General Equations for Determining Species Involved in Solution Equilibria and Equilibrium Constants from Solution Molecular Mass Measurements

A simple way by which equilibrium species can be identified with reasonable certainty, and equilibrium constants and thermodynamic data can be thereby extracted, from variable-concentration cryoscopic molecular mass measurements in solution is reported. The method relies on the assumption that the individual molecular species involved in such solution equilibria exert independent and additive contributions to the depression in freezing point from that of the pure solvent. Given this assumption, individual equations can be developed for a given dynamic equilibrium, relating the equilibrium constant (K_s) to the molecular mass (M_r) and hence to the degree of association (n) of the species involved. The measurements will fit best one such equation and can thereby be used to pinpoint the equilibrium species present. To first illustrate this method, the variation of the degree of association (n) has been examined over a range of concentrations (up to ca. 0.1 mol L⁻¹) for two lithium-containing complexes in benzene solutions; the inorganic complex, lithium bromide pentamethyldiethylenetriamine, (LiBr*PMDETA)n (1) [PMDETA = MeN(CH₂CH₂NMe₂)₂], and the amidolithium reagent, (dicyclohexylamido)lithium hexamethylphosphoramide, [(C₆H₁₁)₂NLi*HMPA]_n (2) [HMPA = (Me₂N)₃P═O]. While both complexes have known dimeric structures in the solid state, the cryoscopic work presented here, and its fitting to specific equations, implies that both 1 and 2 are involved in essentially dimer ⇌ monomer equilibria in such solutions. On this basis, the equilibrium constants for both complexes were determined [for 1, K_s = 1.41 × 10⁻² mol L⁻¹, ΔG^o = 9.8 kJ mol⁻¹; and for 2, K_s = 4.0 × 10⁻³ mol L⁻¹, ΔG^o = 12.8 kJ mol⁻¹].