Simplified Simultaneous Determination of Concentrations of V2+, V3+ and V4+: An Example of Inverse Problems

Abstract

A simplified spectrophotometric method was developed for the simultaneous determination of
concentrations (c_n) of vanadium ions at different oxidation states, which has applications in
vanadium redox-flow batteries. This method is of particular use for mixed V³⁺/VO²⁺ and V²⁺/V³⁺
solutions. A singular value decomposition was applied to the spectra to resolve the severe overlaps
of the spectra of these cations and to obtain the selected wavelength ^k. By using absorbance
obtained at the selected wavelength in the single- and mixed-valence solutions, the coefficient
matrices G = {g_k, n} were determined via the least squares method. The determined formulae
are:  where Ik is the absorbance at ¹ = 520 nm, ² = 401 nm, ³ = 770 nm, or ⁴ =
320 nm. The determined concentrations agree well (within 5 %) with those determined using
other methods.