A theoretical treatment of changes in energy and entropy of solids caused by additives or impurities in solid solution

Abstract

A dimensionless entropy increment ratio, termed the “disruption index” (d.i.), has previously been defined and evaluated from heats of fusion, ΔH f ( Int. J. Pharm., 25 (1985) 57–72), and from heats of solution, ΔH s ( Int. J. Pharm., 28 (1986) 103–112). The d.i. quantifies the disruptive influence of an additive or impurity (the guest substance), when present at low mole fractions ( x 2 < 0.05), in solid solution in the crystal lattice of a host substance. The present report places the thermodynamics in the previous papers on a more rigorous foundation. (1) The previously defined “enthalpy of fusion of the solid”, ΔH f, and “enthalpy of solution of the solid”, Δ H s θ , are defined in terms of the partial molar enthalpies of the two components. (2) The d.i., previously calculated from ΔH s θ , is expressed in terms of the excess entropy of the solid, S E. The original definition of d.i. is found to involve the assumption that the partial molar excess entropy of the guest, S E 2, is directly proportional to the ideal entropy of mixing. (3) The pseudo-d.i. (p.d.i.), originally defined by assuming enthalpy-entropy compensation, is expressed in terms of the excess enthalpy of the solid, H E. (4) The d.i. previously calculated from ΔS f is expressed in terms of S E 2 . The d.i. derived from ΔS f and tha from ΔS s θ differ by about a unit in theory and are approximately equal when d.i. ⪢ 1; however, the experimentally observed differences express the temperature dependence of S E, which is reflected in the annealing of the solid between ambient temperatures and the melting point. (5) The d.i. is compared with an alternative approach based on the limiting value of ( ∂ΔS/ ∂Δx 2) at low x 2, i.e. ( S E 2) 0 . The empirical relationship between these two approaches is d.i. = 0.035 · [( S E 2) 0] 0.912 with ( S E 2) 0 in JK −1 mol −1 and d.i. from 10 −1 to 4 × 10 3. It is concluded that the present rigorous thermodynamic treatment does not alter the main conclusions derived from the previous intuitive treatment but defines better the approximation inherent in the evaluation of d.i. and shows that d.i. is directly related to S E.