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. 2021 Nov 12;11:22150. doi: 10.1038/s41598-021-00899-y

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Transitivity of Gibbs–Donnan factors: compartments 1 and 2, 2 and 4, and 1 and 3 are separated by identical permselective membranes; non-permeating charged species (np) are present in compartments 1 and 2 only. In equilibrium, the concentrations of permeating ions in compartments 3 and 4 are equal, C_i,4 = C_i,3. The Gibbs–Donnan factors for equilibrium between compartments with non-permeating species, compartment 1 vs. 2, can be calculated using Gibbs–Donnan factors for compartments without non-permeating species, 1 vs. 3 and 2 vs. 4: DF_i,21 = DF_i,31 /DF_i,42. Symbols: Z charge number, C concentration, DF Gibbs–Donnan factor.