Rate constants definition

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

Ablation rate constant definition, 418 determination, 414,417f Ablative photodecomposition description, 411 See also Photoablation Ace ty la ted m-cresol—novolac copolymers, preparation, 193... 

Using the contact transfer rates (7.16) in the rate constant definitions (3.278), we reduce Eq. (3.420b) to its contact analog ... 

In addition to not addressing the need for individual tumbling rate constants for radical pair combinations leading to 3b, 2-BN, and 4-BN, Scheme 13.5 does not take into account the fact the values of probably differ and the translational rate constants definitely differ within amorphous and interfacial cages of either LDPE or HDPE. 

Second, there is a surface concentration of the ligands or capturing sites immobilized on a functionalized surface, with a concentration of [FJo. Finally, there is a surface concentration of the adsorbed targets (products of the reaction), with a concentration of [F]. The units of [F] and [F]q are in mol/m, whereas [S] is expressed in units of mol/m. In case of adsorption, the definitions of the reaction rates are somewhat modified from the usual rate constant definitions mentioned above, primarily because of the fact that the immobilization of the substrate S not only depends on the volume concentration at the wall but also depends on the available sites for adsorption. Accordingly, one can write —d[S]/dt = a([F]o - [F])[S]w and -d[F]/dt = ka[F], where and k are the adsorption and dissociation rates, respectively. The concentration of F is increased by the former and is decreased by the latter, and the net rate of change is given by their balance, i.e.,... 

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

A plot against Hammett s cr-constants of the logarithms of the rate constants for the solvolysis of a series of Mz-substituted dimethylphenylcarbinyl chlorides, in which compounds direct resonance interaction with the substituent is not possible, yielded a reasonably straight line and gave a value for the reaction constant (p) of — 4 54. Using this value of the reaction constant, and with the data for the rates of solvolysis, a new set of substituent parameters (cr+) was defined. The procedure described above for the definition of cr+, was adopted for... 

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. 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Taft began the LFER attack on steric effects as part of his separation of electronic and steric effects in aliphatic compounds, which is discussed in Section 7.3. For our present purposes we abstract from that treatment the portion relevant to aromatic substrates. Hammett p values for alkaline ester hydrolysis are in the range +2.2 to +2.8, whereas for acid ester hydrolysis p is close to zero (see Table 7-2). Taft, therefore, concluded that electronic effects of substituents are much greater in the alkaline than in the acid series and. in fact, that they are negligible in the acid series. This left the steric effect alone controlling relative reactivity in the acid series. A steric substituent constant was defined [by analogy with the definition of cr in Eq. (7-22)] by Eq. (7-43), where k is the rate constant for acid-catalyzed hydrolysis of an orr/to-substituted benzoate ester and k is the corresponding rate constant for the on/to-methyl ester note that CH3, not H, is the reference substituent. ... 

In this definition ko is the rate constant for CH3COOR and k is the constant for RCOOR thus = 0 for R = CH3. Table 7-11 lists some values. Taft s Es steric constants are in some instances based on averages of several different reactions, so MacPhee et al. have defined a steric constant Es by Eq. (7-52) for a single reaction, namely, the acid-catalyzed esterification of carboxylic acids in methanol at 40°C. Es values are also given in Table 7-11. Additional Es and Es values are available. 

The careful worker explicitly states the usage chosen for v. This choice defines the rate constant, which may otherwise be uncertain by a small numerical factor. When the definition is obvious, as in a reaction in which all the stoichiometric coefficients are unity, the definition of the numerical factor can be omitted. 

The two rate constants are related by a numerical factor ki = 2fc2. (Also, see Problem 1.) Obviously one should not report that the rate constant for the dimerization of methyl radicals is X, unless a definition of k is given. 

When a reaction follows first-order kinetics, [A] decreases exponentially with time. According to Eq. (2-3), a plot of In [A], against time is a straight line with a slope of -k. This slope and the rate constant are independent of [A]o- This straight line also means that a definite fraction of the reaction occurs over a given time interval, independent of [A]0. If this fraction is taken as 50 percent, substitution of [A] 1/2 = [A]q/2 into Eq. (2-2) gives... 

Generally speaking, the investigator carries out such simulations to show whether the postulated scheme matches the data. To carry out the simulations, the researcher fixes the rate constants of some steps at their established values and evaluates the ones that are unknown by iterative adjustment, the criterion being the quality of the match. Obviously, one is not likely to produce a definitive answer if too many steps have rate constants that are allowed to be adjusted at will. One may also be tempted to adjust one (more ) of the supposedly established rate constants. This action is perilous, without independent cause for suspicion. In that event, one should very seriously consider replicating the original experiments that defined that step. Not to do so invites the construction of an unsound house of cards. ... 

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