The pseudo-binary approximation

In a solid, there may be different kinds of bonds between the atoms, meaning that we can group together several atoms bound together in a single structure element. Such is the case with ionic compounds containing complex ions the atoms which make them up are linked by covalent bonds, and each complex ion will behave Uke a unique element on a unique site. 

Such is the case, for example, with metal carbonates in which we can define the metal cations, on the one hand, and the carbonate anions on the other hand, as normal stmeture elements. We will not distinguish the behavior of the individual oxygen atoms. Thus, these compounds can be considered to be binaries, so this approximation is referred to as pseudo-binary . These compounds may, of course, contain defects, such as an oxygen ion on an anionic site which is normally occupied by a carbonate ion. 

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The empirical predictions described above for pseudo-binary mixtures are compared with rigorous calculations for multicomponent mixtmes in Section 21-3.3 to justify the pseudo-binary approximation. 

Take the exanqtle of a metal carbonate, MCO3. In the ideal crystal, we consider that there are only two types of occupied structure elements the metal ion in cation position and the complex anion carbonate in anion positiow We will thus not distinguish the individual behavior of oxygen or carbon atoms. These compounds are thus regarded as binary ones, from where the name of the pseudo-binary approximation comes. These compounds can have defects, for example, an oxygen ion in the place normally occupied by a carbonate ion (this substitution does not involve any charge deficiency). 

In reactive flow analysis the Pick s law for binary systems (2.285) is frequently used as an extremely simple attempt to approximate the multicomponent molecular mass fluxes. This method is based on the hypothesis that the pseudo-binary mass flux approximations are fairly accurate for solute gas species in the particular cases when one of the species in the gas is in excess and acts as a solvent. However, this approach is generally not recommend-able for chemical reactor analysis because reactive mixtures are normally not sufficiently dilute. Nevertheless, many industrial reactor systems can be characterized as convection dominated reactive flows thus the Pickian diffusion model predictions might still look acceptable at first, but this interpretation is usually false because in reality the diffusive fluxes are then neglectable compared to the convective fluxes. 

Approximate calculations are initially made for the purpose of estimating the liquid compositions at different points in the column. These compositions are used to calculate activity coefficients, K-values and equilibrium vapor compositions as required for the pseudo-binary method. The initial calculations are based on the following assumptions ... 

The phase diagrams of polymer blends, the pseudo-binary polymer/polymer systems, are much scarcer. Furthermore, owing to the recognized difficulties in determination of the equilibrium properties, the diagrams are either partial, approximate, or built using low molecular weight polymers. Examples are fisted in Table 2.19. In the Table, CST stands for critical solution temperature — L indicates lower CST, U indicates upper CST (see Figure 2.15). 

AT=Tci-T = 8 °C where T is the temperature of the measurements and Tci is the doud point. DOP is a neutral solvent for PS and PB and weakens repulsive segmental interactions between PS and PB. ° The system can be approximately treated as a pseudo-binciry system where a phase separation between PS and PB occurs in the medium of DOP and the phase separation between the polymers and the solvent is insignificant. The pseudo-binary system is regarded to be equivalent to bulk systems when the segments of polymers in bulk are replaced by the blobs, as already pointed out at the beginning of Section 2.30.2.1. 

For example, at 278.2 K, hydrates form at a pressure of approximately 5 bar and dissociate upon pressurization at approximately 600 bar. A more detailed explanation of the pseudo-retrograde hydrate phenomena can be found in the binary hydrates section which follows. Note that the hydrate formation pressure of propane hydrates along the Aq-sII-V line at 277.6 K is predicted to be 4.3 bar. 

Again, it is mentioned that the binary mass flux definitions given above are commonly used also for pseudo-binary systems. In these particular cases the above relationships are only approximate. 

The graphical-based shortcut methods for binary batch distillation may be applied to multicomponent distillation only when the separation is between two key components to produce one distillate product and the residue. In this case the calculations may be approximated by lumping the other components with either of the key components and treating the system as a pseudo-binary. 

Where is the axial velocity in laminar flow, a function of the radial position and represented by the classical non-symmetric parabolic profile characteristic of annular spaces (Bird etal., 2002, p. 55). The use of a pseudo-binary diffiisivity is only an approximation if more accuracy is needed the Maxwell-Stefan relationships should be used. The initial ... 

Our calculations go beyond all the earlier work because SIC-LSDA treats all the f-states on equal footing and the KKR-CPA allows for a consistent description of spin and valence disorder. Johansson et al. (1995) used a binary pseudo-alloy concept, but needed one adjustable parameter to put on a common energy scale both the y-phase, described by LSDA with one f-state included into the core, and the a-phase, described by the standard LSD approximation, with all the f-states treated as valence bands. Svane (1996) performed SIC-LSD calculations using a supercell geometry which limited him to the study of a few concentrations only. 

Fig. 47a. Pseudo-binary T-x phase diagram of YBa2Cu30 calculated with the CVM approximation. The small solid circles are the experimental data of Andersen et al. (1990). The inserts are schematic illustrations of the 2D superstructures of the basal plane obtained from Monte Carlo simulations. Large solid circles, oxygen small solid circles, Cul (large and small solid circles form the black chains) open circles, vacant sites. After de Fontaine et al. (1992). The measurements of the phase boundary of the T-0 transition are in good agreement with the theoiy. However, comparison with ftg. 45 shows that the calculated critical temperature of the ortho-II phase (-570K) is much higher than the value recently found [-370 K, von Zimmermann et al. (1999)].

The net diffusive mass flux for each phase still vanishes for binary systems as s- k s using Pick s law, whereas for dilute pseudo-binary systems the latter relationship is only approximate. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

MFLG DESCRIPTION OF BINARY SYSTEMS A two component system is in the MFLG approximation treated as a pseudo-ternary mixture of constituents 1, 2 and holes (index 0). The appropriate thermodynamic function for the description of fluid phase equilibria is the Helmholtz free energy of mixing vacant and occupied sites, and reads in the simplest version of the model ). 

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