Effective Charge Density Determined by the Osmotic Coefficient

Effective Charge Density Determined by the Osmotic Coefficient 

The osmotic pressure of a solution of non-interacting molecules or ions at the zero concentration limit is determined by the total molar concentration of solute species. In this hypothetical state, a solution of a polyelectrolyte may contain highly charged polyions at concentration c and monovalent counterions at a concentration where c is the molar concentration of the polyelectrolyte and z is the number of counterions per polyion. Since the total molar concentration is c (z +1) c z for z l, the ideal osmotic pressure is 

The product zc equals Qc where Q is the fraction of ionizable monomers in the polyelectrolyte and c is the monomolecular concentration of the polyelectrolyte. To account for nonideality due to electrostatic interaction etc., the osmotic coefficient J is introduced and defined as 0=1 l/rfa with 

Comparison of equations 19 and 20 shows that we can consider the product zO as an effective charge number zeff, hence O=zeffz. The osmotic coefficient is therefore a measure of the extent to which the dissociated counterions are not independent osmotic entities due to Coulombic interactions [61]. The effective charge density fos is then derived as 

Manning derived the following expression for f based on counterion condensation for infinite dilution 

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