Non-ideal

Various other non-ideal-gas-type two-dimensional equations of state have been proposed, generally by analogy with gases. Volmer and Mahnert [128,... 

Pressure-area isotherms for many polymer films lack the well-defined phase regions shown in Fig. IV-16 such films give the appearance of being rather amorphous and plastic in nature. At low pressures, non-ideal-gas behavior is approached as seen in Fig. XV-1 for polyfmethyl acrylate) (PMA). The limiting slope is given by a viiial equation... 

On the other hand, as applied to the submonolayer region, the same comment can be made as for the localized model. That is, the two-dimensional non-ideal-gas equation of state is a perfectly acceptable concept, but one that, in practice, is remarkably difficult to distinguish from the localized adsorption picture. If there can be even a small amount of surface heterogeneity the distinction becomes virtually impossible (see Section XVll-14). Even the cases of phase change are susceptible to explanation on either basis. 

This can be illustrated by showing the net work involved in various adiabatic paths by which one mole of helium gas (4.00 g) is brought from an initial state in whichp = 1.000 atm, V= 24.62 1 [T= 300.0 K], to a final state in whichp = 1.200 atm, V= 30.7791 [T= 450.0 K]. Ideal-gas behaviour is assumed (actual experimental measurements on a slightly non-ideal real gas would be slightly different). Infomiation shown in brackets could be measured or calculated, but is not essential to the experimental verification of the first law. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

Just as increasing the pressure of a gas or a gas mixture introduces non-ideal corrections, so does increasing the concentration. As before, one can introduce an activity a- and an activity coefficient y and write a- = cr-[. and... 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

This is an example of a classical non-ideal system for which the PF can be deduced exactly [13]. Consider N hard rods of length [Pg.459]

Finally, by considering increasing the number of particles by one in the canonical ensemble (looking at the excess, non-ideal, part), it is easy to derive the Widom [34] test-particle fomuila... 

So little systematic information is available about transport in liquids, or strongly non-ideal gaseous mixtures, that attention will be limited throughout to the behavior of ideal gas mixtures. It is not intende thereby, to minimize the importance of non-ideal behavior in practice. 

However, in the study of thermodynamics and transport phenomena, the behavior of ideal gases and gas mixtures has historically provided a norm against which their more unruly brethren could be measured, and a signpost to the systematic treatment of departures from ideality. In view of the complexity of transport phenomena in multicomponent mixtures a thorough understanding of the behavior of ideal mixtures is certainly a prerequisite for any progress in understanding non-ideal systems. 

Georgakopoulos and Broucek (1987) investigated the effect of recycle ratio on non-ideality, both mathematically and experimentally. They investigated two cases from which the bypass case b was completely uninteresting, because total bypass of the catalyst bed could be avoided by feeding the makeup directly to the location of highest sheerfield, at the tip of the impeller blade. For their case a they showed on their Fig. 3. that from a recycle ratio of about 10 = 32 there was no observable falsification effect. This matched well the conclusion of Pirjamali et al. 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

Commercial computer services are available to do rigorous distillation calculations. Perhaps the licensor will provide copies of rigorous computer runs to validate his balances. Alternately, the operating company can make such runs. For highly non-ideal systems, literature data for binary pairs may have to be sought. In some cases, laboratory equilibrium data may have to be obtained in-house or contracted out to one of several organizations or universities that are in this business. 

At the end of the 1930s, the only generally available method for determining mean MWs of polymers was by chemical analysis of the concentration of chain end-groups this was not very accurate and not applicable to all polymers. The difficulty of applying well tried physical chemical methods to this problem has been well put in a reminiscence of early days in polymer science by Stockmayer and Zimm (1984). The determination of MWs of a solute in dilute solution depends on the ideal, Raoult s Law term (which diminishes as the reciprocal of the MW), but to eliminate the non-ideal terms which can be substantial for polymers and which are independent of MW, one has to go to ever lower concentrations, and eventually one runs out of measurement accuracy . The methods which were introduced in the 1940s and 1950s are analysed in Chapter 11 of Morawetz s book. 

One may distinguish at least five categories of non-ideal solutions in terms of the nature of the intermolecular interactions that dominate their behavior ... 

In the first category of solutions ( regular solutions ), it is the enthalpic contribution (the heat of mixing) which dominates the non-ideality, i.e. In such solutions, the characteristic intermolecular potentials between unlike species differ significantly from the average of the interactions between Uke species, i.e. 

It is evident that many solutions fall between these limiting categories, with both energetic and entropic effects contributing to solution non-ideality. For example, if the energy of interaction between unlike species in a solution is highly favored over like-like interactions, it is obvious that these interactions will be preferred, a fact which in itself will lead to non-randomness of the packing in the solution. 

The simplest type of solutions which exhibit non-randomness are those in which the non-randomness is attributable solely to geometric factors, i.e. it does not come from non-ideal energetic effects, which are assumed equal to zero. This is the model of an athermal solution, for which... 

Figure 9 is the VLE plot for a binary mixture that has essentially a uniform equilibrium, and therefore represents a relatively easy separation. However, there are many cases where non-ideal separations are encountered. These more difficult distillations are defined by the examples shown in Figure 10. 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

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