Critical binary pairs

Our experience with multicomponent vapor-liquid equilibria suggests that for system temperatures well below the critical of every component, good multicomponent results are usually obtained, especially where binary parameters are chosen with care. However, when the system temperature is near or above the critical of one (or more) of the components, multicomponent predictions may be in error, even though all binary pairs are fit well. 

Equations (76) and (77) contain two constants, A and B, which, for any binary pair, are functions of temperature only. These equations appear to be satisfactory for accurately representing activity coefficients of nonpolar binary mixtures from the dilute region up to the critical composition. As examples, Figs. 12 and 13 present typical results of data reduction for two systems in these calculations, the reference pressure Pr was set equal to zero. 

We consider a binary liquid mixture of components 1 and 3 to be consistent with our previous notation, we reserve the subscript 2 for the gaseous component. Components 1 and 3 are completely miscible at room temperature the (upper) critical solution temperature Tc is far below room temperature, as indicated by the lower curve in Fig. 27. Suppose now that we dissolve a small amount of component 2 in the binary mixture what happens to the critical solution temperature This question was considered by Prigogine (P14), who assumed that for any binary pair which can be formed from the three components 1, 2 and 3, the excess Gibbs energy (symmetric convention) is given by... 

This introduces two "interaction parameters" per binary pair. The pure component coefficients, a and b i, are evaluated from critical data and the acentricity, as proposed by Soave in his original paper (1). The pure component aii varies with reduced temperature so as to match vapor pressure. (Soave s recently revised expression for a (17) has not been used.)... 

Before describing variations in the critical points in the four-component water-gas shift mixture it is instructive to examine the critical points in the various binary mixtures. There are six binary pairs to consider. 

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. 

Because of the uncertainty as to the data and because of the sensitivity to the parameters it should be understood that calculated critical points reported in this paper need not represent actual behavior, even of the binary pairs. 

The calculated critical lines for the binary pairs are shown in Figure 1. All these lines are discontinuous, indicating high density phase separations. For each binary pair the principal part of the critical line begins at the critical point for the component with the higher critical temperature. There is a second branch of each of the critical lines, beginning at the critical point of the component with the lower critical temperature, which terminates on intersecting a liquid-liquid-vapor three-phase line. 

Figure 1. Critical lines in water-gas shift binary pairs...

The mole fraction of the component whose critical point is the origin of the critical line has been indicated as a parameter along each of the lines. It should be noted from these numbers that there are compositions in each of the binary pairs for which there is no critical point. It should also be noted that some C02 - CO mixtures have two critical points. 

This program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which calculations are to be done (for the number of sets of calculations, as the the user wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the constants of the PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and vapor phase compositions (these last entries are optional, and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply model parameters for each pair of components in the multicomponent mixture. These model parameters can be obtained using the program WS (see Appendix D.5) if experimental data are available for each of the binary pairs. Alternatively, the user can select an already existing file (for tliese files we use the extensions WSN, WSW, and WSU, respectively, for the W,S-NRTL, WS-WILSON, and WS-UNIQUAC options, and some examples are provided on the accompanying disk) and calculate the multicomponent VLE for the mixture of that input file. 

The fit of the last binary pair, the methane-octane system, is shown in figure 5.3. This fit was obtained with a value of kij equal to 0.01. A few words of caution are warranted in this case. As noted in chapter 3, methane-hydrocarbon mixtures are expected to deviate from type-I behavior if the methane-solute carbon ratio is greater than 5. The P-x data shown in figure 5.3 are far above the temperatures where a three-phase LLV line is expected for this binary system. However, a three-phase LLV line is predicted near the critical point of methane using A ,y equal to 0.01. 

One example will serve to underscore the reason for the advantage over Chao-Seader at high pressure. Figure 1 shows the convergence of RKJZ K-values to unity as the mixture critical pressure is approached, for a temperature and composition on the mixture critical locus for the methane-ethane-butane ternary (20). This mixture was chosen in order to check RKJZ apparent critical pressure vs. the 1972 corresponding-states correlation of Teja and Rowlinson (21), which presumably has a better theoretical basis than the RKJZ method. In these comparisons, the Teja and Rowlinson correlation uses two interaction parameters per binary pair, based primarily on fits to binary critical loci the RKJZ method uses Cij = 0 for all binaries, based on binary VLE data. 

For the third binary pair, methanol (1) and cyclopentane (3), a critical solution temperature of 16.6°C with X3 = 0.58 was reported by Kiser, Johnson, and Shetlar." Thus... 

In so-called closed systems (case (a)), which are observed for about 75% of the systems, only one binary pair shows a miscibility gap. For these systems, a critical point C arises, where both liquid phases show the same concentration. Case (b) presents a system where two binary pairs show partial miscibility (open system). This behavior occurs in about 20% of all cases. Besides these most important cases, however, there are a large number of other possibilities [47]. For example, there are systems where all binary subsystems are homogeneous, but a miscibility gap (island) is found in the ternary system (see Figure 5.76). Additionally, there is the chance that three liquid phases are formed. 

For 8 < A gV/RT, Fig. 7c, there are no critical points all three of the binary pairs show a miscibility gap, and the three phase triangle remains. 

Pure component data which cannot be easily measured are estimated—including the critical properties of ethylene glycol. In many cases, the estimation techniques are crude, or they have not been developed for the type of compound under consideration. If VLB data are missing for any of the important binary pairs, they are measured. 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

Expression (3.23) allows us to determine the partial pair distribution functions at a point by measurement of the intensity of the energy losses in a simple two-component system, particularly across the gas-liquid critical lines. Also for a statistically determined system, (3.23) allows us to determine the charge density distribution in the neighborhood of a localized excitation in a binary complex. 

Figure 1. Variation of with density (full lines) fordilTerent values of R a using Chesnoy. The strong variation ofy (i ) with R at a given density leads to the conclusion that ifcannot be precisely determined, then the test of the binary interaction equation [Eq. (66)] is not critical. Dotted lines show that a hard-sphere pair distribution function fi(us(

We have studied binary blends dxx/hx2 of random olefinic copolymers x=(Ex x EEx)n, with one blend constituent protonated (hx) and the other deuterated (dx). The blends examined were grouped in four pairs of structurally identical mixtures xx/x2 but with a swapped isotope labeled component (dxx/hx2 and hxx/dx2). For such blend pairs the bulk interaction parameter % (and hence also the critical point Tc) has been found (see Sect. 2.2.3 and references therein) to be higher when the more branched (say xx>x2) component is deuterated, i.e., X(dx /hx2)>x(hx /dx2) or Tc(dxx/hx2)>Tc(hxx/dx2) (see Fig. 9). An identical pattern is exhibited here by the force driving the segregation at the free surface. This is illustrated in Fig. 26a,b where the composition vs depth profiles of the more branched (xx) component are shown for blend pairs with swapped isotope... 

The most basic question when considering a polymer blend concerns the thermodynamic miscibility. Many polymer pairs are now known to be miscible or partially miscible, and many have become commercially Important. Considerable attention has been focussed on the origins of miscibility and binary polymer/polymer phase diagrams. In the latter case, it has usually been observed that high molar mass polymer pairs showing partial miscibility usually exhibit phase diagrams with lower critical solution temperatures (LCST). 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

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