Unlike pair interaction parameter

In addition to pure component parameters, mixture calculations require the estimation of the unlike-pair interaction parameters. These were obtained in this study using the Lorenz-Berthelot rules ... 

Effect of Unlike-Pair Interactions on Phase Behavior. No adjustment of the unlike-pair interaction parameter was necessary for this system to obtain agreement between experimental data and simulation results (this is, however, also true of the cubic equation-of-state that reproduces the properties of this system with an interaction parameter interesting question that is ideally suited for study by simulation is the relationship between observed macroscopic phase equilibrium behavior and the intermolecular interactions in a model system. Acetone and carbon dioxide are mutually miscible above a pressure of approximately 80 bar at this temperature. Many systems of interest for supercritical extraction processes are immiscible up to much higher pressures. In order to investigate the transition to an immiscible system as a function of the strength of the intermolecular forces, we performed a series of calculations with lower strengths of the unlike-pair interactions. Values of - 0.90, 0.80, 0.70 were investigated. 

With unlike pair interaction parameter From (16)... 

The unlike pair interaction parameter is determined using binary vapor-liquid equilibrium data as described by Zudkevitch and Joffe (2). The systems used in this study are given in Table I. The interaction... 

A parameter, appears in the combining rule for the energy parameter. It describes the deviations of the unlike-pair interactions from the geometric-mean rule. Values less than 1 indicate that the unlike pair interactions are less favorable compared to the like-pair interactions. 

The difficulty in the application of Equation 29 is in the assignment of values to tt when i /. While such a parameter is said to represent the effects of unlike-pair interactions, it is often difficult to give this idea a clear physical or mathematical statement. Thus one usually resorts to combination rules, which relate nto the pure-component parameters and (possibly) to an empirical interaction parameter, a number (by optimistic definition) usually either of order zero or of order unity. 

FIGURE 8.14 Comparison between the density dependence of (dlnA /d/>)j.j and for the reactive system BP + 2 + COj TS along the near-critical reduced isotherm = 1.01, when all unlike-pair interactions are described by the Lorentz-Berthelot (top) or perturbed Lorentz-Berthelot combining rules (bottom). All quantities are made dimensionless using the solvent s Lennard-Jones parameters from Table 8.3. 

For the like-pair interaction the remaining parameters (e, o, n, 5x, or 2) were determined as those giving the best fit to the saturated liquid density and vapor pressure data (13). The like-pair parameters are shown in Table 2. For the unlike-pair interactions one must also know the parcuneters a3 03 > %3 where 1 is 1 or 2. The last two parameters were estimated from the combining rules... 

The previous section discussed techniques for obtaining the molecular potential interaction parameters <7 and e based on pure species physical properties of molecule i. Interactions between unlike molecules (i.e., all i-j pairs) must also be considered in the calculation of transport properties (notably, binary diffusion coefficients). The following is a set of combining rules to estimate the i- j interaction parameters, assuming that the pure species values are known. 

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. 

This method is to be used to estimate the activity coefficient of a low molecular weight solvent in a solution with a polymer. This procedure, unlike the other procedures in this chapter, is a correlation method because it requires the Flory-Huggins interaction parameter for the polymer-solvent pair which must be obtained from an independent tabulation or regressed from experimental data. In addition, the specific volumes and the molecular weights of the pure solvent and the pure polymer are needed. The number average molecular weight of the polymer is recommended. The method cannot be used to estimate the activity of the polymer in the solution. 

Equation (1.7) requires fly for unlike pairs. The method of calculating fly from and Ujj is called a combining rule. It is in the combining rule that a binary interaction parameter is typically introduced ... 

The method used here for considering conformal solution models for fluids with molecular anisotropies is based on the method used by Smith (4) for treating isotropic one-fluid conformal solution methods as a class of perturbation methods. The objective of the method is to closely approximate the properties of a mixture by calculating the properties of a hypothetical pure reference fluid. The characterization parameters (in this case, intermolecular potential parameters) of the reference fluid are chosen to be functions of composition (i.e., mole fractions) and the characterization parameters for the various possible molecular pair interactions (like-like and unlike-unlike). In principle, all molecular anisotropies (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, and higher multipole interactions, as well as overlap and dispersion interactions ) can be included in the method. Here, the various molecular anisotropies are lumped into a single term, so that the intermolecular potential energy uy(ri2, on, a>2) between Molecules 1 and 2 of Species i and / can be written in the form... 

Mixing rules are available to extend the models to multicomponent systems, with the use of adjustable binary parameters. In the absence of associating sites, we will consider only the binary interaction parameter which enters the mixing mle for the characteristic interaction energy between pairs of unlike segments i and j ... 

The definition of different sites, particularly in the case of mixtures, implies that interaction parameter sets for unlike site pairs are required. If sufficient data - either experimental or theoretical (ab initio) - are available, these rules can be abandoned completely. The main reason for using combining rules is to avoid a huge number... 

The A-units and B-units of the two pol3mier species both have the same volume Vc and occupy the cells of a regular lattice with coordination number z. It is assumed that the interaction energies are purely enthalpic and effective between nearest neighbors only. Excess contributions kTx, which add to the interaction energies in the separated state, arise for all pairs of unlike monomers. The parameter x = ( 2)x was devised to deal with this model and therefore depends on the size of the cell. Flory evaluated this model with the tools of statistical thermod3mamics. Using approximations, he arrived at Eqs. (4.5) and (4.6). 

Here, z is the lattice coordination number, Aw 2 is the interchange energy due to unlike pair contacts, Xi is the number of segments in a solvent molecule, ni is the number of solvent molecules in a solution, and U2.s is the equilibrium polymer volume fraction. The Rory interaction parameter may be written as... 

Dispersive Interactions. For pairs of nonpolar polymers, the intermolecular forces are primarily of the dispersive type, and in such cases the energy of interaction between unlike segments is expected to be closely approximated by the geometric mean of the energies of interaction between the two like pairs (98). In this case, the Flory-Huggins interaction energy between this polymer pair can be expressed in terms of the solubiUty parameters 5 of the pure components. 

For interactions between different atom pairs, we should ideally try to deduce experimental parameters for the interactions of these atoms. What is done instead is to use combination rules, which allow us to relate the parameters for unlike atom pairs A- -B to those for the two like pairs A- A and B- -B. If we use symbols i and j to label the two atoms, then there are a number of different such... 

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