J. Phys. Chem. 1983, 87,2786-2789
2786
Degeneracy of two arguments collapses the three-vector function to a multiple of a two-vector function: A h
h
P , ~ , ( a b b ) =( - - i ) b + ’ t @
The explicit form for the rotationally invariant twovector functions P, results upon setting any index in eq
A.5 to zero: 4T
Py(2b)= C(-)a-YYa(2) a 2v+l
YY-~(L = )P,(P*b)
(AB)
Here the P, on the right is a Legendre function, and the right-hand equality is simply the spherical harmonic addition theorem.
Thermodynamic and Kinetic Coupling of Chain and Catalytic Reactions M. Boudart Department of Chemical Engineering, Stanford University, Stanford, California 94305 (Received; January 3 1, 1983)
It is argued that the most general useful relation relating kinetics and thermodynamics is not the De Donder inequality Av 2 0 but the equation of De Doner In (C/L) = A / R T where A is the affinity of reaction, v denotes net rate, and u’ and 6 represent forward and reverse rates of an elementary process. What matters is not so much the formal thermodynamic coupling between simultaneousoverall reactions but the kinetic coupling between the elementary processes of a chain or catalytic sequence. Examples are discussed.
Introduction Any guidance from thermodynamics in the art and science of chemical kinetics has a value which is particularly appreciated as it is limited but general. An intriguing example of such thermodynamic assistance to chemical kinetics comes from the De Donder inequality Au b 0 linking the affinity A = -AG of a reaction and its rate u.l Indeed, if two reactions (subscripts 1 and 2) occur simultaneously, it is conceivable that the following inequalities Aiul € 0 AZU~> 0 might hold provided that A l ~+ l A2~2> 0
It is said that the first reaction is coupled to the second one, the coupling reaction. It is also said that “thermodynamic coupling thus allows the coupled reaction to proceed in a direction opposite to that dictated by its affinity”. Unfortunately, the coupling mechanism, clearly a notion from chemical kinetics, needs to be known before the inequality Alu, C 0 can be accepted. It is quite possible that, when the coupling mechanism is understood, the need for the thermodynamic coupling as defined above disappears, together with the uncomfortable inequality A,vl € 0, which is purely formal in the sense that the coupled reaction does not take place at all but is only a stoichiometric statement devoid of kinetic (i.e., u l ) meaning. Let (1) I. Prigogine and R. Defay, “Chemical Thermodynamics”, translated by D. H. Everett, Longmans, London, 1954, p 38. (2) Reference 1, p 42.
0022-3654/83/2087-2786$0 1.50/0
us show two examples of this situation. Thermodynamic us. Kinetic Coupling. The first example deals with a chain reaction, while the second one relates to a catalytic process. Consider first the two gas-phase reactions
H2 = 2H (1) 2Hz + 02 = 2H20 (2) taking place at 1000 K between reactants and products at their standard state (1atm 101.3 kPa). The standard affinity for the first one is Aol = -331 kJ mol-l while A o z = 385 kJ mol-’. For the overall reaction 3H2 + 0 2 = 2H20 + 2H (3) the standard affinity is 385 - 331 = 54 kJ mol-’, so that it would appear that reaction 1 is the reaction coupled by reaction 2, the coupling reaction, and that indeed, for reaction 1, Alvl < 0. But this is not so, because, in fact, reaction 1 does not take place a t all. Rather, according to S e m e n ~ v the , ~ branching chain reaction propagates through three elementary processes as follows: OH + H2 H 2 0 + H (4) H
+ O2
--
+0 OH + H OH
(5)
0 + H2 (6) If we sum up these three elementary processes, taking the first one twice, so that its stoichiometric number u is given a value equal to 2, we reproduce the overall reaction 3. The overall De Donder inequality applies to it, but “reaction (3) N. N. Semenov, ‘Some Problems in Chemical Kinetics and Reactivity”, Vol. 11, translated by M. Boudart, Princeton University Press, Princeton, NJ, 1959, p 151.
62 1983 American Chemical
Society
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Coupling of Chain and Catalytic Reactions
1"has nothing to do with what happens at the mechanistic level. Indeed, reaction 1 does not take place at all. There is no thermodynamic coupling in the sense mentioned above. The multiplication of hydrogen atoms is due not to reaction 1but to the branching chain reaction H + O2 OH + 0. Thus, the concept of thermodynamic coupling disappears at the kinetic level of interpretation. The situation is similar for a catalytic reaction selected by Dowden as an example of thermodynamic ~ o u p l i n g . ~ According to Dowden, oxidative dehydrogenation, a widespread industrial process taking place on a number of commercial solid oxide catalysts, proceeds as follows, in the much-studied case of n-butene RCHzCH3(or C4Hs) to butadiene RCHz=CH2 (or C4H6): RCHZCH, = RCH=CHZ + Hz (Aiu, < 0) (7)
-
Hz + 1/z02 = HzO
+ '/202
RCHZCH,
(AZuz> 0)
= RCH=CHz
+ HzO
(Aiui + Azuz > 0)
(8) (9)
However, this macroscopic description, which specifies the first reaction as the coupled reaction made possible by the second coupling reaction, is not substantiated by molecular descriptions which suggest that the reactant first reacts with lattice oxygen of the catalyst, e.g., Moo3 The catalyst is then reoxidized by dioxygen supplied in the reaction mi~ture:~
+ 0 + 02-i- C4H7-+ OHMo6++ C4H7- [Mo=C4H7I5' [Mo=C4H7I5' + 02- Mo4' + 0 + OH- + C4H6 C,H8
-
-
20H-
Oz
i-
02-+
-
+ 20 + 2M04+
+ HzO 202- + 2M06+
The notations are as follows: 0,an anion vacancy at the surface; [Mo=C4H7I5+,a T complex of Mo6+and the allyl carbanion. The molecular details of this mechanism may be revised by further work. But what is important here is that, again, the coupled reaction does not take place. In fact, it is quite certain that dihydrogen is not evolved during oxidative dehydrogenation. As a result, the coupling reaction does not take place either. Hence, there is no evidence that the reaction is an example of thermodynamic coupling. Instead, from a thermodynamic standpoint, the case seems to be the classical one of an adverse equilibrium (the improperly called coupled reaction) circumvented by interception of the precursor of a reaction product before it is formed. This is one way to shift equilibria following the principle of Le Chatelier Br aun . At any rate, the two cases just discussed indicate that great care should be exerted before a case of thermodynamic coupling can be accepted. Unfortunately, a mechanistic knowledge of the reactions is a prerequisite before any conclusion can be reached. This in turn severely curtails the thermodynamic assistance to kinetics provided by the De Donder inequality invoked in thermodynamic coupling. All that can be said is this: Everything happens as if the inequality Alul < 0 were holding for the coupled reaction if it existed. However, the De Donder inequality (4) D. A. Dowden, "Catalysis", Val. 3, C. Kemball and D. A. Dowden, Eds., The Chemical Society, London, 1980, p 137. (5) P. A. Batist, C. J. Kapteijns, B. C. Lippens, and G.C. A. Schuit, J. Catal., 7, 33-49 (1967). (6) Reference 1, p 262.
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provides unambiguous information on the sign of the affinity for all steps of a catalytic or chain sequence, as will now be shown. Affinity of Steps in a Closed Sequence. Consider any sequence of elementary steps that is closed as in the propagation steps of a chain reaction or in a catalytic reaction. The sum of the steps, each one multiplied.by its stoichiometric number ui, reproduces the stoichiometric equation for reaction. Since the latter takes place at a finite overall positive net rate u as a result of the available chain or catalytic path, the De Donder inequality involving the overall affinity reads Au > 0 with both A and u having positive values. But at the steady state, for each of the steps f
u p = vi
vi > 0 C
-
(10)
where Ci and Giare the forward and reverse rates of the ith step. Moreover, for any step with an affinity Ai we can always write, following De Donder7 In (&/:i) = A J R T
(11)
This general relation follows most simply from the formalism of transition-state theory.8 Because of this relation and the above inequality, it follows from eq 10 and 11that
Ai > 0 u ~ A> ~ 0u (12) for all steps. Note that these inequalities hold also for equilibrated or quasi-equilibrated steps for which Ci and Ciare much larger than u so that Ai > 0, but Ai N 0. Thus, the affinity of all steps in a chain or catalytic sequence at the steady state is positive, and their sum properly weighted by the appropriate stoichiometric number is equal to the overall affinity A: CuiAi = A
(13)
1
It is concluded that the phenomenon of thermodynamic coupling cannot possibly occur among the steps of a closed sequence at the steady state, as implied by Dowdene4 Nevertheless, useful information can be obtained from relation 11of De Donder, which links the thermodynamic driving force of the reaction Ai to its kinetic irreversibility, Le., the ratio of forward and reverse rates Zi/Ui. Equilibrium and Steady-State Concentrations of Intermediates in Chain and Catalytic Reactions. Thus, thermodynamic coupling is not going to help the situation that may occur when the standard affinity of a step in a chain or catalytic sequence is uncomfortably negative, bringing the catalytic or chain reaction to a halt. But kinetic coupling can help because it can change substantially the steady-state concentration of active intermediates, above their equilibrium value if they are reactants, or below their equilibrium value if they are products. In both cases the equilibrium of the blocking step can be displaced in a favorable direction by the principle of Le Chatelier Braun. The kinetic consequences will be shown first generally and then illustrated by means of three examples. Consider an ordinary reversible step in a chain or catalytic sequence, involving two active intermediates XI and X2 and two molecular species M, and M2: Xi + Mi 2 M2 + X2 (14) Suppose that the standard affinity of step 14 is negative so that it will stop almost immediately after starting be(7) Th. De Donder, "L'Affinitg", Gauthier-Villars, Paris, 1927, p 43. (8) J. Horiuti and T. Nakamura, Adu. Catal. Rel. Subj., 17, 38 (1967).
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The Journal of Physical Chemistty, Vol. 87,No. 15, 1983
cause of a very unfavorable equilibrium. How can the affinity of the step become positive so that the step proceeds from left to right a t a positive rate? Two nonmutually exclusive solutions come to mind: the steady-state concentration of X1 can be made substantially higher than its equilibrium value, or the steady-state concentration of X2 can be decreased enough below its equilibrium value. In both cases, this can be done by kinetic coupling between the step envisaged and preceding or following ones, respectively. For simplicity we will assume that (M,) = (M,). Suppose first that the steady-state concentration (XJm of the reactant active intermediate is at equilibrium (X1)ss = (Xl)e (15) If the standard free energy Ao of the step is negative, its affinity A may still be positive if (XJSs k-l expresses the kinetic coupling that favors the thermodynamically unfavorable propagation step of the chain. As a second alternative illustration of eq 16, consider now the low-pressure, high-temperature, catalytic decomposition of ammonia on clean foils of tungsten13 or mo1~bdenum.l~ This is an illustration of eq 16 with condition 19. The steady-state concentration of adsorbed nitrogen atoms N* during the reaction, namely, (N*),, and the value of (N*), in equilibrium with dinitrogen N2 were measured by Auger electron spectroscopy. I t was found that (N*),, >> (N*),, and also N* was clearly by far the most abundant reaction intermediate. It was thus possible to apply eq 16 with the result, at 1000 K and an ammonia pressure of 1.1 X Pa, in the case of m ~ l y b d e n u m ' ~
-
+
-
+
c
e x p ( A , / R T )= u , l v ,
Indeed, for the elementary step 20 = kl(Br)e(H2)
(24)
where K' is the equilibrium constant of the reaction
with A' = R T In {(Xz)e/(X1)el
(23)
c, = k.,(H),,(HBr)
For the sake of simplicity, let us write (H2) = (Brp) = (HBr) so that
where K , = kl/k-l is the equilibrium constant of step 20. On the other hand, a t the kinetic steady state u1 - u - ~= k,(Br), - k-l(H)s8= u2 = k,(H),, Thus (9) M. I. Temkin, Adu. Catal. Rel. Subj., 28, 173-291 (1979). (10) C. Kemball, Discuss. Faraday SOC.,41, 190 (1966). (11) M. Bodenstein and S. C. Lind, 2. Phys. Chem., 57, 168 (1907).
exp(A/RT) = ;/
for the desorption step 2N*
-+
= 1.4 X 10'
2*
+ Nz
Thus, in spite of an unfavorable negative standard affinity which is characteristic of desorption processes, not only does desorption take place with a positive affinity, but the latter is high enough that the desorption step is essentially irreversible (u' >> 5). Indeed, the concentration of dinitrogen does not appear a t all in the rate equation for decomposition of ammonia under these conditions. The kinetic coupling here "pumps up" the concentration of (12) K. Tamaru, "Dynamic Heterogeneous Catalysis", Academic Press, New York, 1978, p 30. (13) H. Shindo, C. Egawa, T. Onishi, and K. Tamaru, J. Chem. SOC., Faraday Trans. 1, 76, 280 (1980). (14) M. Boudart, C. Egawa, S. T. Oyana, and K. Tamaru, J . Chim. Phys., 78, 987 (1981). (15) J. H. Sinfelt, H. Hunvitz, and R. A. Shulman, J . Phys. Chem., 64, 1559 (1960).
The Journal of Physical Chemistry, Vol. 87,
Coupling of Chain and Catalytic Reactions
surface N* above its equilibrium value because of the showering of the surface with N* in easy steps preceding the final unfavorable one. The same phenomenon prevails in the last example. Finally, consider the gas-phase catalytic dehydrogenation of methylcyclohexane M to toluene T M =T
+ 3H2
(28)
taking place on platinum far away from the global equi1 i b r i ~ m . lThe ~ kinetics indicated that, just as in the preceding case of high-temperature decomposition of ammonia, the rate equation can be accounted for by assuming two irreversible steps, an opening adsorption step and a closing one similar to the one above for desorption of dinitrogen: T*-T+* Here also, it appears that (T*),, >> (T*),, although this inequality was not measured. But it follows directly from the fact that toluene does not appear in the rate equation, although it appears again to be the most abundant reaction intermediate again that u’ >> 6. Moreover, when benzene B was added to the feed, its inhibiting effect on the rate was very mild, a situation that can be understood again if (T*), >> (T*),, since it would be expected that (B*), (T*),, so that the effect of benzene on the rate can be neglected, since the rate is determined by (T*),,.
Conclusions While thermodynamic coupling plays no role in catalytic or chain reactions at the steady state, kinetic coupling helps to displace the equilibrium of thermodynamically unfavorable steps. It is expected that, for such steps, the corresponding stable product, e.g., HBr in the case of Br + H2 HBr + H, appears as an inhibitor in the rate expression. This is also found experimentally, especially with catalytic reactions for which a reaction product is an inhibitor as it is produced near equilibrium in a thermodynamically unfavorable step, e.g., SOsin oxidation of SO2,
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No. 15,
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2789
NH, in ammonia synthesis, C02 in oxidation of CO on oxide catalysts. When a stable reaction product coming from a thermodynamically unfavorable step does not appear as a reaction inhibitor, it may mean two things. First, the active reaction intermediate leading to that product is present in insignificant concentrations a t the surface. Or second, it is the most abundant reaction intermediate, but its steady-state concentration is made by kinetic coupling so much higher than the corresponding equilibrium concentration that the corresponding step is essentially irreversible. Both cases are illustrated by ammonia decomposition to dihydrogen and dinitrogen on tungsten13 and m01ybdenum.l~ The first case pertains to dihydrogen, which does not appear in the rate expression because of negligible surface concentrations of surface hydrogen a t the steady state. The second case pertains to dinitrogen, which does not appear in the rate expression either because it comes from very high surface concentrations of nitrogen a t the steady state. These rules appear general and useful. They help in understanding broadly the kinetics of chain and catalytic reactions. They are based on the general relation eq 11 of De Donder that provides the bridge between thermodynamics and kinetics. Although the relation can be obtained without recourse to transition-state theory,7J6it can be simply derivedsJs by means of the thermodynamic formulation of the transition-state theory of reaction rates due to Henry Eyring.l7Jg
Acknowledgment. This work was supported by NSF Grant No. CPE 79-09141. It is dedicated to the memory of Henry Eyring. I thank Profs. Tamaru, Nakamura, and Inge Koenig for their help in tracking down ref 7. (16) Denver G.Hall, 2.Phys. Chem. (Frankfurt am Main), 129,109 (1982). (17) H. Eyring, J. Chem. Phys., 3, 107 (1935). (18) J. Horiuti, Ann. N . Y. Acad. Sci., 213, 5 (1973). (19) H. Eyring and E. M. Eyring, “Modern Chemical Kinetics”, Reinhold, New York, 1963.
