Van der Waals co-volume

Fig. 8. Variation of activation energy with kinetic molecular diameter for diffusion in 4A 2eohte (A), 5A 2eohte (0)> carbon molecular sieve (MSC-5A) (A). Kinetic diameters are estimated from the van der Waals co-volumes. From ref. 7. To convert kj to kcal divide by 4.184.

In most cases, the van der Waals co-volume b of a mixture is evaluated with a simple linear mixing rule ... 

Setting di /dr = 0 in Eq. (2.3) gives, for the equilibrium separation between centers of an isolated pair of molecules, = 2 a. Thus, the excluded volume (the van der Waals co-volume b) may be seen to be four times the actual volume of the molecules ... 

FIGURE 5.12. Variation of diffusional activation energy with van der Waals diameter for diffusion in 4A and 5A zeolites and 5A molecular sieve carbon. Van der Waals diameters are estimated according to Eq. (2.5) from values of the van der Waals co-volume b) given in the Handbook of Physics and Chemistry, 55lh ed. C.R.C Press 1974, (Diffusivity data are from refs. 1-3, 48, 49,51-53, and 81.)... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

Previous studies on paraffins, rhodamine dyes, and l,3-bis(N-carbozoyl) propane excimers have concluded that there is a relationship between km and polymer viscosity and free volume [103-105], Indeed, this dependence has been investigated in the context of decreasing free volume during methyl methacrylate polymerization [83,84], It has been shown that the nonradiative decay processes follow an exponential relationship with polymer free volume (vf), in which kra reduces as free volume is decreased [see Eq. (5)]. Here, k. represents the intrinsic rate of molecular nonradiative relaxation, v0 is the van der Waals volume of the probe molecule, and b is a constant that is particular to the probe species. Clearly, the experimentally observed changes in both emission intensity and lifetime for/ac-ClRe(CO)3(4,7-Ph2-phen) in the TMPTA/PMMA thin film are entirely consistent with this rationale. 

Ottewill and co-workers106,200 have used a compression method to measure the double-layer repulsion between the plate-like particles of sodium montmorillonite. This is a particularly suitable system for such studies, since the particles are sufficiently thin (c. 1 nm) for van der Waals forces to be unimportant and surface roughness is not a problem. The dispersion was confined between a semipermeable filter and an impermeable elastic membrane and an external pressure was applied via a hydraulic fluid so that the volume concentration of particles and, hence, the distance of separation between the particles could be measured as a function of applied pressure. 

Schuster calculated the molecular co-volume at the b.p. from 0b= Vmb--bb =RTblBby where bb is van der Waals s constant at the b.p. and Bb is the internal pressure. For normal substances, fe6=0 646c, 6=0 236 =0 147Z>c-Unlike Traube s co-volume, Schuster s depends on the nature of the substance. If the degree of association at the critical point is assumed to be 1, that for an abnormal substance was calculated from the observed and normal 0b values as in the case of Traube s equation. 

The use of the well-known van der Waals one-fluid or related mixing rules for the energy and co-volume parameters. These mixing rules require interaction parameters (especially in the energy term) and are employed for correlating experimental data. 

They have been employed in the use of the van der Waals equation of state for polymers. They require combining rules for both the cross energy and cross co-volume parameters. Kontogeorgis et al. have employed the typically used geometric mean for the co-volume parameter, but they used the Berthelot rule for the cross-energy parameter ... 

By using standard thermodynamics, the activity coefficient expression that corresponds to the van der Waals equation of state can be derived. When the van der Waals one-fluid mixing rules for the energy and co-volume parameters are employed, the van der Waals equation can be expressed as a sum of a combinatorial-FV and a regular solution term ... 

Such an expression provides partial justification for the success of van der Waals-type equations for asymmetric polymer systems the van der Waals equation is a sum of an improved FH (actually an FV) term and a regular solution energetic term. Furthermore, analysis of the contributions (attractive and repulsive) of cubic equations of state showed that the classical (linear) combining rule for the co-volume parameter (Equation 16.66 for 12) is the best choice for size asymmetric systems. This combining rule performs much better than the Lorentz and other combining rules for the cross-co volume parameter. Using this successful arithmetic mean rule for l i2, the combinatorial-FV term of the van der Waals equation of state is functionally identical to that of the Entropic-FV model. 

In this section the experimental liquid phase compositions and liquid molar volumes determined for the P-diketone/C02 systems studied will be presented coupled with the modeling results using the Peng-Robinson EOS with van der Waals-1 mixing rules. Table II lists the estimated critical constants, Tc and Pc, and the acentric factor, co, needed for modeling these systems. 

The internal pressure and the co-volume change their signs, when turning from k = 5/3 to K = 7/3. Thus, for /c > 2 the p —V plot with S as parameter shows an unusual behavior. The situation becomes different, when we try to use the values for a and b obtained from the isothermal van der Waals equation. A drawback is that the reduced form looks somewhat complicated. 

Figure 8.10 demonstrates that to a good approximation the reduced cell volume at F = 0 is constant for all copolymers, W = co/v 0.96. In Figures 8.11 and 8.12 the van der Waals parameters of Eq. (8.9) are plotted versus copolymer composition. One observes marked changes in the internal pressure parameter, AIV, with copolymer composition for EVOH and SAN. The ethylene-based copolymers have approximately the same value of the parameter B. For SAN, it does not change markedly in the range where styrene is in excess. Selected examples of the internal pressure parameter and characteristic specific volume B of copolymers are listed in Table 8.4 and... 

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