The Forward Rate Constant

From the kinetic theory of gases the number of collisions occurring between molecular entities A and B in a gas per unit time and unit volume of the gas is given by (3.6), with slightly different notation 

The /-mer mass is equal to i times the monomer mass (m, = imi). Assuming the monomer and i-mer densities are equal, v, = iv. Thus 

The number of collisions occurring between monomer and a single z -mer per unit time (the forward rate constant) is 

The expression for p, is sometimes multiplied by an accommodation coefficient that is the fraction of monomer molecules impinging on a cluster that stick. The accommodation coefficient is generally an unknown function of cluster size. Its effect on the nucleation rate is relatively small since, as will be shown later, the nucleation rate depends on the ratio of forward to reverse rates and in this ratio the accommodation coefficient cancels. Henceforth, we will assume the accommodation coefficient is unity. 

The saturation ratio, S = p /p, where p is the same aspsA in (11.1). Replacing p by p in (11.19) gives the forward rate constant at saturation, pj. Then, the forward rate constant under nucleation conditions (S 1) can be written as 

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As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

Substitution of this for the golden-rule expression (1.14) together with the renormalized tunneling matrix element from (5.60) gives (5.64), after thermally averaging over the initial energies E-,. In the biased case the expression for the forward rate constant is... 

According to Eq. (3-7), a plot of In [A], - [AL will be linear. The plot has, as the negative of its slope, the sum k + k-. The implication that this data treatment yields a sum is at first surprising, because this rate constant characteristic of the equilibration is clearly larger than the forward rate constant alone. The net rate itself, on the other hand, is smaller than the forward rate, since the reverse rate is subtracted from it, as in Eq. (3-2). These statements are not contradictory, and they illustrate the need to distinguish between a rate and a rate constant. 

Both formulations give the correct equilibrium condition. Clearly, however, this is a special case. In nearly all real examples the reverse rate law and rate constant can be deduced correctly from the forward rate constant and the equilibria condition. To illustrate this characteristic, consider a two-step reaction and the expressions for the rates ... 

FIGURE 13.21 The equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse reactions, (a) A forward rate constant (A) that is relatively large compared with the reverse rate constant means that the forward rate matches the reverse rate when the reaction has neared completion and the concentration of reactants is low. (b) Conversely, if the reverse rate constant (A ) is larger than the forward rate constant, then the forward and reverse rates are equal when little reaction has taken place and the concentration of products is low. 

Kontturi et al. studied TEA ion transfer across water-1,2-DCE microinterfaces covered by different PCs using short potential step techniques [12]. The enhancement in the forward rate constant was observed for all lipids and increased with the surface coverage (Fig. 6). 

FIG. 5 Enhancement factor observed in the forward rate constant for TMA ( ) and TEA ( ) ion transfer at the water-nitrobenzene interface due to the presence of different PCs. (Experimental data are taken from Ref. 11 and correspond to 30°C.)... 

The complete kinetic expression in eq 16 relates the experimental rate constant ke with the forward rate constant ki, as a direct function of the decomposition rate constant k2 and the standard reduction potential E °. Since an independent measurement of k2f = 1.2 cm s"1 (or k2 = 105 s"1) is available for Me2Co(M)+, it can be used in conjunction with E° = 0.53 V to convert ke to ki, shown in Figure 11 (14). 

Kc, the conventional equilibrium constant, is equal to the ratio of the forward rate constant divided by the... 

Denoting the forward rate constant for the Tafel reaction by k2 and that for the back reaction by fc 2, we can write the current density in the form ... 

Here, i is the faradaic current, n is the number of electrons transferred per molecule, F is the Faraday constant, A is the electrode surface area, k is the rate constant, and Cr is the bulk concentration of the reactant in units of mol cm-3. In general, the rate constant depends on the applied potential, and an important parameter is ke, the standard rate constant (more typically designated as k°), which is the forward rate constant when the applied potential equals the formal potential. Since there is zero driving force at the formal potential, the standard rate constant is analogous to the self-exchange rate constant of a homogeneous electron-transfer reaction. 

Worked Example 8.18 Consider the reaction between pyridine and heptyl bromide, to make 1-heptylpyridinium bromide. It is an equilibrium reaction with an equilibrium constant K = 40. What is the rate constant of back reaction k if the value of the forward rate constant k = 2.4 x 103 dm3 mol 1 s ... 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

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... 

The free energy of activation or the forward rate constant may thus be obtained as a function of Ep for each scan rate. The nonlinear character of the rate law, if any, will then become apparent in the way in which AG p varies with the peak potential, which provides a point-by-point description of the activation-driving force relationship (one point per scan rate). The nonlinear character of the rate law will also transpire in the variation of ap, derived from ctp = 2A2 1ZT/F)(Ep/2 — Ep) with the scan rate. 

If a single reaction order must be selected, an examination of the 95 % confidence intervals (not shown) indicates that the two-thirds order is a reasonable choice. For this order, however, estimates of the forward rate constants deviate somewhat from an Arrhenius relationship. Finally, some trend of the residuals (Section IV) of the transformed dependent variable with time exists for this reaction order. 

The idea that an activated complex or transition state controls the progress of a chemical reaction between the reactant state and the product state goes back to the study of the inversion of sucrose by S. Arrhenius, who found that the temperature dependence of the rate of reaction could be expressed as k = A exp (—AE /RT), a form now referred to as the Arrhenius equation. In the Arrhenius equation k is the forward rate constant, AE is an energy parameter, and A is a constant specific to the particular reaction under study. Arrhenius postulated thermal equilibrium between inert and active molecules and reasoned that only active molecules (i.e. those of energy Eo + AE ) could react. For the full development of the theory which is only sketched here, the reader is referred to the classic work by Glasstone, Laidler and Eyring cited at the end of this chapter. It was Eyring who carried out many of the... 

CHEMRev The Comparison of Detailed Chemical Kinetic Mechanisms Forward Versus Reverse Rates with CHEMRev, Rolland, S. and Simmie, J. M. Int. J. Chem. Kinet. 37(3), 119-125 (2005). This program makes use of CHEMKIN input files and computes the reverse rate constant, kit), from the forward rate constant and the equilibrium constant at a specific temperature and the corresponding Arrhenius equation is statistically fitted, either over a user-supplied temperature range or, else over temperatures defined by the range of temperatures in the thermodynamic database for the relevant species. Refer to the website http //www.nuigalway.ie/chem/c3/software.htm for more information. 

MECHMOD A utility program written by Turanyi, T. (Eotvos University, Budapest, Hungary) that manipulates reaction mechanisms to convert rate parameters from one unit to another, to calculate reverse rate parameters from the forward rate constant parameters and thermodynamic data, or to systematically eliminate select species from the mechanism. Thermodynamic data can be printed at the beginning of the mechanism, and the room-temperature heat of formation and entropy data may be modified in the NASA polynomials. MECHMOD requires the usage of either CHEMK1N-TT or CHEMKIN-III software. Details of the software may be obtained at either of two websites http //www.chem.leeds.ac.uk/Combustion/Combustion.html or http //garfield. chem.elte.hu/Combustion/Combustion. html. 

The ratio of rate constants is another constant. The forward rate constant, kf, divided hy the reverse rate constant, is called the equilibrium... 

The thermokinetic method [107,108] uses the measurement of the forward rate constant of the equilibrium... 

Fig. 8 (a) Variations of the forward rate constant, k, with the standard potential difference, — rx/rx -> a function of the rate constant of the follow-up reaction, k (values on each curve in s" ) for typical values of kj (10 m ... 

Note that k+ is the forward rate constant for the nucleation event. If one chooses a particular value of Ci/cq (e.g., one-half), then the left-hand side of the equation can be held constant, and one may measure the time, t, required to obtain this particular cj/co ratio for different values of cq. Indeed, Eq. (3) can be rearranged under these conditions to obtain the following form ... 

Now, it is the forward rate constant that is almost invariant (because there is a common nucleophile H2O), whereas k is very sensitive to the nucleophilic character of X, i.e. it is an associative type reaction. Selectivity would be expected to lead to curved free energy plots but these have not yet been observed. 

The forward rate constant of the above reaction was found to be kf, -10" mol Ls in0.7mol/LNaC104. 

Unusual behavior was reported" for the complexation of the ions IO4 and CIO4 with beta cyclodextrin. Rohrbach and coworkers found" that these ions show values of the forward rate-constant that approach the diffusion-controlled limit that is, 10 to 10 dm mol". s , whereas all of the other ions studied show values significantly less than this limit. In order to explain this observation, Rohrbach suggested" that, because of their larger radii, the CIO and IO4 ions may be too large to fit into the alpha cyclodextrin cavity, but, instead, may form a complex in which the ions straddle one of the entrances to the cavity. 

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