Single-Transition System Equilibrium Theory

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

The present discussion of equilibrium theory has been concerned mainly with constant separation factor Langmuir systems and has been restricted to the analysis of the effect of a single step change in feed composition on a previously equilibrated bed. The Langmuir assumption greatly simplifies the analysis since, for such systems, the characteristics are linear and the same for gradual and shock transitions. 

As is implied by the name, a unimolecular reaction is one in which a single molecule of reactant decomposes or rearranges to give rise to product molecules. Ordinary thermal reactions can be modeled by a process which considers the reactant to be in thermal equilibrium with a transition state which then decomposes (rearranges) to give products. One can theoretically describe the process and its isotope effects using transition state theory. For unimolecular reactions, on the other hand, while there is still a transition state, it is not in thermal equilibrium with the reactant except for systems at high pressure. Consequently, a more elaborate theoretical framework is required to understand unimolecular reactions and their isotope effects. 

According to the theory of absolute reaction rates [4-9], the rate is the product of a universal frequency factor and the concentration of the activated complex or transition state, M (the system in the transient state of highest potential energy), crossing the energy barrier in the direction toward the products. The activated complex, in turn, is postulated to be in equilibrium with the reactants. Say, for a single-step reaction A + B — P ... 

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

The present phase transition mechanism interpretations are based on the fluctuation theory [1,2]. The process of phase destmction in a system usually occurs when its stability is lost. Chaotic heat motion of molecules in single-phase solutions or coexisting components of liquid blends leads to a spontaneous change of concentration A on cpA = cp -local domain cp/, where cpav is the average volume concentration. These fluctuations of concentration (heterogeneities) can have relatively broad scale distribution (A/) and root-mean-square amplitude (Acp ) depending on system state parameters. A system is stable if spontaneous concentration fluctuations with any scale and amplitude disappear (they may appear in another area of the system) in the eourse of time. This effect occurs when system is not in binodal. If fluctuations remain when the state parameters change, further evolution of the system to an equilibrium state can lead to phase destruction. 

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