A generalized thermodynamic model has been developed for the simultaneous extraction of rare earth elements (REEs) of the cerium subgroup, as well as Eu, Gd, and Y, in the systems
Ln(NO₃)₃–NH₄NO₃–HNO₃–H₂O–TBP (100%) and HCl–NaCl–H₂O–Cyanex 272–P507–TBP.

The distinguishing features of the model include consideration of extractant association and incorporation of an experimentally derived dependence of nitric and hydrochloric acid dissociation constants on the ionic strength of the aqueous phase, without the use of activity coefficients.

Unified equations are proposed to calculate concentration constants for extraction, nitric acid dissociation, and complex formation of REEs and yttrium with nitrate ions as functions of aqueous-phase ionic strength. For ionic strength below 0.5, concentration constants are calculated using the Debye–Hückel equation; for ionic strength above 0.5, a modified Vasiliev (Debye–Hückel–type) equation developed by the authors is applied.

Extraction constants and non-ideality parameters were determined using the MULCON program by searching for the global minimum of an objective function based on discrepancies between experimental and calculated distribution coefficients. The optimization employed second-type constraints, ensuring compliance with mass balance, charge balance, and the law of mass action.

The developed model and calculation methodology provide an accurate description, within experimental uncertainty, of co-extraction isotherms for nitrates and chlorides of La, Ce, Pr, Nd, Sm, Eu, and Gd using organophosphorus extractants (TBP, P507, Cyanex 272), as well as extraction of yttrium chloride using P507-based extractants.