Mechanical mixture equilibrium

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

Figure 7.3 depicts phase stability relations in the pseudobinary system CaMgSi206-CaAl2Si208 (diopside-anorthite). The original study of Bowen (1915) described crystallization behavior identical to the previously discussed case a mechanical mixture (Di-An) in equilibrium with a completely miscible melt. A later investigation (Osborn, 1942) showed that the system is not strictly binary... 

The initial products of organic reactions are formed under conditions of kinetic control - the products are formed in proportions governed by the relative rates of the parallel (forward) reactions leading to their formation. Subsequently, product composition may become thermodynamically controlled (equilibrium controlled), i.e. when products are in proportions governed by the equilibrium constants for their interconversion under the reaction conditions. The reaction conditions for equilibrium control could involve longer reaction times than those for kinetic control, or addition of a catalyst. The mechanism of equilibrium control could simply involve reversal of the initial product-forming reactions (as in Scheme 2.4, see below), or the products could interconvert by another process (e.g. hydrolysis of an alkyl chloride could produce a mixture of an alcohol and an alkene, and the HsO"1" by-product could catalyse their interconversion). 

Hindin et al. (18) published data showing benzene formation to proceed readily from methylcyclopentane over mechanical mixtures of platinum bearing particles and silica-alumina, at atmospheric pressure and near 500°C. temperature. Under these conditions the equilibrium constant for conversion of a cyclopentane to a cyclopentene is of the order of unity. Consequently, the first step, if it is catalyzed by X, can itself proceed with... 

These equations define the families of ( ba ba) and ( cA> ncA) conditions for which a solution containing A-, B+ and C ions will be saturated with respect to pure BA and pure CA solids. Thermodynamic equilibrium with respect to a mechanical mixture of the two pure BA and CA solids, in contrast to a solid-solution of BA and CA, will be represented on a Lippmann diagram by a single point, namely the intersection of the pure BA and pure CA saturation curves. The coordinates of this intersection are ... 

In the following sections we will quantify some of the thermodynamic properties of mechanical mixtures and ideal and non-ideal solutions. As we detail the properties of ideal solutions, it will become clear that they are strictly hypothetical another thermodynamic concept, like true equilibrium , which is a limiting state for real systems. Ideal solutions, in other words, are another part of the thermodynamic model, not of reality. It is a useful concept, because real solutions can be compared to the hypothetical ideal solution and any differences described by using correction factors (activity coefficients) in the equations describing ideal behavior. These correction factors can either be estimated theoretically or determined by actually measuring the difference between the predicted (ideal) and actual behavior of real solutions. 

Figure 1(A) shows a phase diagram of a typical eutectic mixture system, which has a minimum melting temperature, i.e. a eutectic point. The eutectic pwint of a binary condensed mixture is defined as the temperature at which a solid mixture phase is in equilibrium with the liquid phase and a eutectic is generally considered to be a simple mechanical mixture of the solid and liquid (Rastogi and Bassi, 1964).

Physisorption-based adsorption and separation processes are of primary interest for various applications, including H2 storage, carbon capture and sequestration, and hydrocarbon separations. Physisorption based separation of adsorbate mixtures can occur through a variety of mechanisms, including equilibrium, steric, kinetic, or some combination of these in more complex systems. 

ATHLET offers the possibility of choosing between different models for the simulation of fluid dynamics.In the current released code version, the basic fluid-dynamic option is a five-equation model, with separate conservation equations for liquid and vapour mass and energy, and a mixture momentum equation, accounting for thermal and mechanical non-equilibrium, and including a mixture level tracking capability. 

The equilibrium of the last step (3), which is not actually part of the condensation mechanism, is far to the right because of the greater basic strength of the ethoxide ion as compared to (IV), and this largely assists the forward reactions in (1) and (2). The reaction mixture contains the sodium derivative of the keto-ester, and the free ester is obtained upon acidification. 

The two dimers of (CH3)2C=CH2 are formed by the mechanism shown m Figure 6 16 In step 1 protonation of the double bond generates a small amount of tert butyl cation m equilibrium with the alkene The carbocation is an electrophile and attacks a second molecule of 2 methylpropene m step 2 forming a new carbon-carbon bond and generating a carbocation This new carbocation loses a proton m step 3 to form a mixture of 2 4 4 tnmethyl 1 pentene and 2 4 4 tnmethyl 2 pentene... 

In a series of organic acids of similar type, not much tendency exists for one acid to be more reactive than another. For example, in the replacement of stearic acid in methyl stearate by acetic acid, the equilibrium constant is 1.0. However, acidolysis in formic acid is usually much faster than in acetic acid, due to higher acidity and better ionizing properties of the former (115). Branched-chain acids, and some aromatic acids, especially stericaHy hindered acids such as ortho-substituted benzoic acids, would be expected to be less active in replacing other acids. Mixtures of esters are obtained when acidolysis is carried out without forcing the replacement to completion by removing one of the products. The acidolysis equilibrium and mechanism are discussed in detail in Reference 115. 

Acyl-pyrroles, -furans and -thiophenes in general have a similar pattern of reactivity to benzenoid ketones. Acyl groups in 2,5-disubstituted derivatives are sometimes displaced during the course of electrophilic substitution reactions. iV-Alkyl-2-acylpyrroles are converted by strong anhydrous acid to A-alkyl-3-acylpyrroles. Similar treatment of N-unsubstituted 2- or 3-acyIpyrroles yields an equilibrium mixture of 2- and 3-acylpyrroles pyrrolecarbaldehydes also afford isomeric mixtures 81JOC839). The probable mechanism of these rearrangements is shown in Scheme 65. A similar mechanism has been proposed for the isomerization of acetylindoles. 

Penicillin sulfoxides can be epimerized by heat to afford thermal equilibrium mixtures of a- and /3-sulfoxides, the position of the equilibrium depending on the C(6) side chain (Scheme 5). Deuterium incorporation studies support a sulfenic acid, e.g. (18), as the intermediate in these transformations. This mechanism is also supported by the finding that when an a-sulfoxide epimerizes to a /3-sulfoxide there is a simultaneous epimerization at C(2) (71JCS(C)3540). With irradiation by UV light it is possible to convert a more thermodynamically stable /3-sulfoxide to the a-sulfoxide (69JA1530). 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

Further evidence to support this mechanism was the fact that solutions of alcohol and hydrogen chloride of varying initial relative concentrations, gave, after five minutes, equilibrium mixtures of identical relative concentrations. 

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