Separation effectiveness of ideal ion exchange membranes: Application of the Gibbs-Donnan theory.

Ion exchange membranes (IEMs) are permselective membranes that, in principle, only allow the flow of ions with a specific charge sign, opposite to that of the fixed membrane ionic groups (counter-ions). This charge-based selectivity, like the size-based selectivity of classic semipermeable membranes, leads to an uneven distribution of permeating ions on the two sides of the membrane, which allows for ion separation or recovery in various processes in industry or environmental protection. Here, we apply the principles of mass balance, charge neutrality, and equality of electrochemical potentials in the state of thermodynamic equilibrium to provide a simple method for estimating the Gibbs-Donnan factors and the equilibrium concentrations of permeating ions in two compartments separated by an ideal IEM, i.e. an IEM that is not permeable to co-ions. We present the method for the case when the equilibrium concentrations are known in one compartment and need to be estimated in the other compartment as well as for the case when the total masses of ions in both compartments are known and their equilibrium concentrations need to be predicted. For both cases, the presented nonlinear algebraic equations require in general the use of numerical methods to approximate their mathematical solutions, although we present as well some closed solutions for simple cases with ideal ionic mixtures. Based on the extended Debye-Hückel theory, we also provide analogous equations (general and for specific cases) for systems with non-ideal ionic mixtures. The presented method can provide the expected ideal separation effectiveness of an IEM, which can then be used to assess the relative separation effectiveness of a real membrane.