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JUIH. WANG
Vol. 58
The straight lines represent the limiting expressions for the self-diffusion12 and salt-diffusion. The curve designated 0-F represents the Onsager and 2 Fuoss theory. It is apparent from these plots that only in very dilute solutions do the values of self-diffusion coefficients approach the limiting law of the theory, although they do converge toward the limiting values. As stated by Wang it is interesting to 1.5 note that the ratios of the self-diffusion coefficients a t infinite dilution t o those a t 5.36 111 are 13 and 8 for the chloride ion and calcium ion, respectively, while the ratio of the viscosity of a 5.36 M calcium 5 chloride solution to that of pure water is 11. X Our observations on numerous electrolytes Q 1 show that, a t concentrations below 0.01 M, the gradient of the chemical potential is the major factor in computing the concentration dependence of diffusion coefficients. Above this concentration other factors which influence the mobility become important if not predominant as illustrated by 0.5 comparison of the curve of the experimental values for the diffusion coefficient of calcium chloride with the computed curve, 0-F. I n addition to the effect of relative viscosity, there are large effects of forces which tend to counteract the influence of the gradient of the chemical potential in the con01 I I I centrated solutions and thus decrease the mobility. 0 1 2 3 C‘/2. This investigation was supported in part by the Fig. 1.-Diffusion coefficient of calcium chloride and the Atomic Energy Commission under Contract ATself-diffusion coefficients of the c$cium and chloride ions in (30-1)-1375. calcium chloride solutions a t 25 . Straight lines represent
”I
(12) L. J. Gosting and H. S. Hirned, I.Am. Chem. Sac., 73, 161 (1951)
Y
limiting laws. The curve designated 0-F was obtained from the Onsager-Fuoss theory.
EFFECT OF IONS ON THE SELF-DIFFUSION An’D STRUCTURE OF WATER I N AQUEOUS ELECTROLYTIC SOLUTIONS1 BY JUIH. WAA-G Contribution No. 1226 from the Department of Cheniistry of Yale University, N e w Haven, Conn. Received June 7, 1864
Liquid water has a semi-crystalline structure. Dissolved ions cause distortion or partial destruction of the structure of water. Such a distortion effect can be used to interpret (1) the temperature coefficients of the limiting equivalent conductances of spherical ions in water, and (2) the observed concentration dependence of the tracer-diffusion coefficient of ions with salt concentration in moderately concentrated salt solutions. Direct evidence of such a distortion effect has been obtained in the present work where the self-diffusion coefficient of water in a series of electrolytic solutions has been measured as a Function of the concentration of the electrolyte. In some cases the measured self-diffusion coefficient of water in salt solution is much higher than that in pure water, indicating that in these solutions the structure OF water is distorted or partially destroyed to a considerable extent. The effect of such a distorted water structure on the equilibrium properties of the ions is discussed. The relationship between the measured viscosity of the solution and the self-diffusion coefficients of the constituent molecular species is also examined.
It is now generally agreed that a t temperatures not much higher than 0” liquid water has a “semi-ice” structure, ie., each water molecule is “hydrogenbonded” to four immediate neighbors in four approximately tetrahedral directions. Inasmuch as the directions are only approximately tetrahedral long range order (ice-structure) disappears rapidly in liquid water although local order may exist a t fairly high temperatures.2 According to this picture, the unusually large heat capacity of water (1) Presented a t the Symposium on the Solutions of Electrolytes sponsored by the Division of Physical and Inorganic Chemistry of the A. C . S. on June 1 6 , 1954, a t New Haven, Conn. (2) J. A. Pople, Proc. Roy. Soc. (London),A205, 163 (1951).
is due to the heats of dissociation of the semicrystalline lattice. The abnormal temperature dependence of the density of water can also be explained by the assumption that on heating water near 0” the extensive “semi-ice” structure becomes more distorted and broken down, makes closer packing possible, and hence the density increases with temperature until about 4” when the effect of thermal expansion just compensates the shrinkage in volume due to the breaking down of the “semi-ice” structure. Since in general the spherical ions with inert gas type of electronic structure do not fit into the water
,
d
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SELF-DIFFUSION OF WATERIN AQUEOUS ELECTROLYTIC SOLUTIONS
lattice, they cause local distortion in the structure of the surrounding water. This kind of distortion will undoubtedly affect many physicochemical properties of the solution. We shall first examine the simplest case, ie., an electrolytic solution at infinite dilution. Temperature Dependence of the Limiting Mobilities of Ions.-The limiting mobility, wio, of ionic or molecular species i in its aqueous solution is defined as wio =
Di”/kT
(1)
where Di0 is the tracer-diffusion coefficient of ion or molecule i in pure water a t temperature T , and IC is Boltzmann’s constant. I n the absence of any excessive distortion in the structure of the surrounding water, we may write w i o = Ai/qo
(2)
where qo is the viscosity of pure water and A ; is a constant characteristic of the diffusing species. For example, if the diffusing molecules are spherical and have equal radius r much larger than the dimensions of a water molecule, Ai is equal to 1/ ( 6 ~ according ) to Stokes-Einstein’s relationship.
Fig. 1.-Schematic representation of the stiucture of water: (a) water near its m.p. and under atmospheric pressure; (b) water under high external pressure; (e) water near a Li+ ion; (d) water near a Cs+ ion.
But since the constant Ai is in general unknown, it is not possible to compute the value of aio by means of equation 2 except for very large spherical or ellipsoidal molecules. However if the shape and size of the diffusing species change only little with temperature, Ai should be practically independent of temperature, and by combining equations 1 and 2 we obtain D,”qo/T = ronstant
(3)
which is valid for all diffusing species provided that there is no excessive distortion in the structure of the surrounding water. The effect of the distortion in the structure of water on its viscosity has been clearly demonstrated by Bridgman’s observation3 that at tem‘(3) P. W. Bridgman, “The Physics of High Pressnre,” The Macmillan Co., New York, N. Y.,1931, p. 347.
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peratures not much higher than 0” the viscosity of water decreases continually when the external pressure is increased from 1 to about 1000 kg./ cm.2 This is because the increased external pressure destroys some of the “semi-ice” structure in water and consequently makes it less viscous in spite of the decrease in volume as depicted in Fig. 1. Figure 1 is a schematic illustration of the 3-dimensional “semi-ice” structure of water under different conditions. The unshaded circles represent water molecules with their net dipole moments oriented in the directions of the respective arrows. We may thus infer from Bridgman’s observatioii that if there is excessive distortion in the structure of water surrounding the diffusing ion or molecule, the “local viscosity” of the surrounding water should be less than 70, and that the ion or molecule should have a mobility higher than that predicted by equation 2 . Furthermore, since the temperature coefficient of the “locd viscosity” will in general be different from that of the viscosity of pure water, one may expect the experimental data in such cases to contradict equation 3. Values of D070/T computed from the self-diffusion data for water obtained in the present work with H201S as tracer, from the conductance data of the alkali metal and the tetraethylammonium (CzH6)4N+,are plotted us, temperature in Fig. 2. It may be seen in Fig. 2 that DO~O/T remains practically constant for the large (CzH5)4N+ a i d the strongly hydrated L i t between 5 and 55”,
-
No’
4.00
3.00
I
Li+
I I
I
I
I
I
25 35 45 55 1, “C. Fig. 2.-Variation of the limiting diffusion coefficients of ions and the viscosity of water with temperature.
5
15
but decreases with increasing temperature for the less hydrated ions. This observation can be readily explained by the picture of liquid water discussed (4) H. S. Harned and B. B. Owen, “Physical Chemistry of Electrolytic Solutions,” 2nd E d . , Reinhold Puhl. Corp., New York, N. Y , , 1950, Apliendix B. ( 5 ) W. E. Voisinet, J r . , Dissertation, Yale University, 1951. (6) B. B. Owen, private corninunication.
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JUI H. WANG
above. For simplicity let us coiisider the two extreme cases Lif and Csf as illustrated by (e) and (d), respectively, in Fig. 1. Let each shaded circle in Fig. 1 represent the anhydrous ion, and each broken circle, together with the anhydrous ion and the water molecules in it, represent the hydrated ion. That the number of water molecules “bound” to each ion in our diagram may not be integer should not cause ambiguity in meaning for we may define the “hydration number” or simply “hydration” as the statistically average number of water molecules carried by the ion when the latter diffuses through solution. The total frictional force suffered by each diffusing ion is due to the viscous drag of all itts surrounding “free” water molecules, ie., water molecules outside the broken circle in (c) or (d) of Fig. 1. Because of the more intense electrostatic field in its vicinity, the Li+ is more highly hydrated than Csf as shown in Fig. 1. Consequently, the amount of distortion in the structure of the (‘free’’ water surrounding the hydrated Li+ should be much less than that surrounding the weakly hydrated Csf as depicted in Fig. 1, although neither of the anhydrous ions fit the structure of water as well as water molecule themselves. This excessive distortion in the structure of water surrounding cs+ makes the “local viscosity” in its neighborhood less than qo, and hence increases the mobility of Cs+. As the temperature increases, qo decreases mainly because of the breaking down of the structure of water. But since the structure of water surrounding the weakly hydrated Cs+ is already distorted more than in the bulk of pure water, the viscosity of water in its immediate neighborhood must decrease less rapidly with increasing temperature than 70. Therefore the limiting diffusion coefficient, DO, of Cs+ will increase less rapidly than that predicted by equation 3, and consequently Doqo/T for Csf will decrease with temperature as shown in Fig. 2. Similarly we may take the constancy of DOqo/T for Lif with temperature as indicating that the amount of excessive distortion in the structure of water surrounding the hydrated Lif ie negligible. The curve for (CzH&N+ in Fig. 3 is similar to that for Li+. Those for Xaf and K + fall between Li+ and Csf. These facts are all consistent with the present picture. Values of DOqo/T for Cl-, Br-, I-, respectively, show the same trend in their temperature dependence, although these may not be directly comparable with those for the cations because the water molecule is not symmetrical with respect to polarity. We have assumed in the above discussion that the hydration of ions is not appreciably affected by changes in temperature. If this is not so, decrease of hydration a t high temperatures would make DOqo/T increase with temperature in the absence of excessive distortion effect. Indeed, careful examination shows that the value of DOqo/T for Li+ does increase slightly when the temperature is increased from 5 to 55”, although this change is not much larger than the experimental uncertainties in the conductance data quoted above. I t may also be noticed from Fig. 2 that as the temperature increases, the value of D0q0/Z’ for
Vol. 58
water remains practically constant. The explanation is that although the labeled water molecule is smaller than all the other hydrated ions considered above, it fits the structure of water almost as perfectly as all the other unlabeled water molecules. Consequently there should be no excessive distortion in the structure of water surrounding the labeled water molecule, and the value of Doqo/T should therefore remain constant. This observation can be used as evidence in favor of the present interpretations. Concentration Dependence of the Tracer-diffusion Coefficients of Ions.-For electrolytic solutions of finite concentration, the effect of interaction between different ions should be taken into consideration. The decrease in mobility of the diffusing ions due to the finite time of relaxation of the ion-atmospheres in very dilute electrolytic solutions has been calculated by Onsager and Fuoss’ and by Onsager.* The result of their treatment of the relaxation effect shows that the tracer-diffusion coefficient, Dj, of ionic species j in a salt solution is given by
where Zi is the charge in electronic units and ci the concentration in moles per liter of ion i, X,o the limiting equivalent conductance of ion j, the dielectric constant of the solvent, R the gas constant, F the Faraday constant, and dj a function given by
For the diffusion of tracer amount of ions of species 1 in salt solution containing ions of species 2 and 3, we have c1
E 0,
CZlZ2l
= CalZaI
hence equation 5 becomes
(6)
The values of the tracer-diffusion coefficients of N a + and C1- in aqueous potassium chloride and sodium chloride solutions a t 25’ computed from conductance data by means of equations 4 and 6 are
and DN*+x 105 = 1.384 - 0.2864C Dci- X l o 5 = 2.033 - 0 . 5 2 6 4 ;
in NaCl solutions
These theoretical values are plotted us. 4 as straight lines in Fig. 3 together with the experiIt may be noticed from Fig. 3 mental (7) L. Onsager and R. AI. Fuoss, THISJOURNAL,36, 2639 (1932). (8) L. Onsager, Ann. N . Y. Acad. Sci., 46, 241 (1945). (Sa) J. H. Wan& J . A m . Chem. S o c . , 7 4 , 1182, 6317 (1952); ibid., 1612 (1952). (Sb) J. H. JVang and 9. A I i l l ~ r ,ibid., 74, 1611 (1052). (9) F. A I . Polestra, Disscrtation, Yale Univer8ity, 1954.
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SELF-DIFFUSION OF WATERIN AQUEOUS ELECTROLYTIC SOLUTIONS
that as the concentration of the salt solution decreases, the tracer-diffusion coefficients of Na+ in KCI and KaC1 solutions, respectively, approach the same Kernst’s limiting value. The same is true for the tracer-diffusion coefficients of the chloride ion. This agreement is an independent check on the reliability of the experimental results. While the shape of the experimental curves in Fig. 3 indicate that a detailed explanation of the concentration dependence of these diffusion coefficients in concentrated solutions is probably very involved, they are at least in qualitative agreement with the present picture. Thus in moderately concentrated electrolytic solutions the “local viscosity” of water in the neighborhood of a diffusing ion may be decreased appreciably by the distortion in the structure of water caused by the presence of other ions. If this effect becomes larger than the relaxation and other retarding effects in moderately concentrated solutions, it could cause the tracerdiffusion coefficient of the ion to increase with salt concentration. According to the discussion in the previous section, there should h e more distortion in the structure of water in a moderately concentrated KC1 solution than in a NaCl solution of equal concentration. If this is true, we would expect the ‘LhumpJJ in the D vs. fi curve for a given ion to be more pronounced in KC1 than in NaCl solution. Figure 3 shows that this is true for both Naf and Cl-. Furthermore since a t infinite dilution the amount of distortion in the structure of water surroundiiig a weakly hydrated chloride ion is already more than that surrounding a strongly hydrated sodium ion, the additional distortion in the neighborhood of the diffusing ion caused by
1.60 ,--.
G
.\%
2 1.40 u
v
2
x
689
other ions must be less for the chloride than for the sodium ion. Consequently we would expect that for a given salt solution the “hump” of the D us.
