Explanation:


Partition coefficient: It is defined as the ratio of the concentration of a substance in a mixture of two immiscible phases at equilibrium.


Tri-n-butylphosphate (TBP): It is an organic solvent which is commonly used in solvent extraction processes for the separation of metal ions from aqueous solutions.


HNO3: It is a strong oxidizing acid and is commonly used in the preparation of various chemicals and fertilizers.


The partition coefficient of a metal ion depends on various factors such as the size and charge of the ion, the nature of the solvent, and the pH of the solution. In the given case, the metal ions are lanthanide(III) ions and the solvent is TBP in concentrated HNO3.


Effect of Lanthanide(III) ion size and charge:


The size of the lanthanide(III) ions increases from La(III) to Lu(III) due to the addition of electrons and protons to the nucleus. However, the charge remains constant at +3. As the size of the ion increases, its ability to interact with the solvent molecules also increases. Therefore, the partition coefficient of the lanthanide(III) ions should increase with increasing size.


Effect of solvent:


TBP is a non-polar solvent and is known to extract metal ions through a solvation mechanism. The solubility of metal ions in TBP is generally higher than in aqueous solutions. Therefore, the partition coefficient of the lanthanide(III) ions should be higher in TBP than in aqueous solutions.


Effect of pH:


The partition coefficient of lanthanide(III) ions is strongly dependent on the pH of the solution. At low pH, the lanthanide(III) ions are protonated and become more hydrophilic, resulting in a lower partition coefficient. At high pH, the lanthanide(III) ions become deprotonated and more hydrophobic, resulting in a higher partition coefficient.


Conclusion:


Based on the above factors, the lanthanide(III) ion having the highest partition coefficient between TBP and concentrated HNO3 is Eu(III) due to its intermediate size and its position in the middle of the lanthanide series. La(III) and Nd(III) are smaller in size and Lu(III) is larger in size, which results in lower partition coefficients.