ELECTRICAL POTENTIALS ACROSS POROUS PLUGS AND MEMBRANES ION-EXCHANGE RESIN-SOLUTION SYSTEMS BY K. S. SPIEGLER, R. L. YOEST AND M. R. J. WYLLIE Gulf Research and Development Co., Pittsburgh, Pennsylvania Received 1st February, 1956 An electrical potential difference arises in a manner analogous to that for a glass elec- trode when a solid membrane made from an ion-exchange resin or a plug composed of ion-exchange resin particles separates two solutions of the same electrolyte. The potential difference depends on the activities of those ions in the two solutions which can exchange with the resin. While ion-exchange membrane potentials can be used to determine ionic activities in many solutions, the simple relationship between membrane potentials and ionic activity is limited to a definite concentration range.Porous plugs made from particles of ion-exchange resin act as “ leaky ” membranes. Both the electrical potentials across ion-exchanging plugs and the electrical conductance of the latter can be quantitatively interpreted if the plug is represented by a simple resistor model. This model consists of three elements in parallel, namely, (i) alternating layers of solution and conductive solid, (ii) ion-conductive solid and (iii) solution. Thus plug potentials can be estimated from conductance data. The implications of this work for the interpretation of pH measurements in ionic suspensions are discussed. When a calomel and a glass electrode are inserted into an ionic suspension or gel, the potential difference between them may be interpreted as the algebraic sum of (i) the potential of the glass electrode in the equilibrium solution of the suspension, (ii) the potential of the calomel electrode, and (iii) the plug potential between equilibrium solution and potassium chloride solution in the calomel electrode.The latter is a plug potential like those studied in this investigation and explains the “ suspension effect ”. SYMBOLS a&, al, a2 mean activity of NaCl in general and in soln. 1 and 2 respectively, UNaCl a, b, c, 4 E E’ EJ E M f l , f 2 , f3, F K KO KR K W N Q R T t+, t- X , Y cc PO, P activity of NaCl = a:, geometrical factors as shown in fig. 2, plug potential (V), potential measured between two solutions separated by a porous plug with a pair of Ag/AgCl eIectrodes (V), junction potential between two sodium chloride solutions of different con- centration, potential of an ideally cation-selective electrode (V), Faraday ’s constant (coulomb equiv.-I), fraction of current carried by resistor elements 1, 2 and 3 respectively (fig.2), specific conductance (mho cm-I), specific conductance of resistor model, specific conductance of solid, specific conductance of solution, expression defined in eqn. (17), expression defined in eqn. (19), gas constant (watt sec deg.-1 mole-I), absolute temperature (“K), transport numbers of cation and anion respectively, geometrical factors equal to (1 - d)/a and d/a respectively, proportionality factor relating the specific conductance of a NaCl solution to the mean activity of NaCl (eqn. (lo)), standard chemical potential and chemical potential respectively (watt sec mole-1).174K . S. SPIEGLER, R . L. YOEST AND M. R. J. WYLLIE 175 It is well-known that electrical potential differences arise when two different solutions are separated by a permselective membrane or plug. These potential differences are termed membrane potential or plug potential respectively. They are different from the potential differences between the two solutions in the absence of the permselective membrane or plug. They depend on the nature of the solutions as well as the material separating them. Permselective materials are defined as media which transfer certain types of ions in preference to others. Cation and anion-exchange resins transfer preferentially cations or anions respectively.This property can be used in industrial electrodialysis.1 Membrane potentials across natural shale bodies containing cation -exchanging clays are very important in the electrical logging of oil wells.2 Membrane potentials across synthetic resinous ion-exchange membranes have been studied extensively in the recent past.3-9 For solutions of the same salt, it was found that the membrane potential EM is roughly proportional to the logarithm of the ratio of the mean activities q of the electrolyte in the solutions separated by the membrane : EM = 0.0592 loglo (u2/al) (at 25" C)* This relationship can be predicted by quasi-thermodynamic reasoning 10 for ideally permselective membranes. It holds over a limited concentration range, depending on the nature of the membrane and the solutions.Examples of such plots for mem- branes made from synthetic ion-exchange resins are shown in fig. 1. Our measure- ments with dilute NaCl solutions are compared to those of Kressman7 with KCl solutions over a wider concentration range. The membranes used were different. The points fall roughly on the line predicted for ideally cation-selective membranes. Similar potentials have been observed previously with certain thin clay films 11 and collodion membranes.12 The relationship shown in fig. 1 between the electrical potential and the ratio of the activities of the solutions is entirely analogous to that between the potential of a glass membrane and the difference of the pH of the solutions separated by it. Glass electrodes act as if they were permeable only to hydrogen ions.Ion exchange membranes are usually permeable to all cations or all anions. Therein lies an ad- vantage and a disadvantage. The resin electrodes are versatile because they do not only measure pH, but also pNa, pCNS and the like. But reliable measurements of the activities of electrolytes can only be made in solutions containing one single electrolyte. The glass electrode measures acidity irrespective of the presence of other electrolytes (unless these are present in abnormally high concentrations when the salt-error is observed), whereas ion-exchange resin membranes develop potentials dependent on all the electrolytes present in the solutions. The interpretation of potentials across membranes separating solutions of different electrolytes is com- plex.13~ 14 No resinous membranes have been described which are specifically permeable only to one particular kind of cation or anion.A further complication results from diffusion processes across the membrane which may cause complex distribution patterns of ions in the membrane and in the adjacent solution layers. This is usually indicated by a variation of the measured potential difference with the rate of stirring of the solutions.9 All ion-exchange membranes lose permselectivity with increasing concentration of the solutions which they separate. This loss is due to the increasing penetration of anions and cations into cation- and anion-exchange resins respectively, and also to water transport.s. 8.13 The loss of permselectivity is often termed ion leakage.Membranes or plugs containing ion-exchange resins are permselective. * This relationship holds for 1 : 1-electrolytes and if the activities of the cation and anion are equal. When reversible electrodes are used, as in almost all experiments reported here, no assumptions about the relationship of the ion activities are necessary. When calomel electrodes are used, the " single " cation or anion activities are conventionally substituted for cation and anion exchange membranes respectively.176 ELECTRICAL POTENTIALS Its source may be two-fold. One is the fact that even mechanically perfect, homo- geneous solid membranes take up electrolytes from solutions in contact with them. The amount of electrolyte taken up increases rapidly with increasing solution con- centration.This phenomenon may be considered as a " Donnan " effect. Theories of the change of membrane potential with solution concentration based on the Donnan effect have been worked out independently by Teorell15 and by Meyer and Sievers.16 These theories apply only for membrane materials which may be con- sidered as single phases. 0 I 2 3 4 loqio ( a ' / a l ) FIG. 1 .-Membrane potential EM against logarithm of activity ratio. Circles represent measurements of Kressman 7 with KCI solutions at 19-20" C. Homogeneous synthetic cation-exchange membrane ; solution concentration on one side of the membrane electrode was kept constant at 0.002 M and varied on other side. Squares represent our measurements with NaCl solutions at 25" C. Hetero- geneous synthetic cation-exchange membrane containing 64 % weight finely powdered Amberlite IR-100 ( R o b and Haas Co., Philadelphia, Pennsylvania) and 36 % polystyrene powder.Solution concentration on one side constant at 0401 M and varied on other side. Electrodes : saturated calomel. In porous plugs of ion-exchange material or in defective membranes, a different type of leak exists in addition to the leak caused by Donnan ions. This additional leak is due to a continuous connection between the two solutions through the liquid phase in the pores. If the specific conductance of the ion-exchanger is high and the solutions are very dilute, this leak does not affect the membrane potential appreciably. But in the reverse situation, the larger proportion of the ion transport phenomena takes place in the conductive liquid phase and the effect of the leak overshadows the effect of the ion-exchange material.Thus porous plugs of many ion-exchange materials give rise to membrane potentials which are almost perfect when the plugs separate very dilute solutions. On the contrary, if the solutions are con- centrated, the solid acts as if it were an inert material and the potential difference between the two solutions approaches the liquid junction potential.K . S . SPIEGLER, R . L . YOEST AND M. R. J . WYLLIE 177 Plug potentials have been studied by a number of investigators in the recent past.17 Their interpretation concerns the soil scientist and petroleum technologist. It is believed that they are of general importance in membrane theory, for there exists hardly a membrane that is mechanically perfect and acts strictly as a single phase.Most membranes show to some extent the characteristics of a porous plug. For an understanding of plug potentials, it is useful to have an electrochemical model for an ion-conducting plug. We have recently postulated such a model l8$ and found that it explains quantitatively the change of the specific conductance of the plug with the specific conductance of the saturating solution. The model is shown in fig. 2. It consists of three conductance elements in parallel represent- ing (1) conduction through alternating layers of resin and solution, (2) conduction A FIG. 2.--Electrochemical model of porous plug composed solution Solution Solid of conducting spheres and A.Schematic representation of current path through plug. (1) represents current through solution and spheres in series, (2) through spheres in contact with each other, (3) current through solution. B. Simplified model representing situation shown in A. a + b + c = 1 cm. through the resin and (3) conduction through the solution. A first attempt has been made to apply this model to plug potentials across a column flushed with solutions, and in some cases containing oil, with a view to interpreting electric logs in shaly sands.20 It was the purpose of the present investigation (i) to measure the potential differences between stationary sodium chloride solutions separated by a plug of cation-exchange resin, (ii) to relate these measurements to the plug conductance, (iii) to develop further the theory of plug potentials based on the proposed model and (iv) to compare the measured potential differences to those predicted from the theory. EXPERIMENTAL AND RESULTS PREPARATION OF msIN.-l000 g Dowex-50 (cross-linked by 8 % divinyl benzene), a cation-exchange resin made by the Dow Chemical Company of Midland, Michigan were used.The resin particles were spherical and represented the size fraction between 100 and 200 mesh (U.S. Standard Screen). The resin was placed in a column, backwashed and " conditioned " by a number of alternating regeneration cycles with 6 % HC1 and 4 % NaOH solutions respectively. The resin was finally treated with 8 % NaCl solution and leached. DETERMINATION OF PLUG CONDUCTANCE A Plexiglas cell was used for the measurement of the conductance of the resin plug, saturated with solutions of sodium chloride of different concentrations.The resins were178 ELECTRICAL POTENTIALS placed between a pair of perforated platinum electrodes, flushed with the solution of the desired concentration until equilibrium was reached and the a.c. conductance of the plug determined at 60 c/s. Then a solution of different concentration was introduced into the column and the procedure repeated. A detailed description of the apparatus and experi- mental procedure for plug conductance measurements has been presented elsewhere.19 The results are plotted as specific conductance of the saturated plug against the specific conductance of the saturating solution (fig. 3). Comparing the results to those of previous experiments 19 with the cation-exchange resin Amberlite IR-120 (a product of the Rohm and Haas Company, Philadelphia, Penna.), it is seen that the points fall approximately Q v) Specific conductance of the interstitial solution (mho c 5 ' ) FIG.3.-Specific conductance of resin] plug saturated with solution against specific conductance of solution Circles represent experimental data with small-size Dowex-50 cation- exchange resin (100-200 mesh, U.S. Standard Screen). Squares represent data from a previous investigation 19 for coarse Amberlite IR-120 (32-35 mesh). Solid line is calculated from theory based on 3-component model (fig. 2) ; cross mark shows isoconductance point ; temp. 25-28" C. on the same line, although the particle size of the latter resin was much larger (32-35 US.Standard Screen). This shows that the amount of current carried by the solid and inter- stitial liquid in a bed of spherical particles is a geometrical effect independent of the particle size ; it is worth remembering that the porosity of a bed of particles of uniform size is also independent of the particle size provided the geometry is the same. From the con- ductance data, the following geometrical parameters for the plug model (fig. 2) were calculated : The specific conductance KR of the resin was found from the isoconductance point, viz. the point where the specific conductance of plug and the solution are equal. KR = 0.030 mho cm-1. LI = 0 6 3 , b = 0.01, c = 0.34, d = 0.95. These parameters are identical with those found for the larger particles,l9 except for KR which was 0.029 mho cm-1.The present value seems to fit the results somewhat better. This slight difference is probably due to the fact that the resins came from different batches and manufacturers. In previous experiments,ls a different batch of resin was used and KR was 0.019 mho cm-1. DETERMINATION OF PLUG POTENTIALS A junction between two resin plugs saturated with NaCl solutions of different concentra- tion was produced in a Tiselius cell (American Instrument Company, Silver Springs, Maryland) and the potential difference was measured between two silver/silver chloride electrodes dipping into the supernatant solutions. The Tiselius cell was selected because the boundary is highly reproducible. It was found that after an initial period of slight change of the order of minutes, the potential differences remained constant for many hours.K .S . SPIEGLER, R . L. YOEST AND M . R . J . WYLLIE 179 The Tiselius cell consists simply of a glass U-tube divided into three sections, of which the top two are held stationary. The bottom section rests on a sliding platform and when the bottom section is slid to one side, the continuity of the U-tube is broken (fig. 4). E : 5 0 4 0 3 0 v W 20 10- 0 Resin t Solutron I Resin t Solution 2 - 1 _-____------ ------- ---- -------- -- - - _ _ _ _ _ _ _ _ _ _ -r - - Inert membrane - I I 1 1 I I l l 1 1 I I I I I I I I I I I I l l ( - 0 0 3 .O I .03 - 1 . 3 FIG. 4.-Schematic representation of the formation of a plug boundary in the Tiselius cell.The solutions used in these experiments were 2-02, 0.673, 0.205, 0.062, 0.0192, 0.00605, and 0.00195 molal NaCl. This corresponds to 1-35, 0.45, 0.15, 0.05, 0.0167, 0.00557, and 0.00186 mean activity. In each experiment, two consecutive solutions were used so that I d e a l membrane a FIG. 5.-Potential difference E' against average activity Z of sodium chloride in solutions separated by ion-exchange resin plug Solution activity ratio 1 : 3. Vertical lines represent potential difference measured between Ag/AgCl electrodes at 25" C ; solid line calculated from theory. the activity ratio between the two solutions was always 3 : 1. The solutions were titrated with silver nitrate after their preparation and corrections made for minor deviations of the activity ratio from the value of 3.Silver/siIver chloride electrodes made by Beckman Instruments, Inc., Pasadena, Cali- fornia, were used. In a number of preliminary experiments, calomel electrodes made by180 ELECTRICAL POTENTIALS the same manufacturer were tried, but it was found that the leakage of saturated potassium chloride solution caused appreciable concentration changes in the dilute solution systems, whereas in concentrated solutions the uncertainty of the junction-potential difference hampers the evaluation of the results. It was verified that the potential difference was less than 0.2 mV when the electrodes were immersed in the same solution. To obtain a continuous record of the potential difference against time, the electrodes were connected to a Leeds and Northrup no.7664 pH Indicator whose output was fed into a Brown Electronik recorder. The accuracy of these measurements is estimated to be 0.5 mV. The recorder trace showed that a few minutes after the formation of the boundary, the potential difference remained constant. The electrodes were then dis- connected and a more accurate reading carried out by means of a Leeds and Northrup manual potentiometer (accuracy 0.2 mV). The potential differences remained almost constant for many hours. Comparing the measured potential differences to those predicted by the theory, a number of additional possible sources of error have to be noted. The ratio of the mean activities of the solutions was 3 f 0.02 corresponding to an error limit of about 0.3 mV. The response of the silver/silver chloride electrodes to the chloride activity may not be ideal over the whole concentration range investigated here.Considering all these possible sources of error, we estimate a limit of error of rt 0.8 mV for the potentials. Each experi- ment was carried out several times and corrections were made for the small fluctuations of the prevailing temperature. At first the junction potential (also called diffusion potential) between pairs of solutions of activity ratio 1 : 3 was determined. The transport number t- can be determined from these measurements (eqn. 20). The resins were equilibrated with the solutions by stirring for at least 30min prior to introduction into the Tiselius cell. The results of the total potential measurements are shown in fig. 5 which also shows the potentials expected from a theory based on the model shown in fig.2. DISCUSSlON DERIVATION OF EQUATION FOR THE PLUG POTENTIAL Consider an equivalent resistor system as shown in fig. 2. This system is defined by three parameters; namely, by either two of the group a, b, c and by d. (The sum a + b + c is theoretically unity.) It is often convenient to use also the parameters x and y derived from a and d : x = (1 - d)/a, y = d/a. This model provides a good explanation for the conductance data obtained in the past 18-19 and in the present investigation (fig. 3). It is now applied to the calcula- tion of the plug potential. The plug is supposed to separate two solutions of activities al, a2 and specific conductances K1 , K2, respectively.It is assumed that the concentration difference between the two solutions is infinitesimal and a2 > al, and also that the boundary layer is large compared to the size of the solid particles. In general, there exists a potential difference between the two solutions. Suppose that we insert silver/ silver chloride electrodes and allow 1 P to pass from solution 2 to solution 1 under reversible conditions. (It is assumed that the plug and the solution volumes are so large that no appreciable concentration change occurs as a result of this current transfer.) We determine the fraction of the current passing and the results produced by the passage of the current through each of these elements. In general, the fractionfof the current passing through an element is equal to f = K/KO, where K and KO are the specific conductances of the element and the total plug respectively.KO is given byK . S . SPIEGLER, R . L . YOEST A N D M . R. J . WYLLIE 181 ELEMENT I.-KI is given by (3) where a, d, x and y are geometrical parameters (fig. 2). KR is the specific resin conductance and KW the average specific conductance of the solution. KRKW - aKRK W K1 = - XKR + p w d K w + (1 - ~ K R ' Hence, the fraction of current carried by element 1 equals When 1 F passes through the whole plug, only f l F coulomb passes through element 1. This element consists of a succession of components alternately of true resin micromembranes and solutions. If the micromembranes are ideally cation-selective, and the dimensions of the plug sufficiently large, then no concentra- tion changes occur in the intermediate solution compartments and the results of the passage of f1F coulombs is the transfer of fi moles NaCl from solution 2 to 1.ELEMENT 2 also acts as an ideal membrane. f2 is the proportion of current carried by element 2 : The fraction of current carried by ELEMENT 3 is f 2 = bKR/KO. ( 5 ) - f 3 = CKWIKO. (6) The sodium chloride transfer through this element is t+ moles per faraday passing through it 10 or f3t+ moles NaCl per faraday passing through the plug. The transfer is from solution 2 to 1. The total amount of sodium chloride transferred is, therefore, f 1 + f2 -I- f 3 t + moles and since the free energy p of sodium chloride is given by (7) it follows that the total free energy change equals (8) dp has been set equal to the reversible electrical work obtainable from this process.If the difference of the activities of the solutions separated by the plug is finite, the plug may be considered as an assembly of infinitesimal elements in series. The activity difference across each of these elements is infinitesimal and so is the potential difference. This reasoning is strictly applicable only if the particle size of the resin is itself infinitesimal. However, the results are reasonably accurate for particles of finite size provided the boundary zone is large by comparison with the size of a resin particle. In the following it will be assumed that this condition is satisfied. To obtain the total potential difference between the electrodes, eqn. (8) is in- tegrated over the whole length of the plug : p = Po + R T h aNaC1, dp = (fi + f 2 + f3t+) RTd In a N a c i = Fa'.This is the complete equation for the potential between two silver/silver chloride electrodes in solutions of activities a1 and a 2 , respectively, separated by the conducting plug. As shown previouslyf1,f;! and f 3 are given in terms of geometrical parameters determined from conductance measurements of the same plug and are functions of the specific conductance. To integrate eqn. (9), a functional relationship between the specific conductance KW and the activity aNaC1 must be found. A plot of KW against the mean activity, ah = (aNaCI)*, shows that KW is roughly proportional to (fig. 6) : KW = aaf, (10) where 0: = 0.126 mho cm-* at 25" C.182 ELECTRICAL POTENTIALS Thus eqn.(9) may be integrated without any assumptions about the magnitude of single ion activity coefficients. However, for the sake of convenience and convention, we shall now split the activity aNaCl into two ion activities aNa+ and aci-. The final result, in terms of E', is independent of the manner in which this separation h - I 5 .-4 I s 0 E z Y v a* FIG. 6.-Spec%c conductance KW against mean activity a& of NaCl solutions, temp. 25" C. The mean activity a& is defined by aNaCl (aNa+>(aCl-) = 6 (1 1) The ionic activities are arbitrarily defined equal to the mean activity : d In aNa+ = d In acl- = d In a&. Hence It is customary to ascribe the first term of this potential difference to the silver/ silver chloride electrodes. The remainder is the plug potential, E : Substituting forfl,.f2 andf3 in eqn.(14) from eqn. (4) to (6) and replacing K W by a a&, we obtain KRCC[I f by f (t- - t + ) C X ] ( l / N ) da* + b X K R 2 - c p 2 (t- - t+) (a*/N) dq}, (16) 0 1 where = cyx2a$ + KRCC (1 + by + cx) a& + bxKR2. (17)K. S. SPIEGLER, R . L. YOEST AND M. R. J . WYLLIE 183 The integrals in eqn. (16) may be found in standard tables.21 The integration yields the following result : E = (RT/F) In (az/al) RT cya2az2 4- KRU(~ + by + cx)a:! 3- bXKR2 - - '- In Cya2a12 $- KRCC(~ + by C X ) q + bXKR2 where Q [(l + by + C X ) ~ - 4 b ~ ~ y ] * . (19) The first term of eqn. (18) represents the potential of an ideally cation-permeable membrane (on the assumption of equal ion activities, aNa+ and acl-).When the plug is in equilibrium with extremely dilute solutions (KW= aka < KR), the algebraic sum of the second and third terms vanishes and the first term alone represents the plug potential. As the concentration of the solution increases, the plug potential drops and the second term approaches -2(RT/F) t- In (a2/al), while the third term vanishes. Thus the plug potential approaches (20) This is the junction potential of the two solutions in the absence of a permselective plug. This value was not reached in our experiments. If we use the numerical values of the geometrical parameters and of KR listed in the previous section, the total potential difference of the cell when using silver/ silver chloride electrodes is obtained from eqn. (14) and (18). For (a2/a1) = 3 at 25" C, E = (1 - 2t-) (RT'F) In (a2/a1).040814 a22 + 0-00395 a2 + 7-65 (10-7) 0.00814 a12 + 0.00395 a1 + 7.65 (10-7) E' = 118.4 log10 3 - 59.2 (t-) log10 ] (21) a2 + 6.18 (10-0 0.129 a1 + 0.0626 0.129 a1 + 6.18 (10-6) I[ 0.129 a2 + 0.0626 + 55.9 (t-) log10 The numerical values in eqn. (21) hold only for the particular materials and con- ditions chosen in our experiments. The computation of the potential differences E' from eqn. (21) is illustrated in table 1 where the values thus calculated are compared with the measured ones. The results are also plotted in fig. 5, using a logarithmic activity scale. TABLE COMPARISON OF OBSERVED AND COMPUTED POTENTIAL DIFFERENCES, E' Solutions of sodium chloride, activity ratio 3 : 1 ; electrodes, Ag/AgCl ; plug, " Dowex-50 " ; temp., 25" C ; t- computed from junction potential mean activity ~~ a2 a1 1.35 0.45 0.45 0.15 0.1 5 0.05 0.05 0.01 67 0.0167 0.00557 0.00557 0.00186 potential computed from eqn.(21), mV I- 1st term 2nd term 3rd term total 0.63 56.5 - 28.7 6.5 34.3 (0.63)* 56.5 - 24.0 10.9 43.4 0.63 56.5 - 20.5 14.2 50.2 0.62 56.5 - 18.3 15.5 53.7 0.61 56.5 - 17.2 15.8 55.1 0.61 56.5 - 16.2 15.9 56.2 * estimated potential obs., mV 34.0 42.9 48.5 51.4 52.9 55.5 It is seen that the agreement between the observed and computed values is within 2.3 mV. In the concentrated solutions the observed potential differences agree well with the theory, whereas in the dilute ones, the observed values are lower.electrode X The potential difference across Xis a plug potential like the potentials described in the present investigation and is the cause of the suspension effect.Jenny et al.22 measured similar potentials before we did and related them to transport numbers across the resin plug. Their investigation gave rise to discussions and other investigations ; these have recently been summarized by Overbeek.17 The potential across X has often been considered as a liquid-junction potential. Since the resin or gel are solids, we prefer to call it a plug potential and have treated it above as the potential of a leaky membrane. Our application of the resistor model (fig. 2) to this problem may perhaps shed some additional light on these earlier investigations. * In a previous publication 20 experiments were reported on plug potentials measured with calomel electrodes in a resin column in which an interstitial solution of sodium chloride was flushed out by another solution of higher activity.The observed potential differences were compared with values computed from a simplified theory, based on average solution conductance values rather than the completely integrated equation presented here. In the equations based on average conductance it is relatively simple to alIow for the non- ideal permselectivity of the resin proper. It was found that the observed potentials agreed fairly well with the measured ones, when such allowance was made. + solution c1 i + saturated KCl KClsolution electrode.K. S. SPIEGLER, R. L . YOEST AND M. R. J . WYLLIE 185 1 Kressman, Ind. Chem., 1954,30, 99. 2 Wyllie, The Fundamentals of Electric Log Interpretation (Academic Press, New York, 3 Juda, Marinsky and Rosenberg, Ann. Rev. Physic. Chem., 1953, 4, 373. 4 Winger, Bodamer and Kunin, J. Electrochem. SOC., 1953, 100, 178. 5 Spiegler, J. Electrochem. SOC., 1953, 100, 303C. 6 Wyllie and Kanaan, J. Physic. Chem., 1954, 58, 73. 7 Kressman, J. Appl. Chem., 1954, 4, 123. 8 Scatchard, in Ion Transport Across Membranes, ed. Clarke (Academic Press, New 9 Helfferich, Thesis (Gottingen, 1955). 10 Spiegler and Wyllie in Physical Techniques in Biological Research, ed. Oster and 11 Marshall, The Colloid Chemistry of the Silicate Minerals, 1st ed. (Academic Press, 12 Sollner, J. Electrochem. SOC., 1950, 97, 139C. 13 Scatchard, J. Amer. Chem. SOC., 1953, 75, 2883. 14 Wyllie, J. Physic. Chem., 1954, 58, 67. 15 Teorell, Proc. SOC. Expt. Biol. Med., 1935, 33, 282. 16 Meyer and Sievers, Helv. chim. Acta, 1936, 19, 649. 17 Overbeek, J. Colloid. Sci., 1953, 8, 593. 18 Wyllie and Southwick, J. Pet. Tech., 1954, 6, 44. 19 Sauer, Southwick, Spiegler and Wyllie, Ind. Eng. Chem., 1955,47, 2187. 20 McKelvey, Southwick, Spiegler and Wyllie, Geophysics, 1955,20, 913. 21 Hodgman (editor), Handbook of Chemistry and Physics (Chemical Rubber Publishing 22 Jenny, Nielsen, Coleman and Williams, Science, 1950, 112, 164. 1st ed., 1954), chap. 2. York, 1st ed. 1954), p. 128. Pollister (Academic Press, New York, 1st ed., 1956). New York, 1949), p. 172. Co., Cleveland, 35th ed., 1953).