R. A. HORNE, R. A. COURANT, D. S. JOHNSON, AND F. F. MARGOSIAN
3988
the membrane surface. This discontinuity is calculated from the ideal Donnan condition that C2 = C-. (C- -4- 0) and has to be taken into account when the integral on the right-hand side of eq. 25 is evaluated. It can be shown that the resulting expression for A(p agrees with that which had been derived by TeorelP and by lleyer and S i e v e r ~ . ~Thus we find that the T.M.S. theory refers to the case in which both positive and negative ions in the membrane behave thermodynamically ideally. It is not quite unlikely that such a situation is realized in charged membranes employed for a usual experimental study. By contrast, the present theory based on eq. 6 has introduced an ex-
treme asymmetry into the activities of positive and negative ions in the membrane. Thus in a charged membrane the co-ion species behaves ideally but the counterion species behaves extremely nonideally. The ion binding by the polyelectrolyte molecule may be primarily responsible for this latter effect. Whatever the mechanism may be, the success of our potential equation is associated primarily with the assumption of eq. 6 for the intramembrane activities of ions. The curves for log [(y - eq))/(eQ - l ) ] us. log X derived from the T.M.S. theory for y = 2 and 10 and a = ‘/z are shown for comparison in Figure 7 . The insufficient nature of the T.M.S. theory is evident.
The Activation Energy of Viscous Flow of Pure Water and Sea Water
in the Temperature Region of Maximum Density
by R. A. Horne, R. A. Courant, D. S. Johnson, and F. F. Margosian Arthur D.Little, Inc., CambridQe,ii’fUS8achuSett8 (Received June 15, 1966)
The visccsities of pure water and sea water have been measured at 1 atm. over the temperature range -2 to 12’. The temperature dependence of the activation energies of viscous flow of water and aqueous electrolytic solutions, unlike that of the activation energies of the electrical conductivities of aqueous solutions of salts, does not exhibit a maximum near the temperature of maximum density. This dissimilarity casts doubt upon the hypothesis that viscous flow and “normal” electrical conductivity both have the same slow step, namely, “hole” formation in the water.
Introduction Viscous flow and ionic conduction in water have long been believed to be closely related processes. Glasstone, Laidler, and Eyring’ argue that “the fact that the activation energies, calculated from the ionic mobilities of most ions in a given solvent are the same, e . g . , 4.0 to 4.2 kcal. in water at 25O, and that the values are almost identical with that for viscous flow of the solvent, suggests that the rate of migration of an ion in an applied field is determined by the solvent molecules jumping from one equilibrium position to the next.” However, The Journal of Phyrrical Chemistry
Wang, Robinson, and Edelman2 concluded that selfdiffusion, dipole-orientation, and viscous flow all have the same mechanism but that H-bond rupture and not vacancy formation is rate-determining. Recent measurements of the electrical conduction of aqueous solutions in the region -2 to 10’ have (1) S.Glasstone, K. J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill Book Co., Inc., New York, N. Y., 1941. p. 557 ff. (2) J. H. Wang, C. V. Robinson, and I. S. Edelman, J . Am. Chem. SOC.,7 5 , 466 (1953).
ACTIVATION ENERGY OF VISCOUSFLOWOF PURE WATERAND SEAWATER
o Pure
iI
Water White and Twining) Pure Water Binghom and Jackson) @Pure Water Dorsey) 0 Pure Water Cohen 1 e 20°100SSea Water A w / s~ Sea Water (Krijmrnel) ~ 4 0 % ~SeaS Water
}
Y’ z 5.0-
W
c
.-w
4.8 -
c
Y
4.6 4.4 -
-10 -8 -6 -4 -2 0 +2+4 +6 +8+10 Tempera turePC Figure 1. Activation energy of viscous flow of pure water and sea water calculated from literature data.
shown that, in the case of strong 1 : 1 electrolytes, the Arrhenius activation energy of conduction, exhibits a pronounced maximum near the temperature of maximum density314 and this maximum was attributed to the facilitation of “hole” formation in the relatively open water structures which are present below about 4’. A parallel maximum in the Arrhenius activation energy of viscous flow, Ea,vis,was expected near the temperature of maximum density. AIiller5 has made Arrhenius plots of the data summarized by Dorsey‘j and found that “there are no discontinuities a t Oo, 4 O , or 100’ . , .” We found this result so surprising that we rechecked Miller’s calculations and also calculated values of Ea,vison the basis of a number of other different sources of data. These calculations revealed discrepancies so great (Figure 1) that we undertook to redetermine the viscosity of pure water and sea water in the temperature range -2 to 12’.
3989
Experimental Section The viscosities were measured in a rolling-ball type viscometer’ which had been previously calibrated a t 25’ with a series of water-glyercol solutions using the data of Sheely,* water-sucrose solutions using the data of Bingham and Jacksonlg and aqueous KaCl solutions using “International Critical Tables” data. The more concentrated water-glyercol solutions gave viscosities differing by as much as 0.5% from the other standards presumably because of water absorption or because of too high a mater content of the original “pure” glycerol. Over the relatively small temperature range of the present experiments, the effects of temperature on the geometry of the viscometer are negligible (less than 0.07%). Deionized water and 19.37~”/,, chlorinity I.A.P.O. Standard Sea Water Pa? were used. At a given temperature the experiment was repeated from four to as many as ten times. The average deviation in roll time for a given set of experiments was only 0.10%. However, the fourth significant figure on the calibration curves could be estimated only to * 5 ; thus the average deviation in the viscosity was somewhat larger being *0.005 cp. or about 0.3yGfor pure water and larger (for unknown reasons) or nearly ~ 0 . 0 cp. 1 for sea water. In the course of a given set of experiments, the average deviation in temperature ranged from *O.O05O (for temperature near room temperature) to as much as 10.02O (for the lowest temperatures). Results from sets of experiments made with successively increasing temperatures agreed with results from sets made with successively decreasing temperatures. Values of E,,,,, mere calculated by the integrated form of the Arrhenius equation Ea,vis
= [log ( 1 / ~ 2 )
- log ( ~ / V I ) I ~ . ~ ~ ~ T ~(1) TI/AT
a t l o intervals from values of viscosity read from a computer-programed second-order least-squares fit through the original viscosity-temperature data (such as Figure 2 but with expanded scales in order to be able to estimate the fourth significant figure) but with the presumably erratic value at -0.346’ for sea water (3) R. A. Horne and R. A. Courant, J . Phys. Chem., 68, 1258 (1964). (4) R. A. Horne and R. A. Courant, J . Geophys. Res., 69, 1152 (1964). (5) A. A. Miller, J. Chem. Phys., 38, 1568 (1963). (6) N. E. Dorsey, “Properties of Ordinary Water Substances,” Reinhold Publishing Corp., New York, N. Y., 1940. (7) The viscometer used is designed for studying the effect of high hydrostatic pressures on viscosity. (8) M. L. Sheely, I d . Eng. Chem., 24, 1060 (1932). (9) E. C. Bingham and R. F. Jackson, Bull. Natl. Bur. Std., 14, 59 (1918).
Volume 69, Number 11 .Vovember 1966
R. A. HORNE,R. A. COURANT, D. S. JOHNSON, AND F. F. MARGOSIAN
3990
8.5 2.0
-
1.9
-
8.0
1 8
1.8 -
-L
.3 0
c
1.7
-
1.6
-
0” .5 ln
I
W0
1.5
-
1.4
-
1.3
-
6.0
v)
5
X Pure Water (Dorsey) o Pure Watedpresent wk) a 19.375%0CI
=*
5.5
F
Q)
15 c 5.0 0 .c
.-z
2
-2-1
6.5
EI 0
0 ln
7.0
.v, >-
0
5
7.5
3 2 0 E
4.5 20%0 CI Sea Water
0 I 2 3 4 5 6 7 8 9 101112 Temperature,%
Figure 2. T h e temperature dependence of t h e viscosities of pure water and 19.375~/,, C1 sea water.
4.0 -
-I -0 + I +2 + 3 + 4 +5 +6 + 7 * 8 +9
t
Temperature, O C Figure 3. Activation energy of viscous flow of pure water and sea water.
deleted. The root-mean-square errors for pure water and sea water were 0.00764 and 0.00721, respectively. The standard deviation of Ea,visin Figure 3 is hO.1 kcal./mole.
Results and Discussion Viscosities and activation energies of viscous flow for pure water and sea water are given in Figures 2 and 3. The present values for the viscosity of pure water are in agreement with the values reported earlier by Bingham and Jacksong and the “best” values of Dorsey.6 Thus the values of Ee,visfor pure water are in agreement with values quoted by or calculated from Miller,5 Bingham and J a c k ~ o nDorsey,6 ,~ and Cohen,lO but not White and Twining.” The present values for the temperature dependence of the activation energy of viscous flow for sea water (Figure 3) are in poor agreement with the values calculated from the results of Krummel (Figure l) as quoted in “Landolt-Bornstein Physikalisch-Chemische Tabellen” (1923). We find, in agreement with Kriimmel’s data, that Ea,via for sea water becomes much greater than the corresponding values for pure water as the temperature approaches Oo, but we find no evidence for a minimum around 4’ (Figure l). The Journal of Physical Chemistry
Figure 3 shows that the presence of KaC1 increases a t low temperatures. This behavior is to be expected from the structure-making tendencies of this salt as evidenced by the positive viscosity B-coefficient.12 In addition, sea water contains ?tlgS04,a very effective structure maker. Miller4 suggests that “the molecular motion required for flow occurs only in the unbonded state.” This is reminiscent of the Grotthuss component of protonic conduction in which the slow step is the rotation of water molecules and which is confined to the “free” rotatable water monomers between the structured clusters.13 The dissimilarity of the energetics of “normal” conduction and viscous flow of aqueous solutions as shown in Figure 3 renders suspect the hypothesis that (10) R. Cohen, Ann. Phys., 45, 666 (1892). (11) G. F. White and R. H. Twining, J . Am. Chem. SOC.,35, 380 (1913). (12) R. W. Gurney, “Ionic Processes in Solution,” McGraw-Hill Book Co., Inc., New York, N. Y., 1953, Chapter 9. (13) R. A. Home and R. A. Courant, J . Phys. Chem., 69, 2224 (1965).
ACTIVATION ENERGY OF VISCOUS FLOW OF PURE WATERAND SEAWATER
these two processes have the same rate-determining step, a t least in the region -2 to 12’. Even in the temperature range 10 to 60°, the activation energies of “normal” conduction, protonic conduction, viscous flow, and self-diffusion appear to parallel one another rather than be numerically equal.14 A recent examination of the temperature dependence of dielectric relaxation in water15 has yielded results very similar to Miller’s6 viscosity studies. On the basis of the foregoing evidence we subscribed tentatively to the view that the slow step of “normal” ionic conduction in aqueous solution is “hole” formation, both in the freeand the clustered water, while the slow step of the viscous flow of aqueous solutions is some sort of rotational tumbling of the “free” mater or of the clusters through the “free” water. Alternatively, the differences in behavior of Es,cond and Ea,,+,could arise from a change in mechanism in the former process resulting from the structural changes that occur near 4’. The possibility that water cluster break-up might
3991
be the rate-determining step in either conductive or viscous processes is unlikely16 in view of the relatively large concentration of unbound water available for transport processes. NBmethy and Scheragal’ estimate that the mole fraction of unbonded or “free” water varies from 0.24 to 0.44 over the range 0 to 100°.18
Acknowledgment. We are indebted to Dr. I. Simon for his assistance. This work was supported in part by the Office of Naval Research. (14) See Figure 9 in R. A. Horne, Water Resources Res., 1, 263 (1965). (15) R. C. Bhandariand and M. L. Sisodia, Indian J . Pur0 A p p l . Phys., 2, 132 (1964). (16) Cf.ref. 2. (17) G. N6methy and H. A. Scheraga, J . Chem. Phys., 36, 3382 (1962). (18) Added note: D. P. Stevenson, J . Phys. Chem., 69, 2145 (1965), has recently argued that NQmethy and Scheraga’s estimate of the concentration of monomeric water is a t least two orders of magnitude too high.
Volume 69. A-umber 11 A’onember 1965
