Stability & Thermodynamics — formula sheet

Stepwise formation constant K_n: the equilibrium constant for adding the nth ligand to the complex that already carries (n − 1) of them. It describes one step only.

Overall (cumulative) formation constant β_n: the equilibrium constant for forming ML_n directly from the free metal ion and n free ligands. Also called the cumulative or gross constant, and sometimes written β_1n.

The three reasons the stepwise constants fall: (1) statistical — fewer vacant sites and more ligands able to leave, worth about 1.6 log units in total for an octahedral centre; (2) electrostatic — the metal’s effective positive charge is progressively neutralised, and ligand–ligand repulsion grows; (3) steric — the coordination sphere becomes crowded.

Bjerrum formation function n̄: the average number of ligand molecules bound per metal ion present, n̄ = (C_L − [L]_not bound to metal) / C_M. It is an experimental quantity — it comes from mass balance and a measured pH, with no assumption about how many complexes exist.

Irving–Williams series: for high-spin octahedral complexes of the first-row divalent metal ions, stability constants increase in the order Mn < Fe < Co < Ni < Cu > Zn, irrespective of the ligand. The maximum at copper and the drop at zinc are as much a part of the series as the rising limb.

Linear free-energy relationship (LFER): a linear correlation between the free energies (equivalently, the logarithms of the equilibrium or rate constants) of two related reaction series. Here, between binding a metal and binding a proton.

Bite angle: the L–M–L angle subtended at the metal by the two donor atoms of a chelating ligand. It is fixed, within a narrow range, by the ligand’s backbone — which is why ring size dictates which metal ions a chelate suits.

Effective molarity (EM): the concentration at which an intermolecular reaction would proceed at the same rate, or lie at the same position of equilibrium, as the corresponding intramolecular one. For chelate ring closure, EM = K(chelate step) / K(analogous intermolecular step), and it has units of concentration.

Hard species are small, of high charge density, and weakly polarisable; their frontier orbitals lie far apart in energy. Soft species are large, of low charge density, and highly polarisable; their frontier orbitals lie close together. The HSAB principle: hard acids prefer hard bases; soft acids prefer soft bases.

Cratic entropy: the part of a solute’s entropy that arises purely from mixing — from the number of ways of distributing solute particles among solvent sites. It depends only on the concentration scale, not on the identity of the solute. Unitary entropy: everything else — the part that reflects the actual structure, vibrations and solvation of the species.

Macrocyclic effect: the additional thermodynamic stability of the complex of a cyclic polydentate ligand over that of its open-chain analogue with the same donor atoms, the same number of donors and the same chelate ring sizes.

Preorganisation: the extent to which a ligand (host) is already in the conformation required for binding, and already stripped of solvent, before it meets the metal (guest). The more preorganised the host, the less free energy must be spent organising it during binding, and the more stable the complex.

Cryptate effect: the further enhancement of stability, and of selectivity, obtained on going from a two-dimensional macrocycle (a crown) to a three-dimensional bicyclic cage (a cryptand) with the same donor set. Typically worth a further three to four orders of magnitude in water.

α_Y: the fraction of the total dissolved EDTA present as the fully deprotonated Y⁴⁻. α_Y = [Y⁴⁻]/C_EDTA. It is a pure number between 0 and 1, it depends only on pH, and it is tabulated.

Conditional (effective) formation constant K′: the formation constant written in terms of the total concentration of each free reagent rather than the concentration of the single reactive species. It is constant at fixed pH and fixed concentrations of any competing ligands, and it is the number that governs the actual titration.

Masking: adding a reagent that binds an interfering ion so strongly that it no longer reacts with the titrant, without removing it from solution. Demasking reverses the process, releasing the ion for a second titration.