The Influence of Electrolyte Concentration on the Ratio of

T H E INFLUENCE OF ELECTROLYTE CONCENTRATION ON T H E RATIO OF ELECTROSMOTIC TO ELECTROPHORETIC MOBILITIES H. L. WHITE, BETTY MONAGHAN,

AND

FRANK URBAN

Department of Physiology and Department of Biological Chemistry, Washington University School of Medicine, Saint Louis, Missouri Received August 30, IQSd

Within the past few years it has become evident that the theoretical treatment of electrophoresis and electrosmosis of Helmholtz, Perrin, and von Smoluchowski must be considerably amplified. The expression developed by these authors is as follows:

where V represents either electrophoretic or electrosmotic velocity, X is the applied field, D the dielectric constant, { the electrokinetic potential, and 9 the viscosity of the medium. According to this formulation, the electrophoretic migration velocity of a particle through a given liquid should equal the velocity of flow of the liquid past a fixed surface of the same material as the particle, irrespective of the size or shape of the particle or the nature of the liquid, provided 5 remains constant. In 1924 Debye and Huckel (10, 16) published a new formula for electrophoretic mobility,

in which K varies as the shape of the particle, having a numerical value of 6 for spherical particles. However, Mooney (19) and Henry (15) pointed out that Debye and Huckel neglected to allow for the distortion of the electric field caused by the particle, a factor which is negligible only when the diameter of the particle j the thickness of the double layer. The von Smoluchowski formula is valid when the ratio of particle diameter to double layer thickness 5 600. (Muller (20) misquotes Henry on this point.) For particles of about 1,u diameter the Debye and Huckel formula, according to Henry, should apply in pure water; in solutions more concentrated than M/100 the von Smoluchowski formula applies, and in intermediate concentrations the factor K should vary between the limits 611

612

WHITE, MONAGHAN AND URBAN

set by the two formulas, Le., between 6 and 4 for spherical particles. It will be shown later’that under the influence of this effect the factor 6 is approached but never attained. A second consideration, the distortion of the charge distribution in the double layer caused by the imposed field, may also affect electrophoretic velocity. Muller (20) points out that if the double layer is very thin, the disturbing effect of an applied field of the order of a few volts per centimeter is negligible, since the electric field of the double layer itself is over 1000 volts per centimeter. However, if the double layer is very thick, the outermost ions are in a weak field and hence may be influenced by the external E.M.F. Muller, assuming that this effect is analogous to the retardation of moving ions due to the asymmetrical surrounding field discussed by Debye and Huckel (9), concludes that electrophoretic velocity may be considerably diminished in dilute solutions because of this factor. However, no quantitative expression for this effect on the retardation of electrophoretic velocity is yet available, and indeed Mooney (18) speaks of polarization increasing the velocity of the particle, although this enhancing effect now seems improbable. Aside from these suggested amplifications of the Helmholtz-Perrin-von Smoluchowski formula, there is the further possibility that the c-potential of a small particle may differ from that of a flat surface. This may occur either by a change of charge density or of double layer thickness or of both.

where u is charge density and X is the effective thickness of the double layer. Following Muller’s treatment in Abramson’s paper (4) A=--- 1

Kr

K1+Kr

where K is a function of ionic coneentration and r is the radius of the particle. According to this treatment the double layer thickness will vary significantly with r when Kr 6 5 . This predicts a decrease of double layer thickness, of r, and of electrophoretic mobility with decreasing radius of particle. It also predicts that with increasing electrolyte concentration “the size effect is shifted to smaller radii or disappears.” The dependence of the adsorbed charge density on the radius of the particle is discussed by Abramson (4). According to his treatment the negative logarithm of the particle velocity should be inversely proportional to the particle radius. It may be pointed out that, unlike the influence of particle radius on double layer thickness, the influence of radius on charge density should operate in all concentrations up to that in which charge

INFLUENCE O F ELECTROLYTE CONCENTRATION

613

density reaches its saturation value, which has been found by Abramson and Muller (7) to be for most materials a t about M/100 for ions not reversing sign of charge. Since, however, the vapor pressure is increased by only 1 per cent above normal with droplets of 2 p diameter and by only 10 per cent with droplets of 20mp diameter, this effect will be negligible with particles in the microscopic range, if the analogy between vapor pressure and solution pressure holds. From the foregoing considerations it may be predicted (1) that in solutions more concentrated than M/100 electrophoretic velocity should be independent of the size and shape of the particle, except for extremely small particles, and should be numerically the same as electrosmotic velocity over a flat surface, and (2) that in solutions less concentrated than M/100, electrophoretic velocity may become less than electrosmotic velocity. For a given solution the ratio, R, of electrosmotic velocity to electrophoretic velocity should increase as the radius of the particle decreases; for a given particle size, R should increase as the concentration of the solution decreases. The absolute magnitude of R for a given particle size in a given solution cannot be predicted, but one should be able to demonstrate qualitatively that changes in R take place under the conditions indicated. Mooney (18) made the first experimental attempt to compare electrophoretic with electrosmotic velocity on the same substance. He found that the mobility of oil droplets in distilled water increased with diameter. The diameter above which no further increase in mobility occurred was established in a later paper (19) as about 1 0 0 ~ . The influence of droplet size on mobility was reduced or abolished by the addition of electrolyte. He compared electrosmotic with electrophoretic velocity in distilled water and in 4 X M copper sulfate. In water the velocity of the small droplets was less than that of electrosmosis; that of the larger (about 401) was the same as or greater than electrosmosis. The finding of electrophoretic > electrosmotic velocity is, even for the larger particles, probably in error. I n the copper sulfate solution the velocities of all droplets were equal and almost as great as that of electrosmosis. Mooney (19) added further data and compared the experimental findings with the limiting slope of his theoretical mobility-diameter curve. In general, he found the expected correlation between mobility and droplet size in the less concentrated solutions, while in the more concentrated solutions droplets of all sizes moved a t approximately the same velocity, as predicted. However, a rather wide scattering of the experimental points and the anomalous effects mentioned by Mooney indicated the desirability of similar experiments on a more stable system. Aside from the work of Mooney there appears to be no direct experimental evidence in support of the theoretical predictions mentioned above. On the contrary, Abramson (4),after reviewing the experimental work of

614


himself and others, concludes that, when the surfaces compared are chemically identical, (1)electrophoretic mobility is independent of particle size, and (2) electrophoretic and electrosmotic velocities are identical. However, a further investigation of the data reviewed by Abramson brings out the fact that in practically every case experimental conditions were such that no size effect would be expected, i.e., either the solutions used were too concentrated or the range in particle size investigated was not sufficiently wide. For example, Abramson (2) found that protein-coated particles of microscopic size have the same migration velocity as the ultra-microscopic protein micelles investigated by Svedberg and Tiselius (23), and Abramson (3, 5) and Dummett and Bowden (11) demonstrated that the migration velocity of protein-covered particles is the same as electrosmotic velocity on a protein-covered surface of zero or small curvature. Since, however, all of these observations were carried out in concentrated solutions (usually M/50 buffers) it is evident that no differences in velocity would be expected. The same objection applies to the experiments of Hardy (14) on protein sols in N/50 acetic acid, of Abramson (1) on blood platelets and their aggregates in plasma, of Freundlich and Abramson (12) on blood cells and their aggregates in serum, of Prideaux and Howitt (21) on egg albumin in M/50 buffer, of Abramson and Michaelis (6) on droplets of Nujol, castor oil, benzyl alcohol, and cocoa butter, bare and protein-covered, in electrolyte solutions, and probably also of Freundlich and Abramson (13) on quartz particles in sugar solutions, since considerable electrolyte must inevitably have been added with the sugar. The observations of McTaggart on gas bubbles (17) in water are confined ) the maximum velocity had probably t o bubbles so large (60 to 1 6 0 ~ that already been attained by the smallest. Burton’s experiments with submicroscopic silver sols (8) are not decisive, since there is little evidence that his variations in methods of preparation actually produced sols of different sizes. van der Grinten (24), whose data have been recalculated by Abramson (4),found the electrophoretic velocity of small particles of glass in distilled water t o be very much less than electrosmotic velocity at a flat surface of the same glass. The value for R, as recalculated by Abramson, varied between 2.2 and 2.8. Abramson, however, dismisses this observation with a suggestion that the surface of a broken glass particle is probably chemically different from a fused or polished surface and hence would not be expected to have the same S-potential, and later (5) points to the measurements of Sumner and Henry (22) as evidence that when the glass surfaces are identical, R is practically unity. Sumner and Henry, comparing the electrophoretic velocity of a fused glass cylinder with electrosmotic velocity on a flat glass surface, found an average value for R of 1.14. However, since these experiments were not performed in water, but in

615

INFLUENCE OF ELECTROLYTE CONCENTRATION

M phosphate buffer, and the cylinder used was 2Op in diameter and 20 cm. long as compared with particles about 3 p in diameter studied by van der Grinten and by Abramson, it is evidently not necessary to suppose that the fused surface studied by Sumner and Henry was chemically different from the broken surfaces of van der Grinten and of Abramson in order t o account for the difference in the value of R. I n view of the fact that with the exception of Mooney’s work there are no published experiments which furnish even a qualitative test of the late theoretical developments, it seemed desirable to compare the electrophoretic mobility of small particles with electrosmotic velocity on a flat surface of the same material, over a wide range of concentrations, using, in place of the oil emulsion studied by Mooney, solid materials with more stable and easily equilibrated surface properties. It would, of course, be desirable to vary the particle size as well as the concentration, but this introduces great experimental difficulties, and hence has not been attempted in the present work. I n the present paper observations on the value of electrosmotic velocity on flat surface 13p in diameter

= (electrophoretic velocity of particles

)

for protein surfaces, found by Abramson and others to be close to unity in concentrated solutions, will be extended to more dilute solutions, and those on the value of R for glass surfaces, found by van der Grinten and Abramson to be greater than unity in water, will be extended to more concentrated solutions. EXPERIMENTAL

All experiments were carried out in flat electrophoresis cells of the Northrop-Kunitz type. It has been shown by von Smoluchowski (25)that in such cells true electrophoretic velocity is obtained by observing the particles at 0.211 of the distance from the bottom to the top of the cell. True electrosmotic velocity is the algebraic difference, particle velocity at wall minus particle velocity at 0.211 cell depth from wall, where movements toward anode and cathode are given opposite signs. When no movement occurs a t the wall the value of R is unity. Since for the purposes of this investigation only the ratio of electrosmotic to electrophoretic velocity was required, the applied field strength was not determined; the voltage drop in any solution was, of course, the same for electrosmosis as for electrophoresis (about 4 volts per centimeter). It was found necessary, in working with unbuffered solutions such as we used, to increase the length of each side arm to about 20 em. in order to avoid contamination of the cell contents from the electrode reactions. The glass powder was prepared by prolonged grinding of clean Pyrex tubing with a porcelain mortar and

616


pestle. A uniform suspension of any desired particle size can be prepared by fractional sedimentation; in the work reported here the diameter range was from 1 to 3 microns. The cell was illuminated through a dark field condenser and the particles observed through an ocular micrometer. There was no appreciable drift while the circuit was broken. Each recorded velocity is the average of five observations in each direction. A concentration of 0.01 g. of gelatin per liter was found adequate to coat completely the glass particles, as evidenced by the fact that the isoelectric point of the coated particles was the same as that of the gelatin (pH 4.75). The same concentration is sufficient to coat the cell also (as evidenced by the fact that electrosmosis reverses a t the gelatin isoelectric point), if this solution is passed through the cell slowly for approximately an hour, We thus confirm the finding of Abramson (2) and of Dummett and Bowden (11) that this procedure is adequate to insure complete coating. Since this concentration is probably about the minimum which will give a complete coating to the cell in a reasonable time, it was considered better to use 0.1 g. per liter for the present experiments. This solution was passed through the cell slowly for an hour before readings were begun. The salt concentrations investigated were all made up in 0.1 g. per liter gelatin solutions, and each solution remained in contact with both particles and cell for a 15-minute period before observations were begun. It is necessary that the cell be carefully cleaned in order that normal electrosmosis be exhibited. The criterion adopted of a normal electrokinetic potential at the cell wall was that the electrosmotic velocity in distilled water without protein be a t least 2.5 times the electrophoretic velocity with glass particles 1 to 3 p in diameter. I n practise, this ratio varied between 2.6 and 3.1,-a range of values which is consistently obtained with proper cleaning. The ratio for gelatin-covered surfaces in distilled water ranged from 1.9 to 2.3. These rather wide variations in the value of R in water and very dilute solutions are probably to be ascribed to differences in electrolyte content of different samples of distilled water, or, in the absence of protein, to variations in the electrical properties of the cell wall, or to both. Table 1 contains the results of experiments with bare Pyrex surfaces in varying concentrations of potassium chloride. These observations were carried out in a flat Pyrex cell. The sign refers to the pole toward which the particles were moving. Since in potassium chloride the particles are always negatively charged, they move always toward the anode at 0.211 of the distance from the wall. As previously stated, the velocity a t this level represents true electrophoretic velocity, V,. Since, a t the bottom of the cell, the particles moved always in the opposite direction if they moved a t all, electrosmotic velocity, V,, is the numerical sum of the observed velocities a t the two levels. The absolute magnitude of V , and V ,

617


in these experiments is not of quantitative significance, since the field strength may vary slightly from one solution to the next, but the value of R is independent of field strength. Similar data for gelatin surfaces are given in table 2. No attempt was made to maintain a rigidly constant pH, since the value of R is independent of pH; however, there was no great variation in pH among the different solutions. Table 2 also shows that the same values for electrosmotic velocity are obtained by observing the particles a t the wall as by the use of the equation V, = 2(Vt - V,). Similar observations on both bare glass and gelatin-coated surfaces in contact with thorium chloride are recorded in tables 3 and 4. The tetravalent thorium ion reverses the sign of charge on glass; the isoelectric concentration for electrosmosis and electrophoresis is approximately M . A t concentrations near the isoelectric point, the value of R 3 X is likely to show irregular variations, but at concentrations other than isoelectric, R approaches unity as the concentration increases. TABLE 1 Electrophoretic and electrosmotic velocities on bare Pyrex surfaces in potassium chloride MOLAR KC1 CONCENTRATION

5 5 1 1

x x x x

10-6 10-4 10-3 10-2

p PER SIlCOND AT 0.211 FROM BOTTOM =

+9.7 +9.9 +8.5 +9.0 +7.9

vp

PER AT BOTTOM

Ve

-16.4 -11.5 -5.9

26.1 21.4 14.4

2.69 2.16 1.69

-4.3

13.3

1.48

0

7.9

1.00

Thorium chloride in unbuffered solutions also reverses the sign of charge of gelatin (probably largely a pH effect) a t concentrations between 1 and 2 X 10-6 M . Here again the isoelectric point is the same for the particles and the cell, additional evidence that all surfaces are completely coated with protein. DISCUSSION

The data show that the unqualified statement that electrophoretic velocity equals electrosmotic velocity provided identical surfaces are used is not correct. The equality holds only in relatively concentrated solutions. Conversely, the view that the electrophoretic velocity of unfused bare glass particles must be different from the electrosmotic velocity a t a fused flat surface is not correct; the velocities are different in dilute solution (which is also true for protein-covered surfaces) and become the same in relatively concentrated solutions. At a concentration between loF3and M the velocities become equal with both bare and protein-covered

618


TABLE 2 Electrophoretic and electrosmotic velocities on gelatin-coated glass surfaces i n potassium chloride 0.1 g . of gelatin per liter

+

MOLAR KCI CONCENTRATION

pPERlECOND AT 0.211 F R O M BOTTOM= v p

He0

+5.38 +5.03 $4.78 +3.89 +2.08

5 x 5 x 1x 1x

10-6 10-4 10-3 10-2

PHIR sEcoND AT BOTTOM

VS

-5.50 -3.79 -1.47 -0.81 0

10.9 8.82 6.25 4.70 2.08

2.02 1.75 1.31 1.21 1.00

Comparison of two methods f o r obtaining electrosmotic velocity

-

MOLAR

KCI

CONCBNTRATION

vs

(Vi -

LEVBL I N C l L L

v,

VS

AVERAQE

R=Ke VP

--He0

-5.74 TOP +7.94 0.211 from top f15.1 +7.65 -5.7 Middle 0.211 from botton +7.36 Bottom -5.74

' 13.4

14.9

14.2

1.86

10-6

-5.71 TOP 0.211 from top +8.44 f14.5 +7.98 -5.7 Middle 0.211 from botton $7.52 Bottom -5.80

13.7

13.0

13.3

1.67

10-4

-4.571 TOP 0.211 from top +8.62 Middle t 1 4 . 5 $8.20 -4.91 0.211 from botton +7.78 Bottom -5.21

13.1

12.6

12.85

1.57

10-5

TOP 0.211 from top $6.43 Middle t10.0 +6.35 -1.5 0.211 from botton +6.27 Bottom -1.58

7.9

7.3

7.6

1.20

10-2

0 TOP 0.211 from top +3.70 +5.26+3.52 Middle 0.211 from botton $3.33 0 Bottom

3.52

3.48

3.50

0.99

I

0

surfaces; a t lower concentrations electrosmotic velocity is greater than electrophoretic. The findings are in accord with the theoretical points discussed in the first part of the paper.

619


We may consider briefly the various factors which might be responsible electrosmotic velocity In the for the departure from unity of the ratio electrophoretic velocity' discussion the dielectric constant and viscosity are assigned normal values. The maximum variation in R, due to variation in K between 4 and 6 as determined by variations in the ratio of particle radius to double layer thickness, could be only between 1 and 1.5, as discussed by Henry (15). Since we have observed R as great as 3.2, this factor alone cannot explain the findings. Moreover, it can be shown that K can never become as great as 6 . Henry states that K = 6 when double layer thickness, X, TABLE 3 Electrophoretic and electrosmotic velocities o n Pyrex surfaces in thorium chloride In this series the long side arms on the electrophoresis cell had not yet been introduced, hence the field strength was greater MOLAR ThCh OONCBNTRATION

p PBR BECOND 0.211 FROM BOTTOM = v p

HzO

+16.9 +14.1 -21.4 -28.7 -23.6

1 1 1 1

x x x x

10-6 10-4 10-3 10-2

p P E R SECOND AT BOTTOM

-34.0 -27.8 +lO.O 0 0

50.9 41.9 31.4 28.7 23.6

3.01 2.99 1.47 1 .oo 1 .oo

TABLE 4 Electrophoretic and electrosmotic velocities o n gelatin surfaces in thorium chloride 0.1 g . of gelatin per liter

+

MOLAR ThClr CONCENTRATION

HzO

1 1 1 1

x 10-5 x 10-4 x 10x 10-2

C P E R SECOND

0.211 FROM BOTTOM = VD

p P E R SECOND AT BOTTOM

VS

+5.80 +6.14 -13.3 -10.8

-7.60 -6.26 $2.9 $1.7

13.4 12.4 16.2 12.5

-

-

0

2.31 2.02 1.22" 1.16 1 .oo

1 equals particle diameter. However, it is not permissible to substitute for X when K r

K

< 5 , according to Muller's treatment

(4). Thus, according 1 to Henry, K = 6 with a particle of 1 p diameter in pure water, since - = K

1p.

But according to the equation

x=-1 -

KT

K 1 + K T

(r

= particle radius), X is not identical with

-K1 but

equals'3.3 X 10-5 cm.

620


The condition that X equals particle diameter can never be attained, since when X = r, K = 0, i.e., in any solution possessing a finite ionic strength, h < r. Thus K must always be less than 6 even with the smallest particles in dilute solutions, although it can become 4 with large particles in more concentrated solutions, and the operation of this effect can vary R only between 1 and something less than 1.5. This leads to consideration of the effect of particle size on A, and therefore on { and on electrophoretic mobility. According to the equation A = - -1

Kr

K 1 + K T

1

h becomes significantly less than its normal value of - when

Kr

K

electrosmosis a t flat surface (r = a),X

=

-.1

7 5. For

1 Substituting - for X in the

K

K

condenser equation 4nuX

(=(u =

charge density, X = double layer thickness) for electrosmosis, and

r

1

+

D

KT

for X for electrophoresis, it follows that the ratio of electrosmotic 1 + K r

1

to electrophoretic mobilities equals -or -. KT

KX

For distilled water a t

25"C., with an estimated ionic strength of 1 X and a particle of 1p radius, K = 3.25 X lo4, K r = 3.25, and the ratio = 1.31. In other

words, electrophoretic mobility, due to the operation of this factor alone, should be 24 per cent lower than electrosmotic or than the electrophoretic mobility of a particle large enough to escape this effect, or than the electrophoretic mobility of a particle of 1p radius in a solution strong enough so that Kr > > 5 . In a 5 X M potassium chloride solution (ionic strength K = 2.3 X lo6, K r = 23.5, and the ratio = 1.04. The ratio =5 X drops to 1.01 or less, with unity as a limit, in solutions with ionic strength or greater. In pure water, with a theoretical ionic strength of 5 X of 1 X lo-', K = 1 X lo4, Kr = 1, and the ratio becomes 2. It is thus seen that the limiting, and in practise unattainable, value of the ratio for a particle of l p radius is 2; for smaller particles the ratio in water of ionic strength 1 X 10-7 could exceed 2. Under the conditions of our experiments the effect of this factor alone is practically negligible with all solutions other than distilled water; with smaller particles the effect should become prominent in solutions of ionic strength' greater than 1 X 10-e. 1 A point which seems obscure may be mentioned here. series expansion involving

K

is obtained (9) by a


621

The third factor, the influence of particle size on charge density and therefore on p , may be dismissed, since ( could be changed by only 1 per cent with a particle of 2 p diameter. The fourth factor, polarization of the double layer, must be invoked to account for the higher values of R we have obtained. The maximum combined influence, as seen in water with an estimated ionic strength of 1 X 10-6, of the first three factors just discussed may be estimated to raise R to not more than 2; further retardation of electrophoretic below electrosmotic mobility in water may be ascribed to polarization of the double layer by the impressed field, and this latter factor apparently is the predominant one in any solution of ionic strength greater than 1 X with particles as large as 2 p diameter. This is contrary to the suggestion of Mooney (18) that such polarization might enhance mobility. A retarding effect seems to us the more probable. SUMMARY

The influence of electrolyte concentration of medium on the ratio of electrosmotic to electrophoretic velocities with bare glass and gelatincovered particles of 1 to 3 p diameter has been investigated. The ratio is unity with either type of surface with a concentration between and M and at all higher concentrations. With lower concentrations the ratio increases progressively with decreasing concentration, reaching values above 3 in distilled water. A discussion of various factors which may contribute to this result is given. The work reported in this paper has been aided by a grant made by the Rockefeller Foundation to Washington University for research in science. 4

where all except the first two terms are neglected, giving - 2 ~ . This approximation

4

r ;

is justified only when - < 1, which is true only when # or < 25 millivolts. Neverkt 1 M potassium theless, when double layer thickness, A, is calculated as for 5 X

r

chloride a t a plane glass surface a t 20°C., where = 120 millivolts, a value of h = 1.47 X 10-6 cm. is obtained. We have recently obtained a specific surface conductivity value of 4.27 X mhos for 5 X lo-' potassium chloride a t a plane glass surface. From this one finds 1.8 X 1013 ions adsorbed per cm.2 of surface. If one substitutes the value of 6, potential, and U , electric charge density at surface, corresponding t o = 120 millivolts and ions per cm.2 = 1.8 X 1Ols, in the condenser equation

r

one obtains h = 1.19 X 10-8 cm. This is an experimentally obtained value and is subject t o no restrictions other than those of experimental error. In view of the above-mentioned restrictions implicit in K , this correspondence seems surprisingly close.

622


REFERENCES (1) (2) (3) (4) (5)

(6) (7)

(8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25)

ABRAMSON, H.: J. Exptl. Med. 41, 445 (1925); 48, 677 (1928). ABRAMSON, H.: Colloid Symposium Monograph, Vol. VI, 115 (1928). ABRAMSON,H.: J. Gen. Physiol. 13, 657 (1930). ABRAMSON,H.: J. Phys. Chem. 36, 289 (1931). ABRAMSON, H.: J. Gen. Physiol. 16, 1 (1932). ABRAMSON,H., AND MICHAELIS,L.: J. Gen. Physiol. 12, 587 (1929). ABRAMSON, H., AND MULLER,H.: Cold Spring Harbor Symposia on Quantitative Biology 1,29 (1933). BURTON, E. F.: Physical Properties of Colloidal Solutions, 2nd edition, p. 144. Longmans, Green and Co., London (1921). DEBYE,P., AND HUCKEL,E.: Physik. Z. 24, 185 (1923). DEBYE,P., AND H ~ C K E E.: L , Physik. Z. 26, 49 (1924). DUMMETT, A., AND BOWDEN, P.: Proc. Roy. SOC.London 142A, 382 (1933). FREUNDLICH, H., AND ABRAMSON, H.: Z. physik. Chem. 128,25 (1927). FREUNDLICH, H., AND ABRAMSON, H.: Z. physik. Chem. 133, 51 (1928). HARDY,W. B.: J. Physiol. 29, xxvi (1903). HENRY,D. C.: Proc. Roy. SOC.London 133A, 106 (1931). HUCKEL,E.: Physik. Z. 26, 204 (1924). MCTAGGART, H. A.: Phil. Mag. 27,297 (1914). MOONEY, M.: Phys. Rev. 23, 396 (1924). MOONEY, M.: J. Phys. Chem. 36, 331 (1931). MULLER,H.: Cold Spring Harbor Symposia on Quantitative Biology 1, 9 (1933). PRIDEAUX, E. B., AND HOWITT,F. 0.: Proc. Roy. SOC.London l26A, 126 (1929). SUMNER, C. G.,AND HENRY,D. C.: Proc. Roy. SOC.London 133A, 130 (1931). SVEDBERG, T., AND TISELIUS, A,: J. Am. Chem. SOC. 48,2272 (1926). VAN DER GRINTEN, K. : J. chim. phys. 23, 209 (1926). VON SMOLUCHOWSKI, M.: I n Graetz Handbuch der Elektrizitat, Vol. 2, p. 382 (1921).

The Influence of Electrolyte Concentration on the Ratio of

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