Solid solutions, ideal solutions

Ballard, A., A Non-Ideal Hydrate Solid Solution Model for a Multi-Phase Equilibria Program, Ph.D. Thesis, Colorado School of Mines, Golden, CO (2002). 

Gases tend to obey the Ideal Gas Law (31.1) well, especially at low pressures where no intermolecular forces are present. In ideal solutions whether solutions of gases in gases (this Frame), solids in liquids (Frame 32) or liquids in liquids (Frame 33) it is not so much interactions which are absent but rather that none are preferential and so dominate the behaviour. Solute-solute, solute-solvent and solvent-solvent interactions are often broadly similar and so there are few preferential molecular orientations favoured above others. Such a situation arises, usually, due to the high state of dilution of the system. 

If BaCOj (A ,p = 8 X 10 ) and SrCOj = 2 x 10 ) form an ideal, homogeneous solid solution, what is the theoretical mole fraction of BaCOa in the precipitate if half the Sr is precipitated from a solution initially equimolar in Ba" " and Sr Answer 0.286. 

The second major, more important success of the theory is that, using the fits to single component data, the theory allows predictions of mixture conditions for any combination of pure components. Due to the ideal solution, solid nature (assumptions 4 and 5) fits of pure component data provide acceptable predictions of all mixture conditions. Since laboratory data are costly both in terms of time and funds (c.g., US 2,000 per data point), it is very efficient to predict rather than to measure each hydrate condition. It is important to emphasize that, in contrast to vapor-liquid equilibria predictions, no interaction parameters are needed for mixture prediction. 

Solubility equilibria resemble the equilibria between volatile liquids (or solids) and their vapors in a closed container. In both cases, particles from a condensed phase tend to escape and spread through a larger, but limited, volume. In both cases, equilibrium is a dynamic compromise in which the rate of escape of particles from the condensed phase is equal to their rate of return. In a vaporization-condensation equilibrium, we assumed that the vapor above the condensed phase was an ideal gas. The analogous starting assumption for a dissolution-precipitation reaction is that the solution above the undissolved solid is an ideal solution. A solution in which sufficient solute has been dissolved to establish a dissolution-precipitation equilibrium between the solid substance and its dissolved form is called a saturated solution. 

The Tg of an ideally mixed solid solution can be predicted from the sum of the weight fractions (w) and Tg values of the individual components of the mixture by using the Gordon-Taylor equation (Gordon and Taylor 1952) ... 

For an ideal liquid/solid solution, it is possible to calculate the solubility of the solid solute. If a saturated solution exists, then saturated solution is in equilibrium... 

In the case of diffusion of foreign atoms in small concentrations into a host lattice, e. g., a solid or liquid metal, this situation is simplified because of the applicability of the laws of ideal diluted solid solutions. The concentration of the electroactive species at the interface with the electrolyte is given directly by Nemst s law. In case of nonideal behavior, further information concerning the activity-concentiation relation of the diffusing species is necessary. This may be acquired from coulometric titration measurements. 

The schematization of Fig. 2.5 and relation [2.1] remain valid for the selective oxidation of ideal disordered solid solutions. Indeed, the total volume of such an ideal solid solution does not depend on the spatial distribution of its constituents, as the partial molar volume of each constituent is equal to its molar volume. Therefore, there is no difference between the pure metal and an ideal solid solution regarding the definition and location of planes il/and Km- In relation [2.1], the value of 0 is then that of the selectively oxidized constituent. 

Figure III-9u shows some data for fairly ideal solutions [81] where the solid lines 2, 3, and 6 show the attempt to fit the data with Eq. III-53 line 4 by taking ff as a purely empirical constant and line 5, by the use of the Hildebrand-Scott equation [81]. As a further example of solution behavior, Fig. III-9b shows some data on fused-salt mixtures [83] the dotted lines show the fit to Eq. III-SS.

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Chemical reaction - formation of intermetollic compounds Diffusion in solid solutions (dilute ideal solutions between solute 300 to 5 X 1 O ... 

When a pure metal A is alloyed with a small amount of element B, the result is ideally a homogeneous random mixture of the two atomic species A and B, which is known as a solid solution of in 4. The solute B atoms may take up either interstitial or substitutional positions with respect to the solvent atoms A, as illustrated in Figs. 20.37a and b, respectively. Interstitial solid solutions are only formed with solute atoms that are much smaller than the solvent atoms, as is obvious from Fig. 20.37a for the purpose of this section only three interstitial solid solutions are of importance, i.e. Fc-C, Fe-N and Fe-H. On the other hand, the solid solutions formed between two metals, as for example in Cu-Ag and Cu-Ni alloys, are always substitutional (Fig. 20.376). Occasionally, substitutional solid solutions are formed in which the... 

At the same time it is recognized that the pairs of substances which, on mixing, are most likely to obey Raoult s law are those whose particles are most nearly alike and therefore interchangeable. Obviously no species of particles is likely to fulfill this condition better than the isotopes of an element. Among the isotopes of any element the only difference between the various particles is, of course, a nuclear difference among the isotopes of a heavy element the mass difference is trivial and the various species of particles are interchangeable. Whether the element is in its liquid or solid form, the isotopes of a heavy element form an ideal solution. Before discussing this problem we shall first consider the solution of a solid solute in a liquid solvent. 

A dependence of w upon composition must also be adduced in the case of the Fe-Ni solid solutions. Over the range from 0 to 56 at. per cent Ni, these solid solutions exhibit essentially ideal behavior,39 so that w 0. Since the FeNi3 superlattice appears at lower temperatures, either w is markedly different at compositions about 75 at. per cent Ni than at lower Ni contents, or w 0 for the solid solutions about the superlattice. Either possibility represents a deviation from the requirements of the quasi-chemical theories. 

Table II also demonstrates the discrepancy existing between E0/RTe calculated by the Yang-Li quasi-chemical theory and the experimental ratio. E0 is the energy difference between a fully ordered superlattice and the corresponding solid solution with an ideally random atom species distribution. It is a quantity that can only be estimated from existing experimental information, but the disparity between theory and experiment is beyond question.

Figure 1. Ideal pressure-composition isotherms showing the hydrogen solid-solution phase, a, and the hydride phase, j3. The plateau marks the region of coexistence of the a and fl phases. As the temperature is increased the plateau narrows and eventually disappears at some consolule temperature...

It is essentially a phase diagram which consists of a family of isotherms that relate the equilibrium pressure of hydrogen to the H content of the metal. Initially the isotherm ascends steeply as hydrogen dissolves in the metal to form a solid solution, which by convention is designated as the a phase. At low concentrations the behaviour is ideal and the isotherm obeys Sievert s Law, i.e.,... 

We will see later that this same equation applies to the mixing of liquids or solids when ideal solutions form. 

Figure 6.7 Vapor pressures for. x CFfiCfTOH +. v2HiO at T- 303.15 K. The symbols represent the experimental vapor pressures as follows , vapor pressure of H2O , vapor pressure of CFfiCFfiOH , total vapor pressure. The solid lines are the fits to the experimental data, and the dashed lines represent the ideal solution prediction.

Since Raoult s law activities become mole fractions in ideal solutions, a simple substitution of.Y, — a, into equation (6.161) yields an equation that can be applied to (solid + liquid) equilibrium where the liquid mixtures are ideal. The result is... 

Figure 8.20 (Solid + liquid) phase equilibria for [.viQHf, +. yl.4-C6H4(CH,)2 - The circles are the experimental results the solid lines are the fit of the experimental results to equation (8.31) the dashed lines are the ideal solution predictions using equation (8.30) the solid horizontal line is at the eutectic temperature and the diamonds are (.v, T) points referred to in the text.

Figure 8.21 gives the ideal solution prediction equation (8.36) of the effect of pressure on the (solid + liquid) phase diagram for. yiC6H6 + xj 1,4-C6H4(CH3)2. The curves for p — OA MPa are the same as those shown in Figure 8.20. As... 

Figure 8.23 (Solid + liquid) phase diagram for (. 1CCI4 +. yiCHjCN), an example of a system with large positive deviations from ideal solution behavior. The solid line represents the experimental results and the dashed line is the ideal solution prediction. Solid-phase transitions (represented by horizontal lines) are present in both CCI4 and CH3CN. The CH3CN transition occurs at a temperature lower than the eutectic temperature. It is shown as a dashed line that intersects the ideal CH3CN (solid + liquid) equilibrium line.

Assume ideal solution behavior with no solid solutions and constant Afus m and... 

We use a different measure of concentration when writing expressions for the equilibrium constants of reactions that involve species other than gases. Thus, for a species J that forms an ideal solution in a liquid solvent, the partial pressure in the expression for K is replaced by the molarity fjl relative to the standard molarity c° = 1 mol-L 1. Although K should be written in terms of the dimensionless ratio UJ/c°, it is common practice to write K in terms of [J] alone and to interpret each [JJ as the molarity with the units struck out. It has been found empirically, and is justified by thermodynamics, that pure liquids or solids should not appear in K. So, even though CaC03(s) and CaO(s) occur in the equilibrium... 

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