STOKES' LAW AND T H E LIMITING CO IONS EMANUEL GONICK'
*\:
"2
Department of Chemistry, Stanford University, k a l @ r Received March 99, 1946
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The limiting conductances of ions derived from the higher aliphatic organic acids and bases cannot be determined by the usual experimental methods with certainty, because micellar association of the salts of these acids and bases in aqueous solution rendeq the usual extrapolation formulas of doubtful applicability. It would, therefore, be useful if the mobilities of organic anions and cations could be related to their constitution, so that the values for the higher members of an homologous series could be estimated by extrapolation from the known values of the lower members, which do not associate and whose conductances can therefore be determined by the usual experimental methods. Such a correlation between conductance and constitution would also be of theoretical interest . J. Weyssenhoff (3) has shown on theoretical grounds that Stokes' law should be applicable to spherical particles of molecular and ionic dimensions provided the mean free path of the solvent molecules is small compared with the radius of the moving particles. Attempts to apply Stokes' law to the conductivity of organic ions have, however, met with but very limited success. These efforts have proceeded on the erroneous assumption that the ionic volume is proportional to the total number of atoms in the ion. Reference t o figure 1, in which a fragment of a saturated carbon chain has been drawn to scale, showing thevan der Waals dimensions, makes the impropriety of such an assumption immediately apparent, as the contribution of the hydrogen atom is almost negligible compared to that of carbon. Consequently, in the following treatment the methylene radical is taken as the unit volume element of the ion; the volumes of other groups, such as carboxyl, are expressed in terms of their methylene equivalents; and Stokes' law is applied to develop simple formulas relating the limiting conductances of the ions in question with their constitution. Where substitution or unsaturation occurs, a simple method of calculating the methylene equivalents of the substituted methylene or ethylene group is demonstrated, and the limiting conductances of several ions are calculated. As a corollary it is shown that hydration through hydrogen-bond formation does not retard but ;ather enhances the mobility of the ions where it occurs. . All the conductances on which the following formulation is based have been taken from Bredig's compilation (1). The original data, which are based on the Siemens unit of resistance, have been multiplied by 1.0630 to convert them to international mhos. 1
Bristol-Myers Company Post Doctorate Fellow in Chemistry. 291
292
EMANUEL GONICK APPLICATION OF STOKES' LAW
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According to Stokes' law, the terminal velocity, u, of a spherical particle of radius, R, moving in a viscous medium of viscosity, q, under the influence of a constant force, f, is: u = f/6nqR cm. per second (1) , For an ion of charge z, moving in a uniform field of 1 volt per centimeter, this becomes : u = xe X lO7/6aqR cm. per second
(2)
where e is the unit ionic charge in coulombs. Since the velocity under these conditions is also related to the equivalent conductance by the equation: u = h/F cm. per second
(3)
FIG.1. Model of hydrocarbon chain viewed end-on, showing van der Waals dimensions and the relatively small contribution of hydrogen to the van der Waals volume of the molecule.
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where F is the Faraday constant, substitution of equation 3 into equation 2 yields the following equation for the ionic conductance: X = zeF X 1O7/67qR mho
per gram-equivalent
(4)
If the ions were true spheres whose volumes were equal to the sum of their volume elements, equation 4 would become
.
X = K/nf
(5)
where all the constant terms have been lumped together into a single constant, K , since one could substitute for.R in equation 4 its equivalent (3nv/47r)*, n being the number of volume elements and v the volume of a single element. Since there is no a priori justification for assuming sphericity, it is useful t o define the effective spherical volume of an ion as the volume of a truesphere having the same charge and mobility. From dimensional considerations, this
LIMITING CONDUCTANCE OF ORGANIC IONS
293
volume must be a linear function of the volume of the element, that is, of the methylene radical, but not necessarily a linear function of the number of such elements. That is, one may write for the effective spherical volume, V ,
V = jnxv
(6)
where j and 2 are empirically determined constants for each homologous series of ions. The radius, R, is then given by
R = (3jnzv/4a)' = (3jv/4a)'nv
(7)
in which y = gx. Substitution of equation 7 in equation 4 yields X = xeF X 107/6a~(3jv/4n)'n~
(8)
Again lumping all the constants together and taking the logarithm of both sides, one obtains log X = log K
- y log n
(9)
a linear equation when log X is plotted against log n, with intercept log K and slope y, thus providing a test of equation 8 and a means of determining the values of jv, x, and y. Plots of equation 9 for aliphatic acid ions and for alkylammonium ions are presented in figures 2 and 3, showing the linear relation between log X and log n when n is properly calculated. Stokes' law is thus shown to hold in modified form. To evaluate n one must know the weight, in methylene equivalents, to be given to the carboxyl, methyl, and amino groups. This was determined by trying various values in largescale plots until the best values were found. Thus, for dibasic acids a linear plot is obtained when each carboxyl is given a weight of 0.725 methylenes; in the monocarboxylic acid series, the same weight was assumed for carboxyl and a weight of 1.185 was assigned to the terminal methyl radical. In the various amine series, linearity was obtained when n was computed in the following ways: primary amines, n = number of carbon atoms; secondary amines, n = number of carbon atoms minus 0.5; tertiary amines, n = number of carbon atoms plus 1; diamines, n = number of carbon atoms plus 1. The conductance of quaternary ammonium ions is determined only to a first approximation by the total size of the ion; a strong secondary factor is the size of the largest alkyl radical. This is exemplified by the following comparison between pairs of tetraalkylammonium ions containing the same number of carbons but differently constituted: dimethyl-diethyl > trimethyl-propyl; methyl-triethyl > trimethyl-isobutyl; tetraethyl > trimethyl-isoamyl. Because of this complicating factor, the quaternary amines have been omitted from this discussion, but are plotted in figure 2 (n = number of carbons plus 1) in order to illustrate the situation. It is quite surprising at first to find that the carboxyl group, containing two oxygen atoms, each comparable in bulk to a carbon atom, nevertheless contributes less to the ionic volume than a methylene; and that the amino group may have zero or even negative value, as' in the case of primary and secondary
294
EMANUEL GONICK
amines, respectively. The significance of some of these facts will be discussed later. In table 1 the values of y and K of equations 8 and 9 and those of x in equation 6 are summarized. Quaternary amines, not listed in table 1, cluster about a line with a slope somewhat in excess of 0.5. Primary amines and diamines are seen to have approximately the theoretical y value of 3 for spherical particles and the K value for diamines is approximately double that for primary amines, in keeping with the double charge carried by the former. The tertiary amines and acids show a moderate deviation from the
x
1.4
-.2
0
.2
.4
1.0
~ o ng FIG.2. Plot of the logarithm of the limiting conductivity, A, versus the logarithm of the number of methylene radicals (or their equivalents), n, for monobasic and dibasic unsubstituted carboxylic acid ions.
theoretical y value for spherical particles. Moreover, the K value for dibasic acids is not double that for monobasic acids, as would be expected from a consideration of equation 2 or 4 alone. Tertiary and quaternary amines show the greatest departure from sphericity. This is surprising. One would expect them to be most nearly like spheres in behavior. The y value of approximately 0.5 characteristic of quaternary amines suggests, rather, a disk-like structure. EFFECT O F HYDRATION
The possible effect of hydration has not been taken into consideration in the foregoing discussion, although. one would expect hydrogen bonding of the type
295
LIMITING CONDUCTANCE O F ORGANIC IONS
1.7
t-
1.5
-
I
I
I
I
1
I
I
I
I
SECONDARY
I
T E R T I A R Y
m 0
1.5-
D I A M I N E S
0
.2
.4
.6
.8
1.0
L o 9 n FIG.3. Plot of the logarithm of the limiting conductivity, A, versus the logarithm of the number of methylene radicals (or their equivalents), n , for unsubstituted alkylammonium ions.
296
EMANUEL GONICK
-0. .H-0-H and N-H. * OH2 between carboxyl and amino groups on the one hand and water on the other. The fact that the conductance can be adequately expressed without allowing for bound water proves that the latter need not be considered. The explanation doubtless lies in the fleeting nature of these bonds, the interchange of water molecules being rapid in comparison with the translational velocity of the ions, so that the water is not carried along with the ion. Hydration through hydrogen bonding does, however, have an important effect. It permits a much closer approach of solvent than would otherwise be possible and thus effectively reduces the van der Waals radius (and volume) of the functional group involved. This explains the low methylene equivalenh of the -CH2
COO-,
-CHzNHf,
and -CH,
\ NH; /
TABLE 1 Values of y, K , and x SERIES
K
Y
Monobasic acids.. . . . . . . . . . . . . . . Dibasic acids. . . . . . . . . . . . . . . . . . . Primary amines. . . . . . . . . . . . . . . . Secondary amines. . . . . . . . . . . . . . Tertiary amines.. . . . . . . . . . . . . . . Diamines . . . . . . . . . . . . . . . . . . . . . .
0.299 0.271 0.329 0.386 0.663 0.344
49.4 83.6 61.2 62.3 125.2 120.9
x = 3y
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1
0.897 0.813 0.987 1.158 1.989 1,032
gr2ups. Thus, the van der Waals radius2 of methylene being approximately 2 A. and that of hydrogen 1.2 B., the minimum internuclear distanoce between the methylene carbon and water hydrogen is approximately 3.2 A. Taking 1.29 A, as the C-0 distance in the carboxyl group and 1.60 8.for the 0 . . .H hydrogen-bond distance arid assuming a C-0. . . H bond angle of 105", the near5st approach of the hydrogen nucleus t o that of the carboxylic carbon is 2.3 A,, or 28 per cent closer. (The corresponding distance, assur$ng the tetrahedral bond angle, is 2.37 A. and for a linear configuration is 2.89 A,) A similar calculation could be made for primary and secondary amines. In the latter case two carbons are involved; hence the greater effect. Hydrogen bonding apparently does not occur with tertiary amines, possibly owing to steric factors. EVALUATION OF j V AND
j
By means of the empirically determined valuesof the constant K listedin table 1, the term jv of equations 6, 7, and 8 may be evaluated for the different homologous series. Recalling that 2
This and other molecular dimensions*aretaken from reference 2.
LIMITING CONDUCTANCE O F ORGANIC IONS
I
