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The Debye-Huckel theory provides an adequate description of the behavior of simple electrolytes at low concentrations. It contains, however, two inherent limitations: spherical symmetry, and the condition that the electrostatic potential energy of a test ion be small compared to kT. Several cases involving non-spherical geometry have been treated; Verwey and Overbeek' summarize the pertinent literature to 1948. In the case of polyelectrolytes2 (especially in the absence of added simple electrolyte), we find high concentrations of charge in small volumes; in the vicinity of such a multiple charge, the potential energy of a counter ion is necessarily many times kT. Furthermore the geometry is in general not spherically symmetric; it has been concluded on empirical grounds2 that a polyelectrolyte becomes an extended rod-like structure at low concentrations, due to intramolecular electrostatic repulsion. We present herewith the mathematical solution of the problem of the potential of an isolated rod-like molecule in the presence of an electrically equivalent number of counter ions; the distribution and density of the latter may then be derived from the potential. The derivation depends on explicit use of the Boltzmann distribution function, and assumes linear superposition of fields in a case where the Poisson equation is non-linear. The justification of these fundamental assumptions is based on a comparison with experiment,3 details of which will be presented later.Let us consider a rigid rod-like molecule of radius a stretched along the axis of a cylinder of radius R. Assume that the length h of the cylinder is large compared to R, so that end effects may be neglected. Suppose the rod carries v ionogenic sites, so that the linear density of sites is v/h per cm.; assume further that these sites are uniformly distributed along the rod. Assume the volume in the annulus to be filled with a continuous medium of dielectric constant D, containing at time zero a solute which will react with the ionogenic sites to produce ions. Let the concentration of solute be n molecules per cm.3, where n is chosen so that v = 7rn(R2 -a2)h.(1)If we allow the reaction to proceed to completion, the rod will then carry a charge ve,/h per cm. and the annulus will contain a charge 7rn(R2-a2)he2, where el = -E2. Here the absolute value of el and 62 will be taken as the unit charge, 4.80 X 10-10 e. s. u. The n counter ions so produced will ob- 'V'OL. 37, 1951 579

Abstract

Equilibrium swelling and salt accumulation were investigated for highly swollen polymethacrylic acid gels in aqueous salt solutions of 0.8 × 10−2 to 0.5 M LiCl, NaCl, and KCl, at varying degrees of ionization of the polymer. The same equilibrium results were obtained with all three cations, thus showing that in the range investigated, specific interactions between the cations and the polyacid network may be disregarded. It was shown that the Donnan distribution, uncorrected for activity factors, cannot be applied to highly charged, highly swollen gels, even as a rough approximation. A general theory was developed which satisfactorily describes these equilibrium results in the entire ionization range of the gel in salt solutions up to 0.5 M. This theory was based on Flory's equation for the contractile free energy of the network, and on the equation of Katchalsky et al. for the electrostatic free energy of polyelectrolyte solutions. It was found that the activity factor of the salt in the gel‐phase is chiefly determined by the electrostatic factor, and closely approximates the activity factor in polyelectrolyte solutions.