S I D N E Y W. B E N S O N
The characteristic of a chain reaction is that the sum of the chain lor propagation1 steps yields the overall reaction stoichiometry-the chain carriers canceling out in the addition. The closed reaction sequence involving the intermediates in a pyrolytic reaction is shown above lsee Equation El. Reactant is consumed and products produced, but the concentration of intermediates which propagate the reaction is controlled only by distinct initiation and termination steps. The number of over-all reactions between the initiation and termination steps is called the chain length. In this article, pyrolytic and metathetical chain reactions are discussed in terms of our recently acquired knowledge of prototypes of their component step-initiation, transfer, propagation, and termination. Predictability of products and product ratios is shown to be fairly good using Rice-Hertzfeld mechanisms and known data for radical reactions. Some semiquantitative generalizations concerning radical step reactions lead to about order-of-magnitude ability to predict overall chain rates.
early days of gas phasekinetics, about 1925-1935, Iandnthethenaivett. field was pervaded with a wonderful exuberance Despite the obvious complexity of the products arising from most cracking reactions-e.g., CHI C&X, from C&,-it was felt that all reactions could be classified as either first or second order. The logic of such a point of view was based on the expectation that despite the chemical complexity of the system, one process leading to radicals or intermediates would be the slow, rate-determining step and all the following steps would be immeasurably rapid. Aiding and abetting this simplicity was the perfectly reasonable theory of unimolecular reactions which showed that the order of a unimolecular reaction could vary anywhere from first to second order depending an the relative rates of collisional deactivation of and band breaking in a critically energized molecule. An excellent example of this approach is provided by the pyrolysis of CHtOCHIwhich gave as products C H O CH, (30-5%) and CO Ht CH, (70-95%). It fallowed a beautiful first order kinetics during a run but the first order rate constants fell by a factor of 7 as
+
+
18
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
+
+
the initial pressure changed from 1500 mm. of Hg to 30 nun. of Hg. We know now of course (8)that this was a completely fortuitous result of the fall-off in rate of a 3/2-order reaction being offset by an autocatalysis produced by products (HZ and CHX)). Since those days the pendulum has swung quite far to the other side. Most of the time it appears an almost hopeless job to interpret the kinetic data for chain reactions in any unique and meaningful way that will incorporate the obvious chemical complexity of the systems involved. It is the purpose of the present manuscript to explore this rather foreboding situation and see what steps have been and can be taken in unravelling the kinetic path of complex chain reactions in terms of our present not meager knowledge of the elementary steps occurring. Cateaories and Jhuctun of Chain Readions
Before beginning our discussion let us note that complex chain reactions fall into a few classifications, the structural elements of which are very few. From the point of view of reactants we can identify two types of chains.
Pyrolytic Chains. One of these is the cracking or pyrolytic chain typified by ethane or neopentane pyrolyses where overall products are largely: Cz", 7C& H, (A) neo-CsH12 i-C,HS CHI (B) Metathetical C h a i n s . In contrast to these are the metathetical chains involving two reactants. Typical examples of these are the halogenation reactions of Hz or hydrocarbons or the sensitized, telomeric additions of halocarbons such as CCq to olefins:
+
7
+
RH+Xs+RX+HX (C) (R)1C=C(R)z CCli -+ Cl(R)G-C(Rs)CCq (D) The reaction of polymerization, though not strictly a chain, can be subsumed as a special case of the metathetical chain. The real distinction between these two classes of reaction lies in the Occurrence of a unimolecular cradring step in the pyrolytic chain. I n the case of C a s pyrolysis the chain is:
+
1
VOL 56
NO, 1
JANUARY 1964
which generally controls the overall concentration of chain carriers. For thermal halogenation reactions, these are frequently simple equilibria such as:
For neopentane the basic chain is: (CHahCCHa
+ CHI
1
(CHI)IC--CH;
2
3
7
( C H ~ S M H Z (CHI),C=CHI
+ C&
(F)
+ CHI
By contrast the halogenation chain contains only bimolecular steps :
X
+ RH f R + H X
R
+ X,+RX + x
1
+ (R)zC=C(R),
1
(H)
3
+ ccb s
(R)~&c(R),ccI,
4
CI(R)rC+C(R)rCCla
secondary chain
+ CH30CHa1,CHI + CHzOCH3 CHzOCHa
H
5 CHzO + CHa
+ CHX) A H~ + CHO CHOAH+ co
+ 01
0
+ HI
---t -+
2 CH,
+ CHO
-+
(1)
(J)
+0 HO + H
HO
One may expect such behavior in all chains involving the cleavage of multiply-bonded atoms or groups. Two additional features characterize chain reactions. One of these is the initiation and termination system
INDUSTRIAL AND ENGINEERING CHEMISTRY
CH,
-+
C&
+ OCH,
-+
CHIO
+ CO)
(L)
(M)
+ CO
The second feature is the transfer reaction. This is an element in chain reactions which appears to have been first appreciated in terms of polymerization reactions. It has however an ancient history in pyrolysis systems. Taking the C a Hpyrolysis S as an example, one finds that in the temperature range 85O0-12OO0 K., the most rapid homogeneous initiation reaction is the fission of the C C bond to form 2 CH, radicals. However the CZH~chain (Reaction E) does not involve CH, radicals. There is instead a transfer of activity from the initial CH, to the carrier C a , :
+ CZHO
1
Transfer.
CHI
CH,
+ CZHI
(N)
Since these reactions are generally reversible and have equilibrium constants not too much greater than unity, the transfer efficiency, given by the radical ratio, (C&)/(CH,) is kept high by the fact that (CZHO)/(CHI) is usually large-i.e., (CI&)/(CH,) = KN(CZ",)/(CHJ, where KN = kl/k;. In special cases, the proper combination of two transfer reactions may constitute an instance of genuine homogeneous catalysis. An old example of this is the generally observed acceleration produced in ppolytic chains by adding Hz gas to the system. These reactions are strongly endothermic (20 to 30 kcal./mole) and so are usually aecompanied by "self-cooling" temperature gradients. Hydrogen, with its high thermal conductivity, will tend to maintain bath temperature and so appear to accelerate the reactions. Aside from this purely physical effect ifalsorenters into the chain via transfer
A METATHETICAL CHAIN
20
-+
CHICHO (or CHI
2 CHO
This is, of course, not the case in the branching chain reaction where in one or more of the propagation steps a single chain carrier will produce two or more carriers. A typical example occurs in the H p O z reaction where there are two such branching steps:
H
CH,
+ CClg
As we shall observe, the unimolecular step adds enormously to the quantitative difficulties of interpretation of the pyrolytic chains: The characteristic of a chain reaction is that the sum of the chain (or propagation) steps yields the overall reaction stoichiometry, the chain carriers cancelling out in the addition. Where there is more than one set of products, there will be more than one set of chain reactions. In the case of CHIOCH3 there are two important chains: primary CHa chain
CH,OCH,
Termination.
(R)&-C(R)~ccla
+M e 2 C1 + M
e2I
Usually they are more complex in that initiation and termination are not the reverse of each other. Again choosing the CHaOCH3 system, most recent evidence (2) is that initiation is the fission of the C-0 bond in the ether while termination involves CHI and CHO radicals: Initiation.
Similarly the telemerization scheme is:
CClt
+M Cl, + M Iz
steps. The CHIOCHa system provides an example of H, autocatalysis in the primary chain (Reactions I):
1+
catalytic (transfer H
i
chain
CHn
+ Ht$CH4 + H
CHIOCHI -+ H I
CH,OCH,
The chains responsible for this are:
n-GHls
+ Me
+ n-Bu \z CHI + sec-Bu
+ CHnOCH,
+ CH,O
+ CH,
(0)
We see that the two transfer steps in this sequence are equivalent to the direct attack of CH, on ether of Reaction I. Hydrogen is not consumed in the sequence, but since both steps are about 10-fold faster than the direct attack of CH, on CH80CH,, even modest amounts of H, produce a marked catalysis of the overall reaction. As we shall note later, catalytic transfer provides an important diagnostic tool for chain reactions.
n-C4Hla
1 -+
CHI
+ n-Bu \z EtH + sec-Bu
+ Et
2
-f
EtH
+ Et +M e
n-Bu 1,C&HI sa-Bu
3‘ -+
C,H,
(Q)
From the stationary state equations for the radicals:
M i c t i o n of Products
Perhaps the simplest feature of chain systems which
has yielded to reasonable analysis is the prediction of products. Rice and Herzfeld, in 1934 (36),formulated an approach to pyrolytic chains which has since become the foundation for the analysis of product distributions. It is based on the premise that the fastest mode of reaction of a radical R ‘ with an H-containing molecule R H is the abstraction of an H atom. If this is followed by a unimolecular decomposition of the product radical R into an unsaturated olefin molecule (designated 0 1 ) and a radical R‘ then we have a pyrolytic chain:
R’
+ RHAR ~ H+ R
chain overall reaction
RH
+0
1
+ R’H
What makes such a sequence a rapid chain is the fact that the bond breaking step (the second step) has a low activation energy because the normally very endothermic bond breaking is compensated by multiple bond formation. Since the energy of multiple bonds is of the order of 6+70 kcal., this means that step 2 of the chain sequence (Reaction P) is endothermic by only 20 f 10 kcal. Activation energies for such steps have been found to be only slightly larger than their endothermicity, making them usually the fast steps in the chain cycle. A good example of this is to be found in the pyrolysis of n-butane at about 450-550’ C. (79, 35,38, 40, 47). Prcducts of the first 25% reaction are principally CH, CsHe and C a s C a 4in about a 2.4:l ratio.
+
+
since one expects that kilki’ E kJkt’-i.e., the CHI and C a H radicals , should have about the same ratios of rates of attack on the primary and secondary hydrogen atoms. If we equate ki‘/kl to the experimental value of 2.4 and assume the Arrhenius A factors are in the statistical ratio A i ’ / A t = 2/3, this yields EI - Ex’ 2 kcal. This value is compatible with independent data on similar abstraction reactions. The nature of chain propagation steps is such that they involve the destruction of one radical and the production of a second by processes which are always first order in radicals. Under conditions where the initiation-termination process is an equilibrium or where chains are long-is., >lO-we can ignore the contributions of initiation and termination steps relative to propagation steps. This simplifies enormously the solution of the usual quasi-steady state algebraic equations which can then always be solved explicitly for the ratio of radical concentrations. In the quasi-steady state technique we set each differential equation for d(R)/dt = 0 (where R is a radical center) and solve the resulting linear set of equations. Since we are omitting any initiation or termination terms in these equations, each remaining term contains one and only one radical concentration. The absolute value of the radical concentrations is then determined by the steady state equation that rates of initiation and termination must be equal. However initial ratios of products and hence initial compositions can be determined independently of the initiation and termination steps.
ACT1V IT Y T R A N S F E R
VOL 5 6
NO. 1 J A N U A R Y 1 9 6 4
21
Measurements of the order of reaction taken during a single run In the case of i-butane, (CH,),CH, a similar line of reasoning would lead us to expect two sets of productsi-butene Hz, and CH4 C3H6. These are in fact the major initial products (42) at 550’ C. in the ratio of about 3 :2. If now we jump to the intuitive conclusion that this must represent the ratio of attack by radicals on the tertiary and primary H atoms in i-C4Hlo then it implies (correcting for the statistical weight of 1 :9) that the activation energy for tertiary H atom abstraction is about 4.4 kcal. less than that for primary H atom abstraction. This is again in accord with independent observations of these activation energies. A great many of the pyrolytic chains investigated in the early decades of this century have yielded readily to this type of treatment. For many, however, unresolved discrepancies and, in some cases, paradoxes remain. One example is provided by the pyrolysis of diethyl ether. A modern Rice-Herzfeld treatment of this system would predict two major sets of products each proceeding through a different intermediate :
+
+
+
A. CHaCHsOCHzCH3 + [CH,CHO] C2H6 4 CH4 CO C2He B. [EtOH] CzH4
L
1
CHd
+
+ [CHzO]
+
--j
Hz
+
+ CO
(R)
+
+
+
+
+
22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
-d(CH,CHO)/dt
=
k, ’(R)A (CH3CHO)
(2)
where ( R ) E Ois the stationary state concentration of radicals in the pure EO system while (R)Ais that for the C H 3 C H 0 system. Since k,’ 30 k,, we are forced to conclude that (R)EO% lo3 (R)A. This would imply a difference in the activation energies for initiation of radicals in the two systems of the order of 20 kcal. or more. The only process fitting this requirement would be the unusual sequence :
r1 1 CH2CH20 @ CHZCHZO 2
CH2CHz07 [CH3CHO]*7CH3
+ CHO
(S)
The equilibrium steps 1, 2 (in Reaction S) are analogous to the similar biradical formation observed in the cyclopropane system (9, 33) and are about equally endothermic ( A H I , z = 52 kcal.). However, step 3 is exothermic by 80 kcal. which is more than enough to produce an excited CH3CH0 capable of dissociating into CH, CHO. The overall process S is thus expected to have an activation energy of about 52-55 kcal. which is indeed less by the required amount than the 78 kcal. for breaking the C-C bond in CH3CH0. The paradox in the ethylene oxide pyrolysis is resolved by the intervention of a thermally excited species. Such species appear to be unusual in chemical reactions but should be common in very exothermic pyrolytic reactions of the type illustrated here. In general the opening of small ring compounds (C, or C4) should result in the formation of highly excited species. In the isomerization of cyclopropane, the product propylene is formed with some 7 2 kcal. of internal vibrational energy. This is not enough to produce any bond-breaking in the excited CaH6 molecule. However in the pyrolysis of methyl cyclopropane, this amount of energy is just sufficient to break a CH3 radical from one of the product species, butene-1 , because of the allylic resonance energy in the accompanying allyl radical. J. P. Chesick (78) has in fact observed about 1-2% of propylene formation in this reaction in amounts which increase with decreasing pressure. This is precisely what would be expected from a “hot” butene-1 molecule, since the competing reaction is collisional deactivation.
+
The A chain is formed by radical attack on the weak secondary H atom of the ether and leads to formation of an ethyl radical in the unimolecular split. The B chain is started by radical attack on the stronger primary H atom and proceeds through a C2H50 radical. However, at the reaction temperatures this may abstract H to give EtOH or decompose into CHz0 CH3. Both EtOH and CHzO then undergo radical induced decompositions. O n this basis one would expect predominantly CO CHI as products. The reported products are in some conflict (27, 26) but appear to indicate a stoichiometry of: CO 2 CH4 ‘/z CzH4 with only traces of c a s . This is in marked contrast to the more recent and probably more reliable results (20) which do indicate about 80% of the A chain and some 20y0 of the B chain with still unresolved discrepancies in C&I4and CzHbOH. The pyrolysis of ethylene oxide (EO) is an example of an interesting and paradoxical chain reaction. This compound is some 28 kcal. less stable than its isomer C H 3 C H 0 and it decomposes thermally at temperatures about 100’ C. lower than the latter; in fact EO can be used to sensitize the thermal decomposition of CH,CHO (22). The products in both cases are very similar, predominantly CHI CO. However it is found that the attack of radicals on CH,CHO (75) is about 30 times faster than their attack on EO (34). If we adopt the Rice-Herzfeld chain for both, then in each case the rate of pyrolysis may be written as:
+
-d ( E O ) / d t = k, ( R ) E O (EO)
Some Experimental limitations
The experimental study of chain reactions has been accompanied by an extraordinary amount of built-in or natural perverseness. Because most chain reactions have chain lengths in the region of l o 2 to lo4, they are inherently sensitive to minor amounts of impurities which can contribute to the initiation or termination of free radicals. An interesting example is provided by 0, which has been shown to double the rate of pyrolysis of
are subject to gross error and therefore are of very limited value CH&HO at 477' C. when present only to the extent of mole % (30, 31). By contrast, 0.01 mole 0 2 in the Hz-ClZ photolysis at room temperature may reduce the quantum yield by 1000-fold (77). The effects of surfaces and temperature gradients are also well documented (7) and we shall not discuss them here. Transient effects of mixing or temperature equilibrium make it extremely difficult to obtain data during the initial stages of reaction in static systems and introduce uncertainties in any calculations of rate or stoichiometry which depend on an accurate knowledge of the initial concentrations. These too we shall not discuss here but shall instead focus our attention on an aspect of chain reactions which has been treated very casually-namely, the order. Although it might appear that the order of a chemical reaction rate is an extremely important experimental parameter, its significance is actually extremely limited. T h e reason for this is at least two-fold. The first has to do with the complexity of chain reactions and the fact that secondary reactions and reactions with the products may play important roles in the development of the chain. In such cases one finds that the rate law is extremely complex and its characterization by a simple mass law expression is almost meaningless. I t should be particularly emphasized that such characterization is of no value at all if one is attempting to interpret the rate constant and its Arrhenius parameters in molecular terms. The second reason, which has not been widely appreciated, has to do with the fact that it is extremely difficult to measure with precision the order of a chain reaction from data taken during a single run. It is only by measuring the rate over a reasonably broad range of initial concentrations that order can be specified with any precision. Let us examine this in a little detail. Suppose we are measuring the disappearance of a species R H in a typical chain reaction and are characterizing the rate law by a simple mass law expression:
(3) The order n is basically determined from changes in the rate with concentration. If we choose two different periods of time during the initial reaction period then n is given by:
(4) where the subscripts refer to the two time intervals chosen. If we wish to avoid the complexity of secondary reactions then we should stay as close as possible to the initial stages of reaction-i.e., the first 20-30010. Thus suppose we measure d(RH)/dt A(RH)/At during the
first 0-5% and during the interval 25-3070. Then In [(RH)I/(RH)z] In (0.975/0.725) = 0.30, while In [(RH)l/(RH)2] will be (0.30n). If now (RH) is determined from differences in (RH) and we ignore errors in measurement of time, we find that the expected error in n from Equation 4 is h0.13 if our analytical accuracy is 4=0.170, and h 1 . 3 if this latter is =kl.OY0. In a typical case of a gas phase reaction in which the total pressure is of the order of 200 mm. of Hg, the stoichiometry is such that the pressure doubles during the course of the reaction and the pressure measurement is accurate to h 0 . 5 mm. of Hg, the uncertainty in n during the first 3Oy0 of the reaction is h0.33. This implies that this interval of reaction could be equally well fitted by values of n ranging over nearly one full unit. We have verified this with some typical data from the pyrolysis of dimethyl ether which we found could be equally well fitted to the orders n = 1.O, 1.5, 2.0, and 2.5 when the total initial pressure was 100 mm. of Hg. This implies that the only meaningful measure of order must be obtained from measures of initial rates taken over a reasonably broad range of initial pressures. But the initial rates themselves must be averaged over at least a 10-2070 interval of reaction to obtain reasonable accuracy in their measurement. However, this in turn is valid only if we know something about the contributions of secondary processes. T h e conclusion of all of this is that until the mechanism is known with some assurance, the order of the reaction and the Arrhenius parameters will have little molecular significance. We can illustrate this conclusion with a few examples from our own recent experiences. The data for the pyrolysis of CH30CH3fit fairly well over 30% decomposition to a first order rate law. However the actual rate expression is very complex and indicates that the rate is actually 3/2 order in ether complicated by secondary autocatalysis by the products H2 and the intermediate CH20. This secondary autocatalysis offsets the normal fall-off of the 3/2 order rate to give a n apparent first order rate over the 3001, interval. However, as noted above, we found (2) that the data could actually be fitted with almost equal precision by rate laws with n = 1, 1.5, 2.0, or even 2.5. Another example is taken from the data on O 8 pyrolysis. The true rate law (4, 5) has the form:
However, it was found (29) that over an astonishing range of conditions the same data could be fitted to a simple mass law expression of the type k ( 0 3 ) 3 / z . The implications of the latter form are clearly quite different from Equation 5. A final example can be taken from the work on the reaction of butene-2 with HI, leading to a single set of products, n-butane 1 2 (73). The data fitted quite
+
VOL 56
NO. 1
JANUARY 1 9 6 4
23
Homogeneous catalysis by consecutive transfer reactions involving well over a range of conditions and composition to a simple second order rate law of the form :
d(lz) -
- ks(olefin)(HI) dt However inspection of the Arrhenius parameters of k~ gave values which were about 3 kcal. too small in E and IO2 too small in A . Further consideration showed the rate law to be of the form:
d(12) -
k&4.6KI,1/2(olefin) (I z)1/2 (HI)
-
dt
1
(12) + K3.2- (HI) + K4,
(7)
(I 2) 'Iz
This has been independently confirmed by spectrophotometric studies on the system under very varied conditions. When the last term in the denominator of Equation 7 is the leading term, which it usually is, the two rate laws are nearly the same.
recombination of radicals while recombination of atoms are third order and have rate constants of about 1O1O liter2/mole2-sec. From the radical recombination rate constants we can deduce the rather unexpected result that bond breaking reactions in molecules proceed with very large first order A factors of about lo1' set.-' and activation energies equal to the bond dissociation energies. For the first order dissociation of atoms from large molecules, the A factors are more normal, being about 1013.6 sec.-l The combination of actually measured rates for many atom and radical reactions, together with the preceeding generalizations permit us to do improved guesswork on pyrolytic mechanisms. Let us consider some examples. The simplest mechanism for the pyrolysis of acetone, neglecting any secondary reactions, is : CH3COCH,
Despite the difficulties we have just discussed, the last two decades have produced a sufficient quantity of information concerning individual propagation and termination steps that we may feel reasonably sure of elucidating more chain mechanisms with about orderof-magnitude reliability. We find, for example, that the atom abstraction reactions by polyatomic radicals from polyatomic molecules all have Arrhenius parameters in the range A = 108.6*0.5liter/mole-sec., while for the exothermic direction E = 8 f 3 kcal./mole. For the abstraction from diatomic molecules, E is about the same while A has the higher value of l O 1 O * O J liter/mole-sec. Abstraction by atoms, on the other hand, tend to have E values somewhat lower (4 f 3 kcal./mole) while A is about 1011*o.6liter/mole-sec. For the addition of atoms to unsaturated systems, the A factors are about 1O1O liter/mole-sec. while E is about 1 to 2 kcal./mole and for addition of polyatomic radicals to unsaturates we find about the same A and E values as given above for the abstraction reactions. The assignment of entropies to various species with precision of about 1-2 Gibbs/ mole enables us to compute A factors for the reverse reactions with a precision of about lO*O.3. Our growing knowledge of heats of formation for atoms and molecules similarly permits us to assign values of E for the back reactions with uncertainties of about f1 kcal. Thus, from the observation that the second order rate constant for the addition reaction (32): CH,
+ C Z H ~ n-C3H7
is k l = 108.410-7J/0liter/mole-sec (e = 2.3 RT in kcal./ mole) we can deduce that kz = 1013.6 1 0 - 3 1 . 3 ' 8 sec.-l [R. K. Brinton (74) reports slightly higher values, kl = 109.4 10--8.8/e
I.
For termination reactions we also find that their rate constants seem to fall in the range 1010.5 liter/mole-sec. for 24
INDUSTRIAL A N D ENGINEERING CHEMISTRY
CH3
+ COCH,
+
CO
Some Simple Generalizations
CH,
+ CH3COCH35 CH, + CHzCOCH3 CHzCOCH3
CH&O
+ 2 CH3
+ CH3
2 CHzCOCH3-4, ( C H Z C O C H ~ ) ~
(TI
Using the usual steady-state approximation for radicals and assuming long chains, the rate becomes: d(CH3COCH3) _ _
= k2
(2)
1/2
(CH3COCH3)'i2 (8)
dt
The actual rate has been reported (27, 28) as nearly first order with ko = 10-6"j,8 set.-' We can abstract from Equation 8 a pseudo-first order rate constant for comparison with the data, namely : (9)
Using our preceeding generalizations and relevant thermodynamic data we can make the following assignment : k z = 1018.5 1 0 - 3 4 / 8 set.-'
kt
=
IO'O.5 liter/mole-sec.
k t = 1017 10--8o./e
sec.-l
(10)
A ? is arrived at from assuming A-2 = 108.5 liter/molesec. and estimating AS2 = 32 e.u. Similarly E2 = AH 8. Et is corrected for the increase in bond dissociation energy with temperature. At (Acetone) = 150 mm. of Hg % 10-*.5 mole/liter : k P -E 10l8.O 1 0 - 7 4 / e sec.-l At 530" C., in the middle of the reaction range 0 = 3.66 kcal. and k, is 20-fold larger than the reported ko. However we have neglected the inhibition by CHzCO (reverse of step 2) which is expected to be competitive with step 1 and we have also neglected chain termination by CH,. Both of these will lower the rate, raise the order, and tend to
+
small molecules is useful in diagnosing major radical species bring the Arrhenius parameters closer to the ones reported. If we include all of these and assume that all termination steps have about the same rate, the new expression becomes :
where A = acetone, K = ketene, and k3 is the rate constant for the reverse of step 2. Using the known value of k l and the assigned value of kz, Equation 11 turns out to be very close to the observed rate. Because, however, the rate is further complicated by secondary decomposition of ketene (25, 43) there is little profit in attempting any more refined analysis. As a final example let us consider the pyrolysis of CzH6C1which leads to C2H4 HC1. These types of systems have given endless kinetic difficulties because radical chains and molecular reactions appear to go on side by side ( 7 I ) . A simple chain is :
+
+ CHzC1 CH, (or CH2C1) + C2H6Cl 1C2H4C1 + CH4 CH3CH2Cl f CH3
(or CH3C1)
7CzH4 + C1 C1 + C2H5C1A C2H4Cl + HCl C1 + C2H4CI A CzH4C12 1
CzHiCl
chain
(W
Using the usual methods we find: d(EtC1) dt
[l
+
(12)
Making the assignments :
k f = 1017 10-*5/e sec.-l k, =
10'0.6
liter/mole-sec.
kl
=
1013.5 10-25/@sec.-l
ka
= 1010.6 1O-6/@liter/mole-sec.
we find for the apparent first order rate constant, neglecting the inhibition term in the denominator that k, = 1016.310-5*.0/0 sec.-l This is to be compared with the observed value (6) of
[email protected]/esec.-l The calculated value is about 30-fold larger at the mean reaction temperature of 450' C.
W. Benson was Professor in the Chemistry Department of the University of Southern California. He is now chairman of the Department of Thermochemistry and Chemical Kinetics, Stanford Research Institute, Menlo Park, Cal;f. He wishcs to acknowledge support from the U. S. Atomic Energy Commission and the National Science Foundation. AUTHOR When this article was written, Sidney
This is slightly too large a difference to be easily accomodated by adjustments either in rate constants or in mechanism. The most obvious source of difficulty here is the step numbered one in Reaction U for which the critical ratio ( E / R T ) is about 17. This is so low that it is very likely that this step is in a pressure dependent region. But if this is the case, then the rate and mechanism could easily be altered by this much. (Reference (7)) pp. 232-234. The relatively low activation energy and few effective internal degrees of freedom [E 101 make it likely that this step is near its low pressure limit at these temperatures.) Thus if k, were at the low pressure limit we would replace it by k3'(M) where A3'(M) would now be of the order of 109.6 liter/mole-sec. Such a result would also tend to eliminate the inhibition term and change the nature of the termination process. In the extreme case the rate would become : d(EtC1) - k,'
(M) (EtC1)lI2
dt
(1 3)
which would appear to be 3/2 order in EtC1. The apparent first order rate constant would now be 10'3.9 1O - 6 4 / * sec.-l which is 40-fold slower than the observed rate. Clearly, putting this first step of Reaction U in the middle of its pressure dependent range would come very close to the proper value. In any case we see here one of the real dilemmas of the pyrolytic chain, the proper assignment of pressure dependence to the unimolecular step. Because there are so little data at hand concerning such systems it is not a simple matter to predict. We also can see why it is hard to believe that the observed reaction does not have an appreciable chain component despite its reported lack of inhibition by propylene. Similar problems arise in almost every study of pyrolytic chains. In the case of C2He (Reaction E) and neopentane (Reaction F) the pressure dependence of the pyrolytic step has still not been resolved. Homogeneous Catalysis; Diagnostic Tool
As we have pointed out in Reaction 0, two rapid transfer reactions can replace a slower propagation step in a chain. The overall result constitutes a case of true homogeneous catalysis. Such effects were first noted early in the fact that H2 exerts a ubiquitous catalytic effect on pyrolytic chains. This arises from the favorable frequency factors for reactions involving Hz and H compared to larger molecules and radicals. A well known example is to be found in the HBrcatalyzed oxidation of i-C4HI0 (37). The chain is: chain catalysis
+ t-BuOO t-BuOO + HBr 5 t-BuOOH + Br Br + t-BuH A HBr + t-Bu t-Bu
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0
2 4
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JANUARY
1964
(VI 25
There is no way in which HC1 can accelerate step 2 of Reaction X, outside of a possible inert gas effect if step 2 is in its low pressure regime. We observed (2) that the overall pyrolysis rate was accelerated 10-fold with 6-10 mole yo of HC1. The acceleration was proportional to HC1 at low HCl but leveled off a t high HC1. The implication of this large effect is that step 2 cannot be the slow step in the chain, but rather that step 1 (Reaction I), CH3 Men0 + CH4 CHz0CHB,must be the slow step. This immediately eliminated CH 20CH3 from being an important radical in termination processes. It appears that this is a quite general phenomenon and one that may be of considerable utility in unraveling major and minor chain carriers in complex systems.
+
+
Initiation Processes
It has been an extremely difficult problem in most cases to decide on the nature of the initiation process. The rule here is very simple. The dominant process will be that one which is most rapid in producing radicals under the ambient conditions. One must not omit product molecules from consideration as initiators, An interesting example of this is provided by our recent studies of iodine atom reactions in systems with chain lengths less than unity. The pyrolysis of primary organic iodides (10) and the reactions of R I with HI are two examples (12). Organic iodides R I react rapidly with H I (250'-300" C.) to form RH and 1 2 quantitatively with no side reactions (72). The mechanism is :
+ RI $ R + R + HI A RH + I I
1
12
(Y)
The initiation however is provided almost exclusively by the product 1% molecule via the equilibrium: Iz+MG2I+M
KI,
The overall rate becomes :
where K z . = ~ kz/k3. Even a t O.Olyoreaction, it can be shown that I2 with its low bond energy provides a more rapid source of radicals than the anticipated R I R I. This is expected to be a quite general phenomenon. Since initiator molecules enter into rate expressions only to the '/a power, the rate will not show very large autoacclerations. In the pyrolysis of toluene the reaction 4CH3 4 f CH3 is very likely (16) to be a more important radical source than the lower energy process +CH3 += +CH2 f
-
+
-
catalytic
26
CH3
+ HC1 5 CH4 + C1
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
H. The reason in this case is that the former is favored by a very loose transition state complex whose entropy of activation is sufficiently large to make up for the unfavorable energy. The A factors are estimated to be in the ratio of about 200 to one.
Disproportionation processes are usually not important as initiating steps even when they have low activation energies. For them to be important their activation energies must be about 30 kcal. less than that for a competing unimolecular process. This, however, appears to be the case in the pyrolysis (7) of C3H8 H2 mixtures to give C2H4 CHI. The process is:
+
+
with an A factor of about 10“ liter/mole-sec. and E = 44 kcal. The competing processes: 1 7
C3H6
+ H - 86 kcal. CzH3 + CH3 - 86 kcal.
C3H5
(AN
both have activation energies which are much too high and A factors not quite high enough to compensate. This appears also to be the case in the pyrolysis of CHZO both with and without added CzH4, reactions which are under current investigation in our laboratories ( 3 ) . In pure CH20, the initiation is : 2 CHZO
CHzOH
with E = 56 kcal. and A competing process:
CH20
+ CHO
101oJ liter/mole-sec.
2 H + CHO
The
at present because there are so few reliable data for the cracking reactions of radicals or for their inverse reactions, the addition of radicals to unsaturated compounds. A major difficulty in this regard comes from the possible pressure dependence of both of these reactions. Uncertainties arising from these considerations introduce major sources of possible error of the order of l o 2 in our a priori estimates of chain rates. I t is unfortunate that the present chemical complexity of pyrolytic chain reactions is such that it is unlikely that their direct study will throw much light on these detailed questions in any unique way. I t is much more likely, as past work has shown, that most progress will come from studies of controlled model systems where individual steps or ratios of steps may be singled out for detailed investigation. One last word of warning is in order here. Where in the past there has been a tendency to interpret all gas reactions from the point of view of molecular steps, ignoring the complexity contributed by radical chains, there may be a tendency now in the reverse direction, namely to ignore the contributions of molecular steps or four-center steps to chains. The recent work of Gordon et al. (24) indicating the occurrence of a four-center elimination reaction from radicals bears much closer investigation as a possible major phenomenon in high temperature systems.
(AC)
has E = 87 kcal. and A 1014J liter/mole-sec. and is about 200 times slower at 500’ C. The addition of CzH4 replaces step 1 by the step: I ,
with E = 47 kcal. and A 10l1 liter/mole-sec. and is about l o 3 faster than step 1 at 500’ C. This is somewhat compensated by retardation of propagation steps. Despite the difficulty of fixing properly the initiation processes, our estimate of the chain rate is not as sensitive to our errors here because the latter usually varies as the 1/2 power of the initiation rate. A 10-fold error in initiation rate estimates causes only %fold error in over-all rate. Conclusions
The extrapolation of radical rate constants based on the ones which have been studied appears to give A factors to within an expected precision of a factor of 3 and E values to within 1 3 kcal. At 700’-1000’ K., these give a joint expected error of about a factor of 6.5 and a possible error of 20. Where the structural features of the reaction are close to one which has been measured, these limits can be considerably reduced, perhaps to an expected error of a factor of 2 and a maximum error of about a factor of 4. O n this basis it is anticipated that we can predict chain rates to within a factor of about 20 in poor cases, 3 in favorable cases. The metathetical chains are in this respect much more amenable to treatment than are the pyrolytic chains. These latter present a difficulty of unknown proportions
LITERATURE CITED (1) Amano, A,, private communication. (2) Anderson, K. H., Benson, S. W , J . Chem. Phys. 39, 1677 (1963). (3) Ibid., unpublished work. (4) Axworthy, A. E., Jr., Benson, S. W., Ibid., 26, 1718 (1957). (5) Axworthy, A. E., Jr., Benson, S. W., “Ozone Chemistry and Technology,” Aduonces in Chcrni.rtry Ser., No. 21, 388, 398 (1959). (6) Barton, D. H. R., Howlett, K. E., J . Cham. Sot. 165 (1949). (7) Benson, S. W., “Foundations of Chemical Kinetics,” McGraw-Hill, New York, N. Y. (1960). (8) Benson, S. W., J. Chem. Phys. 25, 27 (1956). (9) Ibid., 34, 521 (1961). (10) Ibid., 38, 1945 (1963). (11) Benson, S. W., Bose, A. N., Ibid., in press. (12) Benson, S. W., O’Neil, H. E., Ibid., 35, 514 (1961) (13) Bose, A,, Nangia, P., Benson, S. W., unpublished work. (14) Brinton, R. K., J . Chem. Phys. 29, 781 (1958). (15) Brinton, R. K., Volman, D. H., Ibid., 20, 1053 (1952). (16) Buss, J. H., Benson, S . W., J. Phyr. Chem. 61, 104 (1957). (17) Chapman, M. C. C., J . Chem. Soc. 3062 (1923). (18) Chesick, J. P., J. Am. Chcm. SOC.82, 3277 (1960). (19) Crawford, V.A., Steacie, E. W. R., Can. J. Res. 31, 937 (1953). (20) Danby, C. J., Freeman, G. R., Proc. Roy. SOC.A245,40 (1958). (21) Davoud, J. G., Hinshelwood, C. N., Ibid,, A171, 39 (1939); A174, 50 (1940) (22) Fletcher, C. J. M., Rollefson, G . K., J . Am. Chcm. SOC.58, 2129, 2135 (1936). (23) Flowers, M., Batt, L., Benson, S . W., J. Chcm. Phys. 37,2662 (1962). (24) Gordon, A. S., Smith, S. R., Ibid., 34, 331 (1961). (25) Guenther, W. B., Walten, W. D., J . Am. Chem. Soc. 81,1310 (1959). London All4,84 (1927). (26) Hinshelwood, C. N., Proc. Roy. SOC. (27) Hinshelwood, C. N., Hutchison, W. K., Ibid., Alll, 245 (1926). (28) Huffman, J. R., J . Am. Chcm. SOC. 58, 1815 (1936). (29) Kilpatrick, M., private communication. (30) Letort, M., J . Chim. Phys. 34,428 (1937). (31) Letort, M., Niclause, M., Reu. Inrt. Franc. Petrole Ann. Combust. Liquides 4, 319 (1949) (32) Mandelcorn, L., Steacie, E. W. R., Can. J . Chcm. 32,79 (1954). (33) Nangia, P., Benson, S. W., J . Chcm. Phyr. 38, 18 (1963). (34) Phibbs, M. K., Darwent, B. de B., Can. J. Res. 28,395 (1950). (35) Purnell, J. H., Quinn, C. P., Proc. Roy. SOC.London 270A, 267 (1962). (36) Rice, F. O., Henfeld, K. F., J.Am. Chcm. SOC.56,284 (1934). (37) Rust,F. F., Vaughan, W. E., IND. ENQ.CHBH.41,2595 (1949). (38) Sagert, N. H., Laidler, K. J., Can. J . Chem. 41,838 (1963). (39) Smith, S. R., Gordon, A. S., J . Phyr. Chem. 6 5 , 1124 (1961). (40) Steacie, E. W. R., Folkina, H. O., Can. J . Rrs, 18B,1 (1940). (41) Steacie, E, W. R., Puddington, I. E., Ibid., 16B,176 (1938). (42) Ibid., 260 (1938). (43) Young, J. R., J . Chem. SOC.1958, 2909.
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