Binary systems critical conditions

The occurrence of critical phenomena in binary systems (critical vaporiiationor critical solution temperatures) will be studied in Ch. XII where the thermodynamic conditions for phase separation will be considered in detail. Here we shall summarize some basic relations to which we shall refer in many chapters of this book. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Inspection of the table shows that the quotient a/Wj e is in fact nearly constant that I changes much less rapidly than W e] and that the critical depth has doubled when the highest oxide is reached. All three conditions are reflections of the (positive) absorption effect that occurs in this binary system when iron is replaced by oxygen, which has a lower mass absorption coefficient. 

The dilated van Laar model is readily generalized to the multicomponent case, as discussed in detail elsewhere (C3, C4). The important technical advantage of the generalization is that it permits good estimates to be made of multicomponent phase behavior using only experimental data obtained for binary systems. For example, Fig. 14 presents a comparison of calculated and observed -factors for the methane-propane-n-pentane system at conditions close to the critical.7... 

An alloy is said to be of Type II if neither the AC nor the BC component has the structure a as its stable crystal form at the temperature range T]. Instead, another phase (P) is stable at T, whereas the a-phase does exist in the phase diagram of the constituents at some different temperature range. It then appears that the alloy environment stabilizes the high-temperature phase of the constituent binary systems. Type II alloys exhibit a a P phase transition at some critical composition Xc, which generally depends on the preparation conditions and temperature. Correspondingly, the alloy properties (e.g., lattice constant, band gaps) often show a derivative discontinuity at Xc. 

An additional problem arises when the stability of the critical phase involving the gas-liquid equilibrium in a binary system is studied. The conditions of stability of a homogenous system at constant pressure are d2A/dV2)T x> 0 and d25/dx )T P > 0 from Equations (5.136) and (5.141), respectively. The question arises of which of the two conditions becomes zero first as the boundary between stable and unstable phases is approached. 

Problems concerning the conditions of stability of homogenous systems for critical phases in ternary systems are very similar to those for the gas-liquid phenomena in binary systems, because of two independent variables at constant temperature and pressure. The conditions for stability are (82G/dnl)T P 2 3>0 and (820, given by Equations (5.146) and (5.147), respectively. Inspection of the condition equivalent to Equation (5.148) given by Equation (5.152) shows that (82(j>/dnl)TP> 2i 3 and, therefore, it is the condition expressed by Equation (5.147) or (5.150) that determines the boundary between stable... 

At a given pressure and temperature, the total Gibbs free energy of mixing of a one-phase polymer-solvent system of composition 2 should be necessarily minimum, otherwise the system will separate into two phases of different composition, as it is represented in a typical AG versus cp phase diagram of a binary solution (Fig. 25.3). The volume fractions at the minima (dAGIdcp = 0), cp, and (p will vary with temperature (binodal) up to critical conditions (T and (p ) where cp = tp" (Fig. 25.3b). 

The book, which begins with a historical perspective and an introductory chapter, includes a basic derivation for more casual readers. It is then devoted to providing new and very recent applications of FST. The first application chapters focus on simple model, binary, and ternary systems, using FST to explain their thermodynamic properties and the concept of preferential solvation. Later chapters illustrate the use of FST to develop more accurate potential functions for simulation, describe new approaches to elucidate microheterogeneities in solutions, and present an overview of solvation in new and model systems, including those under critical conditions. Expert contributors also discuss the use of FST to model solute solubility in a variety of systems. 

An exhaustive study of high-pressure vapor-liquid equilibria has been repotted by Knapp et at., who give not only a comprehensive literanire survey but also compare calculated and observed results for many systems. In that stu, several popular equations of state were used to perform the calculations but no one equation of state emerged as markedly superior to the others. All the equations of state used gave reasonably g results provided care is exercised in choosing the all-important binary constant k,j. All the equations of state used gave poor results when mixtures were close to critical conditions. 

Fora quasi-binary system, the critical solution point is located in the extremum of the spinodal and hence the necessary condition is... 

Sole et al. (1984) have analyxctl the conditions of existence of multiphase equilibrium and multiple critical points in the system (polymolecular P)+LMWL when a concentration-independent interaction parameter and in binary systems with a concentration-dependent interaction parameter g = /([Pg.494]

Table 7.5 shows the data of Nq, which is calculated using two critical conditions discussed previously. One is the lower limit calculated using the data of A ci Xab = 2.773 when only a single phase occurs for a two-polymer blend. The other is upper limit calculated at xab = 4 when a binary polymer blend favours the formation of a system with multiple phases at almost all concentrations. 

For monodisperse primary chains, we have a strictly two-component system, and the thermodynamic stability limit (spinodal) is given by a cj), T) = 0, where cr is the factor (7.123). Further, for such strictly binary systems, the critical solution point, if it exists in the pregel regime, can be found by the additional condition d A o/dcp = 0. The condition is given explicitly by... 

The mean-field lattice gas model is a molecular model for small-molecular and macromolecular systems that can reliable predict thermodynamic equilibrium properties. The model contains several adaptable empirical parameters for pure substances and for the mixtures, whereas no mixing rules are involved. The use of critical conditions in the adaption of parameters for binary mixtures to experimental data is discussed here, taking as an example the vapour-1iquid critical behaviour of the system ethylene-naphtalene. 

The introduction of a relevant expression for the critical determinant in the mean-field lattice gas model for binary systems is discussed here. It leads to an alternative and thermodynamic consistent method of adjusting two-particle interaction functions to experimental critical binary 1iquid-vapour densities. The present approach might lead to new developments in the determination of MFLG parameters for the mixture in small-molecule mixtures and in polymer solutions and polymer mixtures (blends). These relevant critical conditions appear because of the extra constraint, which is the equation of state, put on the hole model, and are... 

The conditions under which these phosphates exist is discussed in the Critical Evaluation of the respective binary systems. 

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