Solids mixing ideal mixtures

The volume of an ideal gas mixture is given by eq. (3.6). Let us now consider only solid or liquid mixtures. Our starting point is an arbitrary mixture of nA mole of pure A and ng mole of pure B. The mixing process is illustrated in Figure 3.1. We... 

In the case of non-ideal mixtures (e.g. liquid and solid solutions), the activity ai = y has to be used instead of the molar fraction xi of substance i after the logarithmic sign in Eq. 10.23 to express the mixing term of the exergy at the exergy reference temperature T0 and pressure ft as shown in Eq. 10.27 ... 

An ideal mixture is one for which the heat of mixing or solution is negligible and so Hmixture 21 where n, is the amount of mixture component i and D, is the specific enthalpy of the pure component at the temperature and pressure of the mixture. Up to now in this text, we have assumed ideal mixture behavior for all mixtures and solutions. This assumption works well for nearly all gas mixtures and for liquid mixtures of similar compounds (such as mixtures of paraffins or of aromatics), but for other mixtures and solutions—such as aqueous solutions of strong acids or bases or certain gases (such as hydrogen chloride) or solids (such as sodium hydroxide)—heats of solution should be included in energy balance calculations. This section outlines the required procedures. 

Activity Coefficient of a Solute in Multicomponent (Ternary and Higher) Mixed Solvents. Let us consider a n-component mixture containing a solute (component 2), water and (n-2) organic cosolvents. If one considers the (n-1) mixed solvent as an ideal mixture, one can rewrite Equation 4 for the activity coefficient of a solid solute at infinite dilution in a multicomponent (ternary and higher) solvent (24)... 

It is of interest to note that since the mole fraction n of any gas in a mixture must be less than unity, its logarithm is negative hence ASm as defined by equation (19.32) is always positive. In other words, the mixing of two or more gases, e.g., by diffusion, is accompanied by an increase of entropy. Although equation (19.32) has been derived here for a mixture of ideal gases, it can be shown that it applies equally to an ideal mixture of liquids or an ideal solid solution. 

The type of movement required for mixing particulate matter also produces ideal conditions for agglomeration by coalescence. Therefore, unwanted agglomeration is often observed in powder mixers, especially if the particle size of the solids is small and/or a small amount of moisture is present. Considerable problems can arise if components of the bulk mass have different particle sizes because, in that case, the smaller fractions may selectively agglomerate, thus making it impossible to obtain an ideal mixture. Such selective agglomeration is of particular concern in the pharmaceutical industry where an extremely small amount of a finely divided active substance must often be mixed uniformly and reliably with a relatively large amount of inert filler material. 

Most solutions do not exhibit ideal behavior, and the actual curve corresponding to the variation of the molar volume or enthalpy of the mixture deviates from a straight line (e.g., the solid line in Fig. 5.1). When the curve for the molar volume lies above the ideal mixture line, the system expands upon mixing when the curve lies below the line, the system contracts. In the case of the molar enthalpy, a curve that lies above the ideal mixture line corresponds to the system that absorbs heat (e.g., mixing lead bromide and water) a curve that lies below the line corresponds to the system releasing heat (e.g., mixing sulfuric acid and water). This non-ideal mixing in the case of the molar enthalpy is the principle used in cold packs and heat packs. We will develop mathematical models to describe non-ideal mixtures. We use partial molar properties in more detail later. 

When the mixture formed is an ideal mixture (gas, liquid, or solid), and the pure constituents have the same physical state as the mixture, the expressions for various molar mixing quantities are particularly simple. An ideal molar mixing quantity will be indicated... 

Intermetallics also represent an ideal system for study of shock-induced solid state chemical synthesis processes. The materials are technologically important such that a large body of literature on their properties is available. Aluminides are a well known class of intermetallics, and nickel aluminides are of particular interest. Reactants of nickel and aluminum give a mixture with powders of significantly different shock impedances, which should lead to large differential particle velocities at constant pressure. Such localized motion should act to mix the reactants. The mixture also involves a low shock viscosity, deformable material, aluminum, with a harder, high shock viscosity material, nickel, which will not flow as well as the aluminum. 

Semibatch reactors are often used to mn highly exothermic reactions isothermally, to run gas-liquid(-solid) processes isobarically, and to prevent dangerous accumulation of some reactants in the reaction mixture. Contrary to batch of)eration, temperature and pressure in semibatch reactors can be varied independently. The liquid reaction mixture can be considered as ideally mixed, while it is assumed that the introduced gas flows up like a piston (certainly this is not entirely true). Kinetic modelling of semibatch experiments is as difficult as that of batch, non-isotherma experiments. 

Figure 2. Cmc s of mixtures of SDS and CgE4 in distilled water (at 25°C). The plotted points are experimental data, the solid line is the result for the nonideal mixed micelle model with B = -3.3, and the dashed line is the result for ideal mixing.

It is important to emphasize here that, theoretically, if a solid mixture is ideal, intracrystalline distribution is completely random (cf section 3.8.1) and, in these conditions, the intracrystalline distribution constant is always 1 and coincides with the equilibrium constant. If the mixture is nonideal, we may observe some ordering on sites, but intracrystalline distribution may still be described without site interaction parameters. We have seen in section 5.5.4, for instance, that the distribution of Fe and Mg on Ml and M3 sites of riebeckite-glaucophane amphiboles may be approached by an ideal site mixing model—i.e.. 

Results for the various binary mixed surfactant systems are shown in figures 1-7. Here, experimental results for the surface tension at the cmc (points) for the mixtures are compared with calculated results from the nonideal mixed monolayer model (solid line) and results for the ideal model (dashed line). Calculations of the surface tension are based on equation 17 with unit activity coefficients for the ideal case and activity coefficients determined using the net interaction 3 (from the mixed micelle model) and (equations 12 and 13) in the nonideal case. In these calculations the area per mole at the surface for each pure component, tOj, is obtained directly from the slope of the linear region in experimental surface tension data below the cmc (via equation 5) and the maximum surface pressure, from the linear best fit of... 

Figure 1. Mixed cmc s and surface tensions at the cmc for mixtures of C qPO and SDS in 1 mM Na2C0T (at 24°C). The plotted points are experimental results, the solid lines the prediction of the nonideal model for 8 = -3 7 and 8 = -3.5 respectively, and the dashed lines the prediction for ideal mixing in the pseudo-phase.

The cure of thermoset resins involves the transformation of a liquid resin, first with an increase in viscosity to a gel state (rubber consistency), and finally to a hard solid. In chemical terms, the liquid is a mixture of molecules that reacts and successively forms a solid network polymer. In practice the resin is catalyzed and mixed before it is injected into the mold thus, the curing process will be initialized at this point. The resin cure must therefore proceed in such a way that the curing reaction is slow or inhibited in a time period that is dictated by the mold fill time plus a safety factor otherwise, the increase in viscosity will reduce the resin flow rate and prevent a successful mold fill. On completion of the mold filling the rate of cure should ideally accelerate and reach a complete cure in a short time period. There are limitations, however, on how fast the curing can proceed set by the resin itself, and by heat transfer rates to and from the composite part. 

As we have seen in the previous section, the bulk chemical compositions of montmorillonites taken from the literature are dispersed over the field of fully expandable, mixed layered and even extreme illite compositions. Just what the limits of true montmorillonite composition are cannot be established at present. We can, nevertheless, as a basis for discussion, assume that the ideal composition of beidellite with 0.25 charge per 10 oxygens and of montmorillonite with the same structural charge do exist in nature and that they form the end-members of montmorillonite solid solutions. Using this assumption one can suppose either solid solution between these two points or intimate mixtures of these two theoretical end-member fully expandable minerals. In either case the observable phase relations will be similar, since it is very difficult if not impossible to distinguish between the two species by physical or chemical methods should they be mixed together. As the bulk chemistry of the expandable phases suggests a mixture of two phases, we will use this hypothesis and it will be assumed here that the two montmorillonite... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

Photodimerization of 2-pyridone (46) in the presence of the 2,2/-biphenyldi-carboxylic acid host (45) also proceeded via a catalytic process. First, irradiation of the 1 2 inclusion complex of 46 and 45 in the solid state gave the trans-anti dimer (47) in 92 % yield [27], The mechanism of this stereoselective photoreaction was investigated through X-ray analysis of this complex. In the complex, two 46 molecules are arranged in ideal positions for yielding 47 by dimerization [27], Secondly, a catalytic dimerization reaction of 46 was carried out. Photoirradiation for 20 h of a 1 4 mixture of powdered 45 and 46 under occasional mixing in the solid state gave 47 in 81 % yield. These data clearly show that molecules of... 

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