Two Components

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. 

The solvent components usually have a low mutual solubility and are present in reasonably large mole fractions in the system. If solvents are not so designated, we take as the "solvent components" those two components, present in significant mole fraction in the system, that have the lowest binary solubilities. ... 

In principle, extractive distillation is more useful than azeotropic distillation because the process does not depend on the accident of azeotrope formation, and thus a greater choice of mass-separating agent is, in principle, possible. In general, the solvent should have a chemical structure similar to that of the less volatile of the two components. It will then tend to form a near-ideal mixture with the less volatile component and a nonideal mixture with the more volatile component. This has the effect of increasing the volatility of the more volatile component. 

A path is a sequence of distinct lines which are connected to each other. For example, in Fig. 7.1a, AECGD is a path. A graph forms a single component if any two points are joined by a path. Thus Fig. 7.16 has two components and Fig. 7.1a has only one. 

In general, the final network design should be achieved in the minimum number of units to keep down the capital cost (although this is not the only consideration to keep down the capital cost). To minimize the number of imits in Eq. (7.1), L should be zero and C should be a maximum. Assuming L to be zero in the final design is a reasonable assumption. However, what should be assumed about C Consider the network in Fig. 7.16, which has two components. For there to be two components, the heat duties for streams A and B must exactly balance the duties for streams E and F. Also, the heat duties for streams C and D must exactly balance the duties for streams G and H. Such balemces are likely to be unusual and not easy to predict. The safest assumption for C thus appears to be that there will be one component only, i.e., C = 1. This leads to an important special case when the network has a single component and is loop-free. In this case, ... 

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 ... 

So far we have considered only a single component. However, reservoir fluids contain a mixture of hundreds of components, which adds to the complexity of the phase behaviour. Now consider the impact of adding one component to the ethane, say n-heptane (C7H.,g). We are now discussing a binary (two component) mixture, and will concentrate on the pressure-temperature phase diagram. 

When the two components are mixed together (say in a mixture of 10% ethane, 90% n-heptane) the bubble point curve and the dew point curve no longer coincide, and a two-phase envelope appears. Within this two-phase region, a mixture of liquid and gas exist, with both components being present in each phase in proportions dictated by the exact temperature and pressure, i.e. the composition of the liquid and gas phases within the two-phase envelope are not constant. The mixture has its own critical point C g. 

The principal point of interest to be discussed in this section is the manner in which the surface tension of a binary system varies with composition. The effects of other variables such as pressure and temperature are similar to those for pure substances, and the more elaborate treatment for two-component systems is not considered here. Also, the case of immiscible liquids is taken up in Section IV-2. 

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

The adsorption of detergent-type molecules on fabrics and at the solid-solution interface in general shows a complexity that might be mentioned briefly. Some fairly characteristic data are shown in Fig. XlIl-15 [242]. There is a break at point A, marking a sudden increase in slope, followed by a maximum in the amount adsorbed. The problem is that if such data represent true equilibrium in a two-component system, it is possible to argue a second law violation (note Problem Xni-14) (although see Ref. 243). 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

Figure A2.5.3. Typical liquid-gas phase diagram (temperature T versus mole fraction v at constant pressure) for a two-component system in which both the liquid and the gas are ideal mixtures. Note the extent of the two-phase liquid-gas region. The dashed vertical line is the direction x = 1/2) along which the fiinctions in figure A2.5.5 are detemiined.

Figure A2.5.4 shows for this two-component system the same thennodynamic fimctions as in figure A2.5.2, the molar Gibbs free energy (i= + V2P2> the molar enthalpy wand the molar heat capacity C. , again all at...

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

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].

The field-density concept is especially usefiil in recognizing the parallelism of path in different physical situations. The criterion is the number of densities held constant the number of fields is irrelevant. A path to the critical point that holds only fields constant produces a strong divergence a path with one density held constant yields a weak divergence a path with two or more densities held constant is nondivergent. Thus the compressibility Kj,oi a one-component fluid shows a strong divergence, while Cj in the one-component fluid is comparable to (constant pressure and composition) in the two-component fluid and shows a weak... 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

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). 

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

In two classic papers [18, 46], Calm and Flilliard developed a field theoretic extension of early theories of micleation by considering a spatially inliomogeneous system. Their free energy fiinctional, equations (A3.3.52). has already been discussed at length in section A3.3.3. They considered a two-component incompressible fluid. The square gradient approximation implied a slow variation of the concentration on the... 

In the Bom-Oppenlieimer approxunation the vibronic wavefrmction is a product of an electronic wavefimction and a vibrational wavefunction, and its syimnetry is the direct product of the synuuetries of the two components. We have just discussed the synuuetries of the electronic states. We now consider the syimnetry of a vibrational state. In the hanuonic approximation vibrations are described as independent motions along nonual modes Q- and the total vibrational wavefrmction is a product of frmctions, one wavefunction for each nonual mode ... 

If the two constituting molecular volumes are identical in a two component system, we can obtain [37, 38]... 

A similar approach, in spirit, has been proposed [212] for the study of two-component classical systems, for example poly electrolytes, which consist of mesoscopic, highly-charged, poly ions, and microscopic. 

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