Natural Indices for the Chemical Hardness/Softness of Metal Cations and Ligands

Quantitative understanding of reactivity and stability for a chemical species is fundamental to chemistry. The concept has undergone many changes and additions throughout the history of chemistry, stemming from the ideas such as Lewis acids and bases. For a given complexing ligand (Lewis base) and a group of isovalent metal cations (Lewis acids), the stability constants of metal-ligand (ML) complexes can simply correlate to the known properties of metal ions [ionic radii (r _{M ⁿ⁺} ), Gibbs free energy of formation (ΔG°_{f,M ⁿ⁺} ), and solvation energy (ΔG°_{s,M ⁿ⁺} )] by 2.303RT log K _ML = (α*_MLΔG°_{f,M ⁿ⁺} - β*_ML r _{M ⁿ⁺} + γ*_MLΔG°_{s,M ⁿ⁺} - δ*_ML), where the coefficients (α*_ML, β*_ML, γ*_ML, and intercept δ*_ML) are determined by fitting the equation to the existing experimental data. Coefficients β*_ML and γ*_ML have the same sign and are in a linear relationship through the origin. Gibbs free energies of formation of cations (ΔG°_{f,M ⁿ⁺} ) are found to be natural indices for the softness or hardness of metal cations, with positive values corresponding to soft acids and negative values to hard acids. The coefficient α*_ML is an index for the softness or hardness of a complexing ligand. Proton (H⁺) with the softness index of zero is a unique acid that has strong interactions with both soft and hard bases. The stability energy resulting from the acid-base interactions is determined by the term α*_MLΔG°_{f,M ⁿ⁺} ; a positive product of α*_ML and ΔG°_{f,M ⁿ⁺} indicates that the acid-base interaction between the metal cation and the complexing ligand stabilizes the complex. The terms β*_ML r _{M ⁿ⁺} and γ*_MLΔG°_{s,M ⁿ⁺} , which are related to ionic radii of metal cations, represent the steric and solvation effects of the cations. The new softness indices proposed here will help to understand the interactions of ligands (Lewis bases) with metal cations (Lewis acids) and provide guidelines for engineering materials with desired chemical reactivity and selectivity. The new correlation can also enhance our ability for predicting the speciation, mobility, and toxicity of heavy metals in the earth environments and biological systems.