Combining rules hard sphere

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

The justification for using the combining rule for the a-parameter is that this parameter is related to the attractive forces, and from intermolecular potential theory the attractive parameter in the intermolecular potential for the interaction between an unlike pair of molecules is given by a relationship similar to eq. (42). Similarly, the excluded volume or repulsive parameter b for an unlike pair would be given by eq. (43) if molecules were hard spheres. Most of the molecules are non-spherical, and do not have only hard-body interactions. Also there is not a one-to-one relationship between the attractive part of the intermolecular potential and a parameter in an equation of state. Consequently, these combining rules do not have a rigorous basis, and others have been proposed. 

The use of the arithmetic mean for the unlike size parameter was proposed by Lorentz motivated by the colhsion of hard spheres on the other hand, the geometric mean for the unlike energy parameter was proposed with little physical argument by Berthelot. Therefore, it is not surprising that this combining rule often leads to inaccurate mixture properties [34, 37, 38]. 

Theory and computer simulation provides information in addition to the virial equation of state that can be used to develop mixing and combining rules. The equation of state for the pure hard-sphere fluid can be represented by the equation of state of Carnahan-Starling ... 

In all the equations that include the double power-series expansion (or variations thereof), such as the family of perturbed-hard-chain equations, the parameters are related to molecular rather than critical properties, and the mixing and combining rules are quadratic in composition for the attractive term and based on hard-sphere theory for the repulsive term. 

The Group Contribution Equation of State (GC-EoS) has two eonlributions to the residual Helmholtz energy of the system a repulsive hard sphere Camahan-Starling type term and an attractive term, which combines the group contribution approach wifli the local-composition mixing rules. A complete explanation of the model is given by Skjold-Jorgensen. ... 

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