Selectivity coefficients of cation-exchange membranes: Maximizing consistency and minimizing error amplification

Experimental estimation of the selectivity coefficient of ion-exchange membranes involves significant error amplification, but no previous studies quantified the error propagation. Errors in measuring the concentration of the mono-valent cation can be magnified up to 500 times in estimating the selectivity coefficient of mono- to di-valent cations. Further, arithmetic averaging of individual estimations led to systematic overestimation of the selectivity coefficient. To avoid this overestimation, we suggested a least-square method to determine the selectivity coefficient over a range of conditions. In addition, a cohesive explanation for the consistency of the selectivity coefficient with changes in ionic content and equivalent fraction has been lacking in previous literature. We found that the selectivity coefficient of mono-valent potassium to di-valent calcium was consistent with the aqueous-phase ionic content from 5 to 100mN and the aqueous-phase equivalent fraction from 0.2 to 0.8. The valence effect (membranes favoring di-valents over mono-valents) plays an important role in maintaining this consistency. The consistency is also governed by the aqueous-phase activity coefficients and thus depends on the aqueous-phase activity (i.e., unit of the aqueous-phase concentration). The consistency is maintained with the molarity but lost with fractional units, such as mole or equivalent fraction, for the aqueous-phase concentration.