TRANSPORT NUMBERS OF CONCENTRATED SODIUM CHLORIDE SOLUTIONS AT 25"
2747
Transport Numbers of Concentrated Sodium Chloride Solutions at 25"
by L. J. M. Smits and E. M. Duyvis Koninklijke/Shell Exploratie en ProduWie Laboratorium, Rijawijk, The Netherlands (Received November 19, 1966)
Sodium transport numbers in NaCl solutions at 25O, from 0.024 m up to saturation (6.144 m), were obtained by measuring the emf of galvanic cells with transference. The values were higher than those obtained by Caramazaa (up to 5 m using the emf method) and those of Currie and Gordon (up to 2.5 m using the adjusted indicator technique), the latter observations being corrected for volume changes. For this correction an equation was derived that differs from an approximate equation obtained by Beannan, Haase, and Spiro. Our results are in excellent agreement with Stokes' theory over the entire concentration range. The applicability of this theory in concentrated solutions is discussed. Incorporation in Stokes' equation of a recent theory of electrophoretic retardation of Fuoss and Onsager leads to physically impossible results at higher concentrations.
Introduction To our knowledge, few data have been published on transport numbers for sodium chloride solutions at concentrations above 0.2 m (moles/kg of HzO). Currie and Gordon,' using the adjusted indicator technique, which is a modification of the moving boundary method, obtained values of the sodium transport number up to a concentration of about 2.5 m. By measuring the emf of galvanic cells with liquid junction, Caramama2 obtained values of the sodium transport number for concentrations up to 5.0 m. No sets of data extending to saturated solutions are available. We therefore performed measurements of the sodium transport number in NaCl solutions at 25" over a concentration range from 0.024 m to saturation (6.144 m). Like Caramaaaa, we used the emf method, which is most convenient for very concentrated solutions. Experimental Section The cell used was of the flowing junction type, similar to that described by MacInnes and YehSa It was difficult to obtain silver-silver chloride electrodes that were stable enough for measurements in very concentrated NaCl solutions. It was found that spongy electrodes with a large surface area performed satisfactorily. They were prepared in the following manner. Platinum wires 0.5 mm in diameter were coated at one end with a pear-shaped sponge of silver by elec-
trolysis in a 0.4 m AgN08 solution, applying a current of 125-150 ma for 1 min. Every 5-10 sec the current was interrupted by lifting the electrode out of the solution. The silver sponge thus formed was then heated in a blue flame until it just glowed, to give more rigidity to the system. This procedure was repeated three times. The electrode was then heated in an oven at 350" to burn off all possible contamination introduced by the flame. About 15% of the silver was then converted to silver chloride electrolytically in an NaCl solution by applying a current of 2.5 ma. The electrodes were aged by short-circuiting several of them when immersed in the solution until they had an asymmetry potential of less than 0.01 mv. This was normally attained within 1 hr. The electrodes thus prepared gave stable and reproducible emf readings within a few hundredths of a millivolt over a period of at least 2 days. For saturated NaCl solutions, however, the electrodes still deteriorated rapidly, unless the solution was also saturated with AgC1. A sufficient amount of silver chloride could be dissolved by gently heating under reflux a solution, saturated with NaCl at 25", with an ex(1) D.J. Currie and A. R. Gordon, J . Phys. Chem., 64, 1751 (1980). (2) R.Caramazza, Gazz. Chim. Ztal., 90, 1839 (1980). (3) D. A. MacInnes and Y. L. Yeh, J. Am. Chem. SOC.,43, 2563 (1921). For solutions of the same single electrolyte, a flowing junction is not required. However, this apparatus waa already in use for experiments with more than one electrolyte.
Volume 70,Number 9 September 1966
L. J. M.
2748
cess of AgCl only. NaCl must not be present in excess, because then the silver ions in solution precipitate almost quantitatively during cooling, as NaCl and AgCl crystals are isomorpho~s.~Saturation of the solution with silver by adding silver nitrate could not be applied here because the nitrate ion has a measurable effect on the emf.5 Regarding the reliability of silver-silver chloride electrodes in concentrated solutions, Lengyel, Giber, and T a m h e discussed a possible effect of the presence of dissolved AgCl in LiCl solutions, up to 17 m, on the emf of galvanic cells with and without transference. They measured emf values of two kinds of cells with transference, one with silver-silver chloride electrodes and the other with lithium amalgam electrodes. Also, emf values of cells without transference were measured with the usual electrode arrangement of silver-silver chloride electrodes in the two electrolyte solutions and lithium amalgam separating the solutions. Combining the emf values of the different types of cells in three different ways and using isopiestic activities of LiCl, they calculated three sets of transport numbers. If all the electrodes used behave ideally and accurate values of activities a t their measuring temperature of 15” are available, then the three sets of transport numbers calculated from either method should be identical. Lengyel, et al., found this not to be the case at LiCl concentrations above 2.6 m. They concluded that this was due to the presence of dissolved AgCl in the neighborhood of the silver-silver chloride electrodes. Their experimental results, however, can equally well be explained by assuming that the lithium amalgam electrodes show deviations from ideal behavior. Rloreover, Lengyel and Giber6 showed that silver-silver chloride electrodes behave satisfactorily in HC1 solutions even a t extremely high concentrations up to saturation (14.7 m at 25”),where the solubility of AgCl is also very high (more than 0.2 mole % of the HCl’). This proves that any possible deviation of silver-silver chloride electrodes in LiCl solutions does not necessarily mean failure in other solutions. From the work of Harned and Nims,* who measured the emf values of galvanic cells without transference with NaCl solutions up to 4 m, and isopiestic activity coefficients given by Robinson and Stokesj9it appears that sodium amalgam electrodes and silver-silver chloride electrodes in NaCl solutions give emf readings that are correct within a few tenths of 1%. I n view of these considerations, we believe that silver-silver chloride electrodes in NaCl solutions do not give appreciable errors. I n our experiments, we measured the emf of a SUCcession of galvanic cells with liquid junction, keeping The Journal of Physical Chemistry
SMITSAND E. M. DUYVIS
the ratio of the molalities of the NaCl solutions a t each side of the liquid junction a t a constant value of 2, over a concentration range from 6.144 to 0.024 m. Each measurement was made with two electrodes on each side of the cell, the average of the four possible determinations of the emf being taken as the measured value. Two series of experiments were made; for the second one, freshly prepared electrodes and solutions were used. The cell potentials were measured with a compensator, accurate to 0.01 mv (Dr. C. E. Bleeker, N. V., Zeist, Holland, Type 2165). The measurements were carried out at a controlled temperature of 25 0.2”.
*
Results The measured values of the cell potentials were combined as Et(milm,) = E(milmz)
+
+
~(mz/nz3) . . . . .
+ E(m(,-&z,J
in which the E values are the emf values of the successive cells and ml = 0.02400. Thus for each concentration (m,) a value of Et was obtained. The potentials Et are the emf values of galvanic cells with liquid junction of the type
Pt, Ag, AgCl~NaCl(ml)~NaCl(m,) IAgC1, Ag, Pt The equation relating Et to the mean ionic activity (a*) of the electrolyte islOsll dEt =
-ttNa
,2RT - X 2.3026 d log a* F
(1)
where t ~ a his the sodium Hittorf transport number. The experimental data of Et vs. log a* can be ac(4) C. Schierholz, Sitzber. KI. Alcad. Wiss. Wien, 101, 2b, 4 (1892). (5) This was shown by an increase of about 0.1 mv of the liquid junction potential between the saturated solution and a 3 m NaCl solution upon addition of NaN03 to the saturated solution, the number of moles of NaN03 added being equal to the number of moles of silver nitrate already present in the saturated solution. (6) S. Lengyel, J. Giber, and J. Tam&, M a w . K e n . Folyoirat, 6 6 , 161 (1960); S. Lengyel and J. Giber, Acta Chim. Acad. Sci. Hung., 32,235 (1962); 9.Lengyel in “Electrolytes,” B. Pesce, Ed., Pergamon Press Ltd., London, 1962, p 208. (7) A. Seidell, “Solubilities of Inorganic and Metal Organic Compounds,” Vol. I, 4th ed, W. F. Linke, Ed., D. van Nostrand Co., Princeton, N . J., 1958. (8) H. S. Harned and L. F. Nims, J. Am. Chem. SOC.,54,423 (1932). (9) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworth and Co. Ltd., London, 1959. (10) The derivation of eq 1 has been given, for instance, by Guggenheim (ref 11, pp 456-458). It has to be noted that this derivation, especially the transition from his eq 14.06.5 to eq 14.06.11, is valid only if Hittorf transport numbers are used. These transport numbers are defined with ionic velocities taken with respect to the solvent. (11) E.A. Guggenheim, “Thermodynamics,” 4th ed, North-Holland Publishing Co.,Amsterdam, 1959.
TRANSPORT NUMBERS OF CONCENTRATED SODIUMCHLORIDE SOLUTIONS AT 25'
curately represented by a quadratic equation, the coefficients of which are calculated by the method of least squares
-Et = 76.05
+ 44.01 log U A - 0.6977 log2
2749
0.398
0.394
(2)
The results of the measurements and the values calculated from the empirical eq 2 are given in Table I.
Table I : Potentials ( E t )of Galvanic Cells with Liquid Junction between NaCl Solutions a t 25". Comparison with E q 2 Concn, moles/kg of H10, mn
0.02400
0.04800 0.0960 0,1920 0,3840 0.7680 1.536 3.072 6.144
-Et, mv Log aia
First expt
Second expt
-1.683 -1.403 -1.126 -0.848 -0.573 -0.292 0.0036 0.3436 0.7897
0 12.97 25.74 38.31 50.87 63.37 76.35 91.15 110.59
0 12.84 25.59 38.00 50.46 e2.92 76.06 90.88 110.21
---
I
Calod,
t
eq 2
0.005 12.93 25.61 38.23 50.60 63.14 76.21 91.09 110.37
' The values of u*
were calculated from the mean ionic activity coefficients given in ref 9.
0.318l
I
I
'
0.04
I
'
aoe
I
I
012
'
I
0.16
'
I
a20
mNaCl(mohs/kg HIOl
Figure 1. Hittorf transport numbers of the sodium ion in NaCl solutions a t 25' (low concentrations).
0'40r 0.39
From eq I and 2 it follows for the concentration range from m = 0.024 to 6.144 that t ~ a h= 0.3720
- 0.0118 log
(3)
0.31 X
I n Figure 1, the values of tNah are plotted against in for concentrations up to 0.2 m; they show close agreement with the results obtained by Longsworth (ref 9, p 158) and Allgood, et al. (ref 9, p 158). I n Figure 2, the values of tN> are plotted over the concentration range up to 6.144 m together with the values obtained by Currie and Gordon' and Caramazza. We cannot evaluate the significance of the difference between our results and those of Caramazza as the accuracy of his measurements is not known to us. A closer examination (see next section) of the adjusted indicator technique, used by Currie and Gordon,' is required before a comparison with their results can be made.
Discussion of the Adjusted Indicator Technique The adjusted indicator technique used by Gordon, et aZ.,1V1* is a variation of the moving boundary method for measuring transport numbers. I n the latter method, the movement of a boundary between two electrolyte solutions having one ion in common (e.g., KC1 and NaC1) is measured while an electrical field is applied across this boundary. When p coulombs have
0.36
-
0,3S0
I
I
I
2
3
4
m:,Cl(molei/&
I
H,O)
Figure 2. Hittorf transport numbers of the sodium ion in NaCl solutions at 25" (up to saturation).
passed, the boundary has passed a volume V liters of the cell. The following equations then obtain CKV=
ptKa/F
(4)
and C'NaV = p t N a a / F (5) where CK denotes the normality of the K + ions in the KC1 solution originally present in V; C'N* is the normality of the Na+ ions present in V a t the end of the (12) D. R. Muir, J. R. Graham, and A. R. Gordon, J. Am. Chem. SOC.,76, 2157 (1954).
Volume 70,Number 9 September 1966
L. J. M. SMITSAND E. M. DUYVIS
2750
experiment which may differ from the initial Na+ ion concentration. Thus
The transport numbers in eq 4,5, and 6 are apparent transport numbers (ref 13, p 218) and not Hittorf transport numbers, the movement of the boundary being taken with respect to the wall of the cell and not with respect to the water, which is also moving in the cell. Under the experimental conditions the concentration of the following solution (in this case NaC1) adjusts itself in such a way that eq 6, which is known as the Kohlrausch regulating function, is obeyed. This adjusted concentration is measured in the adjusted indicator technique to determine the ratio of the two cation transport numbers. From this ratio and the known value of one of the cation transport numbers, the other cation transport number can be calculated. This technique has been improved by Gordon, et aLI1r1*to such an extent that accurate experimental results can be obtained. Accurate values of potassium Hittorf transport numbers in KCl solutions up to 3 N were obtained by MacInnes and Dolei4 using the Hittorf method. These values were employed by Currie and Gordon' to determine the sodium transport numbers in NaCl solutions. However, using eq 6, they actually obtained the ratio between the apparent transport numbers instead of the ratio between the Hittorf transport numbers. They tested their procedure a t low concentration (0.1 N KCI) against the moving boundary method and found satisfactory agreement. However, a t this low concentration the difference between apparent and Hittorf sodium transport numbers is very small and within their measuring accuracy. Application of eq 6 leads to erroneous results, however, a t higher concentrations. BearmanI5 and Haase,16 treating the theory of the moving boundary technique, and Spiro," dealing with the adjusted indicator technique, corrected the Kohlrausch regulating function (eq 6). They arrived by different routes at the same equation for the ratio of Hittorf transport numbers, which can be written as h
_ 2Kh tNa
C'Ne/C'wS'w CKICWGW
-
P'Na
Px
(7)
where cWand 6, are the molar concentration and the partial molar volume of the water in the KCl solution, respectively, and c t w and gw are the corresponding quantities in the adjusted NaCl solution. Thus P ' N and ~ PKare the electrolyte concentrations in moles per unit partial volume of water. The Journal of Physical Chemktry
Spiro" corrected Currie and Gordon's results' using eq 7 (see Table 11). Table I1 : Sodium Transport Numbers at 25" from the Adjusted Indicator Experiment' tNah
r
"aC1,
moledkg of Ha0
CN~CI,
moles/l.
0,201
0.2024 0.3919 0.7845 1.6018 2.4525
0.388 0.771 1.550 2.333
Currie and Gordon'
Eq 6
Spiro'? Eq 7
Eq 8
0.3825 0.3794 0.3762 0.3771 0,3776
0.3811 0.3766 0.3704 0.3649 0.3586
0.3815 0.3767 0.3706 0.3654 0.3596
Of the approximations made by the above-mentioned authors, the only one that influences the equation for the ratio of Hittorf transport numbers is the neglect of the change in partial molar volume of water when the water, upon passing the boundary, changes from being the solvent of KC1 to that of NaCl. We have derived, however, a rigorous expression in which this change in partial molar volume of water is also taken into account. I n Figure 3, the situations a t the beginning of the experiment (left) and at the end (right) are shown. Let us consider two reference planes I and I1 moving with a velocity such that the water transport through them is zero. During passage of p coulombs, these planes move from positions I and I1 to positions I' and 11', respectively. The reference planes are chosen such that the boundary between the NaCl and KC1 solutions is originally located in I and the final position of the boundary coincides with 11'. By definition, the transport numbers of the ions related to these reference planes are Hittorf transport numbers as no water passes through these planes. During the passage of p coulombs, p t N , h / F moles of sodium ions pass through the first reference plane into the volume between the reference planes and p t ~ ~ / F moles of potassium ions are swept out of this volume through the second reference plane. By definition, the amount of water between the two reference planes remains constant. If this amount is equal to G kg, then (13) H. 8. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed, Reinhold Publishing Corp., New N. 1958* (14) D. A. MacInnes and M. Dole, J . Am. Chem. Soc., 53, 1357 (1931). (15) R. J. Bearman, J . Chem. phys., 36, 2432 (1962). (16) R. Haase, Z. ~ h y s i k Chem. . (Frankfurt), 39,27 (1963). (17) M. Spiro, J . Chem. Phys., 42, 4060 (1965). y.i
TRANSPORT NUMBERS OF CONCENTRATED SODIUM CHLORIDE SOLUTIONS AT 25 O
2751
Figure 2 shows that our observations are higher than those of Caramassa and the corrected data of Currie and Gordon.
NaCl
Comparison with Stokes’ Theory
-----
I’
,KCl
Ns C I
---P‘ K Cl
Start
Finish
Figure 3. Movement of boundary and reference planes, the latter moving a t water velocity.
the initial molal concentration of potassium ions, mg, is given by ptKh mK = FG and the final molal concentration of sodium ions, m‘Na, in the following solution is given by
Thus
Thus the ratio of the Hittorf transport numbers is given by the ratio of the molal concentrations (moles/kg of H20) of NaCl and KCl. From the molar concentrations of KC1 and NaCl reported by Currie and Gordon (ref 1, Table I), we calculated the corresponding molal concentrations using the densities of the solutions given in the “International Critical Tables” (Vol. III,79,87). The sodium Hittorf transport numbers were then calculated from eq 8 and are shown in Figure 2 and in Table 11. As can be seen in Table 11, the approximation in the derivation of eq 7 gives rise to an error that increases with increasing concentration. Up to a concentration of 2.5 m NaCl, this error does not exceed 0.3%.
On the basis of the Fuoss-Onsager theory of the conductance of electrolyte solutions, Stokes (ref 9, p 156 and ref 18) developed a theoretical expression for transport numbers. Stokes found that for 1:1 electrolytes his equation is in excellent agreement with observed data. Generally, however, these observations were made up to moderate concentrations only, but for some concentrated solutions too (HC1, up to 3 N , and LiCl, and KC1, up to 1 N), Stokes found excellent agreement. It is interesting to know how far his equation applies in concentrated NaCl solutions. Simplified for 1:1 electrolytes, Stokes’ equation is
t+ =
X P - l/&zd?/(l A’ - B z a / ( l
+
+
x.)
xa)
(9)
where a! is the mean ionic diameter and Ao and h+O are the limiting equivalent conductivities of the electrolyte and of the cation, respectively. Bz is the coefficient of the electrophoretic term in the theory of conductance of electrolyte solutions; this coefficient is inversely proportional to the viscosity and to the square root of the dielectric constant. The DebyeHiickel parameter, x , is inversely proportional to the square root of the dielectric constant. For the calculation of transport numbers from eq 9, it is of little importance for dilute solutions whether the viscosity and dielectric constant of the pure solvent or those of the solution are used to calculate B2 and x . I n concentrated solutions, however, this remains a problem. By calculation, we found that the variation of the dielectric constant in aqueous sodium chloride solutions affected the results only slightly, but that variations in viscosity produced significant effects.lg The value of a! used in our calculations was taken to be 5.2 A, as Stokes found that this value gave excellent agreement between observed and calculated transport numbers for dilute NaCl solutions up to 0.2 N . The transport numbers calculated with the viscosities of the appropriate solutions are higher than the experimental data. They show a minimum of 0.376 at a concentration of about 1.5 m, and curve upward to a value of 0.383 for the saturated solution (6.144 m). The results of the calculations when using the viscosity of pure (18) R. H. Stokes, J. Am. Chem. SOC.,7 6 , 1988 (1954). (19) Values of dielectric constants of NaCl solutions were taken from J. B. Hasted, D. M. Ritson, and C. H. Collie, J. Chem. Phys., 16, 1 (1948). The viscosities of NaCl solutions were taken from L. L. Ezrokhi, J. Appl. Chem. USSR,2 5 , 917 (1952).
Volume 70, Number 9 September 1966
2752
water agree closely with our experimental data (see Figure 2). For 3 N HCl and 1 N LiC1, the viscosity of the solutions (“International Critical Tables,’’ Vol. 111) is such that their use instead of pure water viscosity in Stokes’ equation does not appreciably influence the calculated transport numbem20 If we apply Stokes’ equation to concentrated solutions, however, some questions arise. The FuossOnsager theory of conductance of electrolyte solutions underlying Stokes’ equation is known to fail even a t moderate concentrations. This failure may be due mainly to the treatment of the relaxation effect, which does not appear in the theoretical expression for the transport number. However, it is hard to see how the electrophoretic effect can be described up to saturation by a theory that regards the medium in which the ions move as a continuum with properties independent of the electrolyte concentration. Regarding the influence of the viscosity of the solution on the calculated transport numbers, it does not seem justifiable to use this viscosity in the electrophoretic retardation terms and still use the limiting equivalent conductivities at infinite dilution. I n the theory of electrolyte conductance, the velocity of an ion is regarded as composed of two elements: (a) the velocity of the unhindered ion and (b) the velocity of the ionic atmosphere surrounding the ion (electrophoretic retardation), Both constituent velocities are affected by variations in the viscosity of the surroundings. If these velocities are both affected to the same extent as a result of varying concentration, all terms on the righthand side of eq 9 will vary by the same factor (cf. ref 18). This may be the reason why a close fit is obtained when the viscosity of pure water is used in Bz, the limiting equivalent conductivities ( x + O and -io)being determined at infinite dilution. At the higher concentrations, the question becomes important whether the transport numbers calculated with Stokes’ equation are in fact Hittorf transport numbers, i e . , whether the velocities of the ions are taken with respect to the solvent. I n the theory of electrolyte conductance, the velocities of the ions and of the ionic atmospheres are considered with respect to their “surroundings” - and it is not certain that water is not moving in this coordinate system. This question is Of course, when conductivities Of solutions are calculated from the ionic mobilities. ~ ~F~~~~and ~ onsagerZ1 ~ published ~ a revised ~ treatment of the electrophoretic retardation in solutions Of symmetrical electrolytes. With the aid of their eq 61, we modified Stokes’ equation for the transport number of 1:1 electrolytes accordingly. The Journal of Physical Chemistry
L. J. M. SMITSAND E. M. DUYVIS
t+ =
x+o - ‘/zS ho - S
where S = B Z d { l - ( x a b / 4 ) F ( b ) ] in which b = e2/aekT,e is the protonic charge, and E is the dielectric constant. S is the term describing the electrophoretic retardation of the ions. It turned out, however, that cation transport numbers calculated with eq 10 for NaCl solutions a t 25” showed very large deviations from the observed values a t concentrations higher than 0.11 m (Table 111). Beyond this concentration, the calculated transport numbers start to increase very sharply. Table 111: Comparison of Calculated and Obseived Transport Numbers of NaCl Solutions a t 25’ -+ t
xa
S, mhocmg/ equiv
Eq 10
Obsd“
Eq9
0.171 0.566 1.13 1.71 3.98
5.14 10 0 -31 -354
0.3918 0.387 0.396 0.417 0.473
0.3918 0.385 0.378 0.375 0.363
0.3918 0.385 0.378 0.374 0.366
m,
moles/kg moledl. of Ha0 e,
0.01 0.1095 0.434 1.00 5.416
0.01 0.110 0.440 1.02 6.144
’For m
=
0.01, ref 9; above 0.01 m, this work (eq 3).
For these calculations we used the viscosity and dielectric constant of pure water and the same mean ionic diameter (a = 5.2 A) as before.22 However, when other reasonable values of these constants are used, the general behavior of the transport numbers calculated with eq 10 is not changed. Table I11 shows that at concentrations above 0.11 m the electrophoretic retardation as represented by S decreases with concentration and even goes through zero a t 0.440 m. Since the latter represents a physical impossibility, eq 10 does not apply a t concentrations above 0.440 m. This is because Fuoss and Onsager considered only cases with x a negligible compared with unity. I n particular, in their approximation of the DebyeHuckel expression for the electrical potential, a factor xa) was omitted. This renders their final e”*/(1
+
(20) KC1 was omitted from this comparison. The agreement between observed and calculated transport numbers in KCl solutions cannot provide evidence for the correctness of the electrophoretic retardation term. This is because AOK is about equal to 1 / d o ~ c i and, as can be seen in eq 9, t~ is then about 0.5, practically indel ~ , and of the magnitude of the electrophoretic pendent of concentration retardation. (21) R. M. Fuoss and L. Onsager, J. P h w . Chem., 67, 628 (1963). (22) For these values b = 1.38; the corresponding value of F ( b ) is 2.57. This value was obtained by graphical extrapolation over a very short range of Table I of ref 21.
COUPLING CONSTANT AND CHEMICAL SHIFTOF TETRAFLUOROBORATE ION
results applicable to very dilute solutions only. I n Table I11 it cnn be seen that at NaCl concentration of 0.11 m, where XCY = 0.566, the comparison between observed transport numbers and those calculated with eq 10 is already less satisfactory. In the two other cases where Stokes' original eq 9 has been verified at high concentrations (3 N HC1 and 1 N LiCl), S is negative and eq 10 again does not apply. It thus appears that the older Fuoss-Onsager theory underlying eq 9 describes the electrophoretic retarda-
2753
tion effect in aqueous solutions much better than the recent revised treatment. The refinements in this revised treatment of electrolyte conductance will show up to full advantage if the restriction XCY
