TITRATIOS OF POLYELECTROLYTES AT HIGHER IONIC STRENGTHS
Aug., 1954 2.2
62 1
-
t I
1.2
I
I
1.2
I
1.4 2.
I
I
1~
1.6
Fig, 2.-Chelate formation constant of copolymer ST/MA with 0.0036 Af alkaline earths: @, CaN03; 0, SrNOs; 9, MgNOs; 0, BaN03.
ten and twenty times as high as the corresponding value for succinic acid, while somewhat lower values of Kf were obtained for copolymer ST/MA. Experimental Preparation of Polymers.-Forty grams of maleic anhydride and 20 g. of the comonomer were dissolved in 400 ml. of butanone and 100 mg. of azo-bis-isobutyronitrile initiat;r was added. The polymerization was carried out at 70 , the copolymer was precipitated in hexane, dissolved in hot water, dialyzed and freeze dried. The vinyl ethyl ether copolymer (VEE/MA) contained 62 mole % of vinyl ethyl
1.6 2. Fig. 3.-Chelate formation constants of copolymer VEEIMA: @, 0.0018 Jl CaNOa; 0, 0.0036 M BaNOs' 0 , 0.0018 M MgN03; 0 , 0.0036 N SrNOa; e, 0.0036 MgNOa.
1.4
id
ether, the styrene copolymer (ST/lMA) 54 mole % of styrene as calculated from titration data. The ma1Fic anhydride copolymers were fully hydrolyzed, since boiling with alkali before dialysis did not alter their titration behavior. Titration .-The pH determinations were carried out with a Cambridge Instrument Co. research model pH meter with external shielded electrodes. During titration, solutions were held a t 25' to within 0.lo, stirred and protected by a stream of nitrogen from atmos heric COZ. Four minutes were allowed for equilibrium after each addition of base; longer time intervals produced no further pH change. All titrations were carried out in the presence of 1 N potassium nitrate.
Acknowledgment.-This study was supported by a research grant from the Monsanto Chemical Company.
TITRATION OF POLYELECTROLYTES A T HIGHER IONIC STRENGTHS BY R. A. MARCUS Polytechnic Institute of Brooklyn, Brooklyn 1, New York Received April 19, 1954
The titration behavior of polyelectrolytes at higher ionic strengths is treated on the basis of the nearest neighbor interaction between the fixed ions. This point of view suggests a quantitative comparison of the behavior of polymeric and dibasic acids. The application of these considerations to existing work on the correlation of viscosity and titration curves of polymeric acids is briefly discussed.
Introduction.-The correlation of various properties of polyelectrolytes, such as viscosity and titration behavior, has been the subject of a number of recent theoretical treatments.' These approaches have had varying degrees of success and while differing in a number of important respects, have in common the assumption that both the fixed and mobile ions in the polyelectrolyte system may be treated as a continuous charge distribution. The purpose of the present note is to examine a 0. Kunale and W. Kuhn, J . Polymer Sci., (I, 283 (1950); J. Hermans and J. T. G. Overbeek, Rec. trou. chim., 67, 761 (1948); G. E. Kirnball, M. Cutler and H. Sanielson, THISJOURNAL, 66. 57 (1948). ( 1 ) A. Katohalsky,
different approach to the titration behavior of these systems, one which does not involve the latter assumption but instead assumes that the important interactions between the fixed ions on the polyelectrolyte are nearest neighbor interactions. It is expected that the repulsions between the immobile ions will vanish rapidly with distance, R , a t the higher ionic strengths, in the first approximation as e-KR/Rwhere K is the reciprocal Debye length. On this basis nearest neighbor interaction will begin to predominate when KR 1 where R is the diftance between nearest neighbors. When R 5 A. this will occur when the salt concentration exceeds 0.4 M.
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-
R. A. MARCUS
G22
If these considerations are valid a correlation between the titration behavior of a polymeric acid and of a suitable dibasic acid is to be anticipated. An examination of available data on polyacrylic and glutaric acids supports this. A treatment of the dependence of pK on the degree of neutralization is given in the following section using, a t first, the Bragg-Williams approximation and later, the more exact king method. Theory of Nearest Neighbor Interaction.The polymer chain is considered to have Na neutralized and N b unneutralized acidic groups. We suppose that the free energy fa of an immobile ion, A, and its counter ion atmosphere are modited by an amount faa/2 for each A nearest neighbor and by fah/2 for each B neighbor, there being only two neighbors in all. Additivity of these effects will be assumed. Corresponding quantities for the uncharged acidic group B will be denoted by fb, fab/2 and fbb/2, respectively. The total free energy of such an assembly depends on the number of pairs, hTab, of nearest AB neighbors and may be written as Fab
= faNa
+
fbNb
+
faaNm
VOl. 58
f i , of the hydrogen ions in a solution in which their activity is U H + , fi = f~ kT In U H + , it follows that
+
pk’ = pH
-l
-
where f
2w -+ 2.3kT(1 - 2a)
o 1 g 2a = 2.&
fa
+ + faa
fH
- fb
(4)
- fbb
Thus the pK is a linear function of CY in this approximation and the change in pK, from a = 0 to CY = 1 is ApK = -4w/2.3kT. The major contribution to w is faa, the repulsion of the immobile ions, and it follows that faa is positive and w negative, as observed. Both Af and w will depend on the counter ion atmosphere and hence upon the salt concentration. If the repulsion of nearest neighbor ions is very pronounced, then the A and B groups are not distributed randomly along the chain. Using the approach employed by Ising in an analogous problem in ferromagnetism, an equation may be derived which takes into account in an exact manner such deviations from randomness. From such a treatment we find2
f .fbbNbb f fabNab kT In d N a , N b , N a b ) (1)
where g(Na, N b , N a b ) is the number of ways of distributing, for a given Na,N b and N a b , the A and B groups along the chain and where the number of AA pairs, N a a , and of BB pairs, N b b , is given by and from this, that 2N,, Nab = 2Na, 2Nbb N a b = 2 N b . The free Af energy of this system, F , equals -kT In (P.F.) = pK=2.31;T+ -kT 111 Z(P.F.)ab = -kT 111 Zexp ( - F a b / k T ) (1/4a(l - a)(e*lU/RT - 1) + 1 + 2a - 1)(1 - a ) log where (P.F.) is the partition function for all config( 4 4 a ( l - a)(e2a’nT- 1 ) + 11 1 - 2a)a urations while (P.F.),b is the partition function (6) for only those configurations corresponding to a where w and Af retain their previous significance. given N a b . The summation is over all N a b consistent with the values of Na and N b . From equation As before, the change in pK from CY = 0 to QI = 1, ApK, is - 4 w / 2 . 3 k T . This is as expected since 1 it then follows that in both cases a dissociating acid group is in an F = Na(fa fsa) f Nb(fb fbb) - kT ln zg(Ara,N b , N a b ) environment of other undissociated groups when CXP (-N.abw/kT) (2) a: = 0 and in an environment of neutralized groups where when CY = 1. When -2w
