376
NOTES
Vol. 61
Acknowledgment.-The assistance of Mrs. Corinne Bonar with the experimental work is gratefully acknowledged. SOME CONSIDERATIONS ON THE GUGGENHEIM AND CONVENTIONAL EQUATIONS FOR ELECTRIC MOMENT CALCULATIONS BY GEORGEK. ESTOK Department of Chemistry and Chemieal Engineering, Tszaa Tschnolooicd Collsgr, Lubbock, Tazas Received October 97, 1868
40
80
120
160
200
.:
[TI
Fig. 3.-The limiting slope of the intrinsic viscosity-rate
of shear curve plotted against the square of the zero shear intrinsic viscosity. The circles correspond to present data
for Heves fractions and the line represents the relation obtained previously for Alfin polyisoprene.
The Guggenheim method for determining electric moments, using dielectric constant and refraction measurements on solutions, has been developed in recent y e a ~ s . ~ - ~ Some confusion, however, appears to have arisen6I6as an outgrowth of the original manner of deriving the equation,a in which an arbitrary symbol representing a fictitious atomic polarizability, ya', was introduced. Another contributing factor was the elimination of two small terms from the original exact equation, yielding an approximate equation. The purpose here is to derive, in a concurrent and completely analogous manner, both the conventional and Guggenheim equations, starting with the same initial equations used by Guggenheim. This aids in making a simple comparison of the two equations. It will further be demonstrated that the derivation made by Palit,b who used a different set of starting equations, leads to exactly the same equation &s that of Guggenheim. Symbols used in the following treatment are: M ,u, V , e, n, P,y and p, which refer, respectively, to: molecular weight, specific volume, molar volume, dielectric constant, index of refraction, molar polarization, polarizability and electric moment. Subscripts are: 1 (pure solvent); 2 (pure solute); a, atomic; and e, electronic. For convenience, symbols without subscripts refer to solution values, except that 2 and w refer to mole fraction solute and weight fraction solute, respectively, in the solution. The original derivation by Guggenheim2used the Debye equations aa a basis
than does the latter, at more or less the same concentrations; similarly for Hevea fraction C and Alfin 3a. Unfortunately, these results cannot be conveniently reduced to any common denominator for comparison short of the molecular weight-independent parameter already mentioned. This, of course, has been shown to be insensitive to the structural differences between the two samples of polyisoprene under considerati on. As pointed out by Goldberg and Fuoss,' the nonNewtonian behavior of polymer solutions can arise from either the shear dependence of interaction between polymer molecules or the shear dependence of the intrinsic viscosity, or both. Since the latter factor is apparently the same for the two olymers in question, it follows that the remaining ifFerence between them must be ascribed to the difference in the shear dependence of polymer interaction. Now, examination of the isoshear plots of Alfin polyisoprenel reveals that the Huggins k' is more or less constant over the shear range 0-600 set.-' (e.g., 0.36 for fraction la), whereas with the natural rubber molecule k' decreases with shear rate (e.g., from 'k = 0.41 a t D = 0 to IC' = 0.25 a t 600 set.-' for Hevea fraction B). The explanation for this disparity in the shear dependence of interaction is presently unknown. It appears from the above that the shear effect in dilute solutions may not be used readily for studying microstructural differences in polymers, where V I and Vs are partial molar volumes of solalthough concentrated solutions may actually show vent and solute and V = VI z(Vz VI) definite differences between such polymers. MoreBy subtracting equation l a from equation 1 over, for most polymers of interest, the dilute solu- Guggenheim obtained tion viscosities are smaller than those considered (1) Presented at the 130th meeting of the Amerioan Chemical Sohere and as a result their shear effects are much oiety, Atlantio City, Sept., 1956. less pronounced, in which case the suggested pa(2) E.A. Ouggenheim, Tmm. Faraday SOC,,46, 714 (1949). rameter would be even less likely to be useful for mi(8) J. W. Smith, {bid., 46, 394 (1960). (4) E. A. Quggenheim, i b i d . , 41, 673 (1951). crostructural studies.
B
+
(8) P. Coldberg and R. M. FUOEB, THIBJOURNAL, 88,648 (1964).
-
(5) 8. R. Palit, J . Am. Chem. Soc., T4, 3952 (1962). (6) E. A. Quggenheim. Proc. P h y s . doc., 68B,186 (1965).
NOTES
March, 1957
377
significant error, thus obviating the need for density measurements on the solution^.^ If the M R D for a solid solute is obtainable without making refraction measurements, then equa(2) tion 7 is much simpler to use, since only dielectric To simplify the following treatment, let 4nN/3 = constant data need be gathered. In the event reK, p2/3kt = F , E - l / e 2 = H , and (e - l / c 4- fraction measurements must be made, there is little 2) - (n2 - l/n2 2) = C. Then equations 1 and to choose between equations 7 or 8, except that the latter is less sensitive to an approximate value of 2 may be written, respectively 02inasmuchas (e1 l/sl 2) - (n12- l/nla 2) is H [ V I + Z(VZ- VI)] HiVi(1 - Z) K(re 7s + F)r normaly much smaller than el - l / e l 2 for non(31 polar solvents. G[Vi S(VZ - VI)] GiVi(1 - Z) K(ra + FIX (4) S. R. Palits criticized and discussed the Guggenheim approximate equation (which neglected the Expanding and rearranging equation 3 gives last two terms of equation 8; Le., QM2s and p2a) HV1 + Hz(VS - VI) - HiVi( 1- Z) K(re F ) z but did not appear to recognize that the Guggenheim exact equation was identical with his own deDifferentiation with respect to x yields rived exact equation. It may not be readily apparent that Palit’s term B is identical with Guggenheim’s GlMzu2(also called [4n=N/3]ya‘by GuggenHIVI e K(re ’Y. F ) heim). This identity may be demonstrated as At infinite dilution HV2 = H1V2,HVi = HIV1,and x(Vs - Vi) (dH/dx) = 0, so that the following conventional equation results where d = density, and = (bd/dw),, also
+
+
+ +
+
3
-
+
+
+ +
+ + YS
+ +
Analogous treatment of equation 4 leads directly to the Guggenheim equation
It has been shown previously’ that BO, at infinite dilution, is equal to dle(ul - oa), and thus Palit’s (1 - BJdl) is equal to ,d1~2, Therefore Palit’s GlM2U3 (or term B is identical with Guaaenheim’s -Equation 6 is analogous to equation 6,except that [4T N / 3 I ?a‘). the electronic polarizability has been eliminated. In conclusion, it is felt that the derivations of the Conversion of the concentration unit from mole conventional and Guggenheim equations as here fraction solute x to weight fraction w may be ac- outlined permit an eat-comparison and better uncomplished by using the following general relation, derstanding of the equations. In general the convalid at infinite dilution: (dA/dz), = M*/Ml. ventional approach is simpler, although occasion(dA/dw) m , where A may be H or G. ally the Guggenheim method could be preferred. The conventional and Guggenheim equations (7) G. R. Eatok. Txie JOURNAL, 60, 1336 (1956). may now be written, respectively MeuI(dH/dw), K(r. r. F ) - HIV; (5s) CHELATE FORMATION BETWEEN Mnvl(dC?/dw)m K ( r s F ) - GIVZ (6a) AND POLYELECTROLYTES CATIONS where VI is the partial specific volume of the solvent. BY GEOFFR~Y ZWAY~ By further differentiating, with respect to w, the Received November 1 , I068 quantities represented by H and G, letting E = €1 Wolcotl Gihbs Msmorial Laboratory, Harvard University, Cambridge, Maee. and n = n1 a t infinite dilution, and resubstituting for K and F, the following working forms are obChelate formation between cations and simple tained dicarboxylic acids may be treated adequately as a binding problem in terms of the mass action law. The situation is more complicated with polyelectrolytes because of the very large electrostatic en(7) ergy term which in general makes the binding much stronger than in the corresponding monomeric case. Considerable progress has been made in treating the case of univalent counterions. The purpose of this note is to point out the additional difficulties that arise in the case of multivalent where Pze = ( 4 ~ N / 3 ) ~ e e electronic polarization, and the inadequacy of a recent attempt usually taken equal to the molar refraction, M R D ; counterions to overcome these difficulties. PW = (4nN/3)ya = atomic polarization, often The binding of copper to oxalic acid2may be contaken as 0.05 M R D . (1) U.8.Public Health Service Predootorate Research Fellow of the I n either equation, if the specific volume of the Heart Institute. pure solute (as a liquid) is available, it may be sub- National (2) The constante are determined from the data in Martell and stitut,ed for the partial specific volume v2 without Calvin, “Chemistry of the Metal Chelate Compounds,” 1952. 5
+ + +
