Catalytic rate constant, definition

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. 

Although the concepts of specific acid and specific base catalysis were useful in the analysis of some early kinetic data, it soon became apparent that any species that could effect a proton transfer with the substrate could exert a catalytic influence on the reaction rate. Consequently, it became desirable to employ the more general Br0nsted-Lowry definition of acids and bases and to write the reaction rate constant as... 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

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... 

The rate constants for such outer-sphere reactions can therefore differ markedly from those corresponding to true weak-overlap pathways, even after correction for electrostatic double-layer effects. This can cause some difficulties with the operational definition of inner-sphere electrocatalysis considered above, whereby outer-sphere reactions are regarded as "non-catalytic processes. In addition, there is evidence that inner- rather than outer-sphere pathways can provide the normally preferred pathways at metal-aqueous interfaces for reactants containing hydrophobic functional groups [116]. 

Figure IB displays relative catalytic activity (RA) - in terms of pseudo first-order rate constants, corrected for coke content, related to the fresh, sulfided catalyst vs carbon content. The individual HDS, HVD and CNH activities all decrease with increasing carbon content, the order of deactivation being HYD < HDS < CNH. (The results for relative HDN activities followed closely those of CNH, and are not shown). Relative activities fall off less sharply as coke content increases. Because of the limited set and scatter of the data, a definitive deactivation correlation could not be obtained. Best fit curves to the data were constructed from a power-deactivation equation in C (1), and are shown by the solid curves in Fig. IB.

The rigorous definition of activity is the ratio of the rate of reaction (or rate constant) on a catalyst after some time, t, to the rate (or rate constant) on a fresh catalyst at the same conditions. Thus, the loss of catalytic ability with time in use is called deactivation. However, the term activity is also often used to merely refer to the rate constant, k, or to the intrinsic ability of a catalyst to carry out a reaction. 

Ref. [4], the corresponding rate constants do not show significant dynamical effects. Furthermore, attempts to define dynamical catalytic effect in a different way and to include in such factor nonequilibrium solvation effects [100] have been shown to be very problematic (e.g. Ref. [4]). Similarly, we have shown that the reasonable definition of dynamical effects by the existence of special vibrations that lead coherently to the TS does correspond to the actual simulation in enzyme and solution. 

Catalysis or Catalytic Power is the ratio between the reaction rate of the catalyzed reaction and that of the uncatalyzed reaction. It is defined as kcat/feun where kcat is the rate of the catalyzed reaction and kun is the rate of the uncatalyzed reaction. By definition, catalysis should be unit-less (a ratio of rate constants), thus care must be practised while determining Catalytic Power that k at and k n have the same units. Alternatively, the second order uncatalyzed reaction s rate (M s units) can be divided by kcat (s ) and the ratio then has units of concentration (M). This concentration is called effective concentration [2] and could be addressed as the concentration of functional groups or substrates in the enzyme s active site. Since that effective concentration is often in the thousands of M range, it is not a physically meaningful concentration, but rather a manifestation of the role of correct orientation, dynamic, and other catalytic effects induced by the enzyme. A similar approach used the substrate concentration in which the enzymatic and uncatalyzed rates are equal as an indicator for catalytic power [8j. The advantage of the first... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

The interpretation follows from the limiting case of Equation 4-8. Consider the limiting case of a high reactant concentration which is so high that the catalytic sites are saturated and [5 ] K - Then, the rate equation reduces to = cat[ ]tot and kcat is recognized as a first-order rate constant. If the rate were written per enzyme molecule rather than per unit volume, then the reaction would be of zero order, and kcat would be the rate at saturation (the maximum number of reactant molecules converted per catalytic site per unit time) this is the definition of the turnover frequency. 

The meaning of this definition is that no chemical component causes the decrease of another one. Nothing has been supposed about the effect of components on themselves. Our definition is in concord with the definitions used in classical chemical kinetics (cf. Bazsa Beck, 1971). A mechanism is called canonically cross-catalytic, if for all the sets of reaction rate constants the canonic complex chemical reaction corresponding to the induced kinetic differential equation of the complex chemical reaction is cross-catalytic. 

But this proposal is not so strict scientifically due to the lack of clear definition of activity. The catalytic activity of ammonia synthesis catalyst can be expressed by outlet ammonia concentration of converter, conversion ratio of ammonia, reaction rate and rate constant of kinetics and turnover frequency of ammonia (TOF). 

Definitions. Early in the history of chemical kinetics a catalyst was defined as a chemical species that changes the rate of a reaction without undergoing an irreversible change /fse//(Ostwald, 1902). Subsequent definitions of a catalyst included (1) a catalyst is a chemical species that may be chemically altered but is tan involved in a whole number stoichiometric relationship among reactants and prodacts and (2) a catalyst is a chemical species that appears in the rate law with a reaction order greater than its stoichiometric coefficient. In the latter case it was realized that either a product of the reaction (autocatalysis) or a reactant may also function as a catalyst. From a practical perspective, a catalyst is a chemical species that influences the rate of a chemical reaction regardless of the fate of the catalytic species. However, a catalyst has no influence on the thermodynamics of n reaction. In other words, the concentration of a catalyst is reflected in the rate law but is not reflected in the equilibrium constant. This latter definition was modified and approved by the International Union of Pure and Applied < hemistry (IUPAC, 1981) to read as follows ... 

At high-mass-transfer Peclet numbers, sketch the relation between average residence time divided by the chemical reaction time constant (i.e., r/co) for a packed catalytic tubular reactor versus the intrapeUet Damkohler number Aa, intrapeiiet for zeroth-, first-, and second-order irreversible chemical kinetics within spherical catalytic pellets. The characteristic length L in the definition of Aa, intrapeiiet is the sphere radius R. The overall objective is to achieve the same conversion in the exit stream for all three kinetic rate laws. Put all three curves on the same set of axes and identify quantitative values for the intrapeiiet Damkohler number on the horizontal axis. 

In catalytic reaction (10.3), the catalyst (K) is defined as the materials that can enhance reaction rate but does not change equilibrium of reaction at definite temperature. Prom the thermodynamic point of view, the difference in free energy between the initial and final states of reaction (10.3) at constant temperature and... 

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