REACTION MECHANISMS FOR ENGINEERING DESIGN H. M. Hulburt
Y . G. Kim
esigners of chemical reactors face a double problem:
D equipment design to meet chemical specifications
and choice of process conditions to optimize or control reaction yields and conversions. The latter has been called “reaction design” in contrast to “reactor design” and in recent years has been receiving increased attention in commercial applications as well as in fundamental studies. From the viewpoint of the reaction designer, a chemical reactor is a device for producing a specified distribution in space and time of concentrations, temperature, pressure, and fluid velocities. These factors control the local opportunities for reaction at each point of the reactor. How these opportunities are utilized, of course, depends upon the molecular properties of the reactants, reaction intermediates, and products. The designer has little control over these intrinsic reactivities, but, worse yet, he usually has little knowledge of them as well. He is, consequently, largely dependent upon luck and gross empiricism in adapting the reactor design to the special requirements of the chemical kinetic system. This paper reviews the available principles and some of the more recent developments of methods for acquiring information on the rates of formation of the significant products of complex reacting systems as functions of the variables that define the local chemical kinetic state of the systeni-Le., concentration, temperature, and pressure. This information is usually referred to as the “mechanism” of the reaction. However, since the 20
INDUSTRIAL A N D ENGINEERING CHEMISTRY
term “mechanism” is often used with different connotations by chemists and design engineers, it will be desirable to attempt a more precise definition for our present purposes. A set of mathematical expressions (normally algebraic) for the rates of formation of each of a set of chemical species expressed as a function of the local temperature, pressure, and composition of the reacting medium will be referred to as the reaction kinetics. A set of chemical equations which, by the laws of stoichiometry and the law of mass action give rise to a set of mathematical expressions for the reaction rates, will be called a mechanism. The engineer needs the kinetics. He is interested in the mechanism to the extent that it supplies useful information on the kinetics. Since the same chemical steps may occur in a variety of mechanisms, there is the possibility of useful generalization in the development of kinetic expressions. The elementary reactions of one mechanism may recur in many others, and kinetic expressions associated with them may be combined to furnish the kinetics of many different complex reaction systems. I t is tantalizing to contemplate the possibility of thus developing a catalog of elementary reactions and their kinetics to be used as building blocks in developing the kinetics of complex reactions. However, progress in this direction is frustratingly slow. Simply the identification of the elementary reactions in a given case usually requirw many years and man-years of effort. There is a logical consequence of
The first scheme can be written in matrix form as
A
dt
kl
0 -kz
0 0
A B
0
k2
0
C
--k1
d B
=
C
(9)
The second scheme is
A
d
B
dt
F
-
ki 0
-
C
dr
- kl[A]
A-B;
--
B+C;
d- [- C - 1- k2[B]
dt
dt
(1)
dt A + B A+B+F
0 0 -k3
0
C = A0
I=[-!
-
k3
[A
0 0 0 0
A B F C
(10)
+B]
-kl' 0-kz
z][:i. ] A+B+F
(13) C=Ao-
[A+B+F]
Where A . = initial concentration of reactant A . assumed that no products are initially present. solutions are, in the first case
(2) A
We deduce
k2'
0
and the law of mass action which imposes a major difficulty. As long as our information is restricted to measured reaction rates only, we are free to postulate reaction intermediates almost at will, since any two successive reaction steps can be replaced by a three-step scheme which leads to indistinguishable kinetic predictions. For example, consider the elementary first-order steps
0 -k2'
-kl
+
A = Aoe-klt B = AOe-k2t
(14) I t is The (15) (16 )
and in the second,
d[B1 = k l [ A ] - kz[BB] dt
(3)
B 4 F
(4)
F+C
(5)
with rates kz ' [ B ]and ks [ F ] ,respectively.
dm dt
dt
+ B = Aoe-kl't A +B +F = A
Now add the steps
dt
A = Aoe-kat
= k3[F]
= k2'[B] = kl[A]
- ka[F]
- kz'[B]
We now find
(6)
(7)
(18) (19 )
Thus, the expression for C is identical in the two cases. The postulated intermediate, F, has no effect on the kinetic expression for C. I t does have an effect on that for B. I n the first case,
[ B ] = [Ao][e-'*t- e-k1t]
(20)
and in the second,
[ B ] = [ A ~ l [ e - ~-" e~- k 1 t ] (8)
(17)
(21)
Thus, only if either B or F were independently observable VOL. 5 8
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S E P T E M B E R 1 9 6 6 21
in addition to A and C would the hypothetical steps involving F be experimentally verifiable. A mechanism is thus open-ended in the sense that one can always add steps which have no experimentally verifiable consequence. We might remove this indeterminacy by requiring the minimum number of elementary reactions, as proposed by Boudart (5), who refers to a minimal reaction sequence. To do this, however, we must define independently what set of experimental results will be employed to test “experimental verifiability.” I t is at this point that chemists and engineers often adopt divergent viewpoints. When a proposed reaction step is not verifiable on the originally given set of data, the chemist will often seek to enlarge the set by adding new types of experiment. Thus, hydrocarbon-free radicals were postulated first to account for metal alkyl pyrolysis. I n paraffin pyrolysis, these hypothetical steps were not uniquely verifiable through kinetics of pyrolysis of pure hydrocarbons alone. New experiments on the effects of chain breakers, such as NO, enlarged the relevant kinetic data, but mass spectrometric analysis of pyrolyzing systems eventually gave direct confirmation of the existence of some of these species. Thus, a major addition to the arsenal of experimental techniques, plus the results of kinetic study of a whole galaxy of peripherally related reactions, was required to establish confidence in the existence and properties of hydrocarbon free radicals. The reactor design problem is much more circumscribed and is at once simpler and more difficult. The primary relevant species are, of course, the reactants and products with known properties and yields. The reactor must be designed to utilize and produce these in a specified way. The range of kinetic variables is limited in any one reactor, and the significant kinetic data need not extend beyond this range. Thus, a postulated intermediate whose influence cannot be detected within this range is of no consequence in the design. The minimal reaction sequence in this case may be much smaller than when a wider range of operating conditions must be considered. The engineer, therefore, usually will analyze a relatively small set of data and will discard “unnecessary” reaction steps in the mechanism to account for the kinetics reported to him, rather than expand the range of experiments to verify the effects of a postulated reaction step. Each approach has its proper place, but each must be used with circumspection. A set of kinetic data is limited in range of variables, and also in the precision of the results. The designer needs predictions with finite confidence limits. Thus, the criterion of kinetically significant consequences will require steps to be included if rates are known or required to *5”& which can be omitted if the rates need be known only to =+=15%. This effect of ’experimental error is often overshadowed by an even more important one. Tt‘e have referred to sets of mathematical expressions for reaction rates as kinetics and to sets of chemical equations as mechanisms. There is a correspondence between them, but it is not one-to-one in each direction. 22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Every mechanism gives rise to a kinetics which is unique in the space of algebraic functions we usually use. However, any given algebraic kinetic expression can be represented arbitrarily and closely by many mechanisms. One is tempted to say by an infinity of mechanisms, but the strict mathematical theorem involved here has not been established. Thus, mechanisms, or even minimal reaction sequences, “deduced” from kinetic data are not unique. The less precisely we require agreement with experiment, the more mechanisms will be indistiguishable. Also, the more restricted the field of experimental data, the more mechanisms will be found to be compatible with it, but not an infinitude of mechanisms as is sometimes suggested. The essence of design capability is to expand the field and range of the kinetic variables. We need to scale up, both geometrically and chemically-to predict with some confidence the effects on the relative rates of the steps of a mechanism when reaction conditions are altered from those of the original test data. It’e must be sure that the minimal reaction sequence which is adequate for our test data remains adequate for the plantoperating data. This means that we must somehow extend our test data to cover the anticipated range in the plant operation. The progression from test tube to bench reactor to pilot plant to semiworks to plant is one time-honored method of doing this. But there are other possibilities. Bench-scale experiments and the whole background of chemistry will suggest several possible mechanisms. These must be explored for their distinguishability under plant conditions. If the possibility of distinguishability exists, it may be possible to devise alternative small-scale experiments which will yield these distinctions and thus permit firmer predictions for plant performance. Test reactors for this purpose may bear little resemblance to the contemplated plant, but should be designed to yield the most significant data. Such “mechanism studies” are expensive and time consuming. Their economic value in a design problem should be fairly clear before undertaking them, but it is a mistake to assume without examination that kinetic data from test reactors not conceivably operable on a large scale arc a priori valueless for design. We may classify reactor and reaction design problems with respect to the importance of a knowledge of the mechanism. I n the first case, the main products of reactant are readily purified and are either pure compounds or simple mixtures. I n this case the reactor is called upon primarily to produce a specified quantity and yield of the principal product. Final control of the composition is achieved in the purification process, which can handle a moderate variation in feed materials. Very little specific mechanism information is required to design a reactor for such a purpose. Composition of the product stream will be defined by the major species only, with the minor species lumped as “impurities.” One or two reaction steps will usually suffice to represent the necessary data on overall yields and conversions. The advocacy of “effective” first-order rate constants
and such concepts as “height of a reaction unit” have some utilitarian justification in this case. A second case arises when the reaction product is a complex mixture which will be only partially separated into commercial products whose utility is a function of the product composition. Crude oil processing, catalytic cracking, and alkylations are typical of this case. Here some control over the composition of the feed to the separation system must be designed into the reactor section. However, it is often found that only a few key species suffice to determine the product distribution and the kinetic expressions for these key components are all the designer requires. A third case arises, for example, in polymerization, where the product is not separated and the reactor must produce a material of specified molecular structure as well as composition. Here extensive knowledge of mechanism is necessary if the reaction is to be designed as well as the reactor. One is often faced with serious problems of duplicating product quality if the reaction is transferred from one type of reaction vessel to another with different temperature distribution or residencetime distribution. I t is in this case that mechanism studies may be able to obviate expensive plant-scale experimentation. Let us now consider useful means of obtaining the requisite kinetic information and the minimal reaction sequences for these cases. All reactors are characterized by the local differential material balance for each species. Reactor types are distinguished by the form of these relationships, and reaction regimes are characterized by the way in which the boundary conditions and process variables depend on the process conditions. Let us denote the differential inflow-outflow relationship by an operator R (C$, T ,p, t ) , which in general depends upon the reaction system variables and time. Thus, for a batch reaction:
For a continuous ideal stirred tank,
For a tubular continuous reactor with constant fluid density,
R3C,
=
bCa
4- U(t, X )
ac a -2- - D(C,) ax
ax
bCa bx
(24)
Analogous expressions can be written for any reactor configuration. The material balance is completed by setting RC, equal to the local reaction rate per unit volume,
An energy balance equation in terms of temperature, T , and mean momentum equations in terms of the fluid velocity vector, y, fluid density, p, and the dynamic pressure, p , together with an equation of state relating p ,
p, T , and the C, complete the mathematical specification of the problem. Boundary and initial conditions, of course, must also be specified (2). The determination of a kinetics involves finding the functions M,, given the Rr and C,(t; CjJ. For the isothermal batch reactor, Equations 22 and 25 prescribe the data treatment directly.
Moreover, for a steady ideal stirred tank with no net accumulation of material, Go = GI = G.
G V
- (Co - CoJ
=
If Ci is a continuous function of 8
Mi =
(27)
V/G, for 8 + 0,
which has the same form as Equation 28. Likewise, for an isothermal tubular reactor of length L in steady radially uniform flow with negligible diffusive transport (Le,, u = constant, D = 0) ; Equation 24 leads to
where0 = L/U. For practical purposes, these three equations, all of the same form, represent the only useful treatment of kinetic data at present for mechanism determination. Since temperature enters the rate expressions, Mi, it is very difficult to treat nonisothermal data in general; although, if the form of M,(C,) is given, the rate constants may be evaluated in favorable cases with some degree of success from nonisothermal data (32). I n principle, if U(R) and D are specified in Equation 24, M,(C,) can be recovered numerically without knowing the form of the relationship in advance (74, 22), but computational exigencies restrict use of this case to laminar flow. Acrivos and ChambrC (7) give relationships for external boundary layers of general geometry which should be invertible to give M a if concentrations of each of the species can be measured accurately throughout the boundary layer. To date, neither of these elaborations has yielded actual results. Having obtained an empirical function, M,(C,, T , p ) , by these means, it is a matter of ingenuity and curvefitting to discover a set of pseudoelementary reactions obeying the law of mass action to reproduce the empirical data. A variety of standard texts and treatments are available for guidance in this procedure (4, 20, 24, 28, 37). We shall not pursue the familiar details here. With only a few exceptions, elementary reactions are either first-order or second-order. Therefore, almost all the functions, M,, we shall need will have the form
where k,, and krjkare the specific reaction rates for proVOL. 5 8
NO. 9
SEPTEMBER 1966
23
duction of species i by reaction of species j or by reaction of species j with species k , respectively. I n most mechanisms, most of the k z j hwill be 0, since most M i will involve only a small number of species. Furthermore, there are always some linear relations between the C, which arise out of the overall conservation of mass and the stoichiometric relationships of the given mechanism. General methods of formulating these relationships are given by a number of authors (2). For first-order or pseudofirst-order systems, Wei (38) has shown how results analogous to Equations 17 to 19 can be formulated for very general mechanisms. Linear combinations of concentrations can always be found which have a simple exponential time dependence in a batch reactor or equivalent. Most important, however, is IVei’s demonstration of how these pseudocomponents can be identified by sequential experimentation without necessarily mapping the whole C,(t) field for the complex system. The so-called “straight-line” reaction paths can be located by judicious choice of initial conditions in a succession of differential rate determinations with a consequent large reduction in the number of experiments necessary to give estimates of the k,,’~ to any assigned degree of precision. These techniques for the design of kinetic experiments have been profitably exploited in recent instances (38) and should become part of standard practice in experimental design where they are applicable. Unfortunately, no such far-reaching methods are as yet available for nonlinear systems, nor is it a priori evident that a given kinetics should have a first-order mechanism. The necessary underlying mathematical study of second-order mechanisms has hardly been begun ( 7 , 38). A deep-seated physical consideration which has not been fully exploited in the mathematical analysis involves the reversibility of reactions. Broad dynamic principles convince us that every molecular process has a reverse and that the specific rates of the forward and the reverse processes are such that for a system in equilibrium the reverse process goes with the same velocity as the forward process. The net rate is thus 0, as required by the equilibrium condition. What is excluded, however, by this principle of detailed balance is the possibility of a chain of essentially irreversible processes which regenerate the original state of the system in a finite number of steps. Such a chain of reversible steps is not forbidden, but the original state in such a case can be regained by a direct retracing of the forward process. Thus, every elementary reaction has a forward rate and a reverse rate which should be included in a complete mechanism. This physical law puts a severe limitation on the kinds of solutions one may expect to Equations 22 to 24, for example. Thus, for a batch reactor, Wei has shown that reversible first-order systems can never develop stable oscillations, but always approach an equilibrium state monotonically after sufficient time has elasped. First-order systems not subject to microscopic reversibility may be unstable or oscillatory, but such systems cannot represent chemical k.inetic mechanisms. 24
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
However, though it is useful for many purposes to recognize that all reactions are reversible, it is also important to recognize that in many cases some of the specific rate constants are extremely small. For practical purposes, the system then becomes irreversible, and one cannot be sure that transient oscillations, once excited, will be rapidly damped. Moreover, Equations 22 to 24 are more general than Equations 26 to 29, to which they reduce for steady operating conditions. When temperatures, pressures, or flow rates are subject to variation during the reaction, it will be possible to find conditions leading to steady oscillations in reaction rate and product composition. Each of us contains such a complex oscillatory reaction system with a frequency of about 75 to 100 cycles per minute. However, we are concerned in this paper with the possibility of imposing oscillatory conditions in such a way as to enhance the kinetic information we can extract from the responses. This will be possible when the steps of a mechanism have differing rates of response to a disturbance. If a sudden disturbance is cyclically repeated often enough so that steady response is never attained, the average outputs will differ from those of steady response. The departure from steady response will depend upon the frequency of the disturbance. At very low frequencies, a new steady state is attained in each cycle, and the average response approaches the average of the steady-state responses attained. At the opposite limit of very high frequency, the system cannot “follow’) the disturbance, and the average response is the same as the steady-state response to the average disturbance. At intermediate frequencies, however, the response cannot be expressed in terms of steady-state responses, and here new information may be obtainable. “Low’) or “high” frequency in this connection is evaluated relative to the half-time for decay of a step disturbance in an ideal plug flow reactor. Thus, in Equation 24 with constant velocity U = UO,negligible dispersive transport, D = 0,
Just before the disturbance, we assume steady conditions. That is, at t = -0,
c, = C,‘
(32) (33)
where 0’ C,’(O; t ) =
=
U’/L
ctol=
C,l(Ol = 0 )
(34) (35)
At t = 0, a step function change is imposed on either U , T , C, or p. Equations 32 to 35 no;v serve to define initial conditions for Equation 31, and we seek the time, t = tip, at which C, = Cj’/2. This half-life or relaxation time is the inverse of the characteristic frequency wR, which serves as a reference in defining “low” and “high” frequency disturbances.
Evidently, the relaxation time, t i p , will depend upon the nature and complexity of the rate function, Mi. I n a complex mechanism, each step, if it could be isolated, would have its own relaxation time. Hence, we would expect a periodic disturbance to affect most strongly those steps of a mechanism with matching characteristic frequencies. When the reaction is not first-order, the relaxation time, tl/z, will vary with the initial conditions, and the detailed analysis of the nonlinear kinetic equations can be quite complicated. I n some nonchemical nonlinear systems, strong interactions are found at frequencies which are fractions of the characteristic exciting frequency, in contrast to linear case. I n nonlinear systems, also, the shape of the response function can be sensitive to the shape and frequency of the disturbance in more complex ways than for linear systems (34). T h e inflow-outflow and transport functions of a reactor have characteristic relaxation times, also. Since a new temperature, pressure, or residence time distribution requires time to become established in the absence of chemical reaction, we may expect to find characteristic frequencies for the reactor transport processes. These are readily excited by manipulation of operating variables and in turn interact with the chemical relaxation processes. I n fact, this interaction is one of the more fruitful sources of information on reaction kinetic processes in heterogeneous catalysis. One of the earliest studies of this sort was by Wilhelm and Deisler (47),who varied the composition of a gas mixture flowing in steady conditions over a catalyst. T h e outflow composition was found to show oscillations of lower amplitude than the input composition and to show its maxima at a later time than expected from a plug-flow residence time delay. This attenuation and phase shift was interpreted in terms of the adsorption and desorption rates of the ethylene in the catalyst controlled by the rate of diffusion in the pores of the catalyst. Numerous later studies have been made on the same principle (23, 27, 33, 36, 37). I n most cases, experimental difficulties restrict observations to relatively low frequencies, which turn out to be related more often to transport relaxation processes than to chemical relaxation processes. T h e reason is not hard to see. An irreversible first-order reaction, by Equation 15 has w R = k1/0.693, the inverse of the half-life of the reaction. Higher order reactions have half-lives which vary with the reactant concentration, but the initial half-life will give the order of magnitude of w R . For typical catalytic reactions, the overall half-life will be about 0.1 to 1.0 sec.-l Estimation of surface rates indicates that the halflives may be much shorter, giving w R about lo3 sec.-l For homogeneous reactions involving reactive intermediates, the values of wR may be as high as 107 sec.-l For a diffusion transport process, the relaxation time will be of the order of the Einstein diffusion time and
where
D is the diffusivity in the pore of the catalyst and
L is the diffusion path length. For typical gases in sq. cm. sec.-l and for 1/8-inch catalyst forces, D w pellets, L = 0.15 cm. This gives w R w 0.1 sec.-l for axial diffusion in a pore, well within the experimentally manageable range. This suggests that it will require special techniques to excite relaxation phenomena involving the intermediate chemical rates. I n a complex system, the approach to equilibrium will be governed finally by the longest relaxation time. However, if there are two or more steps which differ by an order of magnitude in relaxation time, a stepwise response to a step input disturbance may be observable as first the faster and then the slower processes occur. Thus, transient experiments may help in identifying steps with relaxation times which are markedly different. Although batch reactor, stirred tank, and ideal pipeline reactor all become kinetically equivalent in steady state, being characterized completely by the mean residence time, 0, Equations 22 to 24 show that they differ in unsteady state response. These differences are clearly displayed, even for small disturbances from steady operation. For the stirred tank reactor, Equations 23 and 25 may be written (37) where the oscillating feed rate is w 1 = oO(1
+ a sin w t )
(38)
Let us now define the perturbed concentration x, by
c, = c, + x, and choose
(39)
c, to satisfy the steady flow operation.
Then
or
X,(O) = 0 Xd0 5
0
This corresponds to the case of constant feed composition and oscillating feed rate. I n a batch reactor, we may take to be the equilibrium. Then M,(C,) = 0 and
c,
dx, _ -- ( M , dt
-
lit)
(45)
If the x , are small, usually because the disturbing force is small, we may expand the perturbed rate to obtain
M i-
a, =
j
blW 6xj+ higher powers of x , bC,
VOL. 5 8
(46)
NO. 9 S E P T E M B E R 1 9 6 6 25
&a-
bM
dC,
dC,
where -2 indicates the value of
’
evaluated at C, =
C j , j = 1, 2 . . n. Equations 41 and 44 are now linear in the x, but with time-variable coefficients. The flow case, Equation 41, has a time-variable forcing function as well. Several possible applications may be made of these results. I Y e may perturb steady flow conditions by introducing a steady oscillation in the feed rate. If this oscillation has a large amplitude, Equation 41 will be nonlinear when M , is higher than first-order, and numerical integration must be used to obtain specific results. Hulburt and Lamberty have calculated the results of this sort of perturbation for a heterogeneous catalytic reaction in a stirred-tank reactor, assuming a dual-site mechanism.
A+S*A,
B-kS$B, A , f B, @ C, -I- D ,
(47)
c+s*cs
(48) (49) (50)
D+S*Do,
(51)
The catalytic vapor phase esterification of acetic acid with ethanol over silica gel has been claimed to go by this mechanism (25). Since the adsorbed species and the catalyst S do not leave the reactor, Equation 45 must be used for them in placc of Equation 41. This effectively adds a capacity to the system, namely, the inventory of adsorbed species, and introduces an effective lag in the response of the output concentrations of the gaseous species. FVhen the flow rate is varied, the purge rate of the reactor is varied and hence the composition of the gas phase. This modifies the adsorption and desorption rates and then the surface reaction rate, so that the output concentrations lag behind the values they would attain in the absence of reaction. Moreover, the amplitude of the concentration fluctuation will be less for slowly adsorbed or desorbed species than for rapidly adsorbed or desorbed species. This affords a means of identifying these species with less ambiguity than in steady-state kinetic ineasurements. In Figure 1 the calculated predictions of such a measurement are presented for a set of assumed values for the rate constant (25). Whether the conversion per cycle exceeds the steady flow conversion rate depends upon the nonlinear steps of the mechanism. For some types of perturbation, the response becomes quite asymmetric. I n such a case, the average concentration per cycle may exceed that for steady floiv with corresponding enhancement of rate. ]\‘hen temperature fluctuations are employed, this asymmetry is greatly accentuated, and it is possible that self-sustaining oscillations in rate could be excited (27, 30). Denbigh (12), Christiansen (S), and Bak ( 3 ) , among others, have considered kinetic mechanisms in flow systems which could have self-excited oscillations in rates under isothermal conditions. The chemical rnechanisms appear to be rather special, and one cannot expect to induce 26
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 7. action, A
Step function response of Concentration in a multistep re= H I (Equation 47)
+D
+
such oscillations generally. What is required is a chemical feedback in which a reaction product is involved as a reactant in an earlier step of the mechanism leading to its production. This occurs in anticatalytic reactions in open systems where it is possible to regulate independently the concentration of some of the species whose reaction is being catalyzed. I n such a case, oscillating rates can arise, even for steady feed conditions. IVhen the departure from a steady state is small, one may deal with the linearized equatioris. For batch reactions near equilibrium, this approach is known as the “relaxation method” and has been developed in recent years most actively by Eigen (76) in studying fast reactions in solution. Relaxation Method
This method is now widely used in studying fast reactions in solution (75). I t consists of perturbing a reacting system initially at equilibrium and following the response of the system as it approaches a new equilibrium state imposed by the perturbing influence. The perturbation must be small enough so that the differential equations describing the response can be linearized with respect to the driving forces, the difference between the time-dependent concentration and the new equilibrium concentration for each species participating in the reaction. Consider the following siniple example :
Designating the concentration of each species by C1, Cz, and C3, and assuming the reaction to be of the first order with respect to each species, we can write
If the system is slightly perturbed from a state of equilibrium, the time-dependent concentration can be written in the form
c, = c, + A'$
c,
where is the new equilibrium value and X , is the timedependent deviation of concentration from the new equilibrium value. I t will be assumed henceforth that
X,
IIk
(54)
k2, k4
Same as scheme (1) with k l ; k2 > k3,
k4
kz
ka
A2 F! A4 kP
Same scheme as 111 with kl,
(IV)
k2
> Czo, the resulting simplification gives
Thus, it is seen that experiments carried out under well chosen conditions allow one to judge the validity of a mechanism and determine rate constants as well as equilibrium constants from the concentration dependence of relaxation times. For both mechanisms I1 and IV, 7 1 is independent of concentration and 7 2 depend on concentration. But the concentration dependence of T Z is different in the two cases, which makes it possible to distinguish between them. If the fast step involves monomolecular k4 can be reactions, as in mechanism 11, only k~ determined from the measurement of T I ; but, since Ka4 can be determined from the concentration dependence of 7 2 , the individual rate corstants k3 and k4 can be calculated. Experimental determination of relaxation times is accomplished by shifting equilibrium of a reacting system by a change of external parameters and detecting the concentration change of a reactant. Dependent on
+
how the equilibrium shift is induced, the experimental methods are divided into two types, transient and stationary methods. I n the transient methods the forcing function is applied once, and the transient response of the system is followed in real time. The stationary methods use stationary periodic forcing functions and obtain kinetic information from phase lag and attenuation of the frequency response diagram. Concentration change can be detected by a number of methods, the most common ones being those utilizing optical or electrical properties of the reaction mixture. A thorough discussion of various experimental methods is given in a comprehensive treatise by Eigen and De Maeyer (77), and only typical examples will be given below to illustrate the kind of experimental situations involved. (1) Transient Methods. The most common external parameter used to induce equilibrium shift is temperature. Any well defined forcing function can be used, and in the reference cited (77) the possibility of using rectangular step function, step function with linear increase, step function with exponential increase, rectangular pulse, and damped harmonic oscillation is discussed. The simplest of these is the rectangular step function. The commonly used temperature-jump method is based on this type of forcing. For this method the temperature of the reacting system in equilibrium is changed in a time much shorter than the fastest chemical relaxation time, and a fastrecording detector is used to follow the time course of concentrations toward the new equilibrium. This method is developed well enough so that there is a commercial unit available, which combines temperature-jump with detection device based on light transmission or fluorescence measurements. A typical heating time constant for temperature-jump method is of the order of a few microseconds, and any reaction with half-life larger than the heating-time constant can be studied by this method. Pressure-jump and electric-impulse methods have also been used. The range of relaxation times experimentally accessible by the pressure-jump method is sec. to 50 sec. in an apparatus devised by Strehlow and Becker (35). A strong electric field is applied to shift to equilibrium of weak electrolyte solution in the electric-impulse method. The range of relaxation times accessible can be as small as lO-’sec. (2) Stationary Methods. If the external parameter on which the equilibrium of the reacting system depends is caused to oscillate periodically with a frequency f per unit time, the response of the system indicated by concentration of one of the species will depend on the rapidity with which equilibration takes place-i.e., on chemical relaxation time. If the relaxation time is large compared with the period of oscillation of external parameter, the reacting system will not “see” the oscillation and concentration will not fluctuate. O n the other hand, if the relaxation time is much smaller than the period of oscillation, the concentration will oscillate in phase with the external parameter. Between these
u,
4 U c 0 c
71
72
LOGARITHM OF THE TIME Figure 2.
Total signal as a function of time
two extreme behaviors, there will be relaxation times which will cause the concentration to oscillate with the same frequency, but lagging behind the imposed frequency in phase. Such a phase lag is a measure of the relaxation time. An acoustic wave is the perturbing agent in the most commonly used stationary method. Because sound waves are propagated adiabatically, they set up small periodic fluctuations of temperature and pressure in the reacting medium. Reactions whose equilibria are affected by temperature and/or pressure will absorb energy at frequencies whose periods are comparable to the relaxation time of the reaction. As the period of oscillation of the external parameter is decreased from a large value (corresponding to the case where the concentration oscillation is in phase with the imposed oscillation), not only phase lag but amplitude attenuation will also appear. The phase lag represents energy dissipation of the acoustic wave. I t can be shown that this energy dissipation per cycle is proportional (w7)/(1 u % ~ )where 7 is the relaxation time and w is the angular frequency. This dissipation is a maximum a t w7 = 1, and it is this maximum that is experimentally measured. There are several different experimental techniques for measuring ultrasonic absorption (77)) and it is possible to measure relaxation times in the range lop3 to 10-’0 sec. by using two or more of the various techniques. The use of sound absorption for studying chemical relaxation is only one application of the more general study of molecular processes by dispersion and absorption of sound, for which there is an extensive literature (29). Dispersion of acoustic waves gives essentially the same information as the absorption when applied to chemical relaxation phenomena. Let us illustrate the kind of experimental data one might obtain to test the mechanisms in Table I With a typical transient method one might use a temperature jump to induce a shift in equilibrium and follow the concentration change with a spectrophotometer con-
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NO. 9 S E P T E M B E R 1 9 6 6 29
N
c
2.
INITIAL CONCENTRATION, Ci0 Figure 3.
Relaxation time as a function of concentration
nected to a fast recording device. A total signal, As, which is a measure of the concentration of one of the species, changes as a function of time. For any of the mechanisms in Table I, one would obtain three breaks in the total signal us. log of time curve as in Figure 2. The shortest relaxation time, 7 0 , is attributable to the temperature dependence of the transmission coefficient of the reacting system. I t is assumed in temperaturejump experiments that the heating time is much shorter than the chemical relaxation times and, hence, T~ is much shorter than 7 1 or 7 2 . The other two relaxation times, 7 1 and 7 2 , are attributable to the actual reactions. If mechanism I1 is being tested, Equations 71 and 72 indicate that the longest relaxation time, 7 2 , may be measured at different initial concentrations, CIo. If the equilibrium constant, K3*, is known, 7 2 us. CIn measurements can be made under either one of the conditions, Cln = Czo or C1O >>Cz0. From the slope and intercepts (Figures 3 and 4) plus the known equilibrium constant, K34,k l and k~ can be determined independently. If K 3 4 is not known, it will be necessary to measure 7 2 under the condition C1° >> Czofirst and then under Cl0 = CZO. From two such series, one obtains k l , k2, and KW Because the fast relaxation time, 7 1 , is concentration independent, one needs, in principle, only one measurement. I n fact, if mechanism 11 is correct, 7 1 obtained from any of the above experimental runs should be fairly constant. From 1/71 = ka kq and K34 measured previously, k3 and k4 can be determined. An extensive compilation of rate constants obtained by relaxation methods is given by Eigen and his coworkers ( 7 7 , 75, 79). The data range from simple isomerization reactions to fairly complex enzymatic reactions. Because this review is not concerned with particular reactions but rather with discussing a method for possible use by chemical engineers, no further comments will be made on experimental results for individual reactions. A comparison of the relaxation method with the method most commonly used by chemical engineersi.e., the steady flow method-will now be made.
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30
INDUSTRIAL A N D ENGINEERING CHEMISTRY
INITIAL CONCENTRATION, Ci0 Figure 4.
Relaxation tame
as
a function of concentration
-One advantage of the relaxation method over the steady flow method is that the relaxation method is not limited by the mixing time of the reactants. The mixing time is of the order of 1 msec., and a reaction whose halflife is of the order of 1 msec. or less cannot be studied by flow method. With the relaxation method, the experiments start with the system already in equilibrium and therefore half-life of the reaction accessible by this method is not limited by the mixing time. -The relaxation method is concerned with s) stems near equilibrium, and the equations describing the system are linear. This facilitates evaluation of mechanism and constants for complex systems. -The relaxation method gives thermodynamic parameters as a by-product for each observable reaction step. -The relaxation method gives the individual rate constants, whereas with the steady flow method, lumped rate constants are obtained if the reaction in question is a complex one. For example, in the study of enzymatic reactions in terms of the Michaelis-Menten mechanism, or the Briggs-Haldane modification of it, the flow method gives only the Michaelis constant which is a lumped parameter including three rate constants. The relaxation method enables one to determine all three separately. There are two major limitations of the relaxation method. -For complex reactions with several relaxation times, the individual relaxation times can be obtained only if they are separated by a substantial amount in time. Two relaxation times that are close together will not be distinguishable. -Because the relaxation method treats the system near equilibrium, it cannot be used effectively for reacting systems that are virtually irreversible. Perturbed Flow System
Although the relaxation method has been applied so far only to perturbation of an equilibrium state and, hence, is lirnitrd to reversible reactions, Equation 41 affords a similar basis for extending the method to
irreversible reactions in flow systems. The reference concentration is now taken to be the concentration attained under steady flow conditions. A small periodic perturbation in flow rate leads to Equation 41 corresponding to the stationary relaxation method. The change in chemical rates must be small enough to be linear in the concentration perturbations. This equation differs from the relaxation equation, Equation 45, only in the addition of the space velocity, ol(t),to the rate constants for disappearance of species i, and in the forcing function, - a Z , sin at. I n ultrasonic relaxation, every rate constant is multiplied by a periodic function. The perturbed flow case should thus give results which are easier to interpret. If the flow rate is given, a step increase from wo to a new steady value, w1, Equation 41 is replaced by
c,
absent in the steady flow method. This is the requirement of rapid perturbation of the equilibrium. One can think of situations in which this requirement is difficdt to satisfy. NOMENCLATURE = relative fluctuation amplitude = concentration of species i = concentration of species i in steady state = concentration of species i in initial steady state = effective diffusivity = volumetric flow rate = equilibrium constant = specific rate constant = reactor length = rate of production of species i per unit volume = rate of production of species i per unit volume in steady state = pressure = reactor operator, Equation 22 = temperature = elapsed time = half-life = axial velocity = reactor volume Ifluid velocity vector = linear coordinate = deviating concentration
a
Ci Bi
Ci’ D
G Kij
Ki L Mi ltii
P Ri T t til2
Ao -
AT,) - - M ,
U
(73)
On
where Aw = given by
w1
- wo. The new steady state is
x;,
V
Y X
Xi
Greek Letters B
x’
M
X
a1
wo
A M , - .&TuA~
r
(74)
W1
I t is thus given by the increase in moles converted per residence time induced by the change in residence time. As before, the constant space velocity, w1, is added to the rate constants for disappearance of C, and the constant forcing function ( A w / w ~ ) n , acts to make the concentration wo of product species i fall by virtue of the reduced residence time. T h e analysis is, thus, similar to that for transient relaxation in batch equilibrium systems. The flow system has an added first-order process, the purging of the vessel, whose relaxation time must be distinguishable from that of the chemical changes if the latter are to be experimentally observable. Thus, only for reactions with half-lives long or short relative to the purge time will it be possible to get good values of the rate constants. When Aw/wo is small, the effect of the reduced residence time on the average rate will be seen only at long times. T h e observed early relaxations will, thus, be attributed to chemical effects or to simple purging effects, and these will be easily separated. T o date, the authors are not aware that this technique has been applied, but it should have worthwhile possibilities. I n discussing the relative merits of the relaxation method us. the steady flow method, we confined our attention to those aspects pertaining to the principles, and not the practical or experimental ones. The relaxation method in general shares with the steady flow method any difficulties that arise in determination of concentrations. If anything, the relaxation method puts more strenuous requirements on this part of the experimental work because of the need for measuring small change in concentration very rapidly. I n addition to the analytical problems in common with both methods, the relaxation method is burdened with another experimental problem
WR 01,
00
= mean residence time = relaxation time = relaxation frequency = space velocity
REFERENCES (1) Aris, R., IND.ENG.CHEMFUNDAMENTALS 3, 28 (1964). (2) Aris R ‘‘Introduction t o the Analysis of Chemical Reactors,” Preatice.Hal1, New Lor;, 1965. (3) Bak, T. A., “Contributions t o the Theory of Chemical Kinetics,” Benjamin, New York, 1963. (4) Benson, S. W., “The Foundations of Chemical Kinetics,” McGraw-Hdl, New York, 1960. (5) Boudart, M., Second ASEE Conference on Chemical Engineering Education, Boulder, 1963. (6) ChambrC, P. L., Acrivos, A., IND.ENG.CHEM.49, 1025 (1957). (7) Chambrc?, P. L , Acrivos, A., J . APpl. Pkys. 27, 1322 (1956). (8) Christiansen, J. A., “Advances in Enzymology,” Vol. 22, Interscience, New York (1962). (9) Czerlmski, G . , Reu. Scz. Instr 33, 1184 (1962). (10) Czerlinski, G , J . Theoret Btol 7, 435 (1964). (11) Czerlinski, G . , Schreck, G , Btockem. 3, 89 (1964). (12) Denbigh, K G., Hicks, M., Page, F. M., Trans. Faraday Soc. 44, 479 (1948). (13) Douglas, J. M., Rippin, D. W. T., Ckem. Eng. Sa. 21, 305 (1956). (14) Dranoff, J. S , Math. Comp. 15, 403 (1961). (15) Eigen, M ,Advances i n Enzymology 25, 1 (1963). (16) Eigen, M., Discusstons Faraday Soc. 17, 194 (1954). (17) Eigen, M I “Technique of Organic Chemistry,” Vol. 8 , Part 2, A, Weissberger, ed. (1963). (18) Eigen, M Kruse, W., Maass, G., De Maeyer, L., Progr. in Reactton Ktnctics 2,287 (1964j.’ (19) Eigen, M , Kustin, K , J . Am. Ckem. Soc 84, 1355 (1962) (20) Frost, A. A., Pearson, R . G., “Kinetics and Mechanism,” 2nd ed., Wiley, New York, 1961. (21) Higgins, J., ACS S mposium on A plied Reaction Kinetics and Chemical Reaction Engineering, hashington, D. , June 13-15, 1966. (22) Kate, S., Chem. Eng. Sct. 10, 202 (1959). (23) Kramers, H., Alberda, G , Ckem. Eng. Sct. 2, 173 (1953). (24) Laidler, K . J., “Chemical Kinetics,” 2nd ed., McGraw-Hill, New York, 1966. (25) Lambert?, L E ,M.S. Thesis, Northwestern University (1965). (26) Mars, P , Ckem. Eng. Scr. 14, 375 (1961). (27) Mayer, F. X , Rippel, G. R.,Chem. Eng. Progr. Symp. Ser. 59, (46), 84 (1963). (28) Petersen, E. E., “Chemical Reaction Analysis,” Prentice-Hall, New York,
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(29) Proc. of Int. School of Phystcs Course 27 “Dispersion and Absorption of Sound by Molecular Processes,” Acahemic Pres;, New York, 1963. (30) Salnikoff, I. E., Volter, 8 . V., Dokl. Akad. Nauk. SSSR 152, 171 (1963). (31) Satterfield, C. N., Sherwood, T. K.,“ T h e Role of Diffusion in Catalysis,” Prentice-Hall, New York, 1965. (32) Schmidt, J. P., Mickley, H. S., Grotch, S.L., A.I.Q.E. J . 10,149 (1964). (33) Stahel, E. P., Geankoplis, C. J., Ibtd., p. 174. (34) Stoker, R., “Non-Linear Vibrations,” Interscience, New York, 1956. (35) Strehlow, H., Becker, M., 2. Eleklrachem. 63,457 (1959). (36) Taylor,H. M., Leonard, E. F., A.I.Ch.E. J . 11,686 (1965). (37) Turner, G.A., Chem. Eng.Sct. 7,156 (1958); Ibid., 10,14 (1959). (38) Wei, J., IND.ENO.CHEM.FUNDAMENTALS 4,161 (1965). (39) Wei, J., Prater, C. D., Ado. in Catalysts, 13, 203 (1962). (40) Wei, J., Zahner, J. C . , J . Ckem. Pkys.43,3421 (1965). (41) Wiihelm, R. H., Deisler, P. F., Jr., IND.ENG.CHEM.45,1219 (1953).
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