Phase binary mixture

Figure 9 Solvent selectivity triangle approach forthe selectivity optimization in HP-RPC. First, three initial experiments (1-3) with three mobile phases (binary mixtures of ACN/water, MeOH/water, and THF/water, respectively) are performed.

This calculation precludes development of the Einstein diffusion equation for forced diffusion in the presence of a gravitational field. The coefficient of (ge — gA) in equation (25-77) for the diffusional mass flux of species A can be evaluated via thermodynamics. The extensive Gibbs free energy of a one-phase binary mixture with 3 degrees of freedom requires four independent variables for complete description of this thermodynamic state function. Hence, ( (T, p, N/, Nb) is postulated where Ni represents the mole numbers of species i, and the total differential of is... 

Equations (25-77) and (25-87) allow one to express the diffusional mass flux of species A in a homogeneous single-phase binary mixture with respect to the... 

Consider a one-phase binary mixture of components 1 and 2 confined to an isolated vessel, and imagine dividing the fluid into parts A and B. But unlike the pure case, region B is open to A, so that a fluctuation occurring in part B disturbs not only its internal energy and volume V, but also the mole numbers Nf and N . Consequently, the concentration in B fluctuates by transfers of material to and from part A. In addition to the constraints on U, V, and S given by (8.1.4)-(8.1.6), the total amoxmts of each component are conserved. 

In this section we describe the common stability behavior displayed by binary mixtures ( 8.4.1), including a scheme for classifying that behavior ( 8.4.2). Then we show how models can be used to test for the observability of one-phase binary mixtures first we consider PvTx models ( 8.4.3 and 8.4.4) and then models for the excess Gibbs energy ( 8.4.5). 

Nevertheless, it is mathematically possible for some components to violate (8.4.9) while the mixture still might obey (8.4.7). But with the help of (8.4.8) we can show that, in fact, both components of a stable, one-phase binary must satisfy (8.4.9). The proof is given in Appendix F. Consequently, if a single-phase binary mixture has /j >... 

To decide among these possibilities we need a stability criterion for mixtures at fixed T, P, and fugacity Equivalently, we can develop the criterion in terms of T, P, and the chemical potential, then convert it to fugacities at the end. Imagine a one-phase binary mixture surrounded by a reservoir that imposes its temperature, pressure, and chemical potential on the system. The latter is accomplished by a semi-permeable membrane that separates the system from the reservoir. The membrane allows molecules of component 1 to pass, but it blocks passage of molecules of component 2. When diffusional equilibrium is established, the value of the chemical potential Gi is the same in the system and in the reservoir. The extensive state of the system is identified by giving values for the fixed quantities T, P, Gj, and N2. 

Write a computer program that determines the stability of a one-phase binary mixture at a proposed T, P, and Xj. Use the Redlich-Kwong equation of state with the simple mixing rules given in 8.4.4. Test your program by applying it to the situation described in 8.4.4. 

In this appendix we prove that a stable, one-phase, binary mixture must have values for component fugacities that are less than the corresponding pure-component values that is, we prove that a stable, one-phase, binary mixture must have... 

Recently, Kleiner (2) have shown that the moduli of the compatible PS/PPO blends fall outside the upper bounds given by Eq. 2. Instead of the classical composite results, the blend moduli are reported to follow a composition dependency given by the general equation cited by Nielsen (21) for one-phase binary mixtures in the specific form given by Kleiner... 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

CALCULATES VAPOR PHASE PUSACITV COEFFICIENTS FOR PURE ANO BINARY MIXTURES ... 

For a binary mixture of two components A and B in the gas phase, the mutual diffusion coefficient such as defined in 4.3.2.3, does not depend on composition. It can be calculated by the Fuller (1966) method ... 

The example of a binary mixture is used to demonstrate the increased complexity of the phase diagram through the introduction of a second component in the system. Typical reservoir fluids contain hundreds of components, which makes the laboratory measurement or mathematical prediction of the phase behaviour more complex still. However, the principles established above will be useful in understanding the differences in phase behaviour for the main types of hydrocarbon identified. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

Figure A2.5.13. Global phase diagram for a van der Waals binary mixture for whieh The...

A2.5.4.1 LIQUID-LIQUID PHASE SEPARATION IN A SIMPLE BINARY MIXTURE... 

The previous seetion showed how the van der Waals equation was extended to binary mixtures. However, imieh of the early theoretieal treatment of binary mixtures ignored equation-of-state eflfeets (i.e. the eontributions of the expansion beyond the volume of a elose-paeked liquid) and implieitly avoided the distinetion between eonstant pressure and eonstant volume by putting the moleeules, assumed to be equal in size, into a kind of pseudo-lattiee. Figure A2.5.14 shows sohematieally an equimolar mixture of A and B, at a high temperature where the distribution is essentially random, and at a low temperature where the mixture has separated mto two virtually one-eomponent phases. 

Figure A2.5.14. Quasi-lattice representation of an equimolar binary mixture of A and B (a) randomly mixed at high temperature, and (b) phase separated at low temperature.

Few if any binary mixtures are exactly syimnetrical around v = 1/2, and phase diagrams like that sketched in figure A2.5.5(c) are typicd. In particular one can write for mixtures of molecules of different size (different molar volumes and F°g) the approxunate equation... 

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

Within this general framework there have been many different systems modelled and the dynamical, statistical prefactors have been calculated. These are detailed in [42]. For a binary mixture, phase separating from an initially metastable state, the work of Langer and Schwartz [48] using die Langer theory [47] gives the micleation rate as... 

Another important class of materials which can be successfiilly described by mesoscopic and contimiiim models are amphiphilic systems. Amphiphilic molecules consist of two distinct entities that like different enviromnents. Lipid molecules, for instance, comprise a polar head that likes an aqueous enviromnent and one or two hydrocarbon tails that are strongly hydrophobic. Since the two entities are chemically joined together they cannot separate into macroscopically large phases. If these amphiphiles are added to a binary mixture (say, water and oil) they greatly promote the dispersion of one component into the other. At low amphiphile... 

Detailed x-ray diffraction studies on polar liquid crystals have demonstrated tire existence of multiple smectic A and smectic C phases [M, 15 and 16]. The first evidence for a smectic A-smectic A phase transition was provided by tire optical microscopy observations of Sigaud etal [17] on binary mixtures of two smectogens. Different stmctures exist due to tire competing effects of dipolar interactions (which can lead to alternating head-tail or interdigitated stmctures) and steric effects (which lead to a layer period equal to tire molecular lengtli). These... 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

In other cases, association has been demonstrated by means of phase diagrams of binary mixtures (158). 

A hypothetical moving-bed system and a Hquid-phase composition profile are shown in Figure 7. The adsorbent circulates continuously as a dense bed in a closed cycle and moves up the adsorbent chamber from bottom to top. Liquid streams flow down through the bed countercurrently to the soHd. The feed is assumed to be a binary mixture of A and B, with component A being adsorbed selectively. Feed is introduced to the bed as shown. 

The Class I binary diagram is the simplest case (see Fig. 6a). The P—T diagram consists of a vapor—pressure curve (soHd line) for each pure component, ending at the pure component critical point. The loci of critical points for the binary mixtures (shown by the dashed curve) are continuous from the critical point of component one, C , to the critical point of component two,Cp . Additional binary mixtures that exhibit Class I behavior are CO2—/ -hexane and CO2—benzene. More compHcated behavior exists for other classes, including the appearance of upper critical solution temperature (UCST) lines, two-phase (Hquid—Hquid) immiscihility lines, and even three-phase (Hquid—Hquid—gas) immiscihility lines. More complete discussions are available (1,4,22). Additional simple binary system examples for Class III include CO2—hexadecane and CO2—H2O Class IV, CO2—nitrobenzene Class V, ethane—/ -propanol and Class VI, H2O—/ -butanol. 

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