4426
COMMUKICATIONS TO THE EDITOR l
L-O,
u I
2.6
LL
2.4
o
-/ ” 1
f
d=5%
2.2 -
1
d
1
0
0.05
0.10
0.05
0.10
Ib
0
2.6
2.2
i
O
l
lL i--
= d
\
‘ 0.05
0.10
I. Figure 1.
are consistent with the results for any one of their sets of solutions depend on the figure assumed for a and, unless this is zero, differ from the values originally obtained by these authors. Table I lists some values of l / A e ’ (Y and KO’ (defined below) which are consistent with their data for solutions of I = 0.0875 M . Although the values of l / A e corresponding to the three figures for a are very different, each gives rise to a set of K’ values such that log KO’ = log K’ - 2ab is constant and therefore (cf. eq 1) K ’ / ( Y & )is~ constant. Of course if (Y # 0, K’ is not constant and cannot be expected to be constant. We have also used each of the sets of KO)’ (Y and Ae in Table I to calculate the concentrations of C ~ ( e n ) ~ and S ~ the 0 ~ values of D - D’ for all the solutions with I = 0.0875 M . The results (see Table 11) confirm that these four sets of parameters differ little in their consistency with the experimental optical densities of these solutions. It seems that unique values of l / A e and K’ cannot be obtained from these optical densities unless some arbitrary restriction The Journal of Physical Chemistry
’
+
+
(7)
I
0
is placed on yi, e.g., if a is fixed at zero. The position can be shown to be similar in the case of Hemmes and Petrucci’s other solutions for which different values of a again given different figures for l / A e and K’. Thus, if a is not arbitrarily fixed, Hemmes and Petrucci’s optical densities produce a range of values of KO‘ at each I . However KO’ varies with I and there remains the possibility that the extrapolation procedure used by Hemmes and Petrucci may produce a definite figure for K despite the ambiguity in KO1. We have therefore constructed plots of F(K0’) = lo$ KO’ 8Adj/ 1 B d d j vs,I ford = 0,5, and 10 A using values of KO’ calculated from Hemmes and Petrucci’s optical densities for their solutions of I = 0.0335,0.0465, 0.0664, and 0.0875 M8 assuming (a) a = 3.4 103/Ae = 8.1 cm M ; (b) a = 0, 103/Ae = 6.3 cm M; and (c) a = -1.26 M-’, 103/Ae = 4.9 cm M . In each case (see Figures la, lb, and IC) the plots for the three values of d converge satisfactorily as I + 0. However, the limits to which they extrapolate correspond to different values of K (275 to 347 M-l for Figure la, 195 to 246 M-I for lb, and 141 to 178 M-’ for IC). Evidently, this extrapolFtion does not eliminate the ambiguity which results from the dependence of K‘ and Ae upon the activity coeficient assumption used to calculate these quantities from the experimental data. In regard to the agreement between Hemmes and Petrucci’s figure for Gus04 and “the conductance value,” it is in our view misleading to speak of “the conductance value” when, for electrolytes like CuSO4, the value of K which is obtained from conductances depends on the assumptions made in the calculation^.^ KO,’ = lim K’ for the constant ionic strength series in question. b-+O
(8) T h e set of solutions with I = 0.109 was not considered because the points for this set diverged from the original graphs (Figure 2, ref 2). (9) J. E. Prue, “Ionic Equilibria,” Pergamon Press, Ltd., London 1966, Chapter 3. D E P A R T M E N T O F CHEMISTRY,
THEU N I V E R S I T Y
R. A. MATHESON
O F OTAQO
I)UNEDIN, N E W ZEALAND
RECEIVED APRIL28, 1969
On the Reliability of the Spectrophotometric Determination of Association Constants. The Case of CuSO4 and Cu(en)&Oi
Xir: We wish to start our answer from the concluding point of Dr. Matheson’s statements,’&namely from the claimed unsuitability of the conductance method to determine a unique value of association for electrolytes like CuSOa. (1) (a) R. A . Matheson, J . Phys. Chem., 4425 (1969) ; (b) J. E. Prue, “Ionic Equilibria,” Pergamon Press, Ltd., London, 1966.
4427
CONMUNICATIONS TO THE EDITOR By analyzing the data by the Fuoss-Onsager equation2 for associated electrolytes, or more precisely by using the so-called "y-x" method, values of KA = 191 M-l and a = 5.7 A have been obtained.a By starting the calculation with an arbitrary value of a (1 or 14 A, for example) rapid (two or three cycles) convergence to the above values of a and KA is obtained. The claimed ambiguity'b of the calculation was based on the possible fit of the data to the Fuoss-Onsager equation2 by imposing a range of values of a and therefore of J ( a ) a t the expense of KA. This ambiguity arises only by imposing and keeping constant the parameter a. The above should be enough to sustain our spectrophotometric work for CuS04 given the fact that our data extrapolate to the conductance value for I = 0. Since, however, this would not justify our result for C ~ ( e n ) ~ Sand, ~ 0 ~more important, would leave the generality of the use of the spectrophotometric method open to question, the following discussion is presented. It is only because of the arbitrary assumption4 of the validity of the Harned rule for a three-component system log y
=
log
yo
+ ab
(1)
Table I : Values of K' ( M - l ) and lOS/A.e ( M , em) a t Different Ionic Strengths I, for the Various Mixtures Investigated 10a/Ae,
I, M
K', M-1
M om
0.041" 0 .O4lb 0.070" 0.070' 0 .070d 0 .0465e 0.0465' 0 0875e 0.0875' 0 . 0273g 0.0273h 0.0338O 0.0664' 0.0664'
36.3" 43.7b 21.8" 28.4c 28.1d 45.4e 50. 8f 29.3e 35.7' 54.2O 64.1h 47.5 ' 31.lg 43.7'
4.2" 4.8b 3.7"
Species cU2+
+ s04'+
Cu(en)z+2 S203-2
I
+
4.0'
4.5d 5.ge 6.3f 4.5" 6.4' 5.2' 6.2h 5.8' 5.30 6.5'
+
a This work, Cu(C104)2 LizS04 LiClO4. * Matheson's results,* C U ( C I O ~ ) ~ NazS04 XaC104. ' Previous work,6 C U ( C ~ O ~ ) NazS04 ~ NaClO4. dThis work, Cu(ClO& CszS04 CsC104. "This work, C ~ ( e n ) ~ ( C l 0 4 ) ~NazSzOa LiC104. Previous work,6 Cu(en)z(ClO& Na~S203 NaC104. ' This work, C~i(en)~(C104)~ Na2S2Oa CsClO4. Matheson's results, C ~ ( e n ) ~ ( C 1 0 ~Na2SZO3 )~ ?JaC104 (R. 9. Matheson, J . Phys. Chem., 71,1302 (1967).
+
+
+
+
'
+
+
+
+
+
+
+
'
+ +
+
that the expression
a+b-x ~ ab _ - _ D --D' Ad
1 +-AelK'
(11)
is transformed into a three-parameter equation in A€,
K'o,and a ab
___ ..-
D - D'
a
+b Ad
2
-k
exp(-4.606ab) ArlKIo
(111)
making impossible its graphical or analytical solution unless some assumption on a is retained. Our contention in this matter is that the above procedure4 (retention of formula I with large values of a and consequent assumption of the failure of the ionic strength principle) is as arbitrary as the assumption of the ionic principle itself. Therefore, in this range of ionic strengths ( I = 0.04 to 0.1) one should prove that the results for Ae and K' strongly depend on the internal composition of the mixture a t the same value of the ionic strength. In order to clarify this matter, we have measured the optical density differences of solutions containing a moles of Cu(C104)2,b moles of Li2SO4 (or Cs2S04),and c moles of LiClO'L (or CsC104) and the reference solution containing a moles of Cu(ClO& and enough LiC104 (or CsC104) to reach the same ionic strength. Small quantities of He104 to repress the hydrolysis of CuaQ2+ have also been added. The difference with previous determinations4s6 has been the substitution of Na+ with Li+ (or Cs+) a t the same ionic strength I . The results are reported in Table I in the form of the calculated K' and 1Oa/A,.6
I 0
\ Ob!
0.'02
Ob3
Ob4
0'05
Ob6
0
I
zT--O:O8
W
IOl
Ob9
;io
0
I
u
Figure 1. CuSOd in water at 25": @, Matheson's results;4 0, previous work by the same authors;s ,'V present work substitutiiig N a + for Lif; El, present work substitut,iilg T\'af for Cs+; 9, coiiductance result by the Shedlowsky method;T 8, ~ ~ J l i d ~ l ~ ~ aresult l l ( Y 2by the Fiioss-Onsager theory3.
It can be seen that differences indeed exist, the values of K'(Li) differing by 17% at I = 0.04 and by 24% at I = 0.07 with respect to the previous determinations (2) R. M. Fuoss and F. Accascina, "Electrolytic Conductance," Interscience Publishers, New York, N. Y . ,1959. (3) G. Atkinson, M.Yokoi, and C. J. Hallada, J . Amer. Chem. SOC., 8 3 , 1570 (1961); C. J. Hallada, Ph.D. Thesis, University of Michigan, Ann Arbor, Mich., 1960. (4) R. A. Matheson, J . Phys. Chem., 69, 1537 (1965). ( 5 ) P. Hemmes and S.Petrucci, ibid., 72,3986 (1968). (6) The original data for optical density differences can be obtained from the authors upon request.
Volume 73, Yzimber 12 December 1969
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).
