Bottleneck equilibrium

Kinetic data provide information only about the rate-determining step and steps preceding it. In the hypothetical reaction under consideration, the final step follows the rate-determining step, and because its rate will not affect the rate of the overall reaction, will not appear in the overall rate expression. The rate of the overall reaction is governed by the second step, which is the bottleneck in the process. The rate of this step is equal to A2 multiplied by the molar concentration of intermediate C, which may not be directly measurable. It is therefore necessary to express the rate in terms of the concentrations of reactants. In the case under consideration, this can be done by recognizing that [C] is related to [A] and [B] by an equilibrium constant ... 

Three characteristics of the MRD profile change when the protein is hydrated with either H2O or D2O. Both terms of Eq. (6) are required to provide an accurate fit to the data. The second or perpendicular term dominates once the transverse modes become important. The power law for the MRD profile is retained, but the exponent takes values between 0.78 and 0.5 depending on the degree of hydration. A low frequency plateau is apparent for samples containing H2O which derives from two sources the field limitation of the local proton dipolar field as mentioned above, and from limitations in the magnetization transfer rates that may be a bottleneck in bringing the liquid spins into equilibrium with the solid spins. 

Figure 9.12 contains sketches for several different models of pores that will be useful in our discussion of capillary condensation. Figure 9.12a is the simplest, attributing the entire effect just described to variations in pore radius with the depth of the pore. That is, when liquid first begins to condense in the pore, the larger radius Ra determines the pressure at which the adsorption-condensation occurs. Once the pore has been filled and the desorption-evaporation branch is being studied, the smaller radius Rd determines the equilibrium pressure. Although bottlenecked pores of this sort may exist in some cases, this model seems far too specialized to account for the widespread occurrence of hysteresis. 

Figure 19.7 Transfer across a two-layer bottleneck boundary between two phases. The situation is analogous to Fig. 19.6 except for the fact that the equilibrium condition between the two layers is now expressed by the relation KB/A = (CB/A / CA/B),.q. The dashed line in zone A gives the concentration in zone B expressed as the cor-responding A-phase equilibrium concentration.

The unsaturated zone can be modeled as a bottleneck boundary of thickness 8 = 4 m. The TCE concentration at the lower end of the boundary layer is given by the equilibrium with the aquifer and at the upper end by the atmospheric concentration of TCE, which is approximately zero. Thus, you need to calculate the nondimensional Henry coefficient of TCE at 10°C, KTCB a/w(10°C). 

In these equations we recognize expressions which by now should have become familiar to us. During the initial phase of the exchange process (t Zcm), boundary concentration and flux at the interface remind us of a (B-side controlled) bottleneck boundary with transfer velocity vbl = Db,/S (see Eq. 19-19). The concentrations on either side are C and CBq=CA/FA/B, where the latter is the B-side concentration in equilibrium with the initial A-side concentration CA. 

By marrying molecular dynamics to transition state theory, these questionable assumptions can be dispensed with, and one can simulate a relaxation process involving bottlenecks rigorously, assuming only 1) classical mechanics, and 2) local equilibrium within the reactant and product zones separately. For simplicity we will first treat a situation in which there is only one bottleneck, whose location is known. Later, we will consider processes involving many bottlenecks, and will discuss computer-assisted heuristic methods for finding bottlenecks when their locations are not known a priori. 

First let us assume that the system has been undisturbed for so long that it is in a macrostate of thermal equilibrium. Trajectories will then pass through the bottleneck region equally often from left to right and from right to left, and the probabilities of different microstates in the bottleneck region, as in any part of phase space, will be given by the formulas of equilibrium statistical mechanics (e.g. the equilibrium microcanonical density,... 

The lack of equilibrium between reactant and product zones leads to a distinctly nonequilibrium distribution in the bottleneck, but fortunately it is one that can be expressed easily (11) in terms of the equilibrium distribution and trajectory information. To do this, the equilibrium probability density Peq(p,q) is split into two nonoverlapping parts, Pa(p,g) and Pc(p,c[), the former originating from an equilibrium distribution in A, the latter from an equilibrium distribution in C. 

Since every phase point (except for uninteresting ones accessible from neither A nor C) satisfies one of the two trajectory conditions above and no phase point satisfies both, the two terms add up to the equilibrium density on the other hand, each term separately represents the situation in which an equilibrium distribution of trajectories attacks the bottleneck from one... 

Definition of Critical and Rate-Limiting Bottlenecks" The hypothesis of local equilibrium within the reservoirs means that the set of transitions from reservoir to reservoir can be described as a Markov process without memory, with the transition probabilities given by eq. 4. Assuming the canonical ensemble and microscopic reversibility, the rate constant Wji, for transitions from reservoir i to reservoir j can be written... 

Suppose one is interested (as Torrie and Valleau were) in the equilibrium probability of an r value, say r=30, outside the observed range alternatively, one may suspect that p(r), the true equilibrium distribution of r, is bimodal, with another peak around r=40, but that a bottleneck around r=30 is preventing this peak from being populated. 

If there is a bottleneck at r=30, the system is much more likely to find it and suddenly leak through if not, one has a least measured the equilibrium distribution of r in a region where it would be too low to measure directly. The normalizing factor Q/Q, necessary to make the connection between p and p, can found be from the histograms via eq. 20 or, more accurately, by eqs. 12a and 12b of reference 17. 

An example of this phase-coexistence is shown in Fig. 13. We will call this macroscopic phase boundary a bottleneck due to its shape. In fact, this coexistence includes three phases, i.e. swollen gel, shrunken gel and pure solvent phases surrounding the gel, and has been called triphasic equilibrium in the... 

Transition-state theory is based on two assumptions, the existence of both a dynamic bottleneck and a preceding equilibrium between a transition-state complex and reactants. Eq. (2.4) results, where k denotes the observed reaction rate constant, k the transmission coefficient, and v the mean frequency of crossing the barrier. 

How does the solvent influence a chemical reaction rate There are three ways [1,2]. The first is by affecting the attainment of equilibrium in the phase space (space of coordinates and momenta of all the atoms) or quantum state space of reactants. The second is by affecting the probability that reactants with a given distribution in phase space or quantum state space will reach the dynamical bottleneck of a chemical reaction, which is the variational transition state. The third is by affecting the probability that a system, having reached the dynamical bottleneck, will proceed to products. We will consider these three factors next. 

A new issue arises when one makes a solute-solvent separation. If the solvent enters the theory only in that V(R) is replaced by TT(R), the treatment is called equilibrium solvation. In such a treatment only the coordinates in the set R can enter into the definition of the transition state. This limits the quality of the dynamical bottleneck that one can define depending on the system, this limitation may cause small quantitative errors or larger more qualitative ones, even possibly missing the most essential part of a reaction coordinate (in a solvent-driven reaction). Going beyond the equilibrium solvation approximation is called nonequilibrium solvation or solvent friction [4,26-28], This is discussed further in Section 3.3.2. 

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