One phase and two components

At once we have to give up the idea of a continuum and turn to a material containing countable numbers of atoms. Let the atoms be of just two species A and B let the number of atoms of A in 1 m be and the volume occupied by each one be Va, and similarly for B then 

The relative amounts of A and B can be expressed in various ways their number-ratio is just pjpa, their volume-ratio is Pa a/PsK and their mass-ratio, which we do not need, would be different again. Also the concentration of A can be taken as pj px + Pj,) and written C of any large number of atoms, a fraction of that number would be expected to be atoms of A. Going back to Chapter 3, especially eqns. (3.4), we may need to distinguish the activity of species A from its concentration, but before embarking on those details, we can already consider the general nature of the problem and its solution. 

Will the state of stress in the worm be affected The radial compression will continue to be atmospheric pressure, and we can imagine the worm to be floating on an inviscid fluid support (such as mercury). Then if the axial compression is uniform at the start, there is no reason for it to become nonuniform. For example, consider the second instance just imagined B is the more mobile species so that a region where the concentration of A was initially a maximum will be invaded by B from both sides. Will B drag on A as it arrives, thus increasing the axial compression on the central element No, attention has to be paid to the fact that the concentration gradient makes B move with respect to A but not with respect to any external reference frame, so no build-up of compression occurs. 

This fundamental point deserves review. One approach is to give attention to momentum. Suppose that at the initial moment, the worm overall has no momentum along the direction of its length, and no subdivision has any momentum either. Atoms of A and B can exchange momentum, but however much they do this, an element of worm that had no momentum at the start cannot pick up momentum from anywhere. 

To work out completely the relation of the stress profile to the composition profile, we would have to find its form as well as its amplitude but to see the nature of the problem, so as to begin seeking a solution, we can focus on the amplitude, assuming the profile to be something like a sine wave. The 

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The basis for the construction of the psychrometnc chart is the Gibbs phase rule (Section 6.3a). which states that specifying a certain number of the intensive variables (temperature, pressure, specific volume, specific enthalpy, component mass or mole fractions, etc.) of a system automatically fixes the value of the remaining intensive variables. Humid air contains one phase and two components, so that from Equation 6.2-1 the number of degrees of freedom is... 

Let us consider a binary solution that is composed of two liquid components that are not interacting chemically. If the volume of the liquid is equal to the size of the system, then we have only one phase and two components, so the Gibbs phase rule says that we have F=2 — 1+2 = 3 degrees of freedom. We can specify temperature, pressure, and mole fraction of one component to completely define our system. Recall from equation 3.24 that the mole fraction of a component equals the moles of some component i, divided by the total number of moles of all components in the system,... 

Reasoning in terms of molar Gibbs free energy G, the energy of one mole of the given phase) and considering a homogeneous system composed of one phase with two components, because... 

Azeotropic Systems. An azeotropic system is one wherein two or more components have a constanl boiling point at a particular composition. Such mixtures cannot be separated by conventional distillation methods. If rhe constant boiling point is a minimum, the system is said lo exhibit negomv azeotropy if it is a maximum, positive azeotropy. Consider a mixture of water and alcohol in the presence of the vapor. This system of two phases and two components is divarianl. Now choose some fixed pressure and study the composition of the system at equilibrium us a function of temperature. The experimental results arc shown schematically in Fig. 5. 

FIGURE 9.1 Theoretical ternary phase diagram outlining the region of existence of one-phase and two-phase systems. Note the illustrative representation of the droplet w/o, droplet o/w, and bicontinuous MEs. O, oil component W, water component S, amphiphile component (surfactant/cosurfactant). 

The basis for the separation is that when two polymers, or a polymer and certain salts, are mixed together in water, they are incompatible, leading to the formation of two immiscible but predominantly aqueous phases, each rich in only one of the two components [Albertsson, op. cit. Kula, in Cooney and Humphrey (eds.), op. cit., pp. 451 71]. A phase diagram for a polyethylene glycol (PEG)-Dextran, two-phase system is shown in Fig. 22-85. Proteins are known to distribute unevenly between these phases. This uneven distribution can be used for the selective concentration and partial purification of the products. Partitioning between the two phases is controlled by the polymer molecular weight and concentration, protein net charge and... 

Teaching yourself phase diagrams part 2 one and two component systems... 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

We have considered thermodynamic equilibrium in homogeneous systems. When two or more phases exist, it is necessary that the requirements for reaction equilibria (i.e., Equations (7.46)) be satisfied simultaneously with the requirements for phase equilibria (i.e., that the component fugacities be equal in each phase). We leave the treatment of chemical equilibria in multiphase systems to the specialized literature, but note that the method of false transients normally works quite well for multiphase systems. The simulation includes reaction—typically confined to one phase—and mass transfer between the phases. The governing equations are given in Chapter 11. 

The concept of supercriticality is more complex if a two-component fluid is used. For most mixtures used, SFE must be carried out above a certain pressure to ensure that the fluid is in one phase. For MeOH-C02 mixtures at 50 °C the fluid is in one phase and can be described as supercritical above 95 bar, whatever the composition [284]. Compounds may also be added to the supercritical phase as a reactant rather than as a simple modifier. 

Though both miscible and immiscible blends are composite materials, their properties are very different. A miscible blend will exhibit a single glass transition temperature that is intermediate between those of the individual polymers. In addition, the physical properties of the blends will also exhibit this intermediate behavior. Immiscible blends, on the other hand, still contain discrete phases of both polymers. This means that they have two glass transition temperatures and that each represents one of the two components of the blend. (A caveat must be added here in that two materials that are immiscible with very small domain sizes will also show a single, intermediate value for Tg.) In addition, the physical properties... 

If we now recall the phase rule, it is evident that, at the P-T conditions represented by point D in figure 2.5, slight variations in the P or T values will not induce any change in the structural state of the phase (there are one phase and one component the variance is 2). At point A in the same figure, any change in one of the two intensive variables will induce a phase transition. To maintain the coexistence of kyanite and andalusite, a dP increment consistent with the slope of the univariant curve (there are two phases and one component the variance... 

In the case of the interfacial tension of two pure liquids we have had to deal with the superficial system in equilibrium with a two phase two component system of three dimensions. If we add to this system a third component the problem becomes still more complicated. The simplest case is that in which the added substance is soluble in one phase and completely insoluble in the other, the original liquids being themselves mutually insoluble. The change of interfacial tension should then run parallel to the change of surface tension of the liquid in which the third component dissolves. 

Thus different phase behaviors of polyrotaxanes induced different thermal transitions. One-phase or two-phase materials can be obtained simply by proper choice of the components. The easy introduction of highly flexible cyclic components such as crown ethers with low T% surely expands the applications of otherwise brittle polymers into the low temperature range and also improves elasticity. The plasticizing effect of the crown ether is different from that of a normal plasticizer, because the cyclic is permanently connected to the backbone and no migration can occur. 

When the interaction rate is measured in this way one studies the course of a chemical reaction which occurs in the dispersed phase between two components. One component (C) is present in the reactor before the experiment starts and either is dissolved in the continuous phase or homogeneously distributed in the dispersed phase. The other component (A) is added at the beginning of the experiment in a highly concentrated form in a very small extra volume of the dispersed phase. The total amount of A must at leaBt be the stoichiometric equivalent of the total amount of component C already present in the reactor. 

SOLIDUS CURVE. A curve representing the equilibrium between the solid phase and the liquid phase m a condensed system of two components. The relationship is reduced to a two-dimensional curve by disregarding the influence, of the vapor phase. The points on the solidus curve are obtained by plotting the temperature at which the last of the liquid phase solidifies, against the composition, usually in terms of the percentage composition of one of the two components. 

Where the two phases are completely compatible, a homogeneous polyblend results which behaves like a plasticized resin (one phase). If two polymers are compatible, the mixture is transparent rather than opaque. If the two phases are incompatible, the product is usually opaque and rather friable. When the two phases are partially compatibilized at their interfaces, the polyblend system may then assume a hard, impact-resistant character. However, incompatible or partially compatible mixtures may be transparent if the individual components are transparent and if both components have nearly the same refractive indices. Furthermore, if the particle size of the dispersed phase is much less than the wavelength of visible light (requiring a particle size of 0.1/a or less), the blends may be transparent. 

The measurement of osmotic pressure and the determination of the excess chemical potential of a component by means of such measurements is representative of a system in which certain restrictions are applied. In this case the system is separated into two parts by means of a diathermic, rigid membrane that is permeable to only one of the components. For the purpose of discussion we consider the case in which the pure solvent is one phase and a binary solution is the other phase. The membrane is permeable only to the solvent. When a solute is added to a solvent at constant temperature and pressure, the chemical potential of the solvent is decreased. The pure solvent would then diffuse into such a solution when the two phases are separated by the semipermeable membrane but are at the same temperature and pressure. The chemical potential of the solvent in the solution can be... 

The added third component, sometimes called the entrainer, may form a ternary azeotrope with the two components being separated. However, it must be sufficiently volatile from the solution so that it is taken overhead with one of the two components in the distillation. If the entrainer and the component taken overhead separate into two liquid phases when the vapor overhead is condensed, the entrainer phase is refluxed back to the column. The other phase can be fractionated to remove the dissolved entrainer and the residual amount of the other component before it is discarded. Alternatively, this second liquid phase is recycled to some appropriate place in the main process scheme. 

T and H components are the same as in the yttrium oxyfluorides etc., but now they occur as strips intergrown in each layer the entire structure is divided by anti-phase boundaries perpendicular to the layers, with a slip vector R equal to half the unit-cell vector in the layer-stacking direction. These structures (often slightly monoclinic ) are CC types, with finite incommensiurate portions. They will be considered more fully in Chap. 6 below. Meanwhile, we will simply point out that anti-phase-boundary structures of this sort are strictly limited to ternaries with two cations and one anion. The related ternaries considered earlier in this section - those containing one cation and two anions - do not have these boundaries they are truly non-commensurate. This difference we take to be significant. 

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