Rate constant backward

A few more comments are in order. The backward rate constant can be computed from the condition of detailed balance... 

The methylolation step, which usually is performed at high formaldehyde (F) to urea (U) molar ratio (F/U = 1.8 to 2.5), consists of the addition of up to three (four in theory) molecules of the bifunctional formaldehyde to one molecule of urea to give the so-called methylolureas. The types of methylolureas formed and their relative proportions depend on the molar ratio, F/U. Each methylolation step has its own rate constant k, with different values for the forward and the backward reaction. The formation of these methylols mostly depends on the molar ratio, F/U, and tends with higher molar ratios to the formation of higher methylolated species. 

Note the rate constant symbolism denoting the forward (fc,) and backward (/c i) steps.] The differential rate equation is written, according to the law of mass action, as... 

Forward and backward reaction-rate constants, respectively [Eq. (109)]... 

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... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

The law of mass action states that the rate of a reaction is proportional to the product of the concentrations of the reactants. Thus the rate of the forward reaction is proportional to [A][R] = k+i[A][R], where k+ is the association rate constant (with units of M s ). Likewise, the rate of the backward reaction is proportional to [AR] = k i[AR], where k- is the dissociation rate constant (with units of s ). At equilibrium, the rates of the forward and backward reactions will be equal so... 

If the intermediate compound XZ is very unstable, Z cannot serve as a catalyst, while if it is very stable then the reaction stops. The intermediate compound XZ must have the right degree of stability for the catalyst to be effective. It must be borne in mind that the catalyst will accelerate the forward as well as the reverse reactions to the same extent, so that the ratio of the specific rate constants for the forward (kf) the backward (kb) reactions will not be affected. As an example a gaseous reaction between sulfur dioxide and oxygen to yield sulfur trioxide may be considered. The reaction, which can be represented by the equation... 

In these equation all microscopic details of the dynamics are condensed into the forward and backward reaction rate constants k ag and k Sr//. 

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 ... 

If [S] = Km, the Michaelis-Menten equation says that the velocity will be one-half of Vmax. (Try substituting [S] for Km in the Michaelis-Menten equation, and you too can see this directly.) It s really the relationship between Km and [S] that determines where you are along the hyperbola. Like most of the rest of biochemistry, Km is backward. The larger the Km, the weaker the interaction between the enzyme and the substrate. Km is also a collection of rate constants. It may not be equal to the true dissociation constant of the ES complex (i.e., the equilibrium constant for ES E + S). 

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. 

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 ... 

We shall also encounter situations where a chemical reaction influences the concentrations. For a volume reaction of the type j - i, the reaction rate for the production of i is given by k(Cj (if we assume it to be an elementary reaction), where k is the forward reaction rate constant, and the rate of consumption of i by a term where kb is the rate constant of the backward... 

In the catalytic WGS reaction on Rh/Ce02, linear OH groups reacted with CO to produce bidentate formates. In vacuum, 65% of the surface formates decomposed backwardly to H20+C0, and 35% of them decomposed forwardly to H2+C02. When water vapor coexisted, 100% of the formates decomposed forwardly to H2+C02 as shown in Table 8.1. The activation energy for the forward decomposition of the formate decreased from 56kJ/mol in vacuum to 33 kJ/mol due to the presence of water(D20) vapor. By addition of a small amount of Rh (0.2 wt%) to Ce02, the rate of the WGS reaction increased tremendously, and the value of the forward decomposition rate constant (k ) was promoted about 100-fold by the coexistence of gas-phase water (Table 8.1). 

Table 8.1 Rate Constants for Forward (Ar+) and Backward (k ) Decompositions of the D-Labeled Formates in Vacuum and Under Ambient D2Q...

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 forward and backward activation free energies and the corresponding rate constants thus depend on an extrinsic factor, the standard free energy of the reaction, AG° = E — E°, and an intrinsic factor, the standard activation free energy, that reflects the solvent and internal reorganization energy, Aq and A [equation (1.31)]. 

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) ... 

One of the most important consequences of taking all electrode electronic states into account is the disappearance of the inverted region that is predicted by the simplified treatment. Equation (1.32) indeed entails that the forward rate constant should increase as E = EP becomes more and more negative, reach its maximal value for E — E° = —1, and decrease further on (Figure 1.16a). Similarly, the backward rate constant should increase as = ° becomes more and more positive, reach its maximal value for E — = 1, and decrease... 

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. 

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