Gibbs determinant

Let us specifically consider a two-component system. Because the Gibbs determinant is simply D = d G/dx )pj, the critical condition is... 

In particular, for the three-component blends P1/P2/P3, the Gibbs determinant is... 

Let us consider the Gibbs determinant to study the stability of the homogeneous phase. Using Gibbs-Diihem relation described in Section 2.1, variation of one component among 0, / = 1,2,..., can be expressed by the others. We take the solvent as the reference component, and consider the difference A/j./ — A/u-o- Its derivative by the composition of the -component... 

For systems with polydisperse primary chains, the spinodal and critical conditions have to be determined from the appropriate Gibbs determinants. 

The critical conditions satisfy eq. (8) simultaneously with the second Gibbs determinant J, obtained by substituting any row from eq. (8) by the row vector [9]... 

The significance of these phenomena to thermodynamic studies has been demonstrated many times over the past decade, -with especial importance attaching to the two loci which can be expressed in closed form by equating to zero the Gibbs determinants, viz ... 

The critical point is not in general located at the lowest concentration Cmin of the spinodal line (this is true only in the symmetric case Na = NB hAA-hsa)- It is located at the point where the spinodal and the binodal line (which gives the two phases at coexistence) are tangent. Mathematically, this can be expressed as the point where a Gibbs determinant vanishes. Explicitly, writing down this condition and... 

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

Smith [113] studied the adsorption of n-pentane on mercury, determining both the surface tension change and the ellipsometric film thickness as a function of the equilibrium pentane pressure. F could then be calculated from the Gibbs equation in the form of Eq. ni-106, and from t. The agreement was excellent. Ellipsometry has also been used to determine the surface compositions of solutions [114,115], as well polymer adsorption at the solution-air interface [116]. 

Panagiotopoulos A Z 1992 Direot determination of fluid phase equilibria by simulation in the Gibbs ensemble a review Mol. SImul. 9 1 -23... 

The heat capacity of thiazole was determined by adiabatic calorimetry from 5 to 340 K by Goursot and Westrum (295,296). A glass-type transition occurs between 145 and 175°K. Melting occurs at 239.53°K (-33-62°C) with an enthalpy increment of 2292 cal mole and an entropy increment of 9-57 cal mole °K . Table 1-44 summarizes the variations as a function of temperature of the most important thermodynamic properties of thiazole molar heat capacity Cp, standard entropy S°, and Gibbs function - G°-H" )IT. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

Because experimental measurements are subject to systematic error, sets of values of In y and In yg determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. 

The standard Gibbs-energy change of reaction AG° is used in the calculation of equilibrium compositions. The standard heat of reaclion AH° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating AH° and AG° with T. Numerical values for AH° and AG° are computed from tabulated formation data, and AC° is determined from empirical expressions for the T dependence of the C° (see, e.g., Eq. [4-142]). 

Complex Clieinical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy G in accord with Eq. (4-271). The treatment here is... 

The preferred erystallographie strueture is determined by the Gibbs free energy, whieh is defined as... 

The change in Gibbs free energy for the reaction is determined from Equation 2 and the Gibbs free energy of formation for the products and reactants ... 

Determination of the equilibrium spreading pressure generally requires measurement and integration of the adsorption isotherm for the adhesive vapors on the adherend from zero coverage to saturation, in accord with the Gibbs adsorption equation [20] ... 

The simple approach described before involves approximations, particularly to obtain the stagnation pressure loss. The full determination of pf) m (7 o)5m from the various equations given above can lead to an approximation for the downstream entropy (.vsni). using the Gibbs relation applied between stagnation states. 

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