Noura Dawass

1 6 I NTRODUCTION P , and N β . v β is the partial molar volume of component β at a constant T , P , and N α . κ T is the compressibility at a constant T . η and ζ are auxiliary quantities that were defined for convenience [13] , η = ρ α + ρ β + ρ α ρ β G f (1.8) ζ = 1 + ρ α G αα + ρ β G ββ + ρ α ρ β ( G αα G ββ − G 2 αβ ) (1.9) In Eq. (1.8) , the term G f = G αα + G ββ − 2 G αβ can be used to indicate the thermody- namic ideality of a binary mixture (i.e. it has the value of zero for ideal solutions). Expressions for ternary and multi-component mixtures of these thermodynamic quantities in terms of KBIs are available in literature [13, 17, 33] . 1.2. I NVERSION OF THE K IRKWOOD –B UFF THEORY Prior to the use of molecular simulation to compute KBIs, the inversion of the KB theory [13, 14] was used to obtain KBIs from experimental data. In this section we will briefly discuss the inversion procedure, and some of its applications. For a binary mixture with components α and β , partial molar volumes, the isothermal compressibility, and partial derivatives of chemical potential with re- spect to number of molecules are related to KBIs G ∞ αα , G ∞ ββ and G ∞ αβ (Eqs. (1.6) , (1.7) , and (1.5) ). Moreover, the Gibbs-Duhem relations apply to these thermody- namic quantities, ρ α µ ∂µ α ∂ N α ¶ T , P , N β + ρ β µ ∂µ β ∂ N α ¶ T , P , N β = 0 ρ β µ ∂µ β ∂ N β ¶ T , P , N α + ρ α µ ∂µ α ∂ N β ¶ T , P , N α = 0 ρ α v α + ρ β v β = 1 (1.10) where v α and v β are the partial molar volumes of components α and β , respec- tively. Using Eqs. (1.5) , (1.6) , (1.7) , and (1.10) , Ben-Naim [14] derived the follow- ing expression for KBIs of binary mixtures, G ∞ αβ = k B T κ T − δ αβ ρ α + ρ k B T (1 − ρ α v α )(1 − ρ β v β ) ρ α ρ β ³ ∂µ α ∂ N β ´ T , P , N α (1.11)