A mathematical model for the liquid-liquid equilibrium of the lanthanide and yttrium oxides in the nitric acid - water - tributyl phosphate (HNO₃-H₂O-TBP) two-phase system has been developed. This model is based on the interrelationships of chemical reactions, phase equilibrium, and material balance. The chemical reactions were modelled using experimentally derived empirical equations for components involved. The distribution coefficients of 16 rare earths which were experimentally determined. by UK Harwell Laboratory were used to evaluate the parameters of this equilibrium model. Once model parameters are determined, material balance for all species in the HNO₃-H₂O-TBP two-phase system can be accomplished simultaneously.

The developed equilibrium model was validated by a published experimental extraction system for the separation and purification of specific component from the mixture of lanthanide and yttrium elements.

This model was used for analyzing the minimum production cost of extraction systems for (1) separation and purification of europium from the lanthanide and yttrium mineral sample; and (2) purification of yttrium from the mixture of lutetium, ytterbium and yttrium.