Mixtures, binary

The simplest mixture is a binary mixture that has two components A and B. In order to understand phase separation in this system, we consider its mixing free energy, which is composed of two parts mixing interaction energy Um and mixing entropy 5.  

Now we consider the mixing entropy. Imagine putting the Na particles of A and Nb particles of B into a lattice with N = Na+Nb lattice sites. The number of distinct states (arrangements of the particles) is [6] 

When the system is completely phase separated, the number of states is close to 1. Therefore the mixing entropy is 

For a macroscopic system, A 1, Aa 1, and Ab 1. Using the Sterling approximation that InN =NQnN- 1), Equation (7) becomes 

The second-order derivative has the minimum value atx = 0.5, which is = -a + AksT. 

Methanol (1) and water (2) are being distilled in a tray column operating at 101.3 kPa. On one of the trays in the column the mole fractions of methanol in the bulk vapor and liquid phases are = 0.5776 and xf = 0.3105. Estimate the rates of interphase mass transfer. 

The vapor-liquid equilibrium behavior of the methanol-water system at 101.3 kPa may be represented by 

The pure component vapor pressures may be estimated from the Antoine equation as follows  

The activity coefficients may be estimated from the Margules equation (Table D.2) with 

ANALYSIS Since there are only two components, the number of equations we must solve is just y. We list these equations below to assist us in taking the derivatives that we need if we are to use Newton s method. 

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Presents a variety of measured thermodynamic properties of binary mixtures these properties are often represented by empirical equations. 

Compilation of data for binary mixtures reports some vapor-liquid equilibrium data as well as other properties such as density and viscosity. 

Appendix C presents properties and parameters for 92 pure fluids and characteristic binary-mixture parameters for 150 binary pairs. 

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. 

An additional option allows the user to fit data for binary mixtures where one of the components is noncondensable. The mixture is treated as an ideal dilute solution. The solute... 

Subroutine BIJS2. This subroutine calculates the pure-component and cross second virial coefficients for binary mixtures according to the method of Hayden and O Connell (1975). 

CALCULATES ACTIVITY COEFFICIENTS FOP A BINARY MIXTURE USING 1 OF 12 POSSIBLE EQUATIONS AS DETERMINED BY ILIO... 

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

Diffusivity measures the tendency for a concentration gradient to dissipate to form a molar flux. The proportionality constant between the flux and the potential is called the diffusivity and is expressed in m /s. If a binary mixture of components A and B is considered, the molar flux of component A with respect to a reference plane through which the exchange is equimolar, is expressed as a function of the diffusivity and of the concentration gradient with respect to aji axis Ox perpendicular to the reference plane by the fpllqvving relatipn 6 /... 

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 ... 

A motor fuel has an octane number X if it behaves under tightly defined experimental conditions the same as a mixture of X volume % of isooctane and (100 - X)% of n-heptane. The isooctane-heptane binary mixtures are called primary reference fuels. Octane numbers higher than 100 can also be defined the reference material is isooctane with small quantities of tetraethyl lead added the way in which this additive acts will be discussed later. 

A diesel fuel has a cetane number X, if it behaves like a binary mixture of X% (by volume) n-cetane and of (100 - A) % a-methylnaphthalene. 

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. 

A logical division is made for the adsorption of nonelectrolytes according to whether they are in dilute or concentrated solution. In dilute solutions, the treatment is very similar to that for gas adsorption, whereas in concentrated binary mixtures the role of the solvent becomes more explicit. An important class of adsorbed materials, self-assembling monolayers, are briefly reviewed along with an overview of the essential features of polymer adsorption. The adsorption of electrolytes is treated briefly, mainly in terms of the exchange of components in an electrical double layer. 

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... 

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

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.

Pegg I L, Knobler C M and Scott R L 1990 Tricritical phenomena in quasibinary mixtures. VIII. Calculations from the van der Waals equation for binary mixtures J. Chem. Phys. 92 5442-53... 

We start with a simple example the decay of concentration fluctuations in a binary mixture which is in equilibrium. Let >C(r,f)=C(r,f) - be the concentration fluctuation field in the system where is the mean concentration. C is a conserved variable and thus satisfies a conthuiity equation ... 

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... 

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