NOTES
August, 1961
1441
lnent of the dependent variable; (d) time constant of the individual dead-elid pores. TABLE I
N o . of
deadrnd pores
0 1
Location
...
Network time constant in relative aec. (ca. Dcnd-end pore component 60% greater _.-__ than real Ca pa ci tance Resistance time in (farads) (ohm-) sec.)
...
...
10.5" 10.4" 25.5" 21.5"
At source 10 -6 2 x 104 At sink 10-6 2 x 104 Uniformly dis- 10-6 10 2 x 105 tributed 10 Uniformly dis- 10 -6 15.7' 2 x 105 tributed 40 15.9" Uniformly dis- 2 . 5 x 10-7 s x 106 tributed a 311Jasiircd nt sink. b Alcnsiircd in middle of nctmork. 1
In all cases m have held the total capacitance of the dead-end pores as a constant fraction of the
total capacitaiice of the central network domain region ; likewise, the tinie-c,onstant of the individual dead-end pores was held fixed (at 0.2 second). The central network itself was entirely isotropic and homogeneous, being made up of a rectangular array of 0.5 megohm resistors with 0.1 microfarad capacitors a t each mesh point. These values arbitrarily were chosen and scaled to be meaningful with reference to the experiment of Goodknight, et aL6; however, it is to be expected that, had a less uniform network form been chosen, an even greater influence on the transients (from deadend porosity) mould have been in evidence. It is clear that the analog network model offers t,lie opportunity to investigate problems of much greater complexity than those envisioned by Goodknight, et al. For example, media inhomogeneity a.nd anisotropy can be scaled without increased difficulty, and micro- as well as macroscopically irregular dead-end pore configurations can be represented. As for the latter, if each mesh point is thought of as the macroscopic differential volume clement to which equation 1 applies, the data of Table I refer to a discrete rather than a continuous distribution of dead-end pores in the macroscopic sense. On the other hand, if the entire netmork of many contiguous mesh points is taken as the macroscopic differential volume element of interest, t,he dat,a of Table I reflect on the influence of a given microscopic distribution of dead-end pores. cm.2, ( 5 ) If in the prototype system, permeability is taken as f i u i d compressibility is taken as per dyne per om.*, porosity is tsken as n . 2 5 fluid viscosity is taken as 10-2 poise, the length to cross-section31 a m i ratio is taken as unity, and the T'E/VI and H IiaramPters of Goodknight. et nl.. are talien as 0.625 and 10-1 reciprocal arconds. r e s p i d r e l y , the scale ratio between time in the electrical analog s p t r m and in the prototype system n-ill be 2 X 10-2.
ESTlnlATZON OF DISSOCIATIOX CONSTANTS OF ELECTROLYTES BY HENRYE. WIRTH Department of Chemrstry, Suracuse Unioersity, Syracuae, N . Y . Received Februarg 1. 1961
I n order to evaluate the dissociation constant,
K,' of a symmetrical electrolyte
C$Y2*-
K =
(1)
I-ff
from conductivity measurements, lcuoss2 defined by the expression ff=
= _ -A
A
- S(Aj
hO(l
h°F(Z)
dV/CyC/Aa)
cy
(2)
where Z = S(n)dG(hO)-*/z and F ( Z ) is the continued fraction
(3)
P(Z) = 1
(4)
- Z(1 - Z( 1
-Z(1
- ..
.)-1/E)-'/z)-'/Y
The activity coefficient y.t is evaluated from the limiting law log Y t =
-S
w 6 c
(5)
Substitution in equation 1 gives the relation
so that Ao and K can be obtained from the intercept and slope of a plot of F ( Z ) / A versus cAy_t2/ F(Z). In the corresponding method developed by Shedlovsky,a a is defined by
where
This leads to the relation
from which A0 and R are evaluated from a plot of l/AS(Z) vs. chg&S(Z). The Shedlovsky method has been stated4 to give better results than the Fuoss method when I< is greater than lo+, but as shown in Fig. 1-3 both methods fail a t relatively low concentrations. The nature of the deviation in the case of the Shedlovsky treatment is such that the linear portion of the curve can be extended to higher concentrations by using the extended law for activity coefficients logy* =
- &*)&/(l+ .4'4/LYc)
(10)
with an appropriate choice of ('a," the distance of closest approach of the ions. Curves labeled S'(2) in Fig. 1-3 were obtained in this way. This is essentially the same procedure used by Owen and Gurryj to interpret the data on ZnS04. They made the product cuyh equal to the experimentally determined stoichiometric activity coefficient. Usually the Fuoss method cannot be modified in this way without introducing negative values for ((U.',
The use of the Fuoss and Shedlovsky methods has been simplified by the publication of tables (1) The notations used in this article are those employed by H. S. Harned and B. B. Owen, "Physical Chemiatry of Electrolytic Solutions," Reinhold Publ. Corp., New York, N. Y., 3rd Ed., 1958. (2) R. hf. Fuoes, J . Am. Cham. Soc.. 67, 488 (1935). (3) T.Shedlovsky, J . Fronklin Inst., 226, 739 (1938). (4) Harned and Owen, ref. 1, p. 290. (5) B. B. Owen and R. W. Gurry, J . Am. Chcm. Soc., 60, 3074 (1938).
NOTES
1442
Vol. 65
0.012
0.050
h
1 9 uz
0.011
- 15
e 0.040
IN
&.
,-. 0
I..
0
SG 0.030 kl
-51
d
kl
0.009
0.020 L
0
I
I
0.005
0.010
0.015
0.008 0
0.10
0.30
0.20
Fig. 1.--Estimation of A0 and K for ZnSOd in water a t 25“. The ordinate scale has been shifted 0.005 unit for W ( 2 )and 0.001 unit for S ( 2 ) to separate the functions. 0.014
i
+
Fig. 3.-Estimate of 110 and K in 82% dioxane-water solution a t 25’. The ordinate scale has been shifted 0.005 unit for W ( 2 )and 0.01 unit for S ( 2 ) . a!=
A --- A Ao(l - 2 )
Aow(z)
(11)
where W ( Z ) = 1 - 2. This substitution partially corrects for the fact that the conductivity of ions a t finite concentrations is less than that calculated by the limiting law. On substitution in equation 1, the relation
0.013 7
4
s
-
‘s,
d
4
S’d 81 0.012
,--.
53.4 kl
0.01 1
I
0
u.050
I
0.010
Fig.Z.---Estirniltton of A0 and K for HCl in 70% diovanemater solution at 25”. The ordinate scale has been shifted 0.0005 unit for ll’(Z) and 0.001 unit for S(2).
for F(2)I and S ( Z ) 6 for values of 2 between 0 and 0.209. For larger values of 2 the calculation of the functions is tedious. h simpler function can be developed if, in equation 2, A/An is substituted for the CY in the denominator of the defining term for 01 giving 16) H. M. Daggett, %bid.. 73 4977 (1951).
is obtained. The activity coefficient y h is obtained from either equation 5 or 10. The three functions F ( Z ) , S’(2)and W ( 2 )were used to estimate A0and K for three examples where the experimental values of A were available for both dilute and moderately concentrated solutions. The examples chosen were ZnS04 in water5 and HCI in 70% and 8270 dioxane-water mixtures? and cover a wide range of dielectric constant of the solvent. The curves are given in Fig. 1-3, and the results are given in Table I. The parameter “a” was chosen to give a straight line over the largest possible concentration range, and should be regarded as an arbitrary constant compensating for some of the errors introduced by using limiting laws at finite concentration. I t was pcssible to find a value of “a” that mould give a linear plot up to the point where l~hS”(2) or l V ( Z ) / h went through a maximum. This was usually close to the point where the calculated value of 01 started increasing with increasing concentration. In the case of HC1 in 82y0 dioxane where it ma‘ necessary to use equation 10 with TT’(2) t o giw :I linear plot, the value of lru” used is close to that estimated by other methods, whereas the valw (7) B. R. Owen and G. W. Waters, ibid.. 60, 2371 (1938).
NOTES
August, 1961 TABLE I COMPARISON OF E Q U l T I O N S FOR ESTIhL4TISG D I S S O C I A T I O N
COXSTANTS F(Z)
Function
S(Z)
S’(Z)
W(Z)
ZndOl in water A0
K (1.
x
132.P 5.25
103
(B.,
....
132.85 132.55 4.95 5.07 .... 5.40
132.90
5.04 0
HCl in 707, dioxane
K 11.
.... .... ....
93.27 93.52 6.76 7.48 0.867 0
57.9 2.0%
.,..
....
. . ..
58.55 1.95 7.31
!J3.18
.io
x
7.F
103
(H.,
,...
HC1 in 827, dioxane ‘10
x 104 (1. (A,) K
58.52 1.99 6.30
required with S’(2)was somewhat larger. The same observation has been made in other cases examined in testing these functions. Use of the functions S ’ ( 2 ) or W ( 2 ) permits the use of conductivity data obtained at higher concentrations in estimating dissociation constants. W(2)is to be preferred since it gives reasonable answers with less calculation. -___-
CRYOSCOPIC DETERMINATION OF XOLECULAR WEIGHTS I N AQUEOUS PERCHLORIC: ACID BY MICHAELXRDONAND AMOSLINENBERG Department of Inorganzc and Analytical Chemistry, The Hebrew Unmuerszty, Jerusalem, Israel Received January 17, 1961
The determination of molecular weights of small polynuclear ionic species, in aqueous solution, from freezing point data, is a difficult task, not only because the mean activity coefficients of the solutes usually are unknown, but mainly because the contribution of the counterions of such ionic species to the depression of the freezing point is often much greater than the contribution of the species investigated, thus making this method highly inaccurate. This difficulty may be overcome by the use, as a solvent, of a eutectic solution of a strong electrolyte in water. If a foreign electrolyte is dissolved in a solvent of this kind, only those ions which do not exist in the pure solvent (the “foreign ions”) will depress the freezing point of the solution.23 One could investigate the molecular weight of a polynuclear catioii by iising a coiinter-ion which already exists in the solwilt, thereby eliminating its effect on the freezing point. By this method, a much higher prwicioii in molecular 1% eight deterininatioiib is Another advantage of this nirthod iq the fact that the variation of the activity coefficient of the solute xvith concentration is small and linear in such a This makes possible a correct extrapolatioi~of K fto infinite diliition. (1) G. I’arissakis and G . Schwarienba~:l~, Helu. Chim. A c t a , 4042, 41 (1968); 4426, 41 (1968). ( 2 ) H. J. Muller, A n n . Chim., I l l 18, 143 (1937).
1443
The systems hitherto used were mainly eutectic solutions of salts in water2 (I
