Calculation of the Gibbs–Donnan factors for multi-ion solutions with non-permeating charge on both sides of a permselective membrane

Separation of two ionic solutions with a permselective membrane that is impermeable to some of the ions leads to an uneven distribution of permeating ions on the two sides of the membrane described by the Gibbs–Donnan (G–D) equilibrium with the G–D factors relating ion concentrations in the two solutions. Here, we present a method of calculating the G–D factors for ideal electroneutral multi-ion solutions with different total charge of non-permeating species on each side of a permselective membrane separating two compartments. We discuss some special cases of G–D equilibrium for which an analytical solution may be found, and we prove the transitivity of G–D factors for multi-ion solutions in several compartments interconnected by permselective membranes. We show a few examples of calculation of the G–D factors for both simple and complex solutions, including the case of human blood plasma and interstitial fluid separated by capillary walls. The article is accompanied by an online tool that enables the calculation of the G–D factors and the equilibrium concentrations for multi-ion solutions with various composition in terms of permeating ions and non-permeating charge, according to the presented method.

Jacek Waniewski, Mauro Pietribiasi, Leszek Pstras "Calculation of the Gibbs-Donnan factors for multi-ion solutions with non-permeating charge on both sides of a permselective membrane"

Appendix A. Calculation of the Gibbs-Donnan factors for non-ideal solutions
From equation (4), the activities of any two ions with charge numbers i z and j z (i, j = 1,2,...n) in the two considered compartments can be related as follows: As before, we assume here that the ion concentrations (ci) represent free (unbound) ion fractions. Let us also assume that the function i f is invertible, i.e., if one knows the activities of all ions, then their concentrations can be calculated as follows: ( ) x a / a = .
From equations (2) and (A2) we have:   Let us assume that we can express ion activity as: where the activity coefficient i  is a function of concentrations of all ions (not to be confused with the previously defined γα describing the relative ionic equivalents).
The Debye-Hückel theory (validated for low ion concentrations) provides the following description of activity coefficient: where i r is the effective ion radius in the solution, A and B are known constants that depend only on the solvent and temperature (not to be confused with Aα and B introduced when describing the equilibrium distribution of permeating ions among two compartments), and I is the so-called ionic strength of the solution.
As is clear from the above, the functions 1 i f − and 1 np f − are highly complex, and hence solving equation (A3) for a multi-ion solution is a very computationally demanding task. Note, however, that in the equation (A1) we have the ratios of ion activities in the two considered compartments, and hence, based on the Debye-Hückel theory, if the ionic strength I and the effective ionic radii i r are not much different in the two solutions (as may be the case), the activity coefficients i  will be similar in both compartments, and therefore the activity ratio i,2 i,1 a / a could be approximated by the concentration ratio i,2 i,1 c / c .

Appendix B. Mass balance with ion speciation
Regardless of the initial distribution of permeating ions, an electroneutral two-compartment system tends to an equilibrium described by equations (1) and (2), and by the balance of mass for each ion i: where Vi denotes the volume of the i-th compartment and Mi denotes the total mass of the i-th ion, ik c is the concentration of the chemical compound built of ion i and its counterion k , the summation is over all counterions to ion i. Here, we assume that c , need to be found.
Assuming the reversibility of the reaction between an ion i and its counterion k that forms a species ik, one has: where  are stoichiometric coefficients of the species in the reaction and ik K is the equilibrium constant for the reaction, ( ) For simplicity, we omit here the reactions that can involve three or more different ions and nonreversible reactions, such as precipitation. Thus: The equations (B1), (B5), and (A3) need to be solved together to obtain the G-D factors for the generalized, non-ideal case of multi-ion solutions with various chemical forms of permeating ions i.

Appendix C. Proof of the transitivity of Gibbs-Donnan factors
To prove the transitivity of the Gibbs-Donnan factors, equations (24) and (25) The separation of the sum in equation (C7) into two terms that include m ions with a positive charge number and s-m ions with a negative charge number yields the following (here we assume, for the sake of simplicity, that the positive charge numbers are at the beginning of the list and the negative charge numbers follow the positive ones):  / v  31  42  21  ,1  31  42  21  31  42  21  ,1  1  m 1 In equation (C9) This proves equation (24).