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In the present study,the modified Sverjensky-Molling equation,derived from a linear-free energy relationship,is used to predict the Gibbs free energies of formation of crystalline phases ofα-MOOH (with a goethite structure)andα-M_2O_3(with a hematite structure)from the known thermodynamic properties of the corresponding aqueous trivalent cations(M~(3+)).The modified equation is expressed asΔG_(f,M_VX)~0=a_(M_VX)ΔG_(0,M~(3+))~(0)+b_(M_VX)+β_(M_VXγM~(3+)),where the coefficients a_(M_VX),b_(M_VX),andβ_(M_VX) characterize a particular structural family of M_VX(M is a trivalent cation[M~(3+)]and X represents the remainder of the composition of solid);γ~(3+)is the ionic radius of trivalent cations(M~(3+));ΔG_(f,M_VX)~0 is the standard Gibbs free energy of formation of M_vX;andΔG_(n,M~(3+))~0 is the non-solvation energy of trivalent cations(M~(3+)).By fitting the equation to the known experimental thermodynamic data,the coefficients for the goethite family(α-MOOH)are a_(M_VX)=0.8838,b_(M_VX)=-424.4431(kcal/mol),andβ_(M_VX)=115(kcal/ mol.(?)),while the coefficients for the hematite family(α-M_2O_3)are a_(M_VX)=1.7468,b_(M_VX)=-814.9573(kcal/ mol),andβ_(M_VX)=278(kcal/mol.(?)).The constrained relationship can be used to predict the standard Gibbs free energies of formation of crystalline phases and fictive phases(i.e.phases that are thermodynamically unstable and do not occur at standard conditions)within the isostructural families of goethite(α-MOOH)and hematite(α-M_2O_3)if the standard Gibbs free energies of formation of the trivalent cations are known.
In the present study, the modified Sverjensky-Molling equation, derived from a linear-free energy relationship, is used to predict the Gibbs free energies of formation of crystalline phases of alpha-MOOH with a goethite structure and alpha-M_2O_3 with a hematite the modified equation is expressed as ΔG_ (f, M_VX) ~ 0 = a_ (M_VX) ΔG_ (0, M ~ (3+)) (M_VX) + β_ (M_VXγM ~ (3 +)) where the coefficients a_ (M_VX), b_ (M_VX), and β_ (M_VX) characterize a particular structural family of M_VX (M is trivalent (3+) and X represent the remainder of the composition of solid; γ ~ (3+) is the ionic radius of trivalent cations (M ~ (3 +)); ΔG_ (f, M_VX) ~ 0 is the standard Gibbs free energy of formation of M_vX; and ΔG_ (n, M ~ (3 +)) to 0 is the non-solvation energy of trivalent cations (M ~ experimental thermodynamic data, the coefficients for the goethite family (α-MOOH) are a_ (M_VX) = 0.883 (Α-M_VX) = - 424.4431 (kcal / mol), and β_ (M_VX) = 115 (kcal / mol.(?)) while the coefficients for the hematite family (α-M_2O_3) are a_ (M_VX) = 1.7468 The constrained relationship can be used to predict the standard Gibbs free energies of formation of crystalline phases and fictives. (k_VX) = - phases (iephases that are thermodynamically unstable and do not occur at standard conditions) within the isostructural families of goethite (α-MOOH) and hematite (α-M_2O_3) if the standard Gibbs free energies of formation of the trivalent cations are known.