Theoretical equilibrium studies

Phelps, Edmund, and Robert Poliak. 1968. "On Second-Best National Saving and Game-Theoretic Equilibrium Growth." Review of Economic Studies 35 185-99. 

Equilibrium concentrations describe the maximum possible concentration of each compound volatilized in the nosespace. Despite the fact that the process of eating takes place under dynamic conditions, many studies of volatilization of flavor compounds are conducted under closed equilibrium conditions. Theoretical equilibrium volatility is described by Raoulf s law and Henry s law for a description of these laws, refer to a basic thermodynamics text such as McMurry and Fay (1998). Raoult s law does not describe the volatility of flavors in eating systems because it is based upon the volatility of a compound in a pure state. In real systems, a flavor compound is present at a low concentration and does not interact with itself. Henry s law is followed for real solutions of nonelectrolytes at low concentrations, and is more applicable than Raoult s law because aroma compounds are almost always present at very dilute levels (i.e., ppm). Unfortunately, Henry s law does not account for interactions with the solvent, which is common with flavors in real systems. The absence of a predictive model for real flavor release necessitates the use of empirical measurements. 

A theoretical study of ring closure as applied to the cyclic linear distribution in polydimethylsiloxanes assuming a three-rotational-state model resulted in good correlations between experimental and theoretical equilibrium constants (54). 

An NMR and theoretical (DFT) study of the ring-chain-ring isomerization of 3-(2-pyridyl)-[l,2,3]triazolo[l,5- ]pyrid-7-yl derivatives (41) to 6-2-pyridyl derivatives (42) has been carried out.65 The effect of substituents on the (41)-(42) equilibrium is reported. 

These basic thermodynamic considerations show that intermediate reactions in combustion processes can be very advantageous and that in some cases most or all of the chemical energy could be harnessed as mechanical energy at least theoretically. Important questions of reaction kinetics, actual design and applicability of such a device of the selected oxygen carriers have not been included in these fundamental thermodynamic equilibrium studies. 

A theoretical analysis of oxalate-carbonate transformation The objective of this theoretical approach is to try to compare the equilibrium diagrams of the CaCOs—CaC204—CO2—H2O system with the biogeochemical data available on the oxalate-carbonate transformation. The initial hypothesis is that the oxido-reduction reaction of oxalate-carbonate occurs biochemically, i.e. due to bacterial activity. In other words, activation energy is present and high enough to initiate the reactions. The theoretical system studied is defined by three different phases present ... 

The objective of this paper is to demonstrate the importance of phase and reaction equilibria considerations in the rational development of SCF reaction schemes. Theoretical analysis of phase and reaction equilibria are presented for two relatively simple reactions, viz., the isomerizations of n-hexane and 1-hexene. Our simulated conversion and yield plots compare well with experimental results reported in the literature for n-hexane isomerization (4) and obtained by us for 1-hexene isomerization. Based on our analysis, the choice of an appropriate SCF reaction medium for each of these reactions is discussed. Properties such as viscosity, surface tension and polarity can affect transport and kinetic behavior and hence should also be considered for complete evaluation of SCF solvents. These rate effects are not considered in our equilibrium study. 

Table 10.3 provides an example of the cyclic steady-state performance of the SERP concept using a 6 1 H20 + CH4 feed gas at a pressure of 11.4psig and a temperature of 490°C.S1 The process can directly produce an essentially COx-free H2 product (-94.4% H2 + 5.6% CH4), which is suitable for H2 fuel-cell use. The conversion of CH4 to H2 was -73.0%. The table also shows the equilibrium compositions of the H2 product from a conventional plug-flow reforming reactor operating under identical conditions. Both the H2 conversion and product purity were rather poor in the latter case, which demonstrates the advantage of the SERP concept. Theoretical models of the above-described SERP concept and its variations for H2 production by SMR have been developed, and theoretical parametric studies of the process have been conducted by various authors.62,63... 

The experimental equilibrium data were fitted by Langmuir, Freundlich, Langmuir-Freundlich and Ridlich-Peterson models. The equilibrium studies have shown that all proposed equilibrium models indicate a good correlation between the theoretical and experimental data for the whole heavy metals concentration range. 

Figure 7 The dependence of carbon conversion to methanol in a COj/CO/Hj synthesis gas containing 70% Hj and a variable ratio of COj and CO. Experimental points are shown for 250°C(o), 235°C(v), and 225°C(a). Lines drawn through the data points are theoretical curves derived from the model described in text. Theoretical equilibrium methanol conversions for the three temperatures studied are also shown as heavy lines in the upper portions of the figure. The point denotes the yield in COj/Ar/H, = 6/24/70 synthesis gas, expressed in terms of equivalent conversions of (CO + Ar) to methanol. The dotted line represents the rate of CO hydrogenation rco = co (Reproduced, by permission, from ref. 159) ...

Chemical reactions establish the same kind of equifibrium. In chemical equilibriiun a chemical reaction and its reverse are occurring at equal rates. Theoretically, all chemical reactions are reversible—able to take place in either direction and, therefore, to come to equilibrium. Studies of many, many chemical reactions have shown that reactions reach equilibrium at a characteristic and predictable point. Understanding what this is provides an answer to the How far question for a reaction. 

The capillary condensation phenomenon often involves vapor and liquid phases in equilibrium. Hence, vapor-liquid and liquid-liquid equilibrium studies become extremely important and essential not only from a scientific point of view but also for the efficient design and improved operation of various industrial processes (Altwasser et al. 2005 Bhatia 1998 Gupta et al. 1997 Jackson and McKenn 1990 Matranga et al. 1992). Several studies done over the last decade using theoretical approaches such as density functional theory (Evans 1992) and molecular simulation (Evans 1990 Koga and Tanaka 2005 Rivera et al. 2002 Zangi 2004), as well as experimental... 

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