1、Chapter 3 Interpretation of Batch Reactor Data,A rate equation characterizes the rate of reaction, and its form may either be suggested by theoretical considerations or simply be the result of an empirical curve-fitting procedure. In any case，the value of the constants of the equation can only be fo
2、und by experiment，predictive(预言性的) methods are inadequate(不充分的, 不适当的) at present.,The determination of the rate equation is usually a two-step procedure; first the concentration dependency is found at fixed temperature and then the temperature dependence of the rate constant is found, yielding the c
3、omplete rate equation.,化 学 反 应 工 程,Equipment by which empirical(经验的) information is obtained can be divided into two types, the batch and flow reactors. The batch reactor is simply a container to hold the contents while they react. All that has to be determined is the extent of reaction at various t
4、imes, and this can be followed in a number of ways, for example:,1. By following the concentration of a given component. 2. By following the change in some physical property of the fluid, such as the electrical conductivity(导电率) or refractive index(折光率). 3. By following the change in total pressure
5、of a constant-volume system. 4. By following the change in volume of a constant-pressure system.,化 学 反 应 工 程,The experimental batch reactor is usually operated isothermally and at constant volume because it is easy to interpret(解释,说明) the results of such runs. This reactor is a relatively simple dev
6、ice adaptable(能适应的) to small-scale laboratory set-ups, and it needs but little auxiliary(辅助的)equipment or instrumentation. Thus, it is used whenever possible for obtaining homogeneous kinetic data.,The flow reactor is used primarily(起初,主要地) in the study of the kinetics of heterogeneous reactions.,化
7、学 反 应 工 程,There are two procedures for analyzing kinetic data, the integral (积分的) and the differential (微分的) methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation (处理), predict that the plot of a certa
8、in concentration function versus time should yield a straight line. The data are plotted, and if a reasonably good straight line is obtained, then the rate equation is said to satisfactorily fit the data.,In the differential method of analysis we test the fit of the rate expression to the data direc
9、tly and without any integration. However, since the rate expression is a differential equation, we must first find (1/V)(dNi/dt) from the data before attempting the fitting procedure.,化 学 反 应 工 程,There are advantages and disadvantages to each method. The integral method is easy to use and is recomme
10、nded when testing specific mechanisms, or relatively simple rate expressions, or when the data are so scattered(分散的) that we cannot reliably find the derivatives (派生物，此指1/VdN/dt) needed in the differential method. The differential method is useful in more complicated situations but requires more acc
11、urate or larger amounts of data.The integral method can only test this or that particular mechanism or rate form; the differential method can be used to develop or build up a rate equation to fit the data.,In general, it is suggested that integral analysis be attempted first, and, if not successful,
12、 that the differential method be tried.,化 学 反 应 工 程,3.1 CONSTANT-VOLUME BATCH REACTOR,When we mention the constant-volume batch reactor we are really referring to the volume of reaction mixture, and not the volume of reactor. Thus, this term actually means a constant-density reaction system. Most li
13、quid-phase reactions as well as all gas-phase reactions occurring in a constant-volume bomb fall in this class.,In a constant-volume system the measure of reaction rate of component i becomes,(1),化 学 反 应 工 程,or for ideal gases, where C = p/RT,(2),Thus, the rate of reaction of any component is given
14、by the rate of change of its concentration or partial pressure; so no matter how we choose to follow the progress of the reaction, we must eventually relate this measure to the concentration or partial pressure if we are to follow the rate of reaction.,化 学 反 应 工 程,For gas reactions with changing num
15、bers of moles, a simple way of finding the reaction rate is to follow the change in total pressure of the system.,Analysis of Total Pressure Data Obtained in a Constant-Volume System. For isothermal gas reactions where the number of moles of material changes during reaction, let us develop the gener
16、al expression which relates the changing total pressure of the system to the changing concentration or partial pressure of any of the reaction components.,化 学 反 应 工 程,Write the general stoichiometric equation, and under each term indicate the number of moles of that component:,Initially the total nu
17、mber of moles present in the system is,but at time t it is,(3),where,化 学 反 应 工 程,Assuming that the ideal gas law holds, we may write for any reactant, say Ain the system of volume V,(4),Combining Eqs.3 and 4 we obtain,or,(5),化 学 反 应 工 程,Equation 5 gives the concentration or partial pressure of react
18、ant A as a function of the total pressure at time t, initial partial pressure of A, , and initial total pressure of the system, .,Similarly, for any product R we can find,(6),Equations 5 and 6 are the desired relationships between total pressure of the system and the partial pressure of reacting mat
19、erials.,化 学 反 应 工 程,It should be emphasized that if the precise stoichiometry is not known, or if more than one stoichiometric equation is needed to represent the reaction, then the “total pressure” procedure cannot be used.,The Conversion. Let us introduce one other useful term, the fractional conv
20、ersion, or the fraction of any reactant, say A, converted to something else, or the fraction of A reacted away. We call this, simply, the conversion of A, with symbol XA.,化 学 反 应 工 程,Suppose that NA0 is the initial amount of A in the reactor at time t = 0, and that NA is the amount present at time t
21、. Then the conversion of A in the constant volume system is given by,(7),and,(8),We will develop the equations in this chapter in terms of concentration of reaction components and also in terms of conversions.,Later we will relate and for the more general case where the volume of the system does not
22、 stay constant.,化 学 反 应 工 程,Integral Method of Analysis of Data,General Procedure. The integral method of analysis always puts a particular rate equation to the test by integrating and comparing the predicted C versus t curve with the experimental C versus t data. If the fit is unsatisfactory, anoth
23、er rate equation is guessed and tested. It should be noted that the integral method is especially useful for fitting simple reaction types corresponding to elementary reactions.,化 学 反 应 工 程,Irreversible Unimolecular-Type First-Order Reactions. Consider the reaction,(9),化 学 反 应 工 程,Suppose we wish to
24、 test the first-order rate equation of the following type.,(10),for this reaction. Separating and integrating we obtain,or,(11),In term of conversion (see Eqs.7 and 8), the rate equation, Eq.10, becomes,化 学 反 应 工 程,which on rearranging and integrating gives,or,(12),化 学 反 应 工 程,A plot of or ln ( ) vs
25、. t, as shown in Fig. 3.1, gives a straight line through the origin for this form of rate of equation. If the experimental data seems to be better fitted by a curve than by a straight line, try another rate form because the first-order reaction does not satisfactorily fit the data.,化 学 反 应 工 程,Cauti
26、on. We should point out that equations such as,are first order but are not amenable(应服从的) to this kind of analysis; hence, not all first-order reactions can be treated as shown above.,Irreversible Bimolecular-Type Second-Order Reactions. Consider the reaction,(13a),with corresponding rate equation,(
27、13b),化 学 反 应 工 程,Noting that the amounts of A and B that have reacted at any time t are equal and given by CA0XA, we may write Eqs.13a and b terms of XA as,Letting M=CB0/CA0, be the initial molar ratio of reactants, we obtain,which on separation and formal integration becomes,化 学 反 应 工 程,After break
28、down into partial fractions, integration, and rearrangement, the final result in a number of different forms is,(14),Figure 3.2 shows two equivalent ways of obtaining a linear plot between the concentration function and time for this second-order rate law.,化 学 反 应 工 程,where M 1,Figure 3.2 Test for t
29、he bimolecular mechanism with CA0=CB0, or for the second-order reaction, Eq.13.,化 学 反 应 工 程,If CB0 is much larger than CA0, CB0 remains approximately constant at all times, and Eq.14 approaches Eq.11 or 12 for the first-order reaction. Thus, the second-order reaction becomes a pseudo first-order rea
30、ction.,Caution 1. In the special case where reactants are introduced in their stoichiometric ratio, the integrated rate expression becomes indeterminate (不确定的) and this requires taking limits of quotients(份额) for evaluation. This difficulty is avoided if we go back to the original differential rate
31、expression and solve it for this particular reactant ratio. Thus, for the second-order reaction with equal initial concentrations of A and B, or for the reaction,化 学 反 应 工 程,(15a),the defining second-order differential equation becomes,(15b),which on integration yields,(16),Plotting the variables as
32、 shown in Fig. 3.3 provides a test for this rate expression.,In practice we should choose reactant ratios either equal to or widely different from the stoichiometric ratio.,化 学 反 应 工 程,Figure 3.3 Test for the bimolecular mechanisms, with CA0=CB0, or for the second-order reaction of Eq.15.,化 学 反 应 工
33、程,Caution 2. The integrated expression depends on the stoichiometry as well as the kinetics. To illustrate, if the reaction,(17a),is first order with respect to both A and B, hence second order overall, or,(17b),The integrated form is,(18),化 学 反 应 工 程,When a stoichiometric reactant ratio is used the
34、 integrated form is,化 学 反 应 工 程,These two cautions apply to all reaction types. Thus, special forms for the integrated expressions appear whenever reactants are used in stoichiometric ratio, or when the reaction is not elementary.,Irreversible Trimolecular-Type Third-Order Reactions. For the reactio
35、n,(20a),化 学 反 应 工 程,Irreversible Trimolecular-Type Third-Order Reactions. For the reaction,(20a),let the rate equation be,(20b),or in term of XA,化 学 反 应 工 程,On separation of variables, breakdown into partial fractions, and integration, we obtain after manipulation,(21),Now if CD0 is much larger than
36、 both and , the reaction becomes second order and Eq.21 reduces to Eq.14.,化 学 反 应 工 程,All trimolecular reactions found so far are of the form of Eq.22 or 25. Thus,(22),In terms of conversions the rate of reaction becomes,where M=CB0/CA0. On integration this gives,(23),化 学 反 应 工 程,or,(24),Similarly,
37、for the reaction,(25),Integration gives,(26),or,(27),化 学 反 应 工 程,Empirical Rate Equations of nth Order. When the mechanism of reaction is not known, we often attempt to fit the data with an nth-order rate equation of the form,(28),which on separation and integration yields,(29),The order n cannot be
38、 found explicitly(明确地) from Eqs.29, so a trial-and-error solution must be made.,化 学 反 应 工 程,One curious(古怪的) feature of this rate form is that reactions with order n 1 can never go to completion in finite (有限的) time. On the other hand, for order n 1 this rate form predicts that the reactant concentr
39、ation will fall to zero and then become negative at some finite time, found from Eq.29, so,Since the real concentration cannot fall below zero we should not carry out the integration beyond this time for n 1. Also, as a consequence of this feature, in real systems the observed fractional order will
40、shift upward to unity as reactant is depleted (耗尽).,化 学 反 应 工 程,Zero-Order Reactions. A reaction is of zero order when the rate of reaction is independent of the concentration of materials; thus,(30),Integrating and noting that CA can never become negative, we obtain directly,(31),Figure 3.4 Test fo
41、r a zero-order reaction, or rate equation, Eq.30.,CA0,CA,Slope = -k,Eq.31,=CA0 /K,t,t,1,XA,Slope =,Eq.31,CA0 /K,Note the deviation from zero-order,kinetics,which means that the conversion is proportional to time,as shown in Fig.3.4.,化 学 反 应 工 程,As a rule, reactions are of zero order only in certain
42、concentration ranges - the higher concentrations. If the concentration is lowered far enough, we usually find that reaction becomes concentration-dependent, in which case the order rises from zero.,In general, zero-order reactions are those whose rates are determined by some factor other than the co
43、ncentration of the reacting materials, e.g., the intensity of radiation within the vat for photochemical reactions, or the surface available in certain solid catalyzed gas reactions. It is important, then, to define the rate of zero-order reactions so that this other factor is included and properly
44、accounted for.,化 学 反 应 工 程,Overall Order of irreversible Reactions from the Half-Life . Sometimes, for the irreversible reaction,we may write,If the reactants are present in their stoichiometric ratios, they will remain at that ratio throughout the reaction. Thus, for reactant A and B at any time CB
45、/CA= , and we may write,化 学 反 应 工 程,-,-,or,(32),Integrating for n gives,化 学 反 应 工 程,Defining the half-life of the reaction, , as the time needed for the concentration of reactants to drop to one-half the original value, we obtain,(33a),This expression shows that a plot of log vs. log CA0 gives a str
46、aight line of slope 1n, as shown in Fig.3.5.,化 学 反 应 工 程,The half-time method requires making a series of runs, each at a different initial concentration, and shows that the fractional conversion in a given time rises with increased concentration for orders greater than one, drops with increased con
47、centration for orders less than one, and is independent of initial concentration for reactions of first order.,化 学 反 应 工 程,Figure 3.5 overall order of reaction from a series of half-time experiments, each at a different initial concentration of reactant.,化 学 反 应 工 程,Numerous variations of this proce
48、dure are possible. For instance, by having all but one component, say A in large excess, we can find the order with respect to that one component. For this situation the general expression reduces to,where,And the next is another variation of the half-time method.,化 学 反 应 工 程,Fractional Life Method
49、tF. The half-life method can be extended to any fractional life method in which the concentration of reactant drops to any fractional value F=CA/CA0 in time tF. The derivation(来源) is a direct extension of the half-life method giving,(33b),Thus, a plot of log tF versus log CA0, as shown in Fig.3.5, will give the reaction order.,Example E3.1 illustrates this approach.,化 学 反 应 工 程,Irreversible Reaction in Parallel. Consider the simplest case. A decomposing by two competing paths, both elementary reactions:,
