Steps irreversible

Processes of tliis type have been analysed [84, 851 by adding an irreversible step, eitlier in parallel ... 

For tliis model tire parameter set p consists of tire rate constants and tire constant pool chemical concentrations l A, 1 (Most chemical rate laws are constmcted phenomenologically and often have cubic or otlier nonlinearities and irreversible steps. Such rate laws are reductions of tire full underlying reaction mechanism.)... 

If a sequence of reaction steps consists only of irreversible steps, then all forward rates must be equal. When this occurs, the intermediates or active centers concentrations will adjust themselves to achieve this. The reaction that consumes the active center or intermediate of the highest concentration is the rate limiting step. Even in this case all rates must be equal. One should be cautious when speaking about the slowest rate perhaps the smallest rate constant would be somewhat better. 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

The irreversible step is irrelevant to the following argument, which is based on the equilibrium state. Proceeding to define equilibrium constants as Ki = k,/k i = [IH]/[ImH][L] and so on, we obtain the identity... 

Compared with uncatalyzed reactions, catalysts introduce alternative pathways that, in nearly all cases, involve two nr more consecutive reaction steps. Each of these steps has a lower activation energy than does the uncatalyzed reaction. We can nse as an example the gas phase reaction of ozone and oxygen atoms. In the homogeneons uncatalyzed case, the reaction is represented to occur in a single irreversible step that has a high activation energy ... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

FIGURE 2-6 Cyclic voltammograms for a reversible electron transfer followed by an irreversible step for various ratios of chemical rate constant to scan rate, k/a, where a = nFv/RT. (Reproduced with permission from reference 1.)... 

In the classical world (and biochemistry textbooks), transition state theory has been used extensively to model enzyme catalysis. The basic premise of transition state theory is that the reaction converting reactants (e.g. A-H + B) to products (e.g. A + B-H) is treated as a two-step reaction over a static potential energy barrier (Figure 2.1). In Figure 2.1, [A - H B] is the transition state, which can interconvert reversibly with the reactants (A-H-l-B). However, formation of the products (A + B-H) from the transition state is an irreversible step. 

Step 1 represents adsorption of ammonia and step 2 its activation. The irreversible step 3 is obviously not elementary in nature, but unfortunately much information on the level of elementary steps is not available. Step 4 describes water formation and step 5 is the reoxidation of the site. Step 6 describes the blocking of sites by adsorption of water. The model thus relies on partially oxidized sites and vacancies on an oxide, similarly to the hydrodesulfurization reaction described in Chapter 9. The reactions are summarized in the cyclic scheme of Fig. 10.15. 

Depending on pH, increasing the acidity of the solution either makes the potential required to yield a fixed turnover frequency more oxidizing by 60 mV/pH or does not affect it. This pH dependence is in most cases the same as that of the Fe /n (.Q jpjg jjj jjjg absence of a substrate. These identical pH dependences suggest a pre-equilibrium between the ferric and ferrous forms of the catalyst followed, by a kinetically irreversible step that does not involve proton or electron transfer (e.g., O2 binding). 

If the forward reaction rate is significantly faster than the reverse reaction rate, the reaction can be considered to be irreversible. For the coordination step in Figure 3.1, the ratio of ki /k.i can be very high (3) hence, the assumption of an irreversible step is reasonable. 

After the precatalyst is completely converted to the active catalyst Xq, three steps are required to form the desired reduction product. The first step is the coordination of dehydroamino acid (A) to the rhodium atom forming adducts (Xi) and (Xi ) through C=C as well as the protecting group carbonyl. The next step is the oxidative addition of hydrogen to form the intermediate (X2). The insertion of solvent (B) is the third step, removing the product (P) from X2 and regenerating Xq. Hence, the establishment of the kinetic model involves these three irreversible steps. 

The hydrogenation of cottonseed oil proceeds by the following irreversible steps. 

Type I MCRs are usually reactions of amines, carbonyl compounds, and weak acids. Since all steps of the reaction are in equilibrium, the products are generally obtained in low purity and low yields. However, if one of the substrates is a bi-funchonal compound the primarily formed products can subsequently be transformed into, for example, heterocycles in an irreversible manner (type II MCRs). Because of this final irreversible step, the equilibrium is forced towards the product side. Such MCRs often give pure products in almost quantitative yields. Similarly, in MCRs employing isocyanides there is also an irreversible step, as the carbon of the isocyanide moiety is formally oxidized to CIV. In the case of type III MCRs, only a few examples are known in preparative organic chemistry, whereas in Nature the majority of biochemical compounds are formed by such transformations [3]. 

Mechanism for Gluconeogenesis. Since the glycolysis involves three energetically irreversible steps at the pyruvate kinase, phosphofructokinase, and hexokinase levels, the production of glucose from simple noncarbohydrate materials, for example, pyruvate or lactate, by a reversal of glycolysis ( from bottom upwards ) is impossible. Therefore, indirect reaction routes are to be sought for. 

Figure 4.8 Coupling with irreversible step synthesis of D-amino acids from racemic mixture applying a batch process coupling the reaction to the formation of the gaseous side-product C02...

Thermodynamic limitation coupling with irreversible step formation of gaseous side-product azeotropic distillation... 

Irreversible adsorption The LH mechanisms assume that the adsorption of all gas-phase species is in equilibrium. Some mechanisms, however, occur by irreversible steps. In these cases, the intermediates are again treated in the same manner as reactive intermediates in homogeneous mechanisms. An example is the Mars-van Krevelen (1954) mechanism for oxidation, illustrated by the following two steps ... 

Pyridine will be released from B in an irreversible step,... 

Note that in the mechanistic schemes presented, the dissolution steps of reactant and products have been omitted for the sake of brevity. These include, for example CO (g) <-> CO (1), C02 (1) <-> C02 (g), and H2 (1) <-> H2 (g). From the standpoint of thermodynamics, when the equilibrium lies far to the right, reactions are deemed to be irreversible and may be denoted with a forward arrow - symbol. In cases where the reaction is considered to be reversible (i.e., equilibrium lies somewhere in the middle), the forward and backward arrows (e.g., <-> ) are employed. In some cases, however, we do not specify reversible/irreversible steps, and therefore arrows (e.g., or <-> ) might be used in a general sense. From a kinetic standpoint, in some cases a step will be defined that is considerably slower than the others (i.e., the rate determining step) in those cases, the remaining steps may be considered to be pseudo-equilibrated. The reader must therefore use discretion in interpreting the mechanistic schemes. The context of the discussion should clue the reader into how to interpret the arrows. 

It is now well understood that fibril formation requires conformational changes, but the assembly steps may differ from one system to another (Kelly, 1998). For example, aggregation into well-ordered structures occurs in multiple steps during the formation of /Mactoglobulin fibrils. First, there is a fast and reversible step followed by an irreversible step involving the formation of nonreversible /1-sheet structures (Arnaudov et al., 2003). Interestingly, the reversible step, which corresponds to a lag in fibril formation, varies from one system to another and most likely depends on the specific kinetic partitioning between the misfolded intermediate and the native state (Dobson, 1999 Jaenicke, 1995 Uversky, 2003). 

In a recent paper a detailed mechanistic study of this reaction was presented (108). The first step is the reversible binding of oxygen by forming a u-peroxo species. This intermediate reacts further via an irreversible step to the hydroxylated product. The kinetic measurements at high pressure were performed at -20°C, since at room temperature no peroxo intermediate can be observed. The forward reaction of the Cu(I) complex with oxygen is characterized by a strongly... 

The rearrangements 67 —> 70, 71 —> 72 and 74 —> 75 include the transformation of conjugated dienes to cumulenes. Nevertheless, these reactions take place with very high yields in some cases, because either an irreversible step of hydrolysis such as 69 —> 70 is involved or the very exothermic transformation from cyanates to isocyanates is used. Comparison of the energies, calculated by ab initio methods [121], shows that, for example, the energy of methyl isocyanate is lower than that of methyl cyanate by 26.8 kcal mol-1 and that of vinyl isocyanate is lower than that of vinyl cyanate by 28.1 kcal mol-1. 

No current ratio /pr//pf exists. In this connection, it must be taken into account that the lack of any reverse response is not sufficient to diagnose an electrochemically irreversible step. As we will see below, the presence of chemical reactions involving the electrogenerated species can make the reverse response disappear. 

Figure 45 shows that in dichloromethane solution, this complex displays both a reversible oxidation process (Eq/+ = 1.14 V vs. Fc/Fc+) and a reversible reduction Eq, = -0.25 V) (in turn followed by a further irreversible step), according to the sequence 62... 

The enantioselectivity is determined in an irreversible step after the chiral atom has been formed. Deuteration experiments have shown that styrene... 

The subscript on kcat in Equation 11.19 abbreviates catalyzed . Vmax is connected with the rate determining step. For desorption much faster than catalysis, ks >> k3, Vmax = k3 [E]0 which is the result found for the simpler Michaelis-Menten mechanism, Section 11.2.1. If, however, ks is commensurate with k3 the intrinsic catalysis is damped by the weighting function ks/(k3+ks). Note that Vmax/KM sees events through the first irreversible step as illustrated in Fig. 11.3. The same is true for the isotope effects. These points are discussed in considerable detail by Northrop (see reading list). 

E = Faraday constant). The equilibrium potential E is dependent on the temperature and on the concentrations (activities) of the oxidized and reduced species of the reactants according to the Nemst equation (see Chapter 1). In practice, electroorganic conversions mostly are not simple reversible reactions. Often, they will include, for example, energy-rich intermediates, complicated reaction mechanisms, and irreversible steps. In this case, it is difficult to define E and it has only poor practical relevance. Then, a suitable value of the redox potential is used as a base for the design of an electroorganic synthesis. It can be estimated from measurements of the peak potential in cyclovoltammetry or of the half-wave potential in polarography (see Chapter 1). Usually, a common RE such as the calomel electrode is applied (see Sect. 2.5.1.6.1). Numerous literature data are available, for example, in [5b, 8, 9]. 

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