Reaction rate coefficient definition

By using the natural logarithm of the reaction rate coefficient definition [10.42], we have ... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

The definitions of the reaction rate coefficients by Ganguly (1982) differ from those in Table 2-1. The difference is explained in Box 2-3. 

The reaction rate coefficients in the above equations may be related to reaction rates per pair of particles 2/, in nuclear physics (e.g., Fowler et al., 1975 Harris et al., 1983) by k = Xj/(1 + 5/ ), where 8 = 0 except for i= , for which 5/ = 1. That is, for Reactions 2-145 and 2-147 in which two identical particles collide to react, the definition of k is half of defined by nuclear physicists and for reactions in which different particles collide, the definition of k is the same as Xij. The reaction rate coefficients depend on temperature in a complicated way (Table 2-3) and may be calculated as the average value of the product of relative velocity times cross section. The concentrations of the intermediate species can be derived as follows. From Equation 2-155, 145 [ H] = ki4e[ H]pH]. That is. 

Note. The unit of k is based on time (s) and concentration (mol/cmi). The reaction rate coefficients as a function of temperature are from Fowler et al. (1975) and Harris et al. (1983). Note that for Reactions 2-145 and 2-147, the definition of k is consistent with chemists definition used in this book and is half of Xij defined by nuclear physicists. That is, k = Xij/(l + 8j,), where Xi, is the reaction rates per pair of particles, and 5 , = 0 except for i=j for which 5 , = 1. The concentration unit is not converted to mol/L. 

According to the above definitions, diffusion-controlled reactions are generally characterized by kdiff kchem- It should be noted though, that for reactions between highly mobile radical species, this condition is not always satisfied [19, 20]. In such cases, both the dififiision and chemical reaction rate coefficient contribute to the value of the observed rate coefficient. Noyes [19] and Rise [20] have reviewed several theoretical aspects of the calculation of diffusion-controlled reaction rates in solution. 

The constant K issned from the partition function wonld be the constant of an equilibrium between the activated complex and reactants. Considering the definitions of equilibrium constants, the free energy related to the reaction can be introduced and the reaction rate coefficient becomes ... 

This is the situation exploited by the so-called isolation method to detennine the order of the reaction with respect to each species (see chapter B2.1). It should be stressed that the rate coefficient k in (A3,4,10) depends upon the definition of the in the stoichiometric equation. It is a conventionally defined quantity to within multiplication of the stoichiometric equation by an arbitrary factor (similar to reaction enthalpy). 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

The reaction rate is properly defined in terms of the time derivative of the extent of reaction. It is necessary to define k in a similar fashion in order to ensure uniqueness. Definitions in terms of the various rt would lead to rate constants that would differ by ratios of their stoichiometric coefficients. 

Kinetic theory indicates that equation (32) should apply to this mechanism. Since the extent of protonation as well as the rate constant will vary with the acidity, the sum of protonated and unprotonated substrate concentrations, (Cs + Csh+), must be used. The observed reaction rate will be pseudo-first-order, rate constant k, since the acid medium is in vast excess compared to the substrate. The medium-independent rate constant is k(), and the activity coefficient of the transition state, /, has to be included to allow equation of concentrations and activities.145 We can use the antilogarithmic definition of h0 in equation (33) and the definition of Ksh+ in equation (34) ... 

As the reaction rate may not be assessed directly, the definition invokes an auxiliary parameter pk (e.g., an enzyme concentration) that is assumed to act only on the rate v. Note that X may stand for an arbitrary steady-state property with the coefficients for concentrations Cs and flux CJ as the most important examples. 

It is noteworthy that the form of the rate (r = / (state of the system) does not actually depend on our choice of reaction rate definition. Only the rate coefficients and their dimensions change with each rate definition (Levenspiel, 1972). 

This discussion highlights the difficulty of deciding at what separation A and B form an encounter pair and then whether this reacts or separates. Noyes [5] and Wilemski and Fixman [51] have taken the encounter distance to be that separation which, if reduced slightly, will lead to reaction. Where these authors disagree is that Noyes [5] only allows reaction to occur in a very narrow range of separation distances about R (which is the usual assumption) and Wilemski and Fixman [51] assume that any separation distance less than the encounter distance, R, can lead to reaction between A and B and that A and B can diffuse through each other till their centres of mass coincide (Chap. 9, Sect. 4). Neither assumption is good, but the differences in predicted rate coefficients are so small that an experimental test of these theories could not be definitive. 

Thus, firstly, the choice of the pure solvent as the reference state for the definition of activities of solutes in fact impairs a fair comparison of the activity of dilute solutes such as general adds to the activity of the solvent itself. Secondly, the observed first-order rate constants k or k0 for the reaction of a solute with the solvent water are usually converted to second-order rate constants by division through the concentration of water, h2o = oA iho, for a comparison with the second-order rate coefficients HA. Again, it is questionable whether the formal h2o coefficients so calculated may be compared with truly bimolecular rate constants kUA for the reactions with dilute general acids HA. It is then no surprise that the values for the rate coefficients determined for the catalytic activity of solvent-derived acids scatter rather widely, often by one or two orders of magnitude, from the regression lines of general adds.74... 

Thus the presence of steps for the interaction between various intermediates in the detailed mechanisms is only a necessary condition for the multiplicity of steady states in catalytic reactions. A qualitative analysis of the dynamic system (5) for mechanism (4) showed that the existence of several stable steady states with a non-zero reaction rate needs the following additional conditions (a) the stoichiometric coefficients of intermediates must fit definite relationships ensuring the kinetic competition of these substances [violation of conditions (6)] (b) the system parameters must satisfy definite inequalities. 

By definition of the reaction rates, rh and the stoichiometric coefficients, vih the term... 

The polarographic measurements permit deduction of rate coefficients for hydrolysis of the intermediate. This reaction is also subject to general acid catalysis. Table 3 lists values which have been obtained for the acids HsO , HCOJ and NH4 at three temperatures, together with the energies of activation derived from them. Russian polarographic work gave the first definite indication of the presence of an intermediate in BH4 hydrolysis. Surprisingly, Russian workers... 

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