Expressions are derived which introduce an appreciable simplification into the calculation of the thermodynamic properties of solutions of polyelectrolytes in certain cases. For example, for a certain class of theoretical models of these systems it is found that the square of the mean ion activity coefficient of a uniunivalent salt in the presence of polymeric ions is V 2 / fe-eif!/kT dV · fe•if!lkT dV, the integration of the potential f being over a region whose volume is the volume of solution per macro-ion, V, and whose symmetry is that assumed for the polyelectrolyte. The osmotic pressure of a salt-polyelectrolyte system is, ignoring the contribution of the macro-ion, estimated to be ~i Ci'kT, where ~i Ci' is the sum of the concentrations of all ions at the surface bounding the previously defined volume V. Other relations and various applications are given. The activity coefficient of salt in the presence of polyelectrolytes, calculated by extending the "parallel rod" picture of polymeric ions, is found to be in reasonable agreement with the experimental data. The use of the Poisson-Boltzmann equation to estimate fin these systems is shown not to render inconsistent several alternative expressions for the electrostatic contribution to the free energy. INTRODUCTIONT HE strong electrostatic fields in the neighborhood of polymeric ions have been established experimentally and their theoretical description 1 -8 has been the subject of a number of recent communications. Relevant thermodynamic data include osmotic pressures, activity coefficients of salts, and titration behavior of polymeric acids and bases. Usually these properties are calculated theoretically by differentiation of a free-energy expression into which parameters of the models have been introduced. Because of the frequent complex dependence of these parameters on the thermodynamic variables such as the moles of the components and the volume of the solution, the differentiation may become a lengthy process. A somewhat different procedure is employed here for a certain class of models, in that the last two steps are reversed-expressions are set up for various thermodynamic properties by differentiation of the free energy, these are then simplified and the parameters are introduced as a final step. This procedure effects a considerable simplification of the calculation in various cases. THEORETICAL GeneralA number of theoretical models of polyelectrolytes have been advanced 1 -8 ; several of these have in common the following assumptions:(1) The interaction between polymeric ions is neglected except insofar as the concentration of these ions 1 Kuhn, Kiinzle, and Katchalsky, Helv. Chim. Acta 31, 1994 (1948).2 J. J. Hermans and J. T. G. Overbeek, Rec. trav. chim. 67, 761 (1948). 3 Alfrey, Berg, and Morawetz, J. Polymer Sci. 7, 543 (1951). 4 Fuoss, Katchalsky, and Lifson, Proc. Nat!. Acad. Sci. U.S. 37, 579 (1951). 6 Kimball, Cutler, and Samelson, J. Phys. Chern. 56, 57 (1952). 6 P. J. Flory, J. Chern. Phys. 21, 162 (1953). 7 Osawa, Imai, and Ka...

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Diphenyl‐p‐styrylphosphine (I), prepared by the reaction of diphenylchlorophosphine with p‐styrylmagnesium chloride, was converted to the corresponding oxide (II) and sulfide (III) by treatment with tert‐butyl hydroperoxide and sulfur, respectively. These substituted styrenes polymerize readily with thermal or free radical initiation to high molecular weight polymers. These polymers are somewhat more thermally stable than polystyrene. Thus their Ti's in air were in the range 320–365°C. and in nitrogen, 375–395°C. Their T10's were in the 390–415°C. range in air and 425–435°C. in nitrogen. Polystyrene in air has a Ti of 263°C. and a T10 of 340°C. The monomers copolymerize readily with styrene and methyl methacrylate, the low conversion polymers always containing more phosphorus than the monomer mixtures. Reactivity ratios were determined, from which Q and e values were calculated. These values for the following monomer pairs are (phosphorus monomer M2): styrene‐I, r1 0.52, r2 1.43, Q2 1.34, e2 −0.30; styrene‐II, r1 0.42, r2 1.40, Q2 1.34, e2 −0.07; styrene‐III, r1 0.43, r2 1.49, Q2 1.37, e2 −0.13; MMA‐I, r1 0.32, r2 0.91, Q2 1.58, e2 0.80; MMA‐II, r1 0.38, r2 1.46, Q2 1.42, e2 −0.35; MMA‐III, r1 0.29, r2 1.22, Q2 1.58, e2 −0.62. Analysis of these data indicates that these phosphorus moieties are activating through strong electronegative effects. The extent of activation is intermediate in intensity between mildly activating groups (e.g. Cl) and strongly activating groups (e.g. CN or NO2). The behavior of diphenyl‐p‐styrylphosphine is of particular interest in this respect, since its electronegativity points to strong electron delocalization.