Surfaces relative free energy

Wang, C. X. Liu, H. Y. Shi, Y. Y. Huang, F. H., Calculations of relative free energy surfaces in configuration space using an integration method, Chem. Phys. Lett. 1991, 179, 475 178... 

Jarque C, Tidor B (1997) Simulated annealing on coupled free energy surfaces relative solvation energies of small molecules, J Phys Chem B, 101 9362-9374... 

In general terms the relations between possible compound phases in any system are determined by the usual tangent relation between their free energy surfaces. The free energy of any phase is a function of the temperature, activity of the components, number of lattice sites and relative numbers of atoms of each kind in the crystal, concentrations of vacancies, interstitials and substitutions of each kind, concentrations of associated defects, energies of lattice disorder, of defect interactions, of valence change, of ionization, etc. ... 

The potential energy surfaces (or free energy surfaces) governing the thermal behavior of such relatively small molecules, one may conjecture, might well be amenable to complete and reliable assessments by modern computational methods, and this possibility has been pursued tenaciously for nearly 30 years. On the experimental side there have been equally determined efforts to define as completely as possible the relative importance of conceptually distinct and possibly distinguishable mechanisms for these reactions. Numerous review articles on the thermal chemistry of cyclopropanes " and of vinylcyclopropanes provide convincing testimony to the significance this topic has been accorded by chemists in recent decades. 

These equations are the basis for relating the contact surface free energy to the gaseous environment. As shown in [21] the following relation between reduction potential and the relative surface contact free energy can be derived ... 

Almost two decades previous to the Doering papers a reasonable model for substituent rate effects was proposed that was based on a geometric model for the MOE-J energy surface for the 3,3-shift. Thus, a hyperbolic paraboloid surface equation could be differentiated to obtain coordinates and the activation free energy for the saddle point (the transition state) cast in terms of the relative free energies for formation of the diyl and the two allyl radicals, the same independent variables of Eqs. (7.1) and (7.2). Equation 7.3, which relates the independent variables by the harmonic mean is based on the simplest hyperbolic paraboloid surface, that is, one with linear edge potentials. Slightly more realistic models were also explored. 

Meier calculated the contributions to the free energy, from the uniaxial expansion factors (a s) [see equation (4.10)], the surface free energy, and the constraint free energy. The latter term arises from the constraints that keep the A and B segments confined to restricted volumes. The relative free energies for the appearance of spheres, cylinders, and lamellar structures are shown... 

Equation (89) defines the change of the relative free energy of the surface, Af.(P), in the pressure domain P-P - Equation (89) is thermodynamically correct if, in the pressure domain P-Pm, the ideal-gas law is applicable and the supposition tf is vahd. The applicability of Eq. (89) may be extended if instead of pressures, the fugacities are apphed (i.e., the limits of integration are / and, corresponding to pressures P and P, respectively). This extension of Eq. (89) is supported by the fact that the supposition if in most cases is still valid when instead of the ideal-gas state equation the relationship (56) should be apphed. 

Because %l > change in relative free energy of the surface always has a finite value. It is evident that if the integration in Eq. (100) is performed between the limits zero and a finite value of 0, then we have... 

In previous subsections, the thermodynamic properties, especially the change in relative free energy of the surface, have been discussed. However, the calculation of the functions Af.( ) and r( r,m) requfre integration according to Eq. (100) therefore, it needs time. This time is necessary when we are interested in the thermodynamic properties of the adsorbed phase. Nevertheless, after measuring an isotherm, the following question should be answered immediately Which isotherm equation can be applied to describe and explain the measured data This problem may be solved by calculation the function i//(P) defined by relationship (90) ... 

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