Interaction Carnahan-Starling

In this relation, N is the number density of the scattering microemulsion droplets and S(q) is the static structure factor. Equation (2.12) is only strictly valid for the case of monodisperse spheres. However, for the case of low polydispersities the occurring error is small [63, 64]. S(q) describes the interactions between and the spatial correlations of the droplets. These are in general well approximated by hard sphere interactions in microemulsion systems [65], The influence of inter-particle interactions as described by S(q) canbe estimated at least for S(0) using the Carnahan-Starling expression [52,64,66]... 

One drawback of the MF1V2 model is the inability of UNIFAC to predict (vapour + liquid) equilibria (VLB) and (liquid + liquid) equilibria (LLE) conditions using the same set of group-interaction parameters. In general, cubic equations of state do not provide precise predictions of the phase equilibria when the mixture is asymmetric in size that is attributed to the large differences in the pure-component co-volumes. The Carnahan -Starling equation for hard spheres is a more realistic model for the repulsive contribution than that proposed by van der Waals. Mansoori et al. proposed an equation for mixtures of hard spheres that has been found to correlate the phase behaviour of non-polar mixtures with large molecular size differences. 

Similarly to the fluid-fluid intermolecular potential, we split the solid-fluid intermolecular potential into repulsive hard-sphere and attractive interactions. Here Fhs Ps P is the excess free energy of the solid-fluid HS mixture, for which we employ Rosenfeld fundamental m ure functional [26] with the recent modifications that mve an accurate Carnahan-Starling equation of state in the bulk limit [27,28] r-r ) is the attractive part of the solid-fluid intermolecular potential. Since the iM>lid-soIid attraction interaction is not included, the solid is effectively modeled as a compound of... 

Nitta et. al. ( 7) extended the group interaction model to thermodynamic properties of pure polar and non-polar liquids and their solutions, including energy of vaporization, pvT relations, excess properties and activity coefficients. The model is based on the cell theory with a cell partition function derived from the Carnahan-Starling equation of state for hard spheres. The lattice energy is made up of group interaction contributions. 

Here, most definitions are the same as in Eq. (7.1) Va, Vb, and Vp denote the volumes of A-homopolymer, B-homopolymer, and particle, respectively. The first line is the ideal mixing entropy contribution, and the second line sums up the enthalpic contributions, taking into account that particles interact with the polymers only at the surfaces. The third line describes two additional contributions due to the nature of the particles. The function Whs(Carnahan-Starling [86] non-ideal contribution to the free energy of hard spheres, given by ... 

Further improvement has been aceomplished by the use of a more realistic equation with respect to the molecular interaction such as the Carnahan-Starling-van der Waals type of equation of state for the description of the solvent [28], However, the cubic equations as well as the Carnahan-Starling kind of equations are not accurate in the critical region [27], The computational method preserved here is based on a Carnahan-Starling-van der Waals kind of equation but expected by a perturbation term that corrects the pVT-behavior in the critical region. This approach is expected to be a promising tool for the correlation of solubility behavior even up to pressures above 100 MPa. 

It is also assumed that, in the case of n-alkane mixtures, the ky-values are independent of the chain length of component j. Dimitrelis and Prausnitz[8] showed that there is a systematic deviation from the Carnahan and Starling[9] repulsive term as the difference in molecular size between two molecules increase. It is thus expected that the value of the interaction parameter ly will be related to the difference in size between the two molecules. It is assumed that the value of ly will approach a constant value when this difference becomes large. The interaction parameters for propane and n-butane were found by fitting this equation of state to the data mentioned above. The parameters are shown in table 2 ... 

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