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Figure 4. Schematic representation of ions concentration and electrical potential for a binary monovalent electrolyte solution in a cation exchange membrane immersed between a dilute and a concentrate solution. By imposing the equivalence between the electrochemical potentials of ions in the electrolyte solution and in the IEM in equilibrium with it, the Donnan potential (𝜑) can be expressed [49, 57] as ∆𝜑 = 𝜑 − 𝜑 = 1 𝑧 𝐹 𝑎 𝑅𝑇 ln + 𝑉 ∆𝜋 (1) where 𝜑 is the electrical potential on the IEM side, 𝜑 is the electrical potential on the solution side, 𝐹 is the Faraday constant, 𝑧 is the valence, 𝑅 is the universal gas constant, 𝑇 is the absolute temperature, 𝑎 is the activity of the ion, 𝑉 is the partial molar volume, ∆𝜋 is the osmotic pressure difference between the two phases, the subscript 𝑖 refers to the salt ion i and the superscripts 𝑖𝑒𝑚 and 𝑆𝑂𝐿 indicate the membrane and the solution, respectively. By equating the Donnan potentials for cation and anion, being the osmotic pressure term 𝑉 ∆𝜋 negligible with respect to the RT-logarithmic one [49], and by assuming that the activity coefficients are equal in both phases [62], the Donnan equilibrium for the concentration can be obtained as [58, 62] (2) is the fixed charge concentration and 𝑎 𝐶𝐶 𝐶= +𝐶− 2 2 where 𝐶 is the co-ion concentration in the membrane, 𝐶 𝐶 is the salt concentration in the solution. over the two sides, referred to as “membrane potential”. According to the well-known theory by Teorell-Meyer-Sievers (TMS), the membrane potential is [62–64] 12 An IEM immersed between two solutions at different concentrations is subject to a voltage differencePDF Image | Electrodialysis for water desalination
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