The Reverse Rate Constant

The reverse rate constant y, governs the rate at which molecules leave a cluster. This quantity is more difficult to evaluate theoretically than the forward rate constant p(. What we do know is that, as long as the number concentration of monomer molecules is much smaller than that of the carrier gas, the rate at which a cluster loses monomers should depend only on the cluster size and temperature and be independent of monomer partial pressure. Thus the evaporation rate constant under nucleation conditions (5 1) should be identical to that in the saturated (51 = 1) vapor  

Two different approaches are commonly employed to obtain y). One is based on the knowledge of the distribution of clusters in the saturated vapor, and the second is based on the Kelvin equation. These two approaches lead to similar results. In what immediately follows we will employ the cluster distribution at saturation to infer y) the approach based on the Kelvin equation is given in Section 11.2. 

It is important at this point to define three different cluster distributions that arise in nucleation theory. First is the saturated (S = 1) equilibrium cluster distribution, N, which we will always denote with a superscript s. Second is the steady-state cluster distribution at 

S 1 and a constant net growth rate of 7, Nj. At saturation (S = 1) all 7i+1 /2 = 0, whereas at steady-state nucleation conditions all 7,+i/2 = 7. There is a third distribution that we will not explicitly introduce until the next section. It is the hypothetical, equilibrium distribution of clusters corresponding to S 1. Thus it corresponds to all 7i+1/2 = 0, but 5 1. Because of the constraint of zero flux, this third distribution is called the constrained equilibrium distribution, Nf. We will distinguish this distribution by a superscript e. 

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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. 

That is, the ratio between the forward and the reverse rate constants is equal to the equilibrium constant Kc. It can be easily shown that the perfect gas model implies... 

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. 

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

A plot of k vs [Y] will have an intercept A , and a slope ki- An example is shown in Fig. 4.8. For different nucleophiles reacting with the same complex, the value of A , is the same, whereas the value of Atj usually will be different. Often, k < At2[Y] and it is then difficult to measure Atj accurately. Care has to be taken that a positive intercept in a plot of the type shown in Fig. 4.8 is not mistakenly assigned to A , in (4.96), when it may in fact represent the reverse rate constant of (4.93) (Sec. 1.5). 

As discussed previously, the reverse rate constants k(J- satisfy microscopic reversibility ... 

Thus, experimentally, one plots the logarithm of the concentration ratio against time under conditions where Eq. (15.7) holds in order to determine ki. The reverse rate constant k i may either be determined analogously, or from Eq. (15.5) once ki is known. 

The Gibbs free energy G is a central thermodynamic quantity in understanding chemistry. The Gibbs free energy determines whether a reaction, or perhaps its reverse reaction, will proceed spontaneously. It provides for the location of chemical equilibrium, at which there is no net forward or reverse reaction. The free-energy change of a reaction determines the equilibrium constant, which also determines the reverse rate constant for a reaction, if the forward rate constant is known. 

The reverse rate constant, of course, can also depend on temperature, and can be specified in a three-parameter modified Arrhenius form analogous to Eq. 9.83. However, the reverse rate constant may also be specified from the reaction thermodynamics. If reaction i were at equilibrium, the forward and reverse rates of progress would be equal, so qt would be 0, and from Eq. 9.73,... 

The ratio of the forward rate constant to the reverse rate constant for reaction 9.100 appearing here is simply the equilibrium constant for that reaction. From the discussion of the equilibrium constant in this chapter (i.e., Eqs. 9.87-9.93), we see that [C ] depends only on the thermochemistry of reaction 9.100 ... 

For reversible reactions the reverse rate constant krj is related to the forward rate constant through the equilibrium constants as... 

Thermochemical properties of gas-phase, surface, and bulk species are assumed to be available. This information is used in the calculation of the equilibrium constant, Eq. 11.110, and thus the reverse rate constant, Eq. 11.108. There is not a great deal of thermochemical data for species on surfaces, but techniques are becoming available for their calculation (e.g., Pederson et al. [310]). If surface reactions are specified to be irreversible, or if Arrhenius coefficients for the reverse rate constant are explicitly supplied, then the thermochemical data are not actually used. 

For each reaction in a surface chemistry mechanism, one must provide a temperature dependent reaction probability or a rate constant for the reaction in both the forward and reverse directions. (The user may specify that a reaction is irreversible or has no temperature dependence, which are special cases of the general statement above.) To simulate the heat consumption or release at a surface due to heterogeneous reactions, the (temperature-dependent) endothermicity or exothermicity of each reaction must also be provided. In developing a surface reaction mechanism, one may choose to specify independently the forward and reverse rate constants for each reaction. An alternative would be to specify the change in free energy (as a function of temperature) for each reaction, and compute the reverse rate constant via the reaction equilibrium constant. 

The forward rate constants are k/ = 1 and kf2 = 1000, the equilibrium constants are K — 0.01 and K2 =, and the reverse rate constants are krt = kjJK and kn = kfr/Ki- The rate constants and species concentrations used here are taken to be unitless. Based on the principles of mass-action kinetics, the transient behavior of the system can be represented in differential-equation form as... 

Kassel also expressed the data in a slightly different form, to give better agreement with the rate constant calculated from the equilibrium constant and the reverse-rate constant... 

The second relaxation time is for the slow step. This is an example of a reversible reaction, and, by analogy with equation 4.21, the reciprocal relaxation time is given by the sum of the forward and the reverse rate constants for the step. However, the effective forward rate constant for this step is given by k2 multiplied by the fraction of the enzyme that is in the form of ES i.e.,... 

In summary, we have seen that the application of microscopic reversibility for the forward and reverse cross-sections and the use of complete equilibrium distributions for the evaluation of the statistical rate constant lead to the usual results known from equilibrium statistical mechanics. If one knows the cross-section for a forward reaction, one can always determine the inverse cross-section through the principle of microscopic reversibility. Also, if one knows the cross-section for the forward reaction, and in addition one knows that the translational and internal distribution functions of reactants and products have reached equilibrium, one can calculate the rate constant. Detailed balance then permits the calculation of the reverse rate constant. 

This result indicates that the sensitivity of the overall rate to the reverse rate constant for a step depends on the reversibility of the step. For example, the sensitivity of the overall rate to the reverse rate constant is equal to zero for an irreversible step (z = 0) and to the negative value of the sensitivity of the overall rate to the forward rate constant for a quasi-equilibrated step (zt = 1). 

The next step in the reaction kinetics analysis is to choose for each family of reactions (i.e., adsorption/desorption, oligomerization//-) -scission, isomerization, and hydride transfer) whether to parameterize the kinetic model in terms of either the forward or the reverse rate constant (kj,for or khrey) since the ratio of the forward and the reverse rate constants must equal the known value of Kit q ... 

A forward rate constant of 3 x 103 sec-1 has been reported for this reaction, although the authors caution that it may only be an upper limit (158). The reverse rate constant, 1.6 x 106 M l sec-1, is relatively well established (121). An equilibrium constant of Keq < 2 x 10-3 M is obtained from the ratio of these rate constants, and from NBS data and AfG° for OH we derive AfG° < —517 kJ/mol for S04- and E° < 2.36 V for the S04 /S042 couple. The mild disagreement between these results and those obtained from homolysis of persulfate is not understood, but the fact that Cl is rapidly oxidized by S04- although its E° is 2.41 V favors the homolysis results. 

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