COMMUNICATIONS TO THE EDITOR
4428
posed. The differences among t.he three sets of results indicate the departure of one solution from this condition with respect to the other solutions at the same ionic strength. The only point left to be considered is that a! be large and similar for the three sets of solutions of digerent internal composition at the same ionic strength. This is the same as saying that the breakdown of the principle of ionic strength occurs t o almost the same extent and in the same direction for the various mixtures investigated. This is extremely unlikely given the different character of Li+, Na+, and Cs+ ions.
X
8
2.4
------
2.3
22
(7) B. B. Owen and R. W. Gurry, J . Amer. Chem. SOC.,60, 3074 (1938).
2.1
o
k
-
*
0103
Ob4
005
2 T l o o 7
oos
w-
obd
I-
Figure 2 . Cu(en),S,OI in water a t 23‘: @, llatheson’s results (R. A . RIatheson, J . Phys. Chent., 71, 1302 (1967)); 0 , previous work by the same authors;5 VI present work substituting NaCIOa for LiCIO4 121, present work substituting NaClO4 for CsClOa.
DEPARTMENT OF CHEMISTRY POLYTECHNIC INSTITUTE OF BROOKLYN BnooIiLYN, NEWYORIC
PAULHEMMES SEROIO PETRUCCI
RECEIVED JUNE 19, 1969
On the Determination of Average Pore K’(Na). The values of K’(Cs) are instead very similar to the previous determined K‘(iYa). (The symbols K’(Na), K’(Li), and K’(Cs) refer to the cation other than Cu2+used.) However, when the same plot as before5 is constructed (Figure l ) , namely the (log K’ - log yh2) is reported vs. I (where yh2 is calculated from the Debye-Huckel relation for various arbitrary values of a ) , the convergence to I = 0 is obtained for K(Li) = 215 f 20 M-’ with respect to the previous5 K(Na) = 226 f 25 M-’. I n Figure 1 the results by conductance calculations a t I = 0 for K are also reported.6 The Shedlowski method gives’ K = 233 M-l, while the Fuoss-Onsager theory gives3 K = 191 M-l. The use of these two forms of the conductance theory gives a range of results comparable with the dispersion of the spectrophotometric results. The same procedure as above has been applied to the Cu(en)2+ S2032-ions. For this case, more complex mixtures have been used, namely optical density differences of solutions of Cu(en)z(ClO4)~,NazSz03, and NazS203, and CsC104) the LiC104 (or C~(en)~(ClO4)2, first and Cu (en), (C104)2, LiC104 (or Cu(en)2 (ClO4)2, CsC104) the reference have been measured. The results are reported in Table I. The differences in K’(Li, Na) are 11% and 18% with respect t o K’(Na) at I = 0.046 and I = 0.0875 M . For K’(Cs, Na) a difference of 15.4% and 29% a t I = 0.0273 and I = 0.0664 with respect t o K’ (Na) is observed (Table I). However also in this case when a plot of (log K‘ log yk2) vs.I is constructed (Figure 2) the graph extrapolates t o K(Li, Na) = 215 f 25 M - I and K(Cs, Na) = 192 f. 14 M-I with respect to the previous determinations K(Na) = 220 f 20 M-l. I t i s quite clear that in the above the condition a: = 0 has been tacitly im-
+
The JOuTnal of Physical Chemistry
Size of Membranes
Xir: Recently, a method for estimating the average pore radius of a membrane from electroosmotic and hydrodynamic permeability measurements has been suggested by Rastogi, et al.’ The phenomenological coefficients L11 and LlZare expressed in terms of classical theory as
(-h/T)= (nrr4)/(8d
(1 1
(LldT) = (nr2Ds?/(4vd)
(2)
where (Lll/T)is the hydrodynamic flow (cm3/sec) induced by unit pressure (dyn/cm2), (L12IT) is the electroosmotic flow (cm3/sec) per volt, n is the number of pores of radius r in the membrane, D is the dielectric constant of the pore liquid of viscosity 7, d is the length of the pore, and { is the potential. Dividing eq 1 by eq 2 gives an equation for r, vix.
r = d(2D{L~)/(aLd
(3)
In the classical work of physiologists described elsewhere2 and reviewed recently by SolomonI3 the pore radius expressed in simple terms without the corrections for steric hindrance and molecular sieving is given by
r
=
d8qL,(d/A),
(4)
where L , = (L11/T),d is the pore length, and A is the pore area for the passage of water, w. Evaluation of r by eq 3 calls for three measurements (LIl/T), (L12/T),and {. On the contrary, derivation (1) R. P. Rastogi, K. Singh, and S. N. Singh, Indian J . Chern., 6,466 (1968). (2) N. Lakshminarayanaiah, Chem. Be%,65, 539 (1965). (3) A. K. Solomon, J . Gen. Physiol., 51,3355 (1968).
COMMUNICATIONS TO THE EDITOR
4429
of a value for r by eq 4 requires only two measurements, L , and (Aid),. I n both cases, values used for D and q for the pore liquid are the values known for the bulk liquid. Rastogi, et al., in a recent paper,4 derived values for r for a number of Pyrex sinters using a value of 0.07 V for [ derived from electrophoretic measurements in which suspensions of Pyrex particles were used. How relevant the value of [ derived this way is to [ appearing in eq 2-3 is unknown. I n classical work, it is considered that the electrophoretic velocity is equal to the velocity of electroosmosis, both obeying the equation of Helmholtz-Smo1uchowski.j This obedience is possible only if the electrokinetic properties of both suspensions and sinters are identical. Protein adsorption on both particles and sinters is recommended to make them electrokinetically equal. No such or other procedure is mentioned in the work of Rastogi and cow o r k e r ~and ~ ~so~ the values of r derived should be considered suspect unless proven otherwise. In order to obviate this lack of relevance and to eliminate the third electrophoretic measurement, the Helmholtz-Smoluchowski equation for electroosmosis, vix.
5
= (4nqkV,3002)/(Di)
(5)
where is in volts, k is the specific conductance of the pore liquid (ohm-’ cm-I), V , is the volume of liquid flowing (cm3/seo), and i is the current (A), can be used to evalua,te [. Substitution of the following values taken from the work of Rastogi, et al., for acetone-Pyrex system gave a value of 1.2 mV for [ has been used for I C ) : q = 0.284 X (5.3 X P6; D = 19.7‘j; L (pure acetone) = 2 X at ohm-l cm-l 18°,7 = 6 X lo-* at 26”,7 = 1.2 X at 35” (calculated); IC (acetone-Pyrex system) = 5.3 X i.e., R = 1.157 X lo5 ohmj6A = 4.9 cm2,6 d = 0.3 C M , V ~ , = 24.25 X cm3 sec-‘ a t 200 V,4 and i = (200/1.157) X 10-5 = 1.73 X A. This value for [ is too low compared to a value of 70 mV derived from electrophoretic measurements.8 The value of 1.2 mV will be further reduced if the value of k for pure acetone is used in the calculations. This discrepancy between the [ values, i.e., 70 and 1.2 mV, can be attributed to the differences in the electrokinetic properties of the particles and the sinter. Therefore eq 3, as suggested and used by Rastogi and coworker^,^^^ is not reliable for estimating the values for r. Instead, electroosmotic and hydrodynamic permeability me+ surements alone are sufficient to derive a value for r. Equation 5 may be written as [ =
(Llz/T)E(4?rqlc)/(Di) (es units)
Substitution of this in eq 3 gives
(6)
Use of the values given above and that of ( L I ~ / Ti.e., ), cm5 dyn-l sec-lj4 gives a value of 1.7 X 0.96 X cm for r as opposed to the value of 3 X cm obtained by Rastogi, et aL4 The k value used in this calculation was 1.2 X T will have a value of 3.7 X is used for k . cm if 5.3 X A trivial manipulation, i.e., (kE/i) = (d/A), shows that eq 7 is equivalent to eq 4 where k is the specific conductance of the pore liquid. It should be emphasized that the equivalence of eq 4 and 7 can only be true when the membrane matrix contributes little to the conductance of the pore liquid. This is so only when the pore radius is very large compared to the thickness of the electrical double layer. So it should be expected that in the case of “tight” membranes whose matrix contains a number of fixed ionic groups, the relation L E / i = kR = d / A cannot be valid. The ratio of pore length to pore area, i.e., ( d / A ) ,has been determined by Lak~hminarayanaiah~ for a polyethylene-polystyrene graft copolymer membrane containing sulfonic acid groups using tritiated water in the manner described in the literaturea2 The value was 0.24 cm-l. The factor kR for the membrane is difficult to evaluate since the pore liquid in the membrane was water (k = ohm-’ cm-I) in which SO3- groups attached to the membrane matrix were submerged. The membrane resistance (thickness = 150 p and area = 0.3 cm2) was about 5-10 ohm. The product kR is therefore about 5-10 X If a value of about 5 X corresponding to an aqueous 0.5 N electrolyte solution is taken fork, then kR will be about 0.24; IC for the membrane in H + form is about loF2. This means that conduction is mostly taking place along the polymer chains and is similar to the phenomenon of surface conduction. Consequently, eq 7 cannot be used for measuring r of “tight” i2n-exchange membranes whose pore radius is about 7 A,9 a little bigger than the thickness of the electrical double layer.
Acknowledgment. The writing of this communication has been supported in part by Public Health Service Grant NB-08163. (4) R. P. Rastogi, K. Singh, and M. L. Srivastava, J . Phys. Chem., 73,46 (1969). (5) J. Th. G. Overbeek in “Colloid Science,” H. R. Kruyt, Ed., Vol. I, Elsevier Publishing Co. Inc., New York, N. Y., 1952, p 224. (6) R. P. Rastogi and K. XI. Jha, Trans. Faraday Soc.. 6 2 , 585 (1966). (7) N. A. Lange, “Handbook of Chemistry,” 10th ed, McGraw-Hill Book Company, Inc., New York, N.Y., 1961, p 1209. (8) R. P. Rastogi and B. M. Misra, Trans. Faraday Soc., 63, 2926 (1967). (9) N. Lakshminarayanaiah, Biophys. J., 7, 511 (1967).
DEPARTMENT OF PHARMACOLOGYK.LAKSHMINARAYANAIAH TjNIVERSITY O F PCNNRYLVAXI-4
PHILADELPHIA, PEXNSYLVANIA 19104 RECEIVED JUNE 11, 1969 Volume 75, Number I d
December 1060
