Forward and backward rate

For each step the rate, r = r+ - r- is obviously the difference between a forward rate, r+, and a backward rate, r- 

For each of the gases we have an formation rate. For many applications, it is sufficient to determine the rate of formation for each of the gases. However, if we want to determine the kinetic parameters for the net reaction, the form of the rate expression must be determined. 

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Figure 3.17 illustrates the processes allowed in Point s model, and defines the rate constants to be used. The approach is very similar to that used in Sect. 3.5.2, and we shall use any results derived there which are applicable without repeating the calculation. The first stem can be of any length, l, and the number of such stems in an ensemble is N,. The net current between Nt and Nl+1 is S, which depends on the forward and backward rate constants for a segment, A and B. Subsequent stems are of the same length and the current between the kth and (k + l)lh stem of length / is Jlk and depends on the rate constants for a complete stem, Alk and Blk, and on the number of such stems, Mlk and Mlk+l. The time dependent equations are ... 

This is Point s [51] equation (1), which he derived by simply postulating a net forward rate for folding, C,. We followed Di Marzio and Guttman s [143] derivation because it illustrates the way in which C, is connected both with the microscopic forward and backward rate constants. 

The constants k1 and k 1 are, respectively, the forward and backward rate constants and their ratio can be expressed by the law of mass action as... 

For part (a), assume that the system is at equilibrium and that the law of mass action holds. Use the procedures described in Chapter 1 to derive an expression forpAB, at equilibrium. At equilibrium, the forward and backward rates for each reaction in the mechanism must be equal. The forward and backward rates are defined using the law of mass action ... 

From the standard thermochemical data ArG° = (—371.3 — 379.9 + 733.9) kJ mol-1 = —17.3 kJmol-1, corresponding to an equilibrium constant K = 1.1 x 103 M-1. This is a worrying result because all peptides in solution at 298 K should spontaneously fall apart to the monomers and hence all proteins are subject to degradation due to spontaneous hydrolysis. Fortunately, the reaction is kinetically hindered, which means that it occurs very slowly. Kinetics always control the rate at which equilibrium is achieved, relating the ratio of the forward and backward rate constants to the equilibrium constant ... 

Equilibrium The balance of forward and backward rates of reactions to produce a steady-state concentration of products and reactants. 

Denote the forward and backward rate constants of this reaction by ka and kb- When the reaction proceeds under stationary conditions, the rates of the chemical and of the electron-transfer reaction are equal. Derive the current-potential relationship for this case. Assume that the concentrations of A and of the oxidized species are constant. 

Let step number j be rate determining-, that is, its forward and backward rates are much smaller than those of the other steps ... 

We ignore complications due to transport and assume that the surface concentrations of A and A+ are constant. Let ki and fc i denote the forward and backward rate constants of the adsorption reaction, so that the adsorption rate is given by ... 

The forward and backward rate constants are related to the corresponding activation free energies, AG and AGf, by equation (1.25) below, introducing koo (and kf ) as the maximal rate constants, reached when A Gf or A Gf vanish. The main laws and models describing the way in which the forward and backward rate constants, or the corresponding free energies of activation, vary with the driving force are discussed in Section 1.4.2. 

AGq is the standard activation free energy, also termed the intrinsic barrier, which may be defined as the common value of the forward and backward activation free energies when the driving force is zero (i.e., when the electrode potential equals the standard potential of the A/B couple). Expression of the forward and backward rate constants ensues ... 

The principle of the computation is to use the expressions of the forward and backward rate constant as being those of individual rate constants and sum these individual rate constants over all electronic states weighting the contribution of each state according to the Fermi-Dirac distribution.44 Assuming that H, and the density of states and therefore Kei, are independent of the energy of the electronic states,45 the results are expressed by the following equations (see Section 6.1.8) ... 

If the nonlinear character of the kinetic law is more pronounced, and/or if more data points than merely the peak are to be used, the following approach, illustrated in Figure 1.18, may be used. The current-time curves are first integrated so as to obtain the surface concentrations of the two reactants. The current and the surface concentrations are then combined to derive the forward and backward rate constants as functions of the electrode potential. Following this strategy, the form of the dependence of the rate constants on the potential need not be known a priori. It is rather an outcome of the cyclic voltammetric experiments and of their treatment. There is therefore no compulsory need, as often believed, to use for this purpose electrochemical techniques in which the electrode potential is independent of time, or nearly independent of time, as in potential step chronoamperometry and impedance measurements. This is another illustration of the equivalence of the various electrochemical techniques, provided that they are used in comparable time windows. 

FIGURE 1.24. Potential-dependent forward and backward rate constants of the ferrocene-ferrocenium couple attached to a gold electrode hy a long-chain alkane thiol assembled together with unsubstituted alkane thiols of similar length. Solid line use of Equations (1.37) to (1.39) with X, = 0.85 eV, ks — 1.25 s 1. Adapted from Figure 4A in reference 65, with permission from the American Association for the Advancement of Science. 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

Hersey and Robinson also foundthat many guest species that show kinetic behavior apparently explicable in terms of a single-step binding, give a discrepancy between the values of the equilibrium constant determined kinetically and those determined from equilibrium studies. It was found that the equilibrium constant, deterrmned spectrophotometrically, was usually greater than the ratio of the forward and backward rate-constants, determined kinetically. They therefore suggested that this discrepancy could be adequately explained if the two-step mechanism just described was used to interpret the results. A similar proposal has also been made by Hall and coworkers, who observed a large discrepancy between AV° values for the inclusion of 1-butanol and 1-pentanol by alpha cyclodextrin, calculated from equilibrium-density measurements and kinetic, ultrasonic-absorption data. 

The kinetic model for this mechanism consists of the forward and backward rates. 

Each step has a forward and backward rate constant. This model is a trivial variation of the previous model. The equilibrium constants for the steps are calculated from the forward and backward rate constants. The stabilities of the molecules are not available. 

For a catalyst operating at 673 K, 100 atm and 28 % approach to equilibrium the turnover frequency is 0.029 s . This can be further interpreted by analyzing the forward and backward rate. Each active site turns over 0.031 times each second, 0.030 times in the forward rate and 0.001 times in the backward direction. 

We do not know the forward and backward rate of the fast steps. For this reason we can only estimate the lifetimes of the intermediates. 

Equation (7.7), in which cR is the concentration of an intermediate, may give an erroneous impression that the current-potential relation is completely determined by the exponential term in d< ). However, species R was the result of a series of charge-transfer mechanisms, and thus its concentration, as shown below, is also potential dependent. To unravel this dependence, it will be recalled that all steps preceding and following the rds can often be assumed to be at equilibrium. Then, one can equate their forward and backward rates, e.g., for the first step A + e B,... 

Let X and Y be the molar concentrations of N02 and NO and Xa and Y0 be these concentrations in the inlet stream, respectively. Then, Z-X+Y is the total NOx concentration. Let Ay be the reaction rate constant of the soot-N02 oxidation reaction and k A and kB the forward and backward rate constants for the NO oxidation, respectively. For the case of constant oxygen concentration across the soot layer the forward reaction can be considered of pseudo-first order with a modified rate constant kA=k [02f5. 

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