Fast equilibrium approximation

Neglecting termination reactions due to persistent radical effect" " and using a fast equilibrium approximation, the radical concentration ([R ]) in the system... 

The pre-equilibrium approximation (PEA also called the partial equilibrium approximation or fast-equilibrium approximation) is applicable when the species participating in a pair of fast-equilibrium reactions are consumed by slow reactions. After the onset of an equilibrium, the rates of the, forward and backward reactions become equal to each other, and therefore the ratios of the concentrations of the participating species can be calculated from the stoichiometry of the reaction steps and the equilibrium constant. According to the pre-equilibrium approximation, if the rates of the equilibrium reactions are much higher than the rates of the other reactions consuming the species participating in the equilibrium reactions, then the concentrations of these species are determined, with good approximation, by the equilibrium reactions only. 

If all steps except one are fast, we can use the quasi-equilibrium approximation For the fast steps we use the corresponding equilibrium equations instead of the kinetic equations. 

The chemical equilibrium assumption often results in modeling predictions similar to those obtained assuming infinitely fast reaction, at least for overall aspects of practical systems such as combustion. However, the increased computational complexity of the chemical equilibrium approach is often justified, since the restrictions that the equilibrium constraint places on the reaction system are accounted for. The fractional conversion of reactants to products at chemical equilibrium typically depends strongly on temperature. For an exothermic reaction system, complete conversion to products is favored thermodynamically at low temperatures, while at high temperatures the equilibrium may shift toward reactants. The restrictions that equilibrium place on the reaction system are obviously not accounted for by the fast chemistry approximation. 

In Section 2.2 we mentioned the impossibility to strictly substantiate the equilibrium descriptions for all cases of life and the need to apply equilibrium approximations in some situations. The vivid examples of the cases, where the strongly nonequilibrium distributions of microscopic variables are established in the studied system and the principal difficulties of its description with the help of intensive macroscopic parameters occur, are fast changes in the states at explosions, hydraulic shocks, short circuits in electric circuits, maintenance of different potentials (chemical, electric, gravity, temperature pressure, etc.) in some spatial regions or components of physicochemical composition interaction with nonequilibrium and sharply nonstationary state environment. 

The problem can be handled using either the equilibrium approximation on the steady state approximation. Experiment shows, however, that true equilibrium is not achieved in the fast step because, the subsequent slow reaction is constantly removing the intermediate enzyme-substrate complex, ES. Generally, the enzyme concentration is far less than the substrate concentration, i.e., [E] [S], so that [ES] [S]. Hence, we can use the steady state approximation for the intermediate, ES. 

In a number of reactions of practical interest, the offending non-simple step is a reversible dissociation of a reactant or intermediate, as in Case V in Table 6.1. Often, such a step is fast compared with the others and thus is at quasi-equilibrium. If so, the quasi-equilibrium approximation (see Section 4.2) can greatly simplify mathematics, in some instances even lead to an explicit rate equation. This has been discussed in Section 5.6. 

This method is called the equilibrium approximation, which assumes that all steps prior to the rate limiting step are in equilibrium. Tire equilibrium approximation requires that the slow step be significantly slower than the fast steps. Since kfc /k is a constant, it is usually replaced by a single constant, inbsm,ri. 

The latter equation includes the effect of an IP3 binding molecule, the IP3 buffer. We assume that binding of the buffer to IP3 is fast compared to other processes. Using a rapid-equilibrium approximation one obtains the first term of Eq. 4.9, where B is the IP3 buffer concentration. The parameter Kplc characterizes the sensitivity of PLC to Ca + and is used to tune the strength of the positive feedback. For the IP3R, we assume that their activation by Ca + and IP3 is fast, whereas the inactivation is slow [13]... 

The reaction order in buta-1,3-diene is close to zero, indicating that the fraction of vacant sites is very low, and at the total consumption of buta-1,3-diene the mole fractions of butenes are not equal to zero. The assumption of equilibrium adsorption of the intermediate compound (but-l-ene) in the case of irreversible butadiene hydrogenation and but-l-ene isomerization and hydrogenation cannot explain the latter observation. Therefore, adsorption/desorption steps for buta-1,3-diene but-l-ene, but-2-ene are thought to be reversible and have an "adsorption-assisted desorption " nature. The desorption of butane step 15 is assumed to be irreversible and fast. For conformational isomerization (step 2) a quasi-equilibrium approximation will be used. 

Kaneko and Oki made studies with a deuterium tracer and found v to be between 1 and 2. Kaneko and Oki and Kaneko et al. also conducted experiments with a 0 tracer, finding an apparent v of approximately 2, for measurements both near and far removed from equilibrium. Oki et al. found that the exchange of 0 between CO2 and H2O was very fast (v approaching infinity) but that exchange between CO and CO2 was not as fast (v approximately 2). Mezaki summarized these and other data (Table 9). 

The Quasi-Surface Equilibrium Approximation If we suppose that the adsorption and desorption processes are fast compared to the surface reaction, we can estimate the surface concentrations from the equilibrium constants. With the Langmuir adsorption isotherm, the following relations result for the simple monomolecular reaction presented in Equation 2.107. 

Assumption of quasiequilibrium or fast equilibrium. If in a sequence of steps both the forward and reverse reactions of some reversible steps are much faster than other reaction steps, the assumption can be made that the forward and reverse reactions of such fast steps occur at approximately equal rates, that is, they are at equilibrium. Typically, this assumption is justified by the fact that the kinetic parameters of these fast steps are much larger than the kinetic parameters of the other, slow steps. For many chemical systems, the assumption of quasiequilibrium is complimentary to the assumption of a rate-limiting step if one step is considered to be rate limiting, other, reversible steps can be assumed to be at equilibrium. 

You should verify for yourself that the three expressions in the first line do combine to give the final expression.) Because step 2 is slow relative to the fast pre-equilibrium, we can make the approximation fc,[01[0 ] [02 [0, or equivalently by canceling the 0, 2fO J [Pg.673]

In this approximation we assume that one elementary step determines the rate while all other steps are sufficiently fast that they can be considered as being in quasi-equilibrium. If we take the surface reaction to AB (step 3, Eq. 134) as the rate-determining step (RDS), we may write the rate equations for steps (1), (2) and (4) as ... 

Alternatively, the synthesis may begin by condensing aniline with the l-chloro-2-carboxy intermediate. Acridone vat dyes of this type have excellent light fastness but only moderate resistance to alkali due to the keto-enol equilibrium. It is interesting that this pentacyclic dye is approximately 30 nm more bathochromic than the closely related tetracyclic 1-amino-2-benzoylanthraquinone. 

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