# Modification in COSMO-SAC

## Introduction

On the basis of the COSMO framework, Lin and Sandler (2002) derived the variation of this model to accommodate nonideality in phase equilibria calculation and predict activity coefficients. In COSMO-SAC, the activity coefficients are calculated segmentwise and the chemical potential of each molecule is determined by summing up contributions of every segment. The solvation free energy (AG^{so1}) was originally taken by Lin and Sandler as the summation of electrostatic contribution and van der Waals free energy. The electrostatic contribution originates due to the electrostatic interaction between solute and solvent. Van der Waals part takes cavity formation and dispersion interaction. Mathematically, it is written as

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The activity coefficient is written in terms of logarithmic summation of restoring, dispersion and cavity formation terms. In the original assumption, the dispersive term was neglected and restoring solvation free energy was taken as electrostatic contribution. The cavity formation term is defined by the Stavermann-Guggenheim combinatorial term to accommodate the free energy change of molecular size and shape differences between species. *
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The picture depicted above does not produce the real picture of solvation of a molecule from vacuum to solvent. According to implicit continuum solvation theory, the solvation free energy is the summation of six free energies:

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The electrostatic free energy is represented by AG^{el}. This originates from the polarization and electrostatic interaction among solute and solvent molecules. A cavity is needed in the solvent molecule to accept a solute molecule and the free energy change is denoted by AG^{cav}. Short range interactions originate from the instantaneous charges on the molecular surface, better known as dispersion free energy (AG^{dlsp}). Van der Waals free energy is the summation of dispersion and cavity terms. From the vibration, rotation and internal structure of a molecule, the last three free energies of Equation 5.5 originate. For phase equilibria calculation, these three terms are negligible, because the differences in their values between the ideal gas and the solvent are assumed to be small. With this, we can write

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Thus we obtain Equation 5.1 (we removed the subscripts for the sake of simplicity). For computation of phase equilibria, the electrostatic free energy change is calculated by the difference of interactions between the molecule in the solvent and in its fluid form. For calculating pure component properties like vapor pressure and enthalpy of vaporization, the electrostatic contribution to the free energy is represented by the sum of ideal solvation (is), charge averaging correction (cc), restoring free energy (res, including hydrogen bond corrections), that is,

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The first two terms on the right-hand side of Equation 5.7 were described in Section 3.2.5. An averaging process is used to obtain only standard segments. The standard segments have 'apparent' charge density distribution о over an area that is larger than that of the original charge density a*. The COSMO-SAC model considers this standard segment surface area as one of the universal parameters. The assumption of independent segments leads to the calculation of AG_{i}^{1}j^{es} assuming a pairwise interaction model. For that the original о-averaging expression is slightly modified and an empirical parameter /_{decay} is introduced. The value of /_{decay} is 3.57. Other notations carry the same meaning as the previous equations. The expression is given by Equation 5.8.

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The AGT^{dw} is in terms of Helmholtz free energy and written as *
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