Stokes' Law and the Limiting Conductance of Organic Ions. - The

---

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

w

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

'

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.

'

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

'

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<

=

zeF X 107/6~q(3jv/4n)'

297 (10)

and solving for j v , one obtains X 10' [zeF6nqK ] 'x 4n

jV =

Substitution of the numerical values of the constants as follows-e = 1.6019 X coulombs, q = 0.000895 poise at 25"C., and F = 96,479 coulombs per gram-equivalent-results in 3.2203 X lo-'' jv =

Ka

The values of jv calculated in this way are tabulated in table 2. The calculated approximate van der Waals volume of a methylene radical, b?sed on a van der Waals radius of 2 A. and an internuclear C-C distance of 1.54 A., is 18.4 X 10-24 cc., in good agreement with the value of j v for primary amines and diamines. Coupling this fact with the fact that the exponents of n in equation 8 (y values of table 1) are close to the theoretical for spheres, the ions of these two series

+

TABLE 2 Values of j v , j, and Vlv SERIES

1

j v X 1024

cc.

Monobasic acids. . . . . . . . . . . . . . . . Dibasic acids.. . . . . . . . . . . . . . . . . . Primary amines. . . . . . . . . . . . . . . . Secondary amines. . . . . . . . . . . . . . Tertiary amines.. . . . . . . . . . . . . . . Diamines. ......................

26.7 44.1 14.0 13.32 1.64 14.6

1.87 3.08 0.48 0.93 0.115 1.02

may be considered as closely approximating true spheres, for which j = 1. T h b is corroborated by the fact that K for diamines is approximately double that for primary amines, as required by theory. The average j v value for primary amines and diamines being taken as the van der Waals volume (= 14.3 X 10-24 cc.), the values of j and the expressions for V/v (the ratio of the effective spherical ionic volume to the methylene volume) are listed in the third and fourth columns of table 2. APPLICATIONS

The limiting conductances of ions containing no substituted methylenes may be calculated directly from equation 9 by inserting the appropriate values for K , y, and n. Thus for palmitate, myristate, and stearate for which n equals 13.91, 15.91, and 17.91, respectively, the following values of the limiting conductance at 25°C. are found: 22.5, 21.6, 20.9 mho cm.2per equivalent. Where substitution or unsaturation occurs, the methylene equivalent of the'

298

EMANUEL GONICK

TABLE 3 Limiting conductances of organic ions after Bredig (corrected to international reciprocal ohms) A

ION

Monobasic acids: Formate ................................. Acetate .................................. Propionate ............................... Butyrate ................................. Valerate ................................. Caproate .................................

n

mho cm.2 per equivalent

54.4 40.7 36.5 32.6 30.6 29.1

0 725 1.91 2.91 3.91 4.91 5.91

75.6 ,66.1 59.7 55.8 52.7 51.0 48.9 45.5

1.45 2.45 3.45 4.45 5.45 6.45 7.45 9.45

.61.2 49.7 42.6 38.6 36.0

1 2 3 4 5

i

Dibasic acids: Malonate . . . . . . . . . . . . Succinate., . . . . . . . . . Adipate .................................. Suberate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebacate . . . . . . . . . . . . .

Primary amines: Met hylammonium ........................ Ethylammonium......................... Propylammonium . . . . . . . . . . . . . . . . . . . . . . . Isobutylammonium ....................... Isoamylammonium . . . . . . . . . . . . . . . . . . . . . . . ~

Secondary amines: Dime t hylammonium ...................... Diethylammonium . . . Dipropylaminonium ...................... Diisobu t ylammoniuni 6 Diisoamylammonium .....................

.

53.3 38.4 32.3 38.6 25.7

Tertiary amines: Trimethylammonium ..................... Methyldiethylammonium . . . . . . . . . . . . . Triethylammonium . . . . . . . . ... Tripropylammonium .....................

50.0 38.1 34.7 27.2

Quaternary amines: Tetramethylammonium. ....... Trimethylethylammonium . . . . . . . . . . . . . . . . Trimethylpropylammonium ............... Dimethyldie thylammonium . . . . . . . . . . . . . . . Trimethylisobutylammonium . . . . . . . . . . . . . Methyltriethylammonium . . . . . . . Trimethylisoamylammonium . . . . . . . . . . . . . Tetraethylammonium.................... Triethyliso but ylammonium . . . . . . . . . . . . . . . Triethylisoamylammonium . . . . . . . . . . . . . . .

46.4 43.0 38.5 40.6 36.0 36.6 32.7 34.2 30.9 28.0

1.5 3.5 5.5 7.5 9.5 4

6 7 10 5 6

7 7 8 8 9 9 11 12

299

LIMITING CONDUCTANCE O F ORGANIC IONS

TABLE 3 -Concluded

x

ION

(1

mho cm.2 per squivaldnl

Diamines:

Dimethyldiammonium .................... Trimethyleneammonium.................. Tetrame thyleneammonium . . . . . . . . . . . . . . . Pentamethyleneammonium . . . . . . . . . . . . . . .

80.7 70.1 69.5

65.3

substituted group can readily be calculated by the method illustrated in the following sample calculations : (a) Methylene equivalent of -CHOH--: Let a be the methylene equivalent group in lactate ion for which h = 35.0 and n,exclusive of the of the -CHOHhydroxymethylene group, is 1,910. Substituting in equation 9: log 35.0 = log 49.41

- 0.299 log (U + 1.910)

whence, a = 1.26. ( b ) Methylene equivalent of -CH=CH--: From the conductance, 34.0, of is calculated by the a-crotonate, the methylene equivalent of -CH=CHsame method as above to be 1.50. (c) Limiting conductance of oleate ion: By use of the above value of the methylene equivalent of the ethylenic group and the appropriate values for the terminal methyl radical and the carboxyl groups, n is found to be 17.41. Substitution into equation 9 leads to the value 21.2 mho cm.2per equivalent. SUMMARY

1. Equations expressing the limiting ionic conductances in terms of the number of methylene groups or their equivalents are given for monobasic and dibasic aliphatic carboxylic acids, aliphatic primary, secondary, and tertiary amines, and aliphatic diamines. 2. Hydration through hydrogen bonding is shown to enhance, rather than diminish, the ionic conductances, owing to the reduction of the van derWaals radius of the functional group involved. 3. Expressions for the “effective spherical volumes” of the ions discussed are developed. The van der Waals volumes of primary amines and diamines calculated on the,basis of Stokes’ law are found to agree closely with the volumes calculated from independent data, and the effective spherical volumes of the members of these two series are concluded to be approximately the same as their true volumes, while those of the other series discussed depart more or less widely from their true volumes. 4. The limiting conductances of several alkyl carboxylate ions are calculated, and a method of determining the approximate methylene equivalents of substituted methylenes or ethylenic groups is illustrated.

The author wishes to thank Prof. J. W. McBain for his interest in this paper.

300

N. F. MILLER

REFERENCES (1) BREDIG, G.: Z. physik. Chem. 13, 191 (1894). (2) PAULING, L. : The Nature of the Chemical Bond, 2nd edition. Cornell University Press, Ithaca, New York (1942). (3) WEYSSENHOFF, J.: Bull. acad. polonaise sci. e t lettres, No. 7a, 219 (1925); Science Abstracts 29A, 218.

THE WETTING OF STEEL SURFACES BY ESTERS OF UNSATURATED FATTY ACIDS N. F. MILLER Reseaich Division, Technical Department, The New Jersey Zinc Company (of Pa.), Palmerton, Pennsylvania Received March 5 , 1946 INTRODUCTION

One of the most important properties of organic protective coatings is their ability to adhere t o the surfaces which they are to protect under a variety of exposure conditions. It is understandable, therefore, that numerous efforts have been made to find the factors which are responsible for the adhesion or the lack of adhesion of organic coatings to various substrates. Progress in this field has been greatly retarded by the fact that there is no satisfactory method for measuring adhesion divorced from all other paint film properties, despite several concerted attempts to design and develop such procedures. A direct frontal attack upon this problem, therefore, did not seem to offer much promise of success, and an indirect approach based on wetting measurements was chosen for the work described in this paper. Although the adhesional force between dry paint film and substrate is probably unmeasurable by available methods, there are reasons for assuming that this adhesional force cannot become large unless the organic molecules of the primer vehicle make intimate contact with the molecules in the surface of the substrate while the paint is a liquid. Thus the degree of wetting of a primer vehicle for the substrate indirectly controls the force of adhesion which will be developed in the dry film. The degree of wetting of solids by liquids is generally evaluated from three free-energy concepts: work of adhesion, W A ,work of spreading, TVs, and work (or heat) of immersion, W r (3). The force of adhesion, FA, of a liquid to a solid will be shown to be quantitatively evaluated by the work of adhesion. However, it will also be shown that the magnitude of the contact angle, particularly the dynamic receding angle of contact, is a much more fundamental criterion of wetting behavior than the magnitude of F A (or W A ) . A scheme for evaluating wetting properties of liquids toward solids can be based almost entirely on contact-angle behavior.