Interaction parameter critical value

It is assumed that there are available NCP experimental binary critical point data. These data include values of the pressure, Pc, the temperature, Tc, and the mole fraction, xc, of one of the components at each of the critical points for the binary mixture. The vector k of interaction parameters is determined by fitting the EoS to the critical data. In explicit formulations the interaction parameters are obtained by the minimization of the following least squares objective function ... 

At each point on the critical locus Equations 40a and b are satisfied when the true values of the binary interaction parameters and the state variables, Tc, Pc and xc are used. As a result, following an implicit formulation, one may attempt to minimize the following residuals. 

Table 14.8 Interaction Parameter Values from Binary Critical Point Data...

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

In the most general case the Lagrangian density of a field suffers a reduction of symmetry at some critical value of an interaction parameter. Suppose that... 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

Most of the interaction parameters employed were taken from other studies (20, 21), and are reportedly obtained by minimizing errors in the match of phase equilibrium data. However, in (21), the SRK equation employed was slightly different from that used here. The parameters for CO2 - H2O were chosen because they had been shown to give a critical line which is qualitatively correct. The H2O - CO interaction parameter is the value given in (20) for H2S - CO. For H2O - H2, kij was taken to be -0.25 in the absence of any literature studies. 

The temperature - dependent interaction parameters were determined from 77°F to 680°F using the data of Culberson and McKetta (20) and of Sultanov et al. (18). This parameter increases with temperature and appears to converge to the value of the constant parameter used for the vapor phase as the critical temperature of water is approached. 

J. As with the alkane - water systems, the interaction parameters for the aqueous liquid phase were found to be temperature - dependent. However, the compositions for the benzene - rich phases could not be accurately represented using any single value for the constant interaction parameter. The calculated water mole fractions in the hydrocarbon - rich phases were always greater than the experimental values as reported by Rebert and Kay (35). The final value for the constant interaction parameter was chosen to fit the three phase locus of this system. Nevertheless, the calculated three-phase critical point was about 9°C lower than the experimental value. 

The critical data and values used for inert components were those given by Ambrose (24). The interaction parameters between the water and the inert component were found by performing a dew-point calculation as described above but with the interaction parameter k.. rather than P taken as the iteration variable. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Let us first consider a network immersed in a melt of polymer chains with degree of polymerization p. In the athermal case, the network should be swollen. As polymer-network interaction parameter Xnp increases, the volume of the network decreases until a practically complete segregation of the gel from polymer melt occurs. It has been found [34, 35] that two qualitatively different regimes can be realized either a smooth contraction of the network (Fig. 8, curve 1) or a jumpwise transition (Fig. 8, curve 2). The discrete first order phase transition takes place only for the networks prepared in the presence of some diluent and when p is larger than a critical value pcr m1/2. The jump of the... 

In principle, all the molecular parameters in Eq. (6) can be determined independently, so that the theory can be quantitatively compared with experimental data. An example of Maxwell s construction in the dependence of x °n critical value of interaction parameter %c of charged PAAm network with the degree of ionization equals to the molar fraction of the sodium methacrylate in the chain i = xMNa = 0.012 are given in Fig. 4 (data of series D from Fig. 5). The compositions of the phases

critical value of Xc were determined by the condition that areas St and S2 defined in Fig. 4 are equal The experimental (p2e is higher and 2 determined by Maxwell s construction (Eq. 13). Thus, the experimental values of (p2e and metastable region the limits of which (p2s and (p2s are determined by the spinodal condition (two values

[Pg.182]

Fig. 4. An example of Maxwell s construction in the dependence of / on

Fig. 6. Dependence of the extent of the collapse A and of the critical value of the interaction parameter yc on the content of sodium methacrylate xMNj (—) course determined by Maxwell s construction of data from Fig. 5 (O), ( ) experimental data. Taken from Ilavsky [11]...

Fig. 9. Dependence of the extent of the collapse A and critical values of the interaction parameter Xc on the effective degree of ionization for networks of the copolymer AAm with sodium methacrylate xMN> ( ) variously aged PAAm networks

The carbon di oxi de/lemon oil P-x behavior shown in Figures 4, 5, and 6 is typical of binary carbon dioxide hydrocarbon systems, such as those containing heptane (Im and Kurata, VO, decane (Kulkarni et al., 1 2), or benzene (Gupta et al., 1 3). Our lemon oil samples contained in excess of 64 mole % limonene so we modeled our data as a reduced binary of limonene and carbon dioxide. The Peng-Robinson (6) equation was used, with critical temperatures, critical pressures, and acentric factors obtained from Daubert and Danner (J 4), and Reid et al. (J 5). For carbon dioxide, u> - 0.225 for limonene, u - 0.327, Tc = 656.4 K, Pc = 2.75 MPa. It was necessary to vary the interaction parameter with temperature in order to correlate the data satisfactorily. The values of d 1 2 are 0.1135 at 303 K, 0.1129 at 308 K, and 0.1013 at 313 K. Comparisons of calculated and experimental results are given in Figures 4, 5, and 6. 

Since both ethylene and ethane have reduced temperatures nearly equal to unity at the extraction conditions of 20 C, (T =. 98) and ethylene (T = 1.04), their respective solvent capacities for butene should be about the same. This is the case as is reflected in the same values for the selectivity against butene for all pure solvent gases. One can conclude that the primary effect of the non-polar solvent is to increase the capacity of the "vapor" phase for the extracted solute near the critical. The influence of the second solvent provides only the option of modifying the physical parameters namely, pressure and temperature, under which the optimal extraction is to be conducted. The evidence for this is the effect of the ammonia on the selectivity as calculated by the EOS in Table V. The higher values for the selectivities in the ethylene mixtures are pronounced. It can be concluded that the solvent mixture interaction parameters must dominate the solubility of butene in the vapor phase. 

---