Determination of Reaction Rate Expression

The main difficulty in using Eq. 7.2.11 is that the extent is not a measurable quantity. Therefore, we have to derive a relationship between Zout and an appropriate measured quantity. We do so by using the design equation and relevant stoichiometric relations. In most applications, we measure the concentration of a species at the reactor outlet and calculate the extent by either Eq. 7.2.5 for liquid-phase reactions or Eq. 7.2.6 for gas-phase reactions. We can then determine the orders of the individual species for power rate expressions. 

Example 7.5 A stream of gaseous reactant A at 3 atm and 30°C (Cao = 120 mmol/L) is fed into a 500-L plug-flow reactor where it decomposes according to the reaction  

The concentration of reactant A is measured at the outlet of the reactor for different feed flow rates. Based on the data helow, determine  

We select the inlet stream as the reference stream hence, yAa = 1, Jbo = yco = 0, and Co = Ca = 120mol/L. Using Eq. 7.2.6, we calculate Zout for each outlet concentration  

Spaced, we calculate the derivative at the midpoints of each of two adjacent points. The calculated values are given in the table below  

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Initial Rate Measurements. Another differential method useful in the determination of reaction rate expressions is the initial rate approach. It involves a series of rate measurements at different initial reactant concentrations but restricted to very small conversions of the limiting reagent (5 to 10% or less). This technique differs from those discussed previ-... 

Determination of Reaction Rate Expressions for Reversible Reactions... 

ILLUSTRATION 5.1 DETERMINATION OF REACTION RATE EXPRESSION FOR THE REACTION BETWEEN SULFURIC ACID AND DIETHYL SULFATE... 

DETERMINATION OF REACTION-RATE EXPRESSIONS FROM PLUG-FLOW-REACTOR DATA 5.24... 

A batch experimental reactor is used for slow reactions since species compositions can be readily measured with time. The determination of reaction rate expression is described in Chapter 6. A tubular (plug-flow) experimental reactor is suitable for fast reactions and high-temperature experiments. The species composition at the reactor outlet is measured for different feed rates. Short packed beds are used as differential reactors to obtain instantaneous reaction rates. The reaction rate is determined from the design equation, as described in Chapter 7. An experimental CSTR is a convenient tool in determining reaction rate since the reaction rate is directly obtained from the design equation, as discussed in Chapter 8. 

The general integral technique for the determination of reaction rate expressions consists of the following trial-and-error procedure. 

In this subsection we have treated a variety of higher-order simple parallel reactions. Only by the proper choice of initial conditions is it possible to obtain closed form solutions for some of the types of reaction rate expressions one is likely to encounter in engineering practice. Consequently, in efforts to determine the kinetic parameters characteristic of such systems, one should carefully choose the experimental conditions so as to ensure that potential simplifications will actually occur. These simplifications may arise from the use of stoichiometric ratios of reactants or from the degeneration of reaction orders arising from the use of a vast excess of one reactant. Such planning is particularly important in the early stages of the research when one has minimum knowledge of the system under study. 

For those cases where the rate expressions for all reactions taking place in the system under study are known, the use of the instantaneous yield in the above equations does not contribute significantly to understanding the system behavior. In such cases it is easier to determine the overall yield by substituting the appropriate ratio of reaction rate expressions for the instan-... 

Below, we analyze the operation of isothermal plug-flow reactors with single reactions for different types of chemical reactions. For convenience, we divide the analysis into two sections (i) design and (ii) determination of the rate expression. In the former, we determine the size of the reactor for a known reaction rate, specified feed rate, and specified extent (or conversion). In the second section, we determine the rate expression and its parameters from reactor operating data. 

Key concepts employed by chemists and chemical engineers in the acquisition, analysis, and interpretation of kinetic data are presented in this chapter. The focus is on determination of empirical rate expressions that can subsequently be utilized in the design of chemical reactors. To begin, we find it convenient to approach the concept of reaction rate by considering a closed isothermal constant pressnre homogeneous system of uniform composition in which a single chemical reaction is taking place. In such a system the rate of the chemical reaction (r) is defined as... 

Pressure measurements can be accomplished by a number of different types of devices without disturbing the system being observed. Another type of reaction system that can be monitored by pressure measurements is one in which one of the products can be quantitatively removed by a solid or liquid reagent that does not otherwise affect the reaction. For example, acids formed by reactions in the gas phase can be removed by absorption in basic solutions. From knowledge of the reaction stoichiometry and measurements of the total pressure as a function of time, one can determine the corresponding extents of reaction and partial pressures or concentrations of the various reactant and product species. An example of how pressure measurements can be used to determine a reaction rate expression is provided in Illustration 3.3. 

In Section 3.1 the mathematical expressions that result from integration of various reaction rate functions were discussed in some detail. Our present problem is the converse of that considered earlier (i.e., given data on the concentration of a reactant or product as a function of time, how does one proceed to determine the reaction rate expression ). 

Determination of the rate expression normally involves a two-step procedure. First, the concentration dependence is determined at a fixed temperature. Then the temperature dependence of the rate constants is evaluated to obtain a complete reaction rate expression. The form of this temperature dependence is given by equation (3.0.14), so our present problem reduces to that of determining the form of the concentration dependence and the value of the rate constant at the temperature of the experiment... 

It is always preferable to use as much of the data as possible to determine the reaction rate expression. This principle often implies that one should use some sort of graphical procedure to analyze the data. Visual inspection of such plots may indicate that certain points are seriously in error and should not be weighted heavily in the determination of the reaction rate expression. The consistency and precision of the data can also be evaluated visually by observing the deviation of the data points from a smooth curve (ideally, their deviations from a straight line). 

If one is interested in the kinetics of reactions that occur at very fast rates, having half-lives on the order of a fraction of a second or less, the methods that we have discussed previously for the determination of reaction rates are no longer applicable. Instead, measurements of the response of an equilibrium system to a perturbation are used to determine its relaxation time. The rate at which the system approaches its new equilibrium condition is observed using special electronic techniques. From an analysis of the system behavior and the equilibrium conditions, the form of the reaction rate expression can be determined. 

The so-called two-step sequence method is that the derivation of reaction rate expression only requires to consider two key steps for a reaction involving multielementary steps. Only the rate constants or equilibrium constants of the two key steps appear in rate expression, which are of clear physical meaning. In order to determine the key steps, a concept of most abundant reaction intermediate (Mari) must be introduced. Mari is an intermediate of maximum concentration among all reactive intermediates invovled in the reaction, and the concentration of other intermediates can be ignored. Based on the concepts of both rate determining step and most abundant reaction intermediate, the mechanisms of many catalytic reactions can be simplified to two-step sequences for the derivation of kinetic equations. In order to explain the rules for the treatment of heterogeneous catalytic reaction kinetics by simplest two-step sequences method, two examples are given as follows ... 

Integrated rate laws have two principal uses. One is to predict the concentration of a species at any time after the start of the reaction. Another is to help find the rate constant and order of the reaction. Indeed, although we have introduced rate laws through a discussion of the determination of reaction rates, these rates are rarely measured directly because slopes are so difficult to determine accurately. Almost all experimental work in chemical kinetics deals with integrated rate laws their great advantage being that they are expressed in terms of the experimental observables of concentration and time. Computers can be used to find numerical solutions of even the most complex rate laws. However, we now see that in a number of simple cases, solutions can be expressed as relatively simple functions and prove to be very useful. 

Arrhenius rate expression and concept of an activation energy provided an important basis for the analysis of the rate of chemical reactions. However, the main difficulty that remained was the absence of a general theory to predict the parameters in the rate expression. Whereas equilibria of reactions could be rigorously defined, the determination of reaction rates remained a branch of science, for which the basic principles still had to be formulated. This was achieved in the 1930s, when Henry Eyring, and independently, Michael Polanyi and M. G. Evans, formulated (and later refined) the transition-state theory. An important aim of this book is to present the current understanding of the Arrhenius equation and its parameters in the context of catalytic reactions. 

These examples illustrate the relationship between kinetic results and the determination of reaction mechanism. Kinetic results can exclude from consideration all mechanisms that require a rate law different from the observed one. It is often true, however, that related mechanisms give rise to identical predicted rate expressions. In this case, the mechanisms are kinetically equivalent, and a choice between them is not possible on the basis of kinetic data. A further limitation on the information that kinetic studies provide should also be recognized. Although the data can give the composition of the activated complex for the rate-determining step and preceding steps, it provides no information about the structure of the intermediate. Sometimes the structure can be inferred from related chemical experience, but it is never established by kinetic data alone. 

Reaction Rate Expression 191 Determine the order of the reaetion and the rate eonstant. Solution... 

Rate constant The proportionality constant in the rate equation for a reaction, 288 Rate-determining step The slowest step in a multistep mechanism, 308 Rate expression A mathematical relationship describing the dependence of reaction rate upon the concentra-tion(s) of reactant(s), 288,308-309 Rayleigh, Lord, 190... 

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). 

The objectives of this research are therefore 1) to see whether rate expressions such as Equations 11 and 12 provide adequate descriptions of reaction rates and, if not, what rate expressions are appropriate, 2) to determine reaction activation energies, heats of adsorption, and pre-exponential factors, and 3) to compare these quantities with those measured under UHV conditions to determine whether the same processes and surface species might be involved. 

Attempts to determine how the activity of the catalyst (or the selectivity which is, in a rough approximation, the ratio of reaction rates) depends upon the metal particle size have been undertaken for many decades. In 1962, one of the most important figures in catalysis research, M. Boudart, proposed a definition for structure sensitivity [4,5]. A heterogeneously catalyzed reaction is considered to be structure sensitive if its rate, referred to the number of active sites and, thus, expressed as turnover-frequency (TOF), depends on the particle size of the active component or a specific crystallographic orientation of the exposed catalyst surface. Boudart later expanded this model proposing that structure sensitivity is related to the number of (metal surface) atoms to which a crucial reaction intermediate is bound [6]. 

It is obvious that to quantify the rate expression, the magnitude of the rate constant k needs to be determined. Proper assignment of the reaction order and accurate determination of the rate constant is important when reaction mechanisms are to be deduced from the kinetic data. The integrated form of the reaction equation is easier to use in handling kinetic data. The integrated kinetic relationships commonly used for zero-, first-, and second-order reactions are summarized in Table 4. [The reader is advised that basic kinetic... 

Basic Concepts in Chemical Kinetics—Determination of the Reaction Rate Expression... 

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