Model of extreme intermediate states

Currently the authors are developing three classes of models of extreme intermediate states (MEIS) (1) with variable parameters (2) with variable flows, and (3) those describing spatially inhomogeneous systems. All these classes of the models are formulated and analyzed in terms of MP, which, in the authors opinion, can be defined as a mathematical theory of equilibrium states. It is natural to start the analysis of the created modifications with the MEIS with variable parameters, which is the closest in character to the traditional equilibrium thermodynamics models. 

Figure 5 A graphical interpretation of the model of extreme intermediate states.

Though in formulations of MEIS of type (7)-(12) or the particular form (53)-(57) the possibility of projecting the space of thermodynamic variables to a tree is not shown, the knowledge of principal possibility to reduce the set Dt(y) to the tree makes the analysis of capabilities and comparative merits of the model of extreme intermediate states essentially easier, clearer, and more convincing. 

The main idea of the research being described is the refusal to use an equation of trajectory and construction of stepwise methods to analyse processes on the basis of the model of extreme intermediate states (MEIS) that was created by B.M.Kaganovich, S.P.Filippov and E.G. Antsiferov (Antsiferov et al., 1988 Kaganovich, 1991 Kaganovich et al., 1989). The features that make MEIS different from the traditional thermodynamic models are 1) statement of the problem to be solved (instead of search for a sole point of final equilibrium the entire set of thermodynamic attainability Dj(y) from the given initial state y is considered and the states with extreme values of modeled system characteristics of interest to a researcher are found) 2) dual interpretation of the equilibrium notion, i.e. both as a state of rest and as an instant of motion in which the equality of action and counteraction is observed and 3) dual interpretation of dynamic quantities (work t, heat q, rate w, flow of substance x, etc.) both... 

Models of extreme intermediate states and construction of trajectories... 

Relations between the theories of states and trajectories and capabilities of equilibrium thermodynamic analysis to study reversible and irreversible kinetics can be more fully revealed by considering another type of models of extreme intermediate states, namely MEIS of hydraulic circuits (Gorban et al., 2001, 2006 Kaganovich et al., 1997, 2007, 2010). Convenience and clearness of using these models to describe the considered problems are determined by the fact that they are intended to study an essentially irreversible process, i.e. motion of a viscous fluid. Besides, they can be treated as models of the mechanism of fluid transportation from the specified source nodes of a hydraulic system to the specified consumption nodes. The major variable of the hydraulic circuit theory (Khasilev, 1957,1964 Merenkov and Khasilev, 1985), i.e. continuous medium flow, has an obvious kinetic sense. 

However, the made groundwork in studying physical-mathematical properties of the models of extreme intermediate states and in their ap>plication enables one to highly estimate the forthcoming achievements in solution of discussed ptroblems and to prove the necessity for continuation and expansion of the scope of kinetic-thermodynamic studies. 

The structure of the intermediate states in Rh7 and Rh8 has been studied recently by theoretical investigation [42], Regarding the proton translocation model, it should be noted that the excitation photon density was extremely high in the low-temperature picosecond experiments [10,35]. Therefore, the non-Arrhenius dependence of the formation rate of bathorhodopsin on temperature and the deuterium isotope effect may be results which could be detected only under intense excitation conditions. In fact, a deuterium isotope effect was not observed in the process from photorhodopsin to bathorhodopsin under weak excitation conditions [43],... 

Since molecular mechanics cannot be used to calculate the energy of transition states, suitable models were adopted. These models are extremely similar to the Jt-olefin complex with an orientation of the growing chain rather similar to that adopted when a a-agostic interaction is present. They were often called pre-insertion intermediates because the insertion transition state could be reached from these intermediates with a minimal displacement of the reacting atoms. 

In the early 50 s, an ion pair model was introduced by Winstein to rationalize the mechanism and stereochemistry of solvolysis of sulfonates72). This research of carbocationic intermediates and the role of ion solvation equilibrium in reaction mechanisms represents a landmark in the study of charged species. These thermodynamically different ionic species were coined as free ions, contact ion-pairs (c.i.p.), and solvent-separated ion pairs (s.s.i.p.). The ion pair situation can be described as an equilibrium between thermodynamically distinct contact (c.i.p.) and solvent-separated ion pairs (s.s.i.p.) 2-l3 16 The situation should be represented by a continuum of ion-solvation equilibria states in which the two extreme states are the c.i.p. and the s.s.i.p. 2 76) (Eq. 12)... 

Nucleophilic attack of a ground-state olefin on the (CT) excited complex leads to a u-bonded intermediate (24), which with a second ground-state olefin as activating ligand" 221) is converted to (22) and a ground state complex molecule. The inspection of molecular models shows no detectable steric hindrance for a trans-exo approach of nor-bomene towards the exo-coordinated 22) norbomene CuX leading to (24). A cis-approach appears extremely unlikely for steric reasons. The trans-endo approach has to overcome some steric interaction, in agreement therewith only 3% (23) is observed besides (22). 

No low level of / = 0 is predicted by these schemes other than the ground state, but it arises naturally as a dilational state in the alpha particle model. This model (Sect. 4) also predicts the 2 state but requires low lying states of J = y and 1" which have not been identified. The 0" state can also be obtained by double nucleon excitation, e.g. in the configuration pfj p i - The transition probability of the O " state for pair emission to the ground state can be estimated from the cross section for excitation of this state with electrons. Schiff claims that the value obtained is too small to be explained by the alpha particle model and too large to be accounted for by a // coupling model with excitation of two -particles he suggests that a collective model intermediate in properties between the two extreme models is necessary. 

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