Table 1.
Examples of the Gibbs-Donnan factors for non-ideal solutions separated by an ideal cation exchange membrane.
The Table presents the Gibbs–Donnan factors (DF1,21 and DF2,21) and the coefficients describing the equilibrium across an ideal cation exchange membrane obtained by solving Eqs (25), (26) and (32) for l2 = w for three different sets of concentrations of permeating ions (counter-ions) in compartment 1 and non-permeating ions (co-ions) in both compartments. The assumed values of A and D were 1.172 (L/mol)1/2 and 1.5 (L/mol)1/2, respectively [23]. The considered system includes two types of permeating cations: monovalent (z1 = 1) with concentration c1,1 in compartment 1 and bivalent (z2 = 2) with concentration c2,1 in compartment 1, and one type of monovalent non-permeating anion (Znp = -1) with concentration Cnp,1 in compartment 1 and Cnp,2 in compartment 2.
Fig 1.
Gibbs-Donnan factors for different ratios of non-permeating ions on the two sides of a cation exchange membrane.
Shown are the relationships between the Gibbs-Donnan factor for a permeating monovalent cation and the ratio of the equivalents of non-permeating ions on the two sides of a cation exchange membrane, , calculated according to Eq (10) for different values of γnp,1 and for two cases: a) permeating monovalent and bivalent cations (left panel); b) permeating monovalent and trivalent cations (right panel).
Fig 2.
Gibbs-Donnan factors for different ratios of non-permeating ions on the two sides of a cation exchange membrane with known total masses of permeating ions.
Shown are the relationships between the Gibbs-Donnan factor for a permeating monovalent cation and the ratio of equivalents of non-permeating ions on the two sides of a cation exchange membrane , calculated according to Eq (45) (for the known total amounts of permeating ions distributed between the two compartments of the same volume, i.e., b = 1) for different values of ϕnp,1 and for two cases: a) permeating monovalent and bivalent cations (left panel); b) permeating monovalent and trivalent cations (right panel).
Table 2.
Examples of predicted Gibbs-Donnan equilibria for different ion solutions separated by a cation exchange membrane vs experimental results.
Shown are exemplary cases of the predicted (calculated) Gibbs-Donnan equilibrium for ideal cation exchange membranes separating different mixtures with monovalent and multivalent cations (n = 2), acting as driving (d) and feed (f) ions, calculated using Eq (45) vs final concentrations of feed ions in the feed compartment obtained in the experimental setup.