Noura Dawass

1.1. T HE K IRKWOOD -B UFF THEORY 1 5 G ∞ αβ = Z ∞ 0 d r 4 π r 2 h g ∞ αβ ( r ) − 1 i = lim V →∞ · V 〈 N α N β 〉−〈 N α 〉〈 N β 〉 〈 N α 〉〈 N β 〉 − V δ αβ 〈 N α 〉 ¸ (1.4) where N α , and N β are the number of molecules of type α and β , inside the vol- ume V . 〈 N α 〉 is the average number of molecules α and 〈 N α N β 〉 is the average product of the number of molecules of components α and β . It is important to note that Eq. (1.4) holds for any isotropic fluid. Fluctuations in the number of molecules relate to several thermodynamic properties [31, 32] . For a binary sys- tem, the following relations can be derived that relate KBIs to [12] : 1. partial derivatives of chemical potential with respect to the number of molecules, µ ∂µ α ∂ N α ¶ T , P , N β = ρ β k B T ρ α V η µ ∂µ α ∂ N β ¶ T , P , N α = µ ∂µ β ∂ N α ¶ T , P , N β = − k B T V η (1.5) 2. partial molar volumes, v α = µ ∂ V ∂ N α ¶ T , P , N β = 1 + ρ β ( G ββ − G αβ ) η v β = µ ∂ V ∂ N β ¶ T , P , N α = 1 + ρ α ( G αα − G αβ ) η (1.6) 3. the isothermal compressibility, κ T = − 1 V µ ∂ V ∂ P ¶ T , N α , N β = ζ k B T η (1.7) where ³ ∂µ α ∂ N α ´ T , P , N β is the partial derivative of the chemical potential of component α with respect to N α at a constant temperature T , pressure P and N β . Similarly, ³ ∂µ β ∂ N β ´ T , P , N α is the partial derivative of the chemical potential of component β with respect to N β at a constant T , P and N α . In Eqs. (1.5) and (1.7) , k B is the Boltz- mann constant. v α is the partial molar volume of component α at a constant T ,