Equilibrium fluxes

Assuming a thennal one-dimensional velocity (Maxwell-Boltzmaim) distribution with average velocity /2k iT/rr/tthe reaction rate is given by the equilibrium flux if (1) the flux from the product side is neglected and (2) the thennal equilibrium is retamed tliroughout the reaction ... 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

In Chapter 5, attention is directed toward the direct calculation of k(T), i.e., a method that bypasses the detailed state-to-state reaction cross-sections. In this approach the rate constant is calculated from the reactive flux of population across a dividing surface on the potential energy surface, an approach that also prepares for subsequent applications to condensed-phase reaction dynamics. In Chapter 6, we continue with the direct calculation of k(T) and the whole chapter is devoted to the approximate but very important approach of transition-state theory. The underlying assumptions of this theory imply that rate constants can be obtained from a stationary equilibrium flux without any explicit consideration of the reaction dynamics. 

Equilibrium flux and kinetics of membrane can be calculated by equations (1) and (2). B,... 

The equilibrium flux at / is related to that at Rj by the Boltzmann factor... 

Consider a system of particles moving in a box at thermal equilibrium, under their mutual interactions. In the absence of any external forces the system will be homogenous, characterized by the equilibrium particle density. From the Maxwell velocity distribution for the particles, we can easily calculate the equilibrium flux in any direction inside the box, say in the positive x direction, Jx = p(vx), where p is the density of particles and dvxVx exp(—fimv /2 ). 

Consider the system at equilibrium. The TST rate is obtained from the equilibrium flux in the outer direction, by dividing it by the well population. It was already argued that this is actually an upper bound to the true rate. One way to see this is to realize that only part of the trajectories that go out of the well are in fact reactive. This is seen in Fig. 14.10 which depicts a single well process as in dissociation or desorption. Obviously, the correction factor a is the fraction of reactive trajectories relative to the total equilibrium flux. 

An easy way to find this correction factor is to look at the history of an exit trajectory. This history is followed by starting atx = xb trajectories with velocity sampled from a Maxwell-Boltzmann distribution in the outward direction—these represent the outgoing equilibrium flux, then inverting the velocity (y -> —v) so that the particle is heading into the well, and integrating the equations of motion... 

This is the total equilibrium flux out of the R region, and if divided by the normalized equilibrium population of R, that is, 1 -/, will give the transition state rate from... 

The tube models a concrete folding mechanism by following the center of the tube and the fluctuations near that center. This concrete mechanism can be examined with respect to the overall funnel picture. For example, if the rate in the tube is exceptionally slow compared to the experimental rate, it is unlikely that the sampled tube is important. Computationally and conceptually, we should partition the funnel to tubes and analyze them one by one. We note that all the tubes meet at the folded state. Therefore the weight of each tube can be measured by overlapping the equilibrium flux into the reactant with the flux from a tube. The calculation of weights of tubes is a topic for future research. 

In the natural systems which are open and in non-equilibrium, flux force or cause-effect relationship controls the entire scenario. Flows and counter flows in the opposite directions dominate the scene. Three cases can arise ... 

The efforts that can be made to restrict the amount of flux decline, i.e. to establish an equilibrium flux rate, can be split into two categories, dependent largely on the scale of operation stirred cells and crossflow filtration. The former is of use in the laboratory for small-scale separations, the latter is more appropriate for process applications. Both stirring and crossflow enq>loy the same princ le hi ear at the sur ce of the deport. Further techniques to restrict deposit thickness, minimise fouling or regenerate flux are discussed in Section. 10.7. All the main membrane arrangements are illustrated in Figure 10.4. 

Figure 10.19 Equilibrium flux with crossflow velocity...

Figures 10.21 and 10.22 stow the correlation of the equilibrium flux rate with wall shear stress for a suspension of talc at 28% by weight, and the deposit thickness predicted by Equation (10.34) as a flmction of transmembrane pressure at three crossflow rates. Further e i erimental details are provided on the figures.

The ratio of equilibrium flux to threshold flux Is low compared to the K Reactors. 

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