Reaction rates—definitions

Following the reaction rate definition of the form given in eq. (3.1), if component i is a reaction product the rate is positive if it is a reactant that is being consumed, the rate is negative thus, the rate of disappearance of the reactant is -r,. In environmental applications, as we are interested in the disappearance of a pollutant, the rate is expressed as -r, which is positive. The rate of disappearance is used in Chapters 3 and 5, where for simplicity it is referred to as the reaction rate. 

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). 

Related to the question of how to define the rate of chemical reaction is the use of the time derivative, dMi/dt or dNi/dt. This implies that things are changing with time and may tempt one to associate the appearance of reaction rate terms in mass conservation equations with unsteady-state processes. This is not necessarily true in general, one must make a distinction between the ongoing time of operation of some process (i.e., that measured by an observer) and individual phenomena such as chemical reactions that occur at a steady rate in an operation that does not vary with time (steady-state). Later we will encounter both steady- and unsteady-state types of processes involving chemical reaction, to be sure, but this depends on the process itself the reaction rate definition has been made without regard to a particular process. 

We can now use these relationships in the basic reaction-rate definition. For the first-order irreversible batch reaction... 

For homogeneous reactions we obtain the conventional definition of the reaction rate u. as rate of conversion per volume... 

The slopes of the fimctions shown provide the reaction rates according to the various definitions under the reaction conditions specified in the figure caption. These slopes are similar, but not identical (nor exactly proportional), in this simple case. In more complex cases, such as oscillatory reactions (chapter A3.14 and chapter C3.6). the simple definition of an overall rate law tluough equation (A3.4.6) loses its usefiilness, whereas equation (A3.4.1) could still be used for an isolated system. 

A catalyst is a substance that iacreases the rate of approach to equiUbrium of a chemical reaction without being substantially consumed itself. A catalyst changes the rate but not the equiUbrium of the reaction. This definition is almost the same as that given by Ostwald ia 1895. The term catalysis was coiaed ia ca 1835 by Ber2eHus, who recognized that many seemingly disparate phenomena could be described by a single concept. For example, ferments added ia small amounts were known to make possible the conversion of plant materials iato alcohol and there were numerous examples of both decomposition and synthesis reactions that were apparendy caused by addition of various Hquids or by contact with various soHds. 

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. 

Chemical engineering inherited the definition for the reaction rate from chemical kinetics. The definition is for closed systems, like batch reactors, in which most of the classical kinetic studies were done. Inside a batch reactor little else besides chemical reaction can change the concentration of reactant A. In a closed system, for the reaction of... 

Usually the Arrhenius plot of In k vs. IIT is linear, or at any rate there is usually no sound basis for coneluding that it is not linear. This behavior is consistent with the conclusion that the activation parameters are constants, independent of temperature, over the experimental temperature range. For some reactions, however, definite curvature is detectable in Arrhenius plots. There seem to be three possible reasons for this curvature. 

So the equilibrium predictions based on ° s do not make all experiments unnecessary. They provide no basis whatsoever for anticipating whether a reaction will be very slow or very fast. Experiments must be performed to learn the reaction rate. The ° s do, however, provide definite and reliable guidance concerning the equilibrium state, thus making many experiments unnecessary the multitude of reactions that are foredoomed to failure by equilibrium considerations need not be performed. 

If, for the purpose of comparison of substrate reactivities, we use the method of competitive reactions we are faced with the problem of whether the reactivities in a certain series of reactants (i.e. selectivities) should be characterized by the ratio of their rates measured separately [relations (12) and (13)], or whether they should be expressed by the rates measured during simultaneous transformation of two compounds which thus compete in adsorption for the free surface of the catalyst [relations (14) and (15)]. How these two definitions of reactivity may differ from one another will be shown later by the example of competitive hydrogenation of alkylphenols (Section IV.E, p. 42). This may also be demonstrated by the classical example of hydrogenation of aromatic hydrocarbons on Raney nickel (48). In this case, the constants obtained by separate measurements of reaction rates for individual compounds lead to the reactivity order which is different from the order found on the basis of factor S, determined by the method of competitive reactions (Table II). Other examples of the change of reactivity, which may even result in the selective reaction of a strongly adsorbed reactant in competitive reactions (49, 50) have already been discussed (see p. 12). 

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... 

From now on, whenever we speak of a reaction rate, we shall always mean an instantaneous rate. The definitions in Eqs.l and 2 can easily be adapted to refer to the instantaneous rate of a reaction. 

The degree of branching by transfer with polymer obviously will increase with the conversion since the relative incidence of branching must depend on the ratio of polymer to monomer in the system. To examine the matter from the point of view of reaction rates, let 6 represent the fraction of monomer molecules which have polymerized out of a total of iVo in the system, and let v represent the total number of branches. (At variance with the definition used elsewhere. No is the total number of units polymerized and unpolymerized as well.) The rates of generation of branches and of polymerization can then be written... 

In order to exemplify the potential of micro-channel reactors for thermal control, consider the oxidation of citraconic anhydride, which, for a specific catalyst material, has a pseudo-homogeneous reaction rate of 1.62 s at a temperature of 300 °C, corresponding to a reaction time-scale of 0.61 s. In a micro channel of 300 pm diameter filled with a mixture composed of N2/02/anhydride (79.9 20 0.1), the characteristic time-scale for heat exchange is 1.4 lO" s. In spite of an adiabatic temperature rise of 60 K related to such a reaction, the temperature increases by less than 0.5 K in the micro channel. Examples such as this show that micro reactors allow one to define temperature conditions very precisely due to fast removal and, in the case of endothermic reactions, addition of heat. On the one hand, this results in an increase in process safety, as discussed above. On the other hand, it allows a better definition of reaction conditions than with macroscopic equipment, thus allowing for a higher selectivity in chemical processes. 

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