25°C.:65% and 70% dioxane . 40 35 34 33 74 2 A)/\'= (/\O fem) (1+ 43(— + AP) 1_ 1 + x 3? “ AX AP B)/\= (/\° ~A/\e) ( 1+ —-- + —-—-) X X L I J I 1 I 1 l 0 1 2 3 4 1 5 6 7 102 CT Figure 12. Various theoretical phoreograms for a 1~1 salt in 60% dioxane-40% water solution at 250C.: A0 z 39. 75, a r: 5. 0 A. also require pairing considerations since the bromate ion possesses a permanent dipole which should give rise to ionradipole interactions. The only other accurate data available over a sufficiently large range of dielectric constant in dioxanenwater mixtures are for tetramethylammoniurn picrate (40). An analysis by Fuoss (12) indicates that nO ion pairing takes place in 0% to 70% dioxane. The necessary a values ranged from 8. 0 A to about 7. 0X, systematically decreasing with decreasing solvent di-- electric constant. The average 3 value was about 4. 5 18 for each ion. Although the calculations using the extended electrophoretic effect were not performed, it is certain that these data can be treated successfully. For the large value Of_a_1_ : 8. 0 X the difference between the extended and Onsager electrophoretic terms is negligible in water, and although it remains relatively small, increases as the dielectric constant decreases. This behavior is exactly the form that would be required tO fit all the data for the picrate with a constant value Of 3. Many more data must be gathered and examined tO assure the generality Of the above results. Studies of simple Spherical ions should help one decide at what radius ions can be considered "sufficiently bulky" to require the use Of the Einstein viscosity correction. Tests Of the correction by viscosity measurements would be of some help. The relation- ship between the minimum distance Of approacha and the hydrodynamic A radius R should also be clarified. Until the generalization Of the Onsager relaxation expression to unsymmetrical salts can be made, work with Symmetrical 2-2 and 3—3 salts should be of great help along these lines. Finally, the above Calculations are not considered to be exact, since higher order terms in the relaxation field Of unknown magnitude resulting from the complete exponential distribution function, remain unevaluated and also because Of errors introduced by the linear superposition approxi— mation. In View Of the complexity of the Onsager treatment outlined above, it would seem that a fresh approach is needed. 76 At the very least, this study Shows that the terms in the electron phoretic effect which are usually neglected are not negligible, and, indeed, their inclusion can in some cases eliminate the need for introducing an association constant. IV. Summary An outline Of the Onsager theory Of conductance has been presented which, it is hoped, will be Of value as an introduction to the study of this difficult treatment. Higher order concentration dependent terms of the electrophoretic effect, which are neglected in the usual treatment, have been evaluated for 1-1 salts in water and in 10% to 70% dioxanenwater mixtures. Significant differences between the Onsager and extended electrophoretic expressions occur which increase rapidly with decreasing ion size and with decreasing dielectric constant. The complete term is always larger than the simpler function and in extreme cases is more than twice the value Of the Onsager correction. The electrophoretic calculations were applied to equivalent cons ductance data for tetraisoamylammcnium nitrate and tetrawnu-butylammoniurn bromide taken from the literature. In both cases significant deviations from the previous theoretical expressions have been attributed to ion pair form» ation. The present work shows that, if higher electrophoretic terms are included, the i~Am4NNO3 data for the four solvents (0% tO 50% dioxane in water) can be interpreted without the concept of ion pairing using two constant distance parameters: the minimum distance Of approacha r: 4. 50 X, and the cation. hydrodynamic radius R r: 7. 5 18.1n a similar manner the data for Bu4NBr in eight solvents (0% to 70% dioxane in water) require no association constant and involve the constant parameters, a 2 5. 00 A andR: 7.0K. 77 It is concluded that much Of the deviation from the Onsager theory, heretofore ascribed to electrostatic aggregation of ions, arises from incomplete treatment of the model used rather than from physical phenomena which cause the model to be inaccurate. PART II TRANSFERENCE NUMBERS AND ACTIVITY COEFFICIENTS OF TRIS—(ETHYLENEDIAMINE) COBALT (III) CHLORIDE IN WATER AT 250C. I. Introduction The interionic attraction theory of electrolytes, as formulated by Debye, Hiickel, and Onsager and Fuoss has met with great success in aqueous solutions Of 1-1 electrolytes. In solution Of higher charge types, however, theoretical predictions Often deviate from experiment. TO permit an adequate evaluation Of the cause Of these discrepancies, an extensive study of the properties Of multicharged electrolytes has been undertaken in this laboratory (43, 44, 4.5, 46, 47). As a part Of this program, the transference numbers and activity coefficients of solutions of iflsvhethylenediamine) cobalt (III) chloride have been measured. II. Transferenc e Number 8 A. Introduction The transference number Of an ion is defined as (158) where i_j_is the current carried by ions of type :1. and T11 is the total current carried by all types Of ions in the solution. Alternatively the transference numbers may be expressed as T- :3 .1321— : .51.... 3 32.1.. .1 Eu, ' Ex, /\ (159) 1 i ' 79 so that a correct theory of conductance should also y.-e1d the correct trans— ! ference number. From equations (30) and (305.) for (‘j and A it. is seen that the relaxation terms cancel and T" is given. by (160) A“ ~ AA. transference measurements, then, help to provide a critical test of the electrophoretic part of conductance theory but suffer the disadvantage that they must be measured at concentrations which are higher than would be desirable for a test Of the theory. Experimentally, transference numbers may he Obtained in three ways: (1) The Hittorf method, (48, 49, 50, 51, 52), which depends upon concentration changes in a cell during electrolysis. (2) The electrornotive force method, (53, 54. 5.5, 56), which requires measurement Of the potentials of cells with and without transference. (3) The moving boundary method. The Hittorf method is tedious, inaccurate, and today, is rarely used. The electromotive force (E. M. F.) method is less accurate and not generally applicable since it requires electrodes which. are reversible to both anion and cation, or else independent activity measurement. The moving boundary method, while limited in concentration range, is capable of a high degree Of accuracy and is now in general use. This method was used in the present work and is described below. B. The Moving Boundary Method Since several excellent; reviews of the history and theory of the moving boundary method (57, 58, 59) are available, only the most important features of the theory will be presented. 80 The method consists in forming a boundary between two solutions which may or may not have an ion in common. For the purpose of the present work it is sufficient to consider the case of solutions of two electrolytes C+A- and B+A — with the common anion .5; . Let the two solutions be placed in an electrolysis cell.r,: ."A line (J: 4.1.x). giyes the initiallpoSition .o'fithe boundary. between. the solutions. If a steady current i_ is passed through the cell for a given time, and the mobility of ions 9: is greater than that of ions at, the boundary between the solutions will move to. the fihal p:o;Sition3asirepmes‘ented by (Eli: Thoughvallnegativeions move toward the anode, and all positive ions toward the cathode, the two solutions remain separated since the 9: ions move faster. If the E: ions lagged far behind, the solution would Ibecome more dilute and the increased resistance and (at constant current) + . . . increased potential gradient would cause the B ion veloc1ty to increase. In this way the boundary is "self- sharpening. " The C+A- solution is designated the "leading solution, " and B+A-the "indicator" or "following solution. " If the boundary moves a distance d cm. in 1; seconds, the + average velocity of positive C ions, v,+, is d/t. Since v+ = Xu+ Where 11+ is the mobility and X is the potential gradient in volts/cm. , u+ = d/Xt (161) Further, X = i/AL (162) where i is the current in amperes, A the cross-sectional area of the cell and I__._ the specific resistance of the solution. Since A: 1000 L/c"‘ = F (1.1+ + u_) (163) where F is the Faraday, then L = c" F(u+ + u_)/1000 (164) 81 Combining the s e statement 3 dAc* F(u+ + u_) u+ = d/Xt = dAL/it 2 1000 it (165) The quantity (dfi) is just the volume Y_, swept out by the boundary, and since i+ 11+ T =. .- —— + 1++1_= u++u_ we obtain for the transference number of leading solution cation c*F V T+ : 1000 it (166) In the first theoretical treatment of moving boundaries, Kohlrausch (60) deduced that, in order to obtain a stable boundary, the condition c T+ "‘ (T+)f (167) must be met. Here, ca‘f is the normality of the following solution and (T+)f its cation transference number. This relationship is known as the Kohlrausch ratio. According to the Kohlrausch treatment, the concen~ tration of the indicator solution will automatically adjust to that given by the Kohlrausch ratio under the influence of an electric field. In an ex- tensive study by MacInnes and Smith (61), it was found that the following solution concentration must be within three to eight per cent of the Kohl- rausch ratio in order that the concentration adjustment can take place properly. The necessary properties of an indicator solution may be summarized as follows: (1) The solution must not react with the ion under investigation. (2) The transference number of the indicator ion must be less than that of leading ion. (3) The density of the following solution must be less than that of the leading solution for falling boundaries and greater for rising boundaries . 82 (4) There must be sufficient difference in some prOperty of the two solutions, such as color or refractive index, to permit the observation of the boundary motion. In the above discussion of the moving boundary method, no mention was made of the electrode processes which necessarily occur. Equation (166) gives the transference number with respect to a fixed mark on a transference cell. Any change in volume caused by an electrode reaction will necessitate a correction to volume swept out by the boundary since the bulk of the solution moves to accommodate the volume change. The necessity of this correction was recognized by Miller (62) and first calcu- lated by Lewis (63). The computation is greatly simplified (64, 65) if one side of the cell is left open to the atmOSphere and the other side is closed since only the volume changes which occur between the closed side and boundary need then be considered. As an example of the computation of the volume correction, consider a cell employing a descending or "falling" boundary between lithium chloride and tris-(ethylenediamine) cobalt (III) chloride (abreviated Co(en)3Cl3). Let the side of the cell containing silvernsilver chloride cathode be closed, and side with a cadmium anode be open to the atmOSphere. The volume changes which take place between the boundary and closed cathode during the passage of one Faraday of electricity are: (1) Loss of one mole of AgCl(s) AV : '"VAgCl (2) Gain of one mole of Ag AV 2 + VAg (3) Gain of one mole of (_L_‘_l:ions AV .2 + VCl“ (4) Gain of TI /3 moles of Co(en)3+++ ions AV : + T+(VCogen)3+++ (5) Loss of T_ moles of El: ions AV : iT_‘\'/:Cl.. Summing the volume changes (1) through (5) the total volume change between the closed side and the boundary is + T+ (VCO(en)3+++ ) 83 If this volume change A}! turns out to be positive (as indeed it does), this means that, effectively, the boundary has swept out a volume (V + AV) rather than the smaller observed volume \_/'_ so that the corrected transference number is larger than the "observed" value. For the passage of one Faraday equation (166) becomes CV _(vv+AV)c"_ T,+C*AV + 1000 1000 — + 1000 (169) A similar analysis,, taki'ngthenadrnium anode as the closed side, shows that for this case AV: VCdCIZ - VCd — T+ ( VCO(en)}Cl3) 2 2 3 (170) An additional correction has been pointed out by Longsworth (64). Realizing that a small fraction of the total current passed through a cell is carried by conducting impurities in the solvent, he derived the expression AT+ = T+ (L ) (171) s olvent/ Ls olution where AT+ is the correction to the transference number 2+; Lsolvent is the specific conductance of the solvent; and Lsolution is the specific conductance of the solution. The final expression for the transference number becomes >'< >',< _ £__s_'___-V_ 2.31 Lsolvent T+‘ 1000 it ‘ 1000 + T+( ) ”72) solution C . Experimental l. Liaterials Tris-(ethylenediamine) cobalt (III) chloride was prepared according to the method of Work (66). The crude product was recrystallized three 84 times from ethanol and dried in a vacuum oven at 550C. Chloride analysis of the semi-pure product gave 30. 59 :1: 0. 01% compared to the theoretical value of 30. 78%. A four step fractional recrystallization of the salt by dissolving in a minimum volume of water and adding an equal volume of ethanol, was then undertaken. - Solutions of constant molali‘ty were prepared as the recrystallization progressed and their conductivity measured. Constant conductances were found for the third and fourth fractions. Chloride analysis of the pure salt gave 30. 79 d: O. 02% chloride. Solutions were made by weight dilutions of a stock solution. Potassium Chloride, used for secondary calibration of the trans- ference cell was prepared by recrystallizing Baker C. P. salt twice recrystallized from conductivity water followed by fusion in platinum ware under an atmosphere of nitrogen. VA Lithium Chloride stock solution was prepared according to the method of Scatchard and Prentice (67). The necessary following solutions were prepared by volume dilution of this stock solution. Conductivity water, used in the preparation of all solutions, was obtained by distillation of demineralized water from alkaline permanganate and subsequent redistillation. The specific conductance of ‘the water was never greater than 2 x 10'6 ohrn'l cm'l. Z . Apparatus The transference numbers reported in this thesis were obtained using the sheared boundary technique (59). The equipment used is a modification of that of Spedding, Porter and Wright (68, 69) and is described in the literature (43). A diagram of the transference cell is shown in Figure 13. The cell was constructed of Pyrex. The measuring tube was constructed from a two millimeter pipette (Corning "redline"). Fine semi-circular l yea/53a, : a 553 c 34/, 2 a f a , , Moving Boundary Transference Cell About '1/2" = 1" Figure 13 . 86 grooves were cut in the tube at constant intervals using a diamond stylus with the tube mounted in a milling machine. Gaps were left in the mark— ings to facilitate timing of the boundary. The tube was twice calibrated with mercury as recommended by Longsworth (64).. This tube was connected to a hollow bore stopcock at which the boundary was formed. The electrodes are connected at the bottom of the cell by a second hollow bore stopcock to permit use with rising boundaries. The anode and cathode compartments were equipped with female ground glass joints to accommodate the male joints into which the electrodes were sealed. Side tubes with stopcocks which could be closed were attached to the electrode compartments for ease in filling the cell. Removable electrode cups were used to prevent the products of the electrode reactions from reaching the measuring tube. The anode consisted of a copper wire sealed into a ground glass joint. The copper wire was immersed in a small test tube of cadmium metal which was melted under a stream of nitrogen. Upon cooling, the test tube was broken away leaving a smooth cadmium plug electrode. The silver-silver chloride cathode was made by sealing a platinum wire into a ground glass joint. Corrugated silver sheet was then fused to the wire and cylindrically wrapped to a diameter of about one-half inch. The electrode was ”plated" with silver chloride by electrolyzing in a one normal solution of hydrochloric acid. The position of the boundary was detected by means of a narrow slit of light placed behind the transference tube with a telescope focused on the tube from the front. The light source was a vertically mounted fluorescent lamp, covered vertically by a movable, slotted cloth blind. The blind was raised and lowered by attachment to the drive shaft 6f a 110 volt reversible D. C. motor. Motor power was provided from the A. C. line voltage, converted by selenium rectifiers. 87 The motion of the boundary was timed with two stopwatches mounted in a stand with a hinged lid. The lid extended over both watches so that one watch could be started and the other stOpped by pressing down on the lid. The watches were checked with the standard w signal and were accurate to three seconds over a twenty—four hour period. A large aquarium-type water both, into which the cell was placed during experiments, was maintained at 25. 00 :i: O. 050C. Constant currents were obtained with an electronic controller and balancing motor. The current was determined from the potential drop across a standard resistor in series with the cell. Compensation for minor fluctuations not eliminated by the electronic apparatus was made by feeding the unbalance from a Leeds and Northrup type K-l potentiometer to a Brown "electronik" 356358-1 amplifier which was used to drive a Brown 76750-3 balancing motor. A diagram of the current controlling apparatus is given in Figure 14. The entire apparatus was checked at intervals by measuring the transference number of potassium chloride followed by lithium chloride. These results agreed with published values to within 0. 05%. 3 . Procedure The transference cell was "quick-rinsed” with warm alkaline cleaning solution followed by acid-chromate cleaner. The cell was thoroughly rinsed and filled with distilled water and allowed to stand for twenty-four hours to insure complete removal of acid from the glass. The dried hollow-bore stopcocks were uniformly coated with a silicone grease and carefully seated. Since only falling boundaries were used, the stopcock at the bottom of the cell was always Open. With the upper stopcock Open, the cathode compartment was rinsed at least five times with the solution to be measured. The cell was then filled with the solution; the electrode cup, and the silver-silver chloride cathode were 817' 0230mm 02:30.“ a It. mhzumummumlfi ) SHBWSOJSNVHJ. DN‘SVHd ‘ SSS-33:1 BdAJ. OLD ONLL Lawns! 0 J... .4; ._1 m9 .. ”L “6 o m mSuenmmm< peonpsv unmpmsoo J3 eufimfim we we are m 8 I . m I mm>oo . 435.2 8 .1 .1 .I .1 .z .2 2m th SozsoauESWmEBmexomh I Z .V 9 Z c6 2 H + + o _ . _ . . . _ 21.9 . a. .I .I .I 7. t .L m A] . C0 C4 8 z 17 0 (T 1% I 3 {m6 . 9 9 .V E 2 WWW Lu . a 11 O W t... A MOO w. n n fl Mus. «a 7. mm W cm! N._ x a x. w. x M Seem .1 3. bunnufiufimo + - -- - - ( 325.th n >o+mm\l 39.10. Fonjwl 50m JI 3m x08 . i , ,. a _ (1a.. eczema. .: .32.. a F aozéya Emu 2.3 m... w m . 116. 9 _ J...) u .tl |I_I II “W N __ _ AUAXU_ MWIIAW ._ tun.” M r. _: moz0 3 - m 26-3. L Tmmmnwmm. mmEZQZ< 33 2,5 amemzhofizfioo .2rz>o>am.m.mmp, z: oo. . . 040.. . ‘ 025:... m mm mDm (.LNVIHOd w I P 33 o 89 inserted, and the side arm stopcock and upper hollowr-bore stopcock. were closed. The anode compartment was rinsed with water and, at least three times, with the lithium chloride following solution. The following solution was made up to the concentration given by the Kohlrausch ratio with an estimated value of the transference number of the complex cobalt cation. The anode compartment was then filled with the lithium chloride solution; the electrode cup and cadmium anode were inserted; and the cathode side arm was opened. The cell was placed in the water bath and checked for electrical leaks to the bath with an ohrnmeter. The cell was then aligned vertically and the light and telescope arranged so as to form a straight line with the cell. After waiting thirty minutes for temperature equilibrium, one side arm was closed to the atmosphere, the other left open. The Leeds and Northrup potentiometer was then balanced against a standard cell, the leads to the cell were connected, the Hollow-bore stopcock was opened, and the current turned on. The current was adjusted to such a value that the boundary required from 200 to 250 seconds to traverse the distance between each pair of tube makings (AV 1: 0. 1 ml. ). The time required for the boundary to pass each mark was measured with the st0p~ watches. The Co(en)§Cl3 solutions were made up by weight dilution of a stock solution. In order to calculate the normality of the solutions, the densities of the solutions were measured with a 50 ml. pycnometer. The densities were also used to calculate values of the partial molar volume of the salt, which were necessary for the evaluation of the volume correction to the observed transference number . 4. Results The cation transference numbers of tris~(ethylenediamine) cobalt (III) chloride solutions were measured using lithium chloride following 90 solutions. The concentration of the following solution was determined by an estimate of the transference number of the leading ion and transference numbers of the lithium ion given by Longsworth (64). The following solution concentration was found to be within the prescribed limits by changing the concentration by several per cent and repeating the determination. Volume corrections were calculated according to equations (168) and (170) using the following values: VCd: 13. 0 ml. (Reference 64) VClClz 2: [23. 24 + 8. 8 (rnolalityfi- ] ml. (Reference 70) VAg 2' 10. 3 m1. (Refer enee 64) VAgCl = 25. 8 ml. (Reference 64) The densities of the Co(en)3C13 solutions were determined using a 50 ml. calibrated pycnometer. The results are described by the expression P = 0. 99707 + 0.1555 (rnolality) The average deviation of five points from. this straight line was :t: 0. 00001 \ g. / cc. Partial molar volumes were calculated from the expression. ——~ _ 1000 , M2 V”¢v '“ WHO” F0 (173) c100 where V 2 partial molar volume of Co(en)3Cl3 (pv : apparent molar volume c = molarity of the solution P0 = 0. 99707 f): density of the solution M2 = molecular weight of Co(en)3Cl3 91 The average value 189 :t 1 cc./"mole was used for the partial molar volume of Co(en)3Cl? in calculating the volume corrections. Solvent corrections were calculated from measured conductances which‘supple— ment those in the literature (71). A Leeds and Northrup "Type A" cell, platinized according to the recommendations of Jones and Bollinger (72), was used. The A. C. bridge employed was designed by Thompson and Rogers (73). The resistances, measured at 400 to 2000 cps. , were independent of frequency and the oil bath employed was maintained at 25. 00 :I: O. 029 C. The results are summarized in Table 13.. Figure 15 is plot of transference number versus cikir, extrapolated to 0.4939, the value calculated from the conductance data of Jenkins and Monk (71). Curves calculated using the OnsageruF’uoss and the extended electrophoretic terms are also shown. Further discussion of the deviation from theory is reserved until after the presentation of activity coefficient data. In View of the disagreement between theory and experiment, it is difficult to estimate the accuracy of these transference data. The four factors which influence the accuracy of the data are: (1) the timing of the boundary; (2) current; (3) volume of the tube; (4) concentration of the solution. The average precision of observed transference numbers and the maximum current fluctuation within each run was :1: 0. 04%. In View of the agreement of transference number arising from the various com- ponents of the apparatus is estimated to be d: 0. 05%. Since the salt was carefully purified and the solution conductances form a smooth extension to literature data, the total maximum error in the reported transference numbers is estimated to be i- 0.1%. III. Activity Coefficients A . Definitions The concept of activity was introduced in 1907' by G. N. .Lewis (74, 75) as a means of precisely treating the thermodynamic behavior of 92 .usoauhmgaoo muons pomoHU mm 5a :35 -- :85 + 835 Show 5 $5585 :35 8555 + 8555 + 22.5 355 $.35 8555 + @855 + 22.5 $215 25255 32.5 5555 + .8555 + $35 7:: 3.3.5 .8555 + .3855 + £35 82 5 @2355 m 4.: 32.5 885 + 585 + 3.2.5 3: 5 9.5355 552 $35 $555 + 8555 + $$5 32.55 3.5855 < J. fzomfieq :63 +e< 733$. «3225on 336882 .063. .3. 952.548 mamoqmo E: eq50m Q73. mmmmzbz mozmmmmwzfiae 20540 2 Mdmxa. 93 0.494 ., ._ 0.492 — — 0.490 _ _ \, 0.488 L \ T+ \ Q 0.486 P 0. 484 a. \ _ \ \G \ Onsager . 8 O. 482 - Equation 0 Extended Equation - a : 6. 0 A a 2: 6. 0 X \ , Circle Radius 2 O, 05% 0.480 .. a 1 I L I 1 f 0 0. 10 0. 20 0. 30 c"< Figure 15. Cation transference number of t_1;i_§.-(ethylenediamine) cobalt (III) chloride versus the square root of normality in aqueous solution at 25°C. 94 non-ideal solutions. The activity a); of component i in a solution may be defined through the relation Hi = H510 + RT ln a, (174) where iii is the ”chemical potential” of the i—component, and p.10 the chemical potential in the standard state. Clearly, the value of the activity depends on the standard state chosen. The chemical potential is a measure of the escaping tendency of the species and a. = ( 3333- = < 9 F ) ani s,v,nk7_£i ani T’pnkfi (175) where E = The internal energy of the system F = The Gibbs free energy S = The entrOpy P, V, T, = The pressure, volume and absolute temperature of the system ni = The number of moles of component _i_ in a system of 15. components An ideal solution is defined as one for which the activity is equal to the mole fraction at all concentrations. Accordingly, the activity coefficient _f_i is defined as f. 2 EL 1 X1 (176) where Xi is the mole fraction of component i. The chemical potential becomes ui=uio+RT1nXi+RTlnfi : Ii (ideal) + RT ln fi (177) It is found that in very dilute solutions the activity approaches the concentration. It is, therefore, customary to chose the standard state 95 in such a way that £1: 32%:— —> 1 ain——->-0. The thermodynamic prOperties of solutions of electrolytes are determined by the properties of the ions and of the solvent. Since the principle of electrical neutrality forbids the forming of solutions of single types of ions, the thermodynamic prOperties of a single type of ion can never be measured. Theoretically, however, it is advantageous to define hypothetical individual ionic quantities which are related to measureable properties. Consider an electrolyte which dissociates according to the scheme: 2+ Z- Individual ionic activities, 3+ and 3-, are defined by the relation _ v+ v_ = v a = (a+ a- ) .. a :1: (178) where v = v+ + v_ a is the activity of the salt; and ai is called the mean ionic activity. On this basis, the rational activity coefficient, fi, is defined as f E (ai)x * Xi (179) where V+ v_ 1 X:,:‘;"-;(X+ X ) /V The standard state is chosen so that fi——>O ain—éo Since mole fraction is not always a convenient concentration unit for 96 ionic solutions, activities and activity coefficients are defined for molal concentrations: (a 1 V V... l/y v- vw 1/" vi = is? .; mi = (m. + m.-. ) ml as C1 ---> 0 Since the chemical potential. must have the same value, regardless of the standard state c1.-.csen, +~ vRT in (7i nrii )1 I : “x0 Jr vRT in (ii Xi) .2- (1.0m 2 (LCD + vRT 1n (yi Ci) (182) At infinite dilution fi- = Vi = Vi "-’ 1 With introduction of the limiting values Xi/ mi =4 ML / 1000 and Xi/ Ci -‘-' Ml/1000 p0 , one obtains ”x0 = (11.1.10 + v RT 1n 13100- : (18 + vRT 1n {993191482 (183) where}? 0 and M1 are the density and molecular weight of the solvent respectively. By combining equations (184) and (185), the various 97 activity coefficients may be related: In ii = 1n 7i + 1n (1 + m le/lOOO) 1n f5: :2 ln yd: +ln (p/Po + C(leuMz)/1000po) ln 7i :- 1n yi+ln ()O/(ooa cMZ _/1000P0) (184) where f: is the density of the solution and M2 is the molecular weight of the solute. The theoretical expression of Debye and Hiickel (3) for ionic activity coefficients is obtained by assuming that all deviations in the chemical potential of electrolyte solutions arise from the charges of the ions. By considering the difference in the energy necessary to charge an isolated ion, and the energy necessary to charge an ion in a potential field \l/jo (Equation 22) the result is ,2 ln f- = 89 X’ J ZDkT(1+ )La) (185) whereij is the activity coefficient of ions of type j. For an electrolyte which dissociates into two kinds of ions, combination of the individual ionic activity coefficients given by equation (185) leads to -S£ N] c f =2 _, i 1+ :1). B N] c (186) log where Sf ‘-‘-‘ 0.5091 to’ 313: a/r\( c 108 “0.32868") and 1 = <1— 2 1212,21? 1 _ l 3 00' = (z V.z.z) 77 1A} 2 i 1 1 o The symbola denotes 108a, or a in Igngstrom units. 98 The various methods of experimentally determining activity coefficients are adequately reviewed in the literature (75, 76, 77, 78). Accordingly, only the use of E.M. F. cells with transference will be discussed below. B. Activity Coefficients from the E.M. F. of Cells with Transference The early theoretical work of Helmholtz (79) and Nernst (80) concern- ing the nature of the E. M. F. of concentration, received partial confirm- ation from the experiments of Moser (81, 82.), Miesler (83, 83) and Jahn (85). With the introduction of the concept of activity by G. N. Lewis (74), the calculations were brought to their modern form. The first adequate treatment of the junction potentials involved in this type of cell was made by Brown and MacInnes (86), using transference numbers obtained from moving boundary measurements. Thus, the accurate evaluation of activity coefficients was made possible. The calculations involved in the determination of activity coefficients from E.M. F. measurements using cells with transference may be illus- trated by considering the following general cell: Z+ 2. Z— ; z J‘ X,_ (c1) : A X,. (C?) I MX-M M—MX I Av+ 12+ (187) where M-MX represents an electrode which is reversible to X- ions, and the molarity c_1 is greater than c_2_z. Consider the changes which occur in left hand side of cell (anode) when one Faraday of electricity passes through the cell: 1;: equivalents of AZ+ will be lost by migration across the junction; '1:_.._ equivalents of XZ' will be gained by migration; and one equivalent of )2. will be removed by the electrode reaction. The sum of all the changes is a loss of T: equivalents of A3: XVZ' . A similar analysis of the right hand side gives a net gain of T‘, 2+ Z- . . Xv , so that the total "reactlon" is a transfer of equ1valents of Av+ _ 99 Id: equivalents of salt from the more concentrated to the dilute solution. The free energy change for this process, if_c_;_ and_<_:_z_ differ infinitesimally is T+ dF = nRT d (ln a) 2' RT d (In a) V+Z+ (188) where n is the number of moles of salt transferred and da is the change in the activity of the electrolyte. Since (1E :2 -anE where E is the potential of the cell d8 RT “ —— 2‘ - [ ] d (In a) T+ V+Z+F (189) For a finite difference in concentration Equation (189) becomes E (C : C2) RT a d8 :- .. —— ln —-2— I 7+ [ v+z+F ] a. (190) 8(C=C1) The integral on the left cannot be evaluated analytically since the trans- ference number is a function of concentration. In order to evaluate the integral, the function 5:_L____1_ T+ T+ (ref) (191) is defined. If gLis taken as the concentration of the reference solution, Cref’ for which the cation transference number 153+(refr then E. 10g yd: : 10g Cref _ V+ZfFE _ 1):le J- (Sd€ C 2. 303 VRTT-l-(ref) Z. 303 VRT Y:I:(ref) 0 (192) since E at c 2' Cref is zero. Converting a to ai by equation (178), equation (192) may be written as E Cref _ viZ+FE _) viziF ISdE 2. 303 vRTT+(ref) 2. 303 vRT log 3,55 = log Yd:(ref) (193) 100 For Co(en)3Cl3; v+ :- 1, v = 4, 2+ = 3. Equation (193) shows that, with the aid of accurate transference data, ratio of the activity coefficients Yi/Yi(ref) may be calculated from the E.M. F. of cells with transference. The ratio fi5/fi(ref) may be calculated from equation (184) . Individual activity coefficients fa: may be obtained using the Debye-Hiickel expression (186). By subtracting log fi(ref) from both sides of equation (186) and multiplying both sides by (1 + g. B N/ c ), one obtains fi lo + S N] c = -— lo f [ g TQref) f ] g :1:(ref) o __ £5: -a B [ (x/ C (log fi(ref) + log )] f:i:(ref) (194) If the left side is denoted by I and the bracketed term on the right by 2:, the intercept of a plot of X vs. 3f gives - log fi(ref) providing the solu- tions obey the Debye-Hiickel equation. In order to obtain this value, a successive approximation method must be used since _)£ contains l_o_g fi(ref)' The value ofg, the minimum distance of approach may be obtained from the lepe of the plot. The requirements for the successful determination of activity of coefficients by this method may be summarized as follows: (1) No changes take place in the cell without passage of current. (2) Every change which takes place during the passage of current may be reversed by reversing the direction of the current. (3) The measured potential must depend only on the concentrations of the solutions in contact with the electrodes and not upon the distribution of concentration gradients at junction of the solutions . 101 Extensive studies of the third criterion have been made (53, 54, 87, 88, 89). It has been found that reproducible potentials are obtained provided the area of junction is at least 12 mm. 2 and that the concentration of the solution in contact with the electrodes remains unchanged. C . Experimental 1. Materials Solutions of Co(en)3Cl3 were prepared in the manner described in the section of this thesis on transference numbers. Traces of bromide in potassium chloride were removed by the method of Pinching and Bates (90). The salt was then recrystallized three times from hot conductivity water and fused in platinum ware under a nitrogen atmosphere. Hydrous silver oxide was prepared by ' the addition C. P. silver nitrate to a boiling solution of potassium hydroxide. The resulting precipitate was washed fifty times with hot conductivity water to remove any potassium carbonate present. Baker reagent grade hydrochloric acid was used without further purification. Z . Apparatus The cell and electrodes used in this work were essentially the same as those used by Spedding, Porter and Wright (68). The cell (shown in Figure 16) consisted of two compartments joined by a high-vacuum hollow-bore stopcock (H. S. Martin Co. ). Each compartment was equipped with two female ground glass joints to hold the electrodes. A trap was placed between one of the compartments and the stopcock to prevent diffusion. The silver-silver chloride electrodes were prepared by the thermal-electrolytic method of Smith and Taylor (91). About three inches of number 26 C. P. platinum wire was sealed into a standard 12/30 male taper. The wire was then coiled by tightly winding it on a two millimeter 102 (I: 1H .— ‘ About 3/4" Concentration Cell with Transference Figure 16. 103 glass rod, and cleaned by heating to redness, plunging into concentrated nitric acid, and rinsing with conductivity water. This cleaning process was repeated several times. The electrodes were then coated with a paste of silver oxide which was ignited to silver at 4000 C. Three or four coats of the oxide, and subsequent ignitions, produced a complete covering of silver. After fill 5 ing with mercury, the electrodes were "plated" with silver chloride by electrolyzing in a one normal solution of hydrochloric acid for forty-five minutes at a current of eight milliamperes per electrode. A convenient source of current was provided by a six volt battery eliminator and a 3, 000 ohm rheostat. A Leeds and Northrup type K-Z potentiometer was used to measure the potentials developed in the cells. The null point was determined with a Leeds and Northrup type R galvanometer (catalog number 2284C) which had a sensitivity of O. 077 uv /mm. The potentiometer was checked against a calibrated Eppley type standard cell. 3. Procedure After electrolysis, the silvernsilver chloride electrodes were connected in parallel and allowed to stand in 0. 1 normal bromideafree‘ potassium chloride solution until they reached a nearly constant potential when measured against one another. If the potential of any electrode in a set of ten differed from the mean of the remainder of the set by more than 0. 02 mv. , it was discarded. Three to four days were usually required for this equilibration. The electrodes were soaked in conductivity water for eight hours followed by three two—hour soakings in the solution to be measured before being placed in the cell. The electrode compartment containing the trap was rinsed four times and filled with the more dilute of the solutions to be measured, 104 and the electrodes were inserted. The remainder of the cell was rinsed and filled in the same way with the more concentrated solution. No stop» cock grease was used, the ground glass plug being tight enough to prevent diffusion. The cell was then placed in a large water bath at 25. 00 :i: 0. 020 C. One hour was allowed for temperature equilibration. The stopcock was then opened and the E.M. F. measured with the potentiometer. To correct for possible differences, between electrodes each determination was repeated using the electrodes in the more concentrated solution which were originally in the more dilute solution and vice versa. Since each compartment contained two electrodes, eight values of the E.M. F. were measured for each pair of solutions. All dilutions were measured against the same reference solution. 4. Results The potentials of cells with transference were measured for solum tion of trism(ethylenediamine) cobalt (III) chloride. A typical set of data is shown in Table 14. Values of log [y :1: / (yi)ref ] were calculated from equation (192) in the form Vi ._. 10g Cref _ 3FE c 9. 212 RT(T+) 8 f 5 daé log (Vi)ref ref 3F 9.212 RT (195) Values of log [ f: /(fi)ref] were obtained from equation (184). In order to determine log (fi)ref, the Debye--Hiickel equation in the form (194) was employed. ti __ lo — 5 ~/ c = — 10 f [ g (fi)ref f ] g ( i)ref f. O :1: - a B [N] c (log (fi)ref +log-(—f—-)- ] :1: ref (194) ;_a C) U! TABLE 14 A TYPICAL SET OF DATA FOR A CONCENTRATION CELL WITH TRANSFERENCE. TRIS-=‘(ETI—IYLENEDIAMINE) COBALT (III) CHLORIDE Conc entration: 0. 0014805 molar Reference Concentration: 0. 0033735 molar Electrodes EMF Average E. M. F. (millivolts) (millivolts) sz 12.502 12.496 4 vs 12. 491 .Q ' 1 . 2v... 12.50 12.494 3 vs 12. 486 1 . .. 1 o F . VS 2 48¢ 12.475 3 VS. 12. 468 1 .‘ 12. ‘73 . VS 4 12.467 4 vs 12.461 Average: 12. 483 my. Average deviation: 0. 012 mv. Maximum deviation: 0.. 016 mv. 106 Figure 17 is a plot of the left side of this equation versus the bracketed term on the right. The smooth extrapolation to zero absissa requires the unreasonably small value of g 2 2. O X . A larger 3.- causes the curve to drop abruptly in the region corresponding to very small concen- trations. The curvature of the plot indicates that solutions of Co(en)3Cl3 do not obey the Debye~Hiickel theory. Attempts to fit the data using larger values of g. and an ion-pairing constant were unsuccessful. The method used, however, may be of interest. Consider the following pairing scheme: CO(en)3+3 + C1- _..__>- [Co(en)3Cl]+ 2 (195) T”. The association constant, K, for this "reaction" is K = -i——a 2 :. Ciz ~ 196 a+3 a_l c+3 c-) (Yi)R ( ) where the signed subscripts denote the species of corresponding charge, a their activities and c their molar concentrations. The symbol yR denotes the activity coefficient ratio 3 (mm 1&— = (Vi).-. (197) 3'1 where (Vi)i—j is the stoichoimetric mean ionic activity coefficient of a ”salt” of Charge type i-j. Let a be the fraction of the salt associated, and c be the stoichiometric concentration Co(en)3Cl3. The concentrations of the various species are, then, C+3 3: C (1 - O.) C+z '—' C G. (198) 3c(l - a) C"l and the association constant is given by (J. K : 3c (1—a)"" ' (“’11 (199) 107 1 .28- C >:< +A (fi)ret fi [ 10g 0 l l 1 I l I l 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 ..._..._. - f—t x/ Cr (log tire, + log 5...... ) (I'ijlref Figure 17.. Debye-Hiickel plot of activity coefficient data for aqueous tris--(ethylenediamine) cobalt (III) chloride solution at 250C. 108 The ionic strength, _I:, of the solution is defined as 1'1 -.-. E ciziz (200) 1 which, for this system becomes F: 8c(3/Z-a) (201) Further, the observed mean activity coefficient, (yi), is related to the value calculated by the Debye-Hiickel theory for a 3--1 salt, (YQDH. by equation (202). (vi) = (1 - <1 ) (vyDH (202) 0r ‘1 = 1 - (Yi1/(Yi1DH (202a) The quantity (Vi)DH may be obtained from 1 log (Vi)DH = - SF 1: T 1+§B'F% (203) A method of successive approximations was then carried out. Choosing a concentration _<_:_, at which the activity coefficient was experimentally obtained, and g for the salt and a trial value of g, (yi )DH was calculated from equation (201) and (203). A second approximation to E. was then obtained from equation (202a), using the experimental value of (yd: ). This cycle of calculations (equations 201, 203, 202a) was repeated until a constant value ofg was obtained. Equations (200) and (204) 1 T [ 4(s )3_1 - 3(§ IN]? (204) [1+3B' FT] serve to evaluate (Vi)R’ , 0 To facilitate calculation, a for the salt and the ion pair were considered to be equal. Using this scheme, then, a K was obtained to ”force-fit" o . . . the data for the chosen a at the first concentration cons1dered. With 109 the value of I: so obtained, (Vi)ref was calculated by estimating (y QR, calculating 9_ref using equations (199) and (204), and iterating until a constant 9-_ref was obtained. Equations (202) and (203) yield (yi)ref° Points at other concentrations were calculated in the same manner so that several values of y:t /(y 4:) ref were compared with experiment. As E1: and 12 were decreased, the calculated values more closely approxim mated the observed quantities. Though unsuccessful, this calculation points out the difficulties involved in the application of the ion pair concept to solutions of unsymmetrical salts. A second attempt was made to explain the data using the extended equation of LaMer, Gronwall and Grieff (92). This expression for the activity coefficient of unsymmetrical electrolytes, arises from the solution of the PoiSSOnuBoltzmann equation (12) with retention of terms to order 463. The calculation proved to be completely unsatisfactory when apIE—md to the data for Co(en)3Cl3 solutions. Since no theoretical expression adequately explains the data, the value of log (fi)ref from the intercept of the DebyeaHiickel plot was used to calculate fi for each solution the mean molar (111:) and molal (3}) activity coefTi-Cients were calculated from equation (184). Table 1-5_—gives the Observed E. M. F. and calculated quantities for the solutions. The measured potentials were precise to dc 0., 1% or better. With the lack of a suitable theoretical function with which to compare the results, it is difficult to obtain a good estimate of the accuracy of the data, however, the values of Vi/(Yi)ref are probably accurate to about :t 0.2%. IV. Discussion of Results Activity coefficients of Co( en)3Cl3 in concentrated solutions have been measured by Brubaker (46) using the isopiestic method. These data 110 TABLE 15 E. M. F. DATA AND ACTIVITY COEFFICIENTS OF TRIS-(ETHYLENEDIAMINE) COBALT (III) CHLORIDE AT 250C. :E.14.En (M olarity) 1 04 (Millivolts) yi fi 6.7897 22.830 .8019 .8022 13.805 12.483 .7318 .7319 19.718 7.428 .6933 .6934 33.735* 0.000 .6328, .6328. 46.677 44.253 .5908 .5904 77.330 -10.732 .5268 .5263 153.37 -18.869 .4344 .4333 236 10 ~23.793 .3802 .3787 Reference Solution 111 were fit with fa: 2 3. 5 X and an empirical linear concentration term. A large plot of log 7' vs. _r_n_ for the E.M. F. and isopiestic data indicates that the chosen reference values are compatible, since, as nearly as one can tell, the two sets of data form a smooth curve. As indicated above, both transference number and activity coefficient data deviate from all available theoretical expressions. The lack of agreement with theory of the activity coefficients is the more funda— mental discrepancy. Since the theory of ionic mobilities is based upon the Debye-Hiickel theory, it is not surprizing that differences between observed and calculated transference numbers occur. It is interesting to note that the transference number calculation improves somewhat as larger 3 values are used and that the extended electrophoretic correction gives a closer fit than the Onsager expression. The activity coefficient data, however, are best approximated with an a value which is certainly too small. A consideration of multiple ion aggregation might improve the situation, however, it seems more likely that the observed deviations are due to other unknown, non-coulombic interactions of the complex cation. Although the interionic attraction theory is limited, it has met with considerable success when applied to systems which more closely approximate the model of hard spherical ions. Solutions of rare earth chlorides serve to illustrate this behavior. The activity coefficients of such salts obey the Debye-Hfickel theory. Further, the difference in observed conductance, and that calculated by the limiting law (usually designated as_/_\__I9 , shows monatonic dependence upon the square root of the concentration. This behavior is anomolous, in the sense that for a great many other unsymmetrical salts of high charge type, including 1 , 1 Co(en)3Cl3 solutions, A0 versus c'T plots exhibit a pronounced minimum. The more usual behavior, then, is deviation from the simple theory which is currently available. 112 V. Summary Experimental data have been presented for aqueous solutions of t_1;i_s-(ethylenediamine) cobalt (III) chloride. Transference numbers were obtained by the moving boundary method and activity coefficients were determined from the electromotive force of concentration cells with transference. 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Phys. Chem., _3_5, 2245 (1931).. K. S. Kunz, H'1\'I"urnerica1 Analysis, " McGraw~Hill Book Co. , Inc. New York, N. Y. (1957). APPENDIX 1 A DESCREPTTON OF THE PROGRAM FOR THE EVALUATION OF THE COMPLETE ELECTROPHORETIC EFFECT The expressions for the electrophoretic contrpprution to the equivalent conductance are git-Jen by equations (151) and (153) and are: of the form 00 A)”, r: M _( p{exp (~B+ ;D/p) -, exp (B:t ewp/p‘, } dp (a) )4 (>0 AN, M j P{e->.