Jan., 1945
THEDIFFUSION OF IONSIN SUPPORTING ELECTROLYTES
31
[CONTRIBUTION FROM STANFORD UNIVERSITY ]
The Diffusion of Ions in Supporting Electrolytes’ BY R. B. DEAN^
It has been recognized since the time of Nernst present a t the same concentration throughout the that electrical attraction between ions of opposite diffusing system. The term is already used in sign forces the separate ions of a single salt to polarography to refer to the electrolyte added to diffuse at the same rate in water regardless of carry the current up to the mercury drops. There differences in the individual ionic mobilities. The should be no confusion in the use of the term since electrical potential difference set up when a salt the supporting electrolytes have similar functions diffuses is the familiar diffusion potential or liquid in the two systems. The equivalent ionic mobilities that apply in j,unction potential. It is also well known that the diffusion potential can be reduced by the addition diffusion are in general greater than the correof a salt in high concentration that of itself pro- sponding equivalent ionic conductances calculated duces little or no diffusion potential such as from transference and conductivity measurepotassium chloride, potassium nitrate or am- ments since the electrophoretic effect and the time monium chloride. The diffusion potential can of relaxation effect are absent when both ions also be reduced by the addition of any electrolyte move in the same direction and a t the same velocin uniform high concentration to both sides of the ity. As a first approximation i t is usual to use the limiting ionic conductancet; a t infinite dilution diffusion layer. The corresponding effects produced by the ad- in diffusion equations such as Hender~on’s.~Howdition of salts to diffusing systems on the diffusion ever, when diffusion takes place in a supporting mobility of the separate ions have been compara- electrolyte the diffusing ions are not necessarily tively neglected. I t is generally realized (for ex- moving a t the same velocity. The effect of ample, see ‘) that the presence of a large excess of interionic attractions and changes in hydration salt will effectively liberate the diffusing ions probably outweigh the electrophoretic and time from electrical effects and allow them to diffuse of relaxation effects a t concentrations of supporta t almost the same velocity they would have if ing electrolyte usually employed. It would seem uncharged. Hartley and Robinson in 19316de- better therefore when possible to calculate the rived an equation for the true (or differential) ionic mobilities directly from transference and diffusion coefficient of an anion diffusing in a salt conductivity measurements of the diffusing salt solution of uniform concentration. They also a t the concentrations in question. The completely general determination of the pointed out that large errors are to be expected when a dye containing an impurity such as sodiuni integrated diffusion coefficient of one ion in a salt chloride diffuses into pure water. Vinograd and mixture cannot be solved without some assumpMcBain6 examined the diffusion of mixed salts tion as to the structure of the diffusing layers. into pure water both theoretically and experi- When diffusion takes place in bulk solution the mentally and derived an equation for the differ- concentration gradicnt varies continuously in ential diffusion coefficient of any ion in terms of time and space. The classical method as used by the mobilities, valences, concentrations and con- Oholnis as well as the modern scale method decentration gradients of all other ions present. vised by Larnmg is not directly applicable to subTheir equation neglects the changes of activity stances whose diffusion coefficients vary with and mobility with concentration but, as is pointed concentration. The Northrup diffusion cell,loimout by Hartley and Robinson,6 these properties proved by McBain and his studentsjBlends itself of the diffusing ions will change very little with more readily to measurements of substances concentration in salt solutions appreciably more whose diffusion coefficients vary with concentration. In these cells diffusion takes place a t concentrated than the diffusing ions. essentially a steady state through a sintered glass I. Theoretical Formulation rnembrane.l1 Under these conditioiis the integral The author proposes to use the term “supphrt- diffusion coefficient is defined as ing electrolyte” to designate electrolytes initially --K D = -.dQ __ (1) The material in this paper was presented at the Colloid Division of the American Chemical Society meeting in Cleveland, April 6,
1944. (2) Bristol Myers Co. Fellow in Chemistry, 1941-44. (3) Maclnnis, “The Principles of Electrochemistry.” Reinhold Publ. Co., New York, Pi. Y , 1930, p. 243. (4) Svedberg and Pederson, “The Ultracentrifuge,” Oxford University Press, New York, N. Y . , 1940, p, 23. (5) Hartley and Robinson, PYOC. Roy. SOC.(Londou), A134, 20
(1931). ( 6 ) Vinograd and McBain, THISJ O U H N A L , 63, 2008 (1941). In this reference, -n page 201 1. N sboiild he in moles. not in equivalents.
dt Ac
where dQ/dt is the rate of movement of the diffus-’ ing substance across the membrane, Ac is the difference in concentration across the membrane and K is a constant depending on the geometry ( 7 ) Henderson, 2. p h y s i k . Chon., 69, 118 (1907). (8) bholm, i b i d . , 70, 378 (1910). (9) Lamm and Poulson, Biochcm. J . , 30, 528 (1936,). (10) Nortlirup and Anson. J. Gcn. Physiol., 12, 543 (102‘5). (11) Barnes, I’hysics. 6, 4 ( 1 934).
R. B. DEAN
32
of the cell and determined from the diffusion of a reference substance such as 0.1 N potassium chloride. The units of concentration cancel if concentration is expressed on a volume basis. When a steady state of diffusioii has been reached across a inernbrane’2
‘J c r where D is the true or differential diffusion coefficient. Cl is usually taken to be zero unless otherwise stated.I3 Likewise
+ c (dD/dc)
D =
(31
The general equation for the diffusion of a given ion in a mixture of salts has been derived by Vinograd and McBain6 following the method of Nernst, and may be written D,G+
RTu, F 2 (G+
=
-
nA
nic
ZuiG+/ni
ZUC
(41 ’
D.+ is the diffusion coefficient of the cation, n+ its valence, u+ its equivalent conductance and G = dc/dx, the gradient of concentration of the ion with respect to distance, Again the units of concentration cancel out. The negative term inside the brackets is positive for anions. 1.0 . I ,
Vol. 67
It is in general impossible to integrate equation 4 when more than one salt is present without making some assumption about the concentration of the supporting electrolyte a t all parts throughout the region in which diffusion is taking place. If we were to assume that a steady state existed within the membrane and that the supporting electrolyte is free to move, then the integration, if possible, would probably resemble that obtained by Planck’! for the diffusion potentials and would probably be even more cumbersome to calculate. As a first approximation we can use the same assumptions that lead t o the relatively simple Henderson equation for diffusion potentials which in this special case amounts to the assumption that the supporting electrolyte does not move and therefore that G, remains zero for all ions of the supporting electrolyte and the necessary corollary that G+ = G- for the two ions of the diffusing electrolyte. G can then be removed from equation 4 leaving
where D+o =
RT U+ the free diffusion coefficient F2 n+
of the cation in the absence of an electric field. The subscript s now refers to the ions of the supporting electrolyte. The term Zu,c, is equal to L the conductance of the supporting electrolyte. The term Zuc is equal to Ao, the equivalent conductance of the diffusing salt. When the value for D+ from equation 6 is introduced into equation 2 we get
Integrating between limits we get
+
0.5 1.o L/(L c.AJ Fig. 1.--Theoretical relation between the conductivity ratio, R, and the electrical contribution to the diffusion of an ion, F. F = 1 - R / ( l - R ) In R. 0
K
=
+
When only one salt is diffusing, equation 4 recluces (as it should) to the Haskel equation14
(12) Hartley aud Kuunicles. Proc. R o y . SOC. (I,oudon), A168, 401 (1Y38). (13) Equation I assumes that the ditiusiou cuetficieut is measured by the diffusion of an infinitesimal amount of solute. I n practice, up t o 10% is allowed to diffuse. According to unpublished calculations by the author the maximum error that can be produced in D , calculated from equatiou 3 on the assumption that is actually being measured by 105% difiusion, is slightly more thau 5% of the highest value of D .except in the immediate iicinity of P sharp c h a n g e in
D. (141 Hn.;krl. I’hvr.
K r : , . ‘21, 1 4 0 ( l Q l l X 1
Let K = L / ( L boCo), the ratio of the conductance of the supporting electrolyte alone to the conductance of the supporting electrolyte plus the diffusing electrolyte and let
-
u +______ /n+ uJnu+
+ u-
- H
which is equal to t+/n+ - t-/n- when u+ and u-. are independent of concentration. Equation S reduces to E + = 30[I - n+H(1 - R(ln R)/(I - R ) ) ] (8’) The factor [l - R(lnR)/(l - R)]depends only upon the relative conductances of the two solutions on either side of the diffusion inembrane and measures the electrical contribution to the diffusion of an ion. This function, which may be replaced by the symbol F , has been calculated for values of R from 0 to 1. Selected values are given in Table I and the function F is plotted in Fig. 1. IT^ the derivation of equatinii 8 the term ( 1 51 Set. rrf ‘1 p a p r 4 0 1
Jan., 1945
THEDIFFUSION OF IONS IN SUPPORTING ELECTROLYTES
33
ended sintered glass cells.’* The sintered glass membranes were carefully flushed with boiled-out distilled water t o remove air bubbles. The more concentrated solution was placed in the upper compartment and diffusion was allowed to proceed in a thermostat at 25” until the rate of diffusion was within 1% of the steady state. This occurs when Dt/hz 2 0.5518where It is the maximum pore length in the membrane. The membranes are about 2 mm. thick and a value of It2 = 0.10 cm. was taken equivalent to a total pore length of 0.316 mm. The solutions in the two compartments were replaced by fresh solutions at 25” after a steady state‘ had been reached. Time was measured from the time the lower compartment was refilled to the time it was emptied. Stock solutions were made up on a volumetric basis and diluted as needed. Cell constants were determined with 0.1 N potassium chloride using the value D = 1.631 cm.s/da: a t 25” obtained from Cohen and TABLE I Bruins’20value at 20 . Analyses were compared with the VALUESOF THE FUNCTIONF = 1 - R(1n R)/(1 - R ) original solution used t o fill the upper compartment diluted 1: 10 with the original solution used to fill the bottom WHERE R IS THE RATIOOF THE CONDUCTIVITY OF THE SUPcompartment. Chlorides were determined by a microPORTING ELECTROLYTE TO THE CONDUCTIVITY OF THE electrometric titration using a micrometer buretz1containDIFPUSINCELECTROLYTE IX THE SUPPORTING ELECTRO- ing a solution of silver nitrate in 0.05 N nitric acid and 0.5 N potassium nitrate. The buret tip contained a silver CONTRIBUTION TO THE LYTE,AND F IS THE ELECTRICAL at its upper end connected to a potentiometer and wire DIFFUSIONOF AN ION dipped into the liquid being titrated. A 5-02. sample of R F R F chloride solution was diluted with 10 cc. of water and 2 cc. 0 1.000 0.6 0.234 of saturated potassium nitrate. A silver wire plated with silver chloride completed the circuit through a potentiome0.1 0.744 .7 .168 ter. After preliminary titrations established the potential .2 .538 .8 .lo9 of the end-point, the potentiometer was set t o that poten.3 .484 .9 .050 tial and silver nitrate added until the galvanometer indi.4 .398 1 .0 ,000 cated zero current flow. This method enables 5 cc. of approximately 0.01 N chlorine to be determined to 0.5c/, .5 ,307 in less than five minutes. Copper was determined The viscosity of the supporting electrolyte, if colorimetrically; 10 cc. df approximately 0.01 N copper was placed in a one ounce screw-top bottle and one cc. of different from pure water, will change the true 33% ammonium hydroxide added. The bottle was shaken diffusion coefficients of the ions in proportion t o and the color determined on a “Lumetron” colorimeter the relative fluidity (inverse viscosity) of the sup- using a narrow band filter having a maximum a t 590 mfi. porting electrolyte. Equation 8 when corrected Citrate was determined by wet oxidation with potassium dichromate in 12 N sulfuric acid a t 100”for fifteen minutes for viscosity becomes followed by electrometric back titration with ferrous ammonium sulfate. B+ ~ O / Do+ T (1 I$+ H F ) (9) High grade analytical reagents were used except for copwhere qo is the viscosity of pure water and q is the per perchlorate which was prepared from perchloric acid viscosity of the supporting electrolyte. Although and an excess of basic copper carbonate. The stock solution was filtered and diluted for use. The concentration there has been some question as to the exact form was determined colorimetrically as above by comparison the viscosity correction should take,” the simple with cbpper wire dissolved in nitric acid. correction factor used is justified by experiments The analytical methods were chosen for their conveniwith potassium chloride in magnesium sulfate ence and rapidity. Conductivities were in general calculated from published data for pure solutions on the assumpdescribed in part I11 b of this paper. tion that conductivities are additive in mixture. Although The theory so far presented is admittedly in- it has been shown16that the conductivities of mixtures are exact, although the main error, introduced by not strictly additive, the errors introduced are not great assuming that the supporting electrolyte does and are probably negligible for large values of the conratio R. The conductivity ratio of copper pernot move, obviously tends to zero a t the limit ductivity chlorate in perchloric acid to perchloric acid was measured R = 1. Therefore it seems impractical a t present in conventional conductivity cells. to allow for minor factors such as change of acViscosities likewise were not measured except for magtivity or ionic mobility with concentration for an nesium sulfate solutions. The fluidities were calculated by making use of Bingham’s priciciple of additive ionic approximate theory of this sort. The second part fluidities22 for solutions not listed as such in the “Interof this paper presents experimental data which riational Critica’i Tables.”
Zuc arises from the conductivity of the system
and therefore in real solutions where the conductivity of the solution is not given accurately by Zuc of the pure components’”.it is correct t o use the measured conductivity ratio L / ( L A&) rather than the calculated value. The function F = [l - R In R/(1 - R ) ] is independent of the relative mobilities of the ions making up the supporting electrolyte. Equation 8 therefore predicts that the diffusion of a given ion should be the same in different supporting electrolytes having the same conductivity but varying relative ionic mobilities.
+
-
follow the predictions of equation 9 to a satisfying degree. Equation 9 should therefore have practical value when determining the diffusion mobility of charged ions in the presence of supporting electrolytes. 11. Experimental Diffusion was measured in Northrup-McBain double(16) Van Rysselberghe and Nutting,
THISJ O U R N A L ,
(1937). (17) Van Ryssrlhrrphr. i h i d . 60, 2326 (1938).
69, 333
111. Results
a. The diffusion of chloride in 0.1 N lithium chloride was measured in various concentrations of potassium nitrate and nitric acid, Fig. 2 and (18) McBain and Dawson, Proc. Roy. SOC.(London), A148, 32 (1935).
(19) Jacobs, Ergeb. B i d , 12, 62 (1935). (20) Cohen and Bruins, 2. physik. Chcm., 11% 154 (1924). (21) Dean and Fetcher, Science, 96, 237 (19421. (22) Ringham. . I . P h v v Chum., 46, XR6 (1941).
R. B. DEAN
34
Vol. 67
TABLE I1 DIFFUSION OF CHLORIDE WHEN
1.0 N Licl L)IFFUSES THROUGH A SUPPORTING OF
2
OR
ELECTROLYTE.
EACH
3 DETERXISATIONS
5 calcd., cm,z/day
D obs., Supporting electrolyte
cm.*/day
D cor. for
r)
0.00 .03 NKNO, .01 NHNO* . 1 NKNOa . 5 A' KNOi * See ref. 6.
1.15" , . 1.36 .. 1.33 .. 1.fio .. 1.63 1.61 1.66 .. Calculated. see text.
R
Ab
(A0
= 9.6)
D
i1
__-
i
1.6 1
1.2
1
,A/---
,/"
,299 ,284 ,553 ,855 1.CHI0
13.0 57
Table TI. F was calculated from R using a large scale graph of Fig. 1. DOC] was obtained from the accurately known values of u+ and u - * ~and D for potassium chloride by means of equation 8. The theoretical curve for D VS. R based on for lithium chloride and Doc! is shown as a solid line in Fig. 2. It will be seen that the value for when nitric acid is used as the supporting electrolyte is not greatly different from the value a t the same R when potassium nitrate is used. The greatest deviation from the theory would be expected in this region where the concentration of the supporting electrolyte is much less than the concentration of the diffusing electrolyte.
i
0
4.1 4. 0
VALUEIS THE MEAN
Fb 1 . 00 .484
D cor.
Dcalcd. (1.001
1.15 1.41 1.41 1.52 1.62 1.66
,489 ,263 ,072 0 0
.96 .91 .99 .9R
(1.00)
rnent between the relative diffusion mobility and the relative fluidity justifies the use of the fluidity correction in the othcr tables of this paper. c. The diffusion of divalent cupric ions as 0.1 N copper nitrate in potassiuni nitrate or nitric acid and as copper perchlorate in perchloric acid has been measured; see Table I11 and Fig. 3. The calculated value for fj0cu++ obtained from and t for 0.1 AT copper sulfate is &Cut+ = 0.48 crn./day. The calculated value for Docu++ obtained from for 0.1 LV copper nitrate and infinite dilution values for ucu++and U N O * - is DOC,,++ = 0.24. Actually Cole and Gordonz4 found that DcUso4is hardly changed by addition of sulfuric acid a t 18'. This means that ucu++= uso,Liii this supporting electrolyte. The value for DcU obtained from their one value at 25' after correcting for viscosity is Dcu++= 0.65. See Table IIIc. The limiting value found in this work on copper nitrate and copper perchlorate is J ~ ~ c , , + += 0.66 in agreetiient with Cole and Gordon.
+
1.00 ______________-
'
7
i.
tY.\
1.0i 0
'
'
1.0
'
0.5
R. Fig. 2.-Diffusion
of C1- as 0 1 5,HSOa.
N LiCl
b. The effect of the viscosity of the supporting electrolyte on the mobility of C1- ion was chloride in 0.2 31 tested using 0.1 ili potassium was found to be 1.17. inagnesium sulfate. D C ~ For these conditions R I- 0.435 and F = 3 5 . Since &c, = 1.63 and = 1.66 (IIIa) the woiild be 1.65 iii this case expected value of if there were no viscosity change. If equation 9 represents the facts then the ratio of 1.47 to 1.G = 0.891 should equal the relative fluidity of 0.2 -If magnesium sulfate. The viscosity of this solution of magnesium sulfate was measured at 2.5' in an Ostwald viscometer and the relative fluidity found to be V O / ~= 0.893 'I'he excellent agree(23) See ref 3 , p :1J(i
0.600
in: 0,KNOa;
'
'
_
L
A
,
0.5
'
-
1
I
1.o
R. Fig. 3.-DiffuGon of C u + +as 0.1 N f h ( N 0 3 ) ~in HNOa, 9, 0.1 NCu(KO& in KKO3, 0; 0.1 iV Cu(ClO&inHClO,,
+.
Table I11 shows that divalent diffusing ions agree with equation 9 to a good approximation. The difference between values of DcU++ obtained in nitric wid and potassiuni nitrate as supporting electrolytes is not large. In general the potassium nitrate values fall closer to the theoretical curve. The niarked disagreement of all the diffusion data with the value for DOC^++ obtnined from trans1m-t nuiiibcrs is perlinps one more indication that C u + associated to form complex ions in solution. (24) Coli. a n d C.r>rdou, J
I'hvs. C k r i n . . 40, 737 (l!J36).
THEDIFFUSION OF IONS IN SVPPORTING ELECTROLYTES
Jan., 1945
39
TABLE I11 DIFFUSION OF 0.1 N Cuff D cor.
Supporting electrolyte
CaAs
0
0
0
b.
0
0 0.5 N HClOi
0.94"
D cor. for q
..
D calcd. (0 * 94)
Eiiz
.90
1.01 1.08 1.03 1.02 1.00
79 .79 .68 .68 .66 .65 ( .65)
Cu(ClO& in HClO,. R determined conductimetrically 0 1.00 * 94 0.974 0.013 -66 .66
( .94) .65
m
m
D obs.
Cu(NO8)s in supporting electrolytes; Cdo = 9.3 .041 0.88 .91 .. .305 .48 .85 .. .303 .48 -81 .805 .lo4 .69 .. .800 .lo8 .69 0.68 .923 .037 .66 .64 .965 ,018 .62 -65 1.00 .00 ..
a. 0.4 4.07 4.05 38.6 37.2 111.9 269
0.001 N HxOs .01 N "01 .03 N KNOs 0.1 "NO, 0 . 3 NKNOs 1.0 N KNOJ 0.8 N HX03
F 1.00
R
*
. I
..
..
0.97 1.00
1.02
c. CuSO, in HZSO,. Data of Cole and Gordon*' 0.52 a t 18' .. 0 a t 18' .532 a t 18' .. 0 . 5 N HSOt a t 18' ,512 a t 18' .. 1 . 0 N H&O4 a t 18' .68 .18 .65 .67 0 . 1 N HzSO, a t 25' Oholm" found D for 0.1 N copper nitrate = 0.94 corrected to 25" from values near 20'.
The diffusion of 0.1 N potassium citrate was measured in water and in 0.1 N potassium chloride and 0.13 N lithium chloride and the results are listed in Table IV. Values of DOare based on the relative ionic mobilities a t infinite dilution.
summary
TABLE IV Supporting electrolyte
0 0.1 NKC1 0.13 NLiCl
0
E 1
0.5 .5
0.3 .3
R
D obo.
0.90 .57 .61
Acknowledgment.-The author wishes to express his appreciation to Professor J. W. McBain for the interest he has shown in this work and for his many helpful suggestions in the course of the preparation of the manuscript.
D
D0b.l
(90)
(1.00)
0.58 .58
0.98 1.06
calcd.
Dcdcd.
I n the case of 0.13 N lithium chloride the change in concentration of chloride was also measured. When an average of 0.0106 mole per liter of the citrate had diffused from ,the upper to the lower chamber the chloride movement averaged 0.0051 mole per liter in the same direction. Thus the chloride movement is half the citrate movement but even in this case the change in the concentration of the supporting electrolyte is only four per cent. This experiment helps to justify the admittedly false assumption that the supporting electrolyte does not move, since it shows that the movement of the supporting electrolyte has only a small effect on the relative conductivities of the diffusing and supporting electrolytes. The fact that the chloride movement w'as half the citrate movement shows how unreliable any nonspecific analysis such as refractive index must be in the presence bf supporting electrolytes. (25) Bbolm, Pinska Kcmislsamfunds Medd., 46, 124 (1937): Chcm. A b s . , S3, 9094 (1939).
1. A "supporting electrolyte" is defined as the electrolyte or mixture of electrolytes initially present a t uniform concentration throughout a system in which diffusion takes place. 2. An equation is developed relating the integral diffusion coefficient of any ion to the conductivities of the supporting and diffusing electrolytes. 3. The diffusion of chloride, copper and citrate ions has been measured in various supporting electrolytes and the results agree satisfactorily with the theory. 4. The diffusion of ions is nearly independent of the relative ionic mobilities of the supporting electrolyte but depends almost entirely on the conductivity of the latter. 5 . When the supporting electrolyte has ten times the conductivity of the diffusing electrolyte, an ion diffuses a t a rate that still differs from its ideal rate if uncharged by 5 per cent. of the difference between the ideal rate and the rate in the absence of supporting electrolyte. For larger conductivity ratios the deviation will be proportionately less. STANFORD UNIVERSITY, CALIFORNIA
RECEIVED JUNE 6, 1944
