118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure.This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point.These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility.The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure.This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point.These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility.The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No.13. REPORT ON A REVISION OF THE CONDUCTIVITY THEORY. B Y L. ONSBGER. (Received PIS/ M a d , 1927.) Introduction.-This paper presents a modification of the Debye-Huckel theory for the conductivity of strong electrolytes. Debye and Huckel start from the assumption, that the fall in molar conductivity which occurs with increasing concentration is chiefly due to the electrostatic forces between the ions. As Debye has told you in his paper, the velocity of an ion in a conduct- ing electrolyte will be affected by its ionic atmosphere in two different ways. Firstly, as the ion moves, the concentration disturbance in its neighbourhood must be forced to move with the same velocity; so the moving ion will be exposed to electric forces from the surrounding ones.Secondly, the surrounding ions, by the action of an external electric field, will cause an additional movement of the solvent, this second effect corresponding to the phenomenon generally known as electrophoresis. For both effects, the resultant decrease in molar conductivity is proportional to the square root of concentration. This gives a limiting formula for the molar conductivity : - - A = h o - a Z / 1 ' . . a ( I ) where h = the molar conductivity, r = twice the ionic strength, This formula was found, many years ago, empirically and thoroughly tested by Kohlrausch. However, according to Debye and Hiickel, it is not possible to determine the value of the factor Q otherwise than from the conductivity measurents themselves, because the formula for the electrophoretic effect involves the radius of the ion.Now, we may try to compute that radius from the mobility of the ion, but the value of a which we find in that way, does not agree with the experiments, the calculated decrease in con- ductivity being always too great, though the order of magnitude comes out right. I. Electrophoresis.-A year ago I published a paper,l showing that it is possible to compute the electrophoretic effect in such a way that the result in the limiting case of small concentrations will be independent of the ionic radius. We shall only consider the result of that deduction ; in the limit of small concentrations, the additional velocity given to the ion b y electrophoresis is found equal to : where X = strength of the electric field, e , = the charge of the ion in question, 7 = the viscosity, and K = the mean inverse radius of the ionic atmosphere.Formula (2) is essentially the same as that previously found by Debye and Huckel. They found for the force caused by electrophoresis : - eXKb . . (3) Physik. Z., 1926, 27, 388. 341342 where b = ionic radius. REVISION OF THE CONDUCTIVITY THEORY Indeed if we take: p = 6 7 4 , according to Stokes’ formula, for the friction coefficient of the ion, then! we get : pAvj 67&Avj = - e,XKb , * (4) for the corresponding force. Of course, if the Stokes’ law does not hold, formula (3) is wrong too, but the validity of formula (2) is not affected. So we have to look elsewhere for an explanation of the reason why the value of a in formula ( I ) , as calculated, by Debye and Huckel, does not fit in with the experimental facts.2. Ionic Forces.l-We are going to show that the method used by Debye and Huckel to compute the first effect mentioned in our introduction needs some modification for fundamental reasons. To begin with, we shall consider a simple condition which the electrical forces between the ions have to fulfil, that is the condition of equilibrium. I t is evident, that the internal forces between the ions form an equilibrium group-as it also follows from Coulomb’s law-so, the resultant of all the forces exerted by the cations upon the anions must be equal to that exerted by the anions upon the cations. Now Debye and Hiickel try to compute the forces arising between the ions in a conducting electrolyte, by considering an ion moving with constant velocity along a straight line through the electrolyte, and calculat- ing the force acting upon it from the surrounding ions.For an ion with the charge ej moving with the velocity v, this force will be (provided the concentration is small enough) : where D = the dielectric constant, T = the absolute temperature, and K = Boltzmann’s constant ; pm is a mean frictional coefficient of the ions in the electrolyte. However, formula (4) fits in with the equilibrium condition mentioned above only in special cases; for instance, in the case of a simple binary electrolyte with one kind of anions and one kind of cations, the equilibrium condition is only fulfilled when the velocities of anions and cations are equal (in opposite directions).A closer analysis of the problem2 leads to the conclusion that, to calculate the resistance upon an anion moving through an electrolyte, we must take into account its own Brownian movements. If we suppose the ion constrained to follow a given path, thus suppressing its Brownian move- ments, we shall always get too high a value for the resistance. A further correction must be introduced for the migration of the ions forming the atmosphere. The diffusion of two ions in relation to one another will cause a combined relaxation of both ionic atmospheres. So, instead of the separate differential equations found by Debye and Huckel for the fields of each ionic atmosphere, we get a system of simultaneous linear differential equations for all atmospheres together. In the most simple case, when only two kinds of ions are present, the separation is carried out very easily, and gives the following expression for the strength of the additional electric field caused by the atmosphere : 2 For an extensive treatment of the problem see : Onsager, Physik.Z., 1927, 28, 277.L. ONSAGER 343 P1 Pi where pl, e, and nl signify frictional coefficient, charge and concentration of the anions, and p2, e2 and n2 are the corresponding characteristics of the cations. According to Debye and Huckel we put : the quantity K' differs from K by a numerical factor; it is given by the equation : nie? + F~&~ ~ PI PP I n accordance with the equilibrium condition AX is equal for both kinds of ions. I n the case of binary electrolytes, equation ( 5 ) becomes simplified as follows : K 2 I n this case K I P always equals -.found by Debye and Huckel : This result may be compared with that for binary electrolytes in which case pm = 'd. The difference between expressions (6) and ( 7 ) is that the former contains the factor 2 - ,/; instead of the factor 3. Mobility Formuh.--In writing formula: for practical use, equivalent or molar conductivities may be used. We will prefer the former; the partial equivalent conductances shall be properly noted as mobilities, because they are directly proportional to the velocities of the ions. Putting numerical values for the universal constants entering into formula: ( 2 ) and (s), we find the limiting formula for the mobility of an ion in a simple electrolyte in practical units (ohm - cm.2) : 2 in (7).P1 Here, p means the equivalent concentration, z;, z2, 110 and 120 valences and zero concentration mobilities of anions and cations.344 REVISION OF THE CONDUCTIVITY THEORY The most important difference from the earlier theory is, that in the new theory, the coefficient of the square root term does not contain the ionic radius, so that this coefficient can be computed from the mobilities of the ions present. For binary electrolytes, formula (8) may be simplified : We note that this formula contains a law found empirically by Kohlrausch, which states that the mobility of an ion is the same in all binary electrolytes of the same con- centration. However, its validity is restricted to simple binary electrolytes and, even for mixtures of these, de- viations are to be expected.Indeeed, conductivity measurements performed by Bray and Hunt on mixtures of HCl and NaCl show with certainty, that the rule of independent mobilities does not hold in this case, the devia- tion being as expected by the theory. According to our deduction, the limiting formulz given here should be expected to hold only in the greatest dilutions ; when I , - I amounts to 10 per cent. of Lo, the terms of the second order which we have neglected in the calculation will reach the order of magnitude of I per cent. Investigations on terms of higher order indicate that they should be of opposite sign to that of the square root term in all cases where association in the sense of Bjerrum3 does not oc- cur.As to association, we should note that it is strongly indicated when the conductivity curves of two electro- lytes of the same type show other dissimilarities than would be expected from the differences in the mobilities of the ions and in the viscosities of t 5 the solutions. However, it would take us to far too treat this subject further. 4. Yenji2atioion 6y Conductivity Measurements.-The limiting formula F I ~ . I. for equivalent conductance may be written simply : A = A , - aJc where r = twice the ionic strength = (zl + z2)p; 8 N. Bjerrum, K K ~ . Danske Vidensk. Selsk. Mat-lfjs. Medd., VII., g (1926).L. ONSAGER 345 w has the value given by (8). For water at 25' C. : For methanol at 25' C. : When the solvent is water of 18" C., we have : a = o*270wA0 f 17.85 (21 + 22).a = 0'274who i- 21*14(z, + z2). a = 1.1 jTOAo + 56.0(2, + 22). Fig. I shows the general behaviour of binary monovalent electrolytes in water, the conductivities of four electrolytes being presented as functions of J.211. The slope at zero concentration is given by the simple equation : a = 35.7 + o ' I ~ ~ A , , The individual differences of these electrolytes are chiefly given by the limiting slopes, HC1 showing the greatest absolute and the smallest relative decrease of A with increasing concentration, LiIO, the greatest relative and the smallestybsolute de- crease. However, in greater concentrations, other influences than that of interionic at- traction are quite ap- preciable, so, for instance when p > 0.5, LiIO, shows a greater A,, - A than KCl.For verification on a larger range of material, we will consider the con- ductivity measurements of Kohlrausch already adapted by Debye and Hiickel, only omitting those salts which are suspect of hydrolysis. From the conductivity values for the greatest dilutions, the coefficients of formula A = A , - aJT+ pr have been calculated by the method of least squares. As Debye and Huckel use somewhat different notations. we 130 125 120 I have partly transformed their figuves into our terms. Table I. gives the a values (uexp.) calculated in the way described from A values at concentrations p = O*OOOI, o*oooz, 0-0005, 0.001, 0.002, and 0.005, compared with the atheor. computed from A,. The deviations are within the possible limit of experimental error.Fig. z illustrates the effect of association. As the influence of associa- tion, in great dilution, is proportional to the concentration, it does hot346 REVISION OF THE CONDUCTIVITY THEORY LiCl . . . . LiIO,. . . . LiNO,. . . . NaCl . . . . NaIO,. . . . NaNO, . . . KCL . . . . KBr . . . . X I . . . . KIO, . . . . KCIO,. . . . KNO,. . . . KCNS. . . . CSCl . . . . AgNO, . . . TINO,. . . . TABLE I. 98.93 67'35 35'24 108.89 77'42 105'34 129.93 132'04 130.52 98-41 119.47 126'41 121'04 133.08 115.82 127'55 I aexp. atheor. ~ 'exp. - "theor. 6.0 1'9 5 '4 1 '7 3 '4 5.8 3'5 5'5 - 5.0 2.8 3'5 9 '9 - 0'9 - 3.2 8.3 7'4 affect the limiting slope, which is practically the same for 11 the four salts .considered ; with increasing concentration, the individual differences become thallium salts is not common among uni-univalent salts in water.Table 11. gives the experimental a values for uni-bivalent salts in water, deduced by Debye and Hiickel from the A values found by Kohlrausch for = O*OOOI, 0'0002, o'oooj, 0'001, and 0.002; atheor. is given by the equation : a = 5 3 . 5 5 i- 1'084-Ao; q I + %fi A0 q = ~ A, + I,' here : when l, is the mobility of the univalent ion. TABLE 11. Ba(NO,), . . . . 'Sr(NO,), . . . . CaCI, . . . . Ca(NO& . . . . MgCI, . . . . K,SO, . . . . 116.95 113.42 116.69 113.56 110.88 132.23 aexp. 92'8 97'8 88.0 97'4 83 -2 81.0 ~~ "theor. h6.9 85 '5 86.1 85.6 84'1 92.1 Diff. + 5'9 + 12.3 + 1.9 + 11.8 - 0'9 - 11.1 The deviations from theory are on the whole somewhat greater than for uni-univalent salts, probably for the reason that only five A values could be used for the calculation of acxp.instead of six values as for the salts in Table I. For clarity, some of the curves have been displaced in the figure. Fig. 3 gives the conductivity curves of the uni-bivalent salts.L. ONSAGER MgSO, . . CdSO, . . 347 '1 0. theor. Diff. 125'6 ! 9s'3 114'70 242'7 j 144.4 115.81 270'7 ~ 145'1 For bivalent binary electrolytes in water Utheor of the formula : A = ho - a,/@ u = 71.4 + o.636ho. i s given by the equation : 130 120 110 100 FIG. 3. Table 111 is analogous to the former ones, a and A being calculated from the A values found for p = O+OOOI, O'OOOZ, o'oooj, 0'001 and 0'002 by Kohlrausch. TABLE 111. The discrepancy is so great that it falls far beyond the limit of experi- Debye and Huckel have suggested that the concentrations mental error.348 REVISION OF THE CONDUCTIVITY THEORY may be too great for extrapolation ; indeed, the A value extrapolated for MgS04 differs considerably from the value I 13.10, given by the mobilities of magnesium and sulphate ions, as found from the A values of MgCI, and K,SO,.Fig. 4 gives the con- ductivity curve of MgSO,, showing that an extrapolation 115 I I0 I05 100 95 A 0'02 0'04 0.06 0.0s FIG. 4. with thetangent 113.1 1 113'4 JG can be carried out with- out violation. The extrapola- tion tangent of Debye and Hiickel is also given for com- parison. The rapidly increasing slope indicates an appreciable associ- ation ; the inversion tangent is given by the dissociation equili- brium. As regards non-aqueous solutions, we shall consider the measurements of methan- olic solutions at 25' C.recently published by Frazer and Hart- ley. They work on dilutions be- tween f i - 10-1 and p= 2-10-3 and find that the square-root formula holds for that range of concentrations. So, they evalu- ate the measurements graphi- - - tally, plotting the A values against & and drawing a straight line. Table IV. gives a comparison between the experimental a values and the theo- retical ones, given by the equation : a = IIZ + 0*674& TABLE IV. LiCI . NaCl . KCI . RbCl . CSCl I KF . KBr . KI . NH,CI . HCl . LiNO, . NaNO, . KNO, . RbNO, . CsNO, . AgNO, . NaBr . NaOCH:: NaClO, . 90'90 96.95 108.65 I 13.60 94'0 '09'35 114.85 193'5 100.25 106'45 114'55 118.15 105'05 111'00 122.95 112'95 101'5 I 15-10 98.40 aexp. - atheor. - I5 - 14 4 - 2 + I4 i- I2 - 8 + I + 5 + o -I- I7 - 3 + 20 + 55 + 59 + 73 4- 131 - I0 - 22 + 8L. ONSAGER 349 Most of the differences are to be con- sidered as real deviations from the limiting law, as no second-order term has been accounted for, and it is probable that most of the electrolytes are associated to a greater or less ex- tent. Fig. j shows the conductivity curves of NaCl and AgNQ extra- polated according to the theory. The curve for AgN03 shows distinct deviation from a straight line course ; it is rather similar to that of TIC1 in aqueous solution. The chief improvement given by the new theory consists in the elimina- tion of all arbitrary constants from the limiting formula. Thus, on one hand, we can put the theory to a much more rigorous test ; on the other hand, we can with considerably greater cer- tainty trace and determine other in- fluences than those of the electric forces which the theory involves. I10 I00 90 0.02 0.04 0.06 FIG. j.