Continuous thermodynamics, stability theory

The orbital phase theory can be applied to the thermodynamic stability of the disubstituted benzene isomers. The cyclic orbital interaction in the benzene substituted with two EDGs is shown in Scheme 21. The orbital phase is continuous in the meta isomer and discontinuous in the ortho and para isomers (Scheme 22, cf. Scheme 4). 

According to the theory of cyclic conjugation, the Hueckel rule is applicable only to a continuous cyclic conjugation, but not to a discontinuous one (Schemes 14 and 15). In the discontinuously conjugated molecules, electron donors and acceptors are alternately disposed along the cyclic chain [25].The thermodynamic stability depends neither on the number of n electrons nor the orbital phase properties, but on the number of neighboring donor-acceptor pairs. Chemical consequences of the continuity-discontinuity of cyclic conjugation are reviewed briefly here. 

The parent azine systems discussed in this chapter, compounds 1-4, have not been prepared experimentally, but there has been continuing interest in their theoretical analysis. The past 10 years have seen the incorporation of computational chemistry into the mainstream of chemical research, facilitated by the advancement of computer hardware, and computational software and methods. Hence, not surprisingly, recent studies have been performed using advanced methods, such as MP2, CCSD, and a considerable number of density functional theory (DFT) calculations. Azines have been investigated in terms of structural features, aromaticity, kinetic and thermodynamic stability, and decomposition reactions. 

For calculating the spinodal curve and the critical point, there are two possible ways in the framework of continuous thermodynamics. The most general one is the application of the stability theory of continuous thermodynamics [45-47]. The other way is based on a power series expansion of the phase equilibrium conditions at the critical point. Following the second procedure. Sole et al. [48] studied multiple critical points in homopolymer solutions. However, in the case of divariate distribution functions the method by Sole has to be modified as outlined in the text below. 

All differentiations have to be performed by using the chain rule and keeping the quantities of phase I constant. The additional equation necessary for the calculation of the critical point can be derived by use of the stability theory of continuous thermodynamics [45-47,73], The derivation is omitted here because of its complexity. In accordance with the stability theory, the power series expansion method is easier to apply and leads to the special case of Eq. (168) to... 

However, many problems and applications still remain unattacked when this review is prepared. Further progress will take place when multivariate distribution functions become available by experiments of higher accuracy than now. More exact and sophisticated G -models have to be developed for the application of continuous thermodynamics to copolymer systems. New insights into the delicate phase behavior of copolymer systems would be gained by further development of the stability theory of continuous thermodynamics [45-47,75]. The polymer fractionation theory by continuous thermodynamics should be extended from homopolymers [100] to copolymers. In short, much remains to be done in the field of copolymer blends and systems containing block copolymers. 

Upon mixing two immiscible liquids, one of the two liquids (i.e., the dispersed phase) is subdivided into smaller droplets. The surface area and the interfacial free energy increase, and the system is then thermodynamically unstable. Without continuous mixing, the droplets will be stabilized throughout the dispersion medium by dissolving the surface-active agent. There are several theories for the stabilization of emulsions but a single theory cannot account for the stabilization of all emulsions. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

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