Handbook of Solvents - George Wypych - ChemTech - Ventech!

104 Valery Yu. Senichev, Vasiliy V. Tereshatov
δis a parameter of intermolecular interaction of an individual liquid. The aim of many
studies was to find relationship between energy of mixing of liquids and their δ. The first attempt
was made by Hildebrand and Scatchard 5,6 who proposed the following equation:
( )
m
ΔU
= x V + x V
1 1 2 2
( xV x V)(
)
2
/ 12 / ⎤
2
ϕϕ 1 2
⎛ΔE
⎞ ΔE
⎜
⎟
⎝ V ⎟
−
⎠ V
⎛
12 ⎡
1
⎢ ⎜
⎣⎢
1 ⎝ 2
= + δ −δ
ϕ ϕ [4.1.10]
1 1 2 2 1 2
2
1 2
where:
ΔU m
internal energy of mixing, that is a residual between energies of a solution and
components,
x1,x2 molar fractions of components
V1,V2 molar volumes of components
ϕ1, ϕ2 volume fractions of components
The Hildebrand-Scatchard equation became the basis of the Hildebrand theory of regular
solutions. 5 They interpreted a regular solution as a solution formed due to the ideal entropy
of mixing and the change of an internal energy. The assumed lack of the volume
change makes an enthalpy or heat of mixing equated with the right members of the equation.
The equation permits calculation heat of mixing of two liquids. It is evident from equation
that these heats can only be positive. Because of the equality of of components, ΔH m =0.
The free energy of mixing of solution can be calculated from the equation
( )
m
ΔG
= x V + x V
1 1 2 2
( )
⎛ΔE
⎞ ΔE
⎜
⎟
⎝ V ⎟
−
⎠ V
⎛
12 ⎡
1
⎢ ⎜
⎣⎢
1 ⎝ 2
− TΔS = V δ −δ ϕ ϕ −TΔS
id
1 2
2
⎞
⎟
⎠
⎞
⎟
⎠
⎥
⎦⎥
2
/ 12 / ⎤
2
ϕϕ 1 2
⎥
⎦⎥
1 2 [4.1.11]
The change of entropy, ΔS id, is calculated from the Gibbs equation for mixing of ideal
gases. The calculated values are always positive.
( )
ΔS =− R x lnx + x ln x
[4.1.12]
id
1 1 2 2
where:
R gas constant
Considering the signs of the parameters ΔSid and ΔH m in Eq. [4.1.10], the ideal entropy
of mixing promotes a negative value of ΔG m , i.e., the dissolution and the value of ΔH m reduces
the ΔG m value. It is pertinent that the most negative ΔG m value is when ΔH m =0, i.e.,
when δ of components are equal. With these general principles in mind, the components
with solubility parameters close to each other have the best mutual solubility. The theory of
regular solutions has essential assumptions and restrictions. 7 The Eq. [4.1.10] is deduced
under assumption of the central role of dispersion forces of interaction between components
of solution that is correct only for the dispersion forces. Only in this case it is possible to accept
that the energy of contacts between heterogeneous molecules is a geometric mean
value of energy of contacts between homogeneous molecules:
=
−