Ind. Eng. Chem. Fundam. 1986, 25, 70-75
70
Some Applications of the Generalized De Donder Equation to Industrial Reactions Mlchel Boudart Department of Chemlcal Engineering, Stanford Unlversify, Stanford, California 94305
The generalized De Donder equation relates the ratio of forward and reverse reaction rates to the exponential of the thermodynamic driving force for reaction. The latter is the affinity divided by 8RT where 8 is the average stoichiometric number for the reaction, equal to the stoichiometric number for the rate-determining step, if there is one, and always equal to unity for an elementary step. I n the latter case,the equation provides useful i n f m t l o n on the kinetic coupling between steps in a chain or catalytic sequence. I f there exists a rate-determining step, the equation may help in revealing its identity.
Introduction Thousands of rate constants are tabulated for elementary reactions involving free radicals (Kondrat’ev, 1970). Many more can be estimated (Benson, 1976). Thus, rates of chain reactions proceeding singly or in networks can be calculated or estimated from kinetic information on elementary steps, Le., a microkinetic database. In this way macrokinetic behavior of many free radical chain reactions can be described from microkinetic information. Many examples can be found in pyrolysis of hydrocarbons (Baronnet and Niclause, 1985), vinyl polymerization (North, 1966), oxidation (Van Tiggelen, 1968), combustion (Glassman, 1977), and chemistry of the troposphere (Hampton, 1973) or the stratosphere (Boudart and DjBga-Mariadasou, 1984). Progress in the numerical solution of stiff differential equations now makes the quasi-steady-state approximation unnecessary in the kinetic analysis of these reactions (Allara and Edelson, 1975). The interplay of micro- and macrokinetics in free radical reactions provides a prime example of the feedback loop between industrial chemistry dealing with, e.g., steam cracking and academic chemical kinetics dealing with rate constants of elementary steps, e.g., of free radical reactions. By contrast, in the case of catalytic kinetics, the old tradition of inferring microkinetic information from macrokinetic observation continues tQ flourish in spite of the well-known ambiguities of mechanisms derived from kinetics, the so-called kinetic mechanisms (Boudart and DjBga-Mariadassou,1984). The reason is clear. Until very recently, there was very little known about the microkinetics of catalytic reactions. The situation is now changing in the case of homogeneous catalysis (Hjortkjaer, 1982). As to heterogeneous catalysis, recent advances in surface science have made available a small but rapidly increasing number of rate constants of elementary steps on so-called well-defined surfaces, mostly metallic single crystals (Madix, 1979). Such data are now available for adsorption, surface reaction, and desorption (Boudart and DjBgaMariadassou, 1984). As these data accumulate, it seems opportune to review the available relations linking microand macrokinetics and to examine the general kinetic structure of catalytic cycles. In particular, some special kinetic features of cycles in heterogeneous catalysis will be discussed. Indeed, it is hoped that the kinetic knowledge made available by studies on single crystals will quickly find its application in the mainstream of catalytic research as a result of a keener awareness of the links between micro- and macrokinetics. It is also hoped that this review may stimulate the discovery of new, more
powerful links between the kinetic of elementary steps and the rate of catalytic cycles. In turn, this would accelerate the transfer of data from industrial catalysis to surface science, and vice versa. Let us now present the main useful relations connecting micro- and macrokinetics or kinetics and mechanism. The Quasi-Steady-State Approximation Any catalytic cycle consists of a sequence of elementary processes. Each one proceeds at a net rate: V i = Ci - gi Steps must be written as they are believed to occur at the molecular level. Thus, for dissociative chemisorption of N2 on a catalytic site denoted by *, one may write two successive steps Nz * -i N*N
+ N*N + * & 2N*
or perhaps in a single step N2 + 2* s 2N* but certainly not 1/2N2+ * e N* If the stoichiometric equations for each step in a catalytic sequence are summed up side by side, the sum reproduces the stoichiometric equation for the overall reaction, provided that each step is taken u, times where ui is the stoichiometric number of that step. The value of cri depends on the arbitrary way in which the stoichiometric equation for the overall reaction has been written, e.g. N2 + 3H2 = 2NH,
or 1/2N2+ 3/2H2= NH,
The quasi-steady-state approximation (QSSA), in the form first clearly enunciated by Christiansen (1953), states that at the kinetic steady state UiU = u’i - fii (1) for all steps in the cycle proceeding at a net overall rate: v=v‘-v’ If the rate, u, is referred to the number of catalytic sites, it becomes the turnover rate, ut. The inverse of ut is a time, the turnover time, T. It seems intuitively clear that the kinetic steady state will not be established before a relaxation time of the order of T. In any event, a substance is not a catalyst unless it turns over more than once. An example of relaxation time (Figure 1)to reach the steady 0 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 71
6
that, for any elementary process, the rate constants, and ki,are related to the equilibrium constant, K ; ,by means of the relation &/Ei = Ki (4)
w
o
0.6
t-
As is well-known, this relation was anticipated by Arrhenius and others, in less fundamental terms. Relation 4 is very useful as it gives access to a rate constant that may be difficult to measure, provided that the rate constant for the step in the reverse direction is available together with the equilibrium constant. The unexpected validity of (4) over 40 orders of magnitude of rate was pointed out by Johnston (1968) for an elementary process O,+M+ O + O + M
1
$ 2
51
lJ?k
u. 3
Zk!
0.4
8
O W
6a u6 [L
O.*
LL
o
20
ao
60
40
100
120
TIME/S
Figure 1. Relaxation to the steady-state decomposition of ammonia (pressure 2.4 X Pa) on a molybdenum foil at 1060 K (Boudart et al., 1982): upper curve, rate vs. time; lower curve, fraction of surface covered with nitrogen vs. time.
state relates to the decomposition of ammonia at low pressure (-lo+ Pa) and high temperature on a molybdenum foil in an ultrahigh-vacuum chamber used as a continuous stirred tank reactor (Boudart et al., 1982). The reaction is run at the steady state. Then, the foil is "flashed" and N* leaves the foil. The upper curve shows the return to the steady-state rate as measured by a mass spectrometer. The relaxation time is of the same order as the value of the turnover time at the steady state. As the steady state is approached, the fraction of surface covered with nitrogen, as measured by Auger electron spectroscopy, also reaches its steady-state value (lower curve of Figure 1)with the same relaxation time as that for the rate of reaction. The Product Rule Following Temkin (Horiuti, 1973),the following identity can always be written for some or for all of the steps in a catalytic sequence (ij1 - v'l)i;2v'3...v', v'1(4 - ii,)v'3 ...v', + ... +
+
v'&73*..(v'n
- fin) =
hi- hi(2)
Pl
i=l
Substitution of (1) for each step into (2) yields
u=
[fi& - fiCi]/D i=l
+
i=l
+ ... +
where D = 1 ~ ~ 4 v ' ~ . . . vv'l~2v'3...Cn '~ 4C2&.,~,,. If the catalytic sequence consists of n steps, u = v' - u is its net rate and, from the above
Hence
(3)
This product rule will be used in what follows. Arrhenius' Relation between Rate Constants and the Equilibrium Constant It is a straightforward consequence of Eyring's thermodynamic formulation of reaction rates (Eyring, 1935)
which may not seem amenable at first glance to the formalism of transition-state theory. The De Donder Irreversibility Relation De Donder's concern with irreversibility and affinity, A, which is identical with -AG, the Gibbs free energy of reaction with the minus sign, led him (De Donder, 1927) to a very useful relation that is similar in form to Eyring's relation above. Ci/Ci = exp(Ai/Rr) (5) Indeed, to obtain Eyring's relation (4) from the relation of De Donder (5), all that is necessary is to specify standard-state conditions for which rates of elementary process become equal to rate constants and affinities become standard affinities so that exp(Aio/RT) = Ki with the superscript denoting standard state. In actual fact, it is ( 5 ) which is obtained most simply from Eyring's relation (4) and transition-state theory (Boudart and DjBga-Mariadassou, 1984). The De Donder relation expresses the kinetic irreversibility of reaction in terms of its thermodynamic driving force. In the form of eq 5, it is valid only for elementary steps. A relation similar to (5)-relat$s forward and reverse currents for an electrode, i and i, to the overvoltage, 7 t i / i = exp(aq/FRT) where F is the Faraday constant and a is an empirical coefficient. This expression gives a measure of the irreversibility of the electrode reaction in terms of its driving force. It was first derived by Volmer and Butler (Bockris and Reddy, 1970). Let us note a simple application of De Donder's relation. Since in the expression (1) of the QSSA Ci > Si as long as the reaction moves forward with u > 0, it follows from (5) that, for a sequence of elementary steps at the steady state, the affinity, Ai, is positive for all steps. This statement may sound evident, but it is useful to stress it (Boudart, 1983) as a result of statements in the literature in which no distinction is made between Ai and its standard value, Ai". Of course, a negative value of Aio does not prevent a step from proceeding forward until it reaches its unfavorable equilibrium. We shall return to this point later as we talk about kinetic coupling of catalytic cycles. The Generalized De Donder Irreversibility Relation For any catalytic chain sequence, let us introduce Temkin's (Temkin, 1971) average stoichiometric number, 8, defined as 5
= CaiAi/ZAi = A / C A i i
i
i
(6)
where the summation extends to all steps and A is the affinity of the overall reaction.
72
Ind. Eng. Chem. Fundam., Vol. 25, No. 1,
1986
Substitution of the De Donder relation (5) into Temkin’s product rule (3) followed by the use of definition (6) leads to the generalized De Donder relation i3/C = exp(A/@Rn
I
I
I
3
(7)
2
which expresses the kinetic irreversibility of the overall reaction proceeding at a net rate u=u’-6
(8)
-
1
Im
c
N L
in terms of its thermodynamic driving force. The affinity is modified by the average stoichiometric number of the reaction. Rearranging (7) with the use of (8) gives simply u = u’[l - exp(-A/@RT)] (9) Thus, the net rate of a reaction is given by its forward rate multiplied by a thermodynamic potential (Happel, 1972) given by the expression between brackets in (9). Linear Relation between Rate and Affinity near Equilibrium Sufficiently near equilibrium, where AIaRT
-I
-2
-0 4
0
04
08
A IRT
Figure 2. Linear relation between net rate u and affinity A for the reaction CsHl2 = CsHs + 3Hz on both sides of equilibrium. Data are from Prigogine et al. (1948), and the least-squares line is from Boudart et al. (1985).
(10)
Further, when AIaRT tends to zero, ?I tends to u,, the rate at equilibrium or the exchange rate (Wagner, 1970) u = u,(A/@RT) (11)
Thus, if there exists an rds, (11)becomes u = u,(A/gdRT)
-0
.
(13) (14)
Thus, it is possible to determine g d by measuring the net rate, u , near equilibrium and the exchange rate, u,, at equilibrium by means of a suitable tracer. It was first reported (Horiuti and Nakamura, 1967) that g d = 2 for ammonia synthesis when the reaction is written as in (12). However, later work suggested that g d = 1 (Mars et al., 1960). If the latter value is correct, it follows from any dissociative mechanism of catalytic ammonia synthesis that the rds is the chemisorption of N2 at the surface of the catalyst. By the words “dissociative mechanism” is meant one in which both N2 and H2 first dissociate at the surface; then successive recombination steps take place between adsorbed species until NH, is released from the surface (Boudart, 1981). This example is just one of the many nice illustrations of the fact that work on ammonia synthesis, the epitome of an industrial catalytic reaction, has fed back many new general concepts to fundamental catalysis science. This point has been discussed in detail elsewhere (Boudart, 1978; Timm, 1974). Validity and Further Use of the Linear Relation between v and A Since u’, a variable in (IO), can be replaced by u,, a constant in ( l l ) , only when the affinity tends to zero, it
is not evident that the linear relation (11)will be obeyed except perhaps so near equilibrium that the h e a r law may be a trivial one (Garfinkle, 1983). It is therefore interesting that linearity between u and A was established experimentally by Prigogine et al. (1948) on both sides of the equilibrium of the reaction
+ 3H2
CCH12 = C&6
(15)
taking place on a nickel catalyst. In fact, it has been noticed (Boudart et al., 1985) that the small standard deviation of the data from the best least-squares straight line (Figure 2) suggests, as follows from (ll),a rather high value of a, equal to at least 3. Let us assume that it is 3 and that there exists an rds. As suggested by Herbo (1942), the rds involved is the dehydrogenation of cyclohexane (C6H12)to cyclohexene (C6H10):
Then, if the reaction proceeds further by the equilibrated disproportionation of C6H10 3C6HIO
C6H6
t 2C6HIZ
(17)
where the symbol =8= denotes zero net rate, or an equilibrated reaction. This reaction proceeds very rapidly on nickel. The value of ad should be equal to 3 since the rds implied in (16) followed by (17) must take place three times for reaction 15 to turn over once. This conclusion is supported by the excellent fit of the data up to high values of A. Details are given in Boudart et al. (1985). Thus, not only is the linear relationship (11)of Horiuti valid near equilibrium but it can provide useful mechanistic information without the need of rate measurements at equilibrium. Virtual Pressure [N,],in Ammonia Decomposition This is another interesting application of the De Donder relation. The overall reaction with equilibrium constant K is K
2NH3 = N2
+ 3H2
On many catalysts, at not too high temperatures and not
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 73
too low pressures, the accepted mechanism is 2NH3
+
2"
-9-
2N"
2"
+
2N*
+
3H2
N,
x2
where the first reaction is equilibrated and the second is the rds on a surface where N is the most abundant reaction intermediate. Define a virtual pressure [N2], as the value of [N,] required to obtain at virtual equilibrium the value [N],, pertaining to the steady state of the reaction v'2
= Ldd(aN*,ss) = 62 = k'ja(a*,ss)[Nzle
(18)
Kinetic Coupling of Elementary Steps in a Catalytic Sequence To drive a thermodynamically unfavorable step, (Aio < 0), two possibilities exist: either the concentration of a reactant should be made large enough, as was the case above in ammonia decomposition, or a product must be kept at a sufficiently low concentration level by removing it is a subsequent step faster than it can return to the original reactants. This can be done by kinetic coupling (Boudart, 1976). Let us explain briefly the nature of this kinetic coupling in the free radical reaction, H2 Br2 = 2HBr, that takes place according to the classical mechanism following an equilibrium between Br and Br,, which, by the way, does not take place through an elementary step
+
where we have carefully used a thermodynamic activity, a, for the surface species and have not specified the form of the functions, fa and fd, that depend on surface activity at the steady state, uN*,,,. Thus, in all generality, we can write with the help of (18)
Applying the De Donder relation and noting that A2 = A, since step 2 is the rds, and expressing A in terms of K and steady-state values of gaseous concentrations or fugacities, we get from (19)
"
- =-= fi2
exp(A,/RT) = exp(A/RT) =
v2Br2
1
Br
propagation
H,
H -I- Br,
Br
-
HBr
+
H
2 HBr
+
Br
kI
T &
(k,/k-,=K,)
The first step in the propagation sequence should be severely limited by equilibrium since it is sizably endothermic and essentially isentropic. At equilibrium the concentration of hydrogen, [HI,, is obtained easily by noting that Kl[Br], = K1K[Br2]1/2= Ko[Br2]1/2 where KOis the equilibrium constant of '/2Br2
[N21,, ~~NH31,,2/[N2lss~H21ss3
+
=&
+
HBr
H,
+
H
so that
Finally, for example
[HI, = Ko[Br2]1/2
[N2Ie= K[NH3],,2/[H2],s3 N 6400 bar at 673 K with ammonia and dihydrogen at 1 bar each. The concept of virtual pressure, presented here as a consequence of the De Donder relation, originated with Temkin and Pyzhev (1940), who introduced the related concept of fugacity of adsorbed species. The concept of virtual pressure was further developed by Kemball(l966). Although it is yet another concept that came about as a result of fundamental studies of ammonia synthesis, it is general and useful. Thus, by using ammonia as a nitriding agent, it is possible to form iron nitride at low pressure and at the high temperatures required so that diffusion of nitrogen into the bulk of the metal proceeds at reasonable speed. Yet, without the high virtual pressure of nitrogen, the nitride would not be stable at those high temperatures. Another consequence of eq 19 is that the irreversibility of the associative desorption of nitrogen is very large under the conditions specified, since 4/17, 6400 during the steady-state decomposition of ammonia. This marked irreversibility of the desorption step means that the steady-state concentration of surface nitrogen must be considerably larger than the concentration of surface nitrogen that would be in equilibrium with the steady-state concentration of gaseous dinitrogen. This conclusion was checked experimentally in the case of decomposition of ammonia on a molybdenum foil at high temperatures but low pressures, at which the steady-state concentration of surface nitrogen could be monitored by Auger electron spectroscopy (Boudart et al., 1982). Another consequence of the irreversibility of the desorption step in ammonia decomposition is that the rate is not inhibited by nitrogen as might be the case if the desorption step was limited by equilibrium. We will return to this point later.
-
Now, if we start from a stoichiometric mixture, we have at half-reaction [H,] = [Br,] = [HBr]. The De Donder relation gives, at half-reaction
Cl/Cl = exp(Al/RT) = [H],/[H],, = 1 + k z / k l
(20)
since
[HI,, = k,[BrI,/(k-, + k2) as readily obtained from the steady-state condition (1) u = v'l - v'l = v'p The meaning of eq 20 is that the thermodynamically unfavorable step proceeds forward with a decided irreversibility even at half-reaction, since kZ/k-1 10 N v'l/C1 because, at steady state, H atoms are "pumped down" from their equilibrium concentration by the subsequent step H + Br, HBr + Br
-
As a result, the affmity for the difficult step at steady state, Al,,,, is positive although the standard affinity, AlO,is negative. Thus, in calculations of the standard affinity, Ai', of steps in a catalytic cycle (Goddard, 1985), the occurrence of negative values for Aio should not prevent the cycle from turning over at an acceptable rate, if kinetic coupling is favorable. Yet, in the above example, because k 2 / k - , is known to be only -10, the step Br + H2 F? HBr + H is nonetheless reversible and HBr is an inhibitor of the overall rate, although the overall reaction itself is not limited by equilibrium. The inhibiting effect of the product HBr on the rate of a reaction which is run under irreversible conditions was indeed a formidable puzzle for Bodenstein and Lind
74
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
(1907), who first observed it but did not know the mechanism of the reaction. Let us now return to the inhibition or lack of inhibition of the rate of a catalytic reaction by a reaction product. Mechanistic Information Obtainable from the Presence or Absence of Product Inhibition In a catalytic reaction, the desorption step or steps that release products are moderately or strongly endothermic. Although desorption proceeds with an increase in standard entropy, a desorption step is normally expected to be limited by equilibrium. Thus, it is likely that, at the steady state, the desorption steps will be in quasi-equilibrium. If the surface species that desorbs occupies a significant fraction of the catalytic sites, it follows that the product of desorption will inhibit the rate of the overall reaction. This is the case of ammonia synthesis on iron that is inhibited by ammonia (Ozaki and Aika, 1981), of oxidation of sulfur dioxide on vanadium oxide that is inhibited by sulfur trioxide (Bodenstein and Fink, 19071, and of the hydrodesulfurization of sulfur-containing hydrocarbons on cobalt-molybdenum sulfide that is inhibited by hydrogen sulfide (Satterfield and Roberts, 1968). When a reaction product is not an inhibitor, three explanations are possible. The first is that the surface species that desorbs occupies a kinetically insignificant fraction of the catalytic sites. This is true for ammonia decomposition at low pressures and high temperatures (Boudart et al., 1982) or at high pressures and low temperatures (Emmett, 1962), where dihydrogen goes not inhibit the rate of ammonia decomposition. Correspondingly, nitrogen is the most abundant reactive intermediate and hydrogen does not compete effectively for catalytic sites. A second explanation applies to dinitrogen in ammonia decomposition. In this case, dinitrogen is not an inhibitor of the rate. As noted above, this is attributed to the fact that, as a result of the high virtual pressure of nitrogen, the surface concentration of adsorbed nitrogen considerably exceeds the equilibrium value corresponding to the steady-state pressure of dinitrogen. Hence, the desorption of nitrogen proceeds essentially irreversibly and no inhibition by nitrogen is observed. A similar situation applks to the dehydrogenation of methylcyclohexane, M, to toluene, T, on a platinum-reforming catalyst (Sinfelt et al., 1960). I t is found that toluene does not inhibit the rate. The preferred mechanism (Sinfelt et al., 1960) is one in which M adsorbs irreversibly in an opening step and T*, the most abundant reactive intermediate, desorbs irreversibly in a final step. Again, why is the desorption step essentially irreversible? Our explanation is that, at the steady state, the surface concentration of toluene is substantially greater than that corresponding to the equilibrium concentration that could be reached at equilibrium between the surface and toluene at the steady-state pressure of the reaction. In other words, the virtual pressure of toluene resulting from kinetic coupling is much higher than its steady-state pressure. This explains clearly why addition of benzene to the feed (Sinfelt et al., 1960) hardly inhibited the rate of M * T + 3Hz. To compete effectivelywith toluene for the surface benzene should be added at a pressure corresponding to the virtual pressure of toluene, not its real pressure. These remarks are qualitative but could be quantified by measuring the steady-state and equilibrium concentrations of adsorbed toluene. Then the irreversibility of the desorption step could be calculated. But even without this knowledge, the mechanistic information inferred from the mere lack of inhibition of the rate of reaction by the product issuing from the most abundant reactive inter-
mediate is quite valuable. Note that the lack of inhibition of the reaction rate by the other reaction product, dihydrogen, follows ipso facto from its inability to compete with the abundant reactive intermediate. In yet another example, when an order of reaction is zero with respect to a certain component, the lack of inhibition of the rate by the product issuing from the species related to the zero-order component is not unexpected. Thus, for ethylene hydrogenation on platinum (Emmett, 1962),the zero order with respect to ethane appears related to the zero order with respect to ethylene. Although the nature of the surface species that saturates the catalytic sites is not known (de Boer, 1957; Cimino et al., 1954),there is no doubt that such a species exists. A third explanation of the lack of inhibition of the forward rate of reaction by a product is that the desorption of the product from the surface of the catalyst is indeed exothermic. This implies, of course, that adsorption of that species would be endothermic and therefore very unfavorable (Aio< 0) since adsorption is normally accompanied by a loss of entropy (de Boer, 1957). This explanation suggests itself in the case of the hydrogenolysis of ethane (Cimino et al., 1954) CZH, + Hz = 2CH4 and the methanation reaction (Vannice, 1982) CO + 3Hz = CH4 + HzO on transition metals. In both cases, methane is not an inhibitor of the rate. That adsorption of methane on metals is endothermic is not surprising in view of the known difficulty of methane activation. It may be noted that methane activation is not assessed correctly by methane-deuterium exchange. Thus, mere inspection of a rudimentary rate equation or even the elementary knowledge whether a product inhibits the rate or not contains nontrivial mechanistic information. Ultimately, the qualitative or quantitative meaning of this information is contained in the De Donder relation that measures the irreversibility of a step. Conclusions The De Donder relation for an elementary step and its generalization for a catalystic or chain sequence contain useful mechanistic information. Thus, the relation provides a useful bridge between macro- and microkinetics. Its measure of irreversibility, far from or near equilibrium, can be used to understand or determine kinetic coupling, virtual pressure, the difference between equilibrium and steady-state concentrations, the inhibition of a catalytic reaction by a reaction product or the lack of it, and the stoichiometric number of the rate-determining step, if there is one. The De Donder relation is obviously a useful bridge between thermodynamics and kinetics. The present paper underscores its usefulness in building bridges between macro- and microkinetics, i.e., between the rates of the overall reaction and the rates of its component steps, or finally, between industrial data and fundamental investigations. Acknowledgment This work was carried out as part of a continuing NSF program, currently under Grant NSF-CBT 8219066. Literature Cited A k a , D. L.; Edelson, D. Int. J . Chem. Kinet. 1975, 7, 479. Baronnet, F.; Niclause, M. Ind. fng. Chem. Fundam.. this issue. Benson, S. W. "Thermochemical Kinetics"; Wlley: New York, 1976. Bockris, J. O.'M.; Reddy, A. K. N. "Modern Electrochemistry"; Plenum Press: New York, 1970; Voi. 2, Chapter 8. Bodenstein, M.; Fink, C. G. 2.fhys. Chem. 1907, 60, 1. Bodenstein, M.; Lind. S. C. Z .fhys. Chem. 1907, 57, 168.
75
Ind. Eng. Chem. Fundam. 1986, 25, 75-84 Boer, J. H. de A&. Catal. Re/. Subj. 1957, 9 , 472. Boudart, M. CHEMTECH 1978. 8 , 231. Boudart, M. Catal. Rev.-Sci. Eng. 1981, 23, 1. Boudart, M. J. Phys. Chem. 1983, 8 7 , 2786. Boudart, M.; Dj6ga-Mariadassou, G. “Kinetics of Heterogeneous Catalytic Reactions”; Princeton University Press: Princeton, NJ, 1984. Boudart, M.; Egawa, S.; Oyama, S. T.; Tamaru, K. J. Phys. Chem. 1982, 78, 987. Boudart, M.; Loffier, D. G.; Gottifredi, J. C. I n t . J. Chem. Kinet. 1985, 77, 1119. Christlansen, J. A&. Catal. Re/. Subj. 1953, 5 , 311. Cimino. A.; Boudart, M.; Taylor, H. S. J. Phys. Chem. 1954, 58, 796. De Donder, Th. “L’AftlnitB”; Gauthier-Viiiars: Paris, 1927; p 43. Emmett, P. H. “New Approaches to the Study of Catalysis”; Phi Lambda Upsilon: University Park, PA, 1962; Chapter 5. Eyring, H. J. Chem. Phys. W35, 3, 107. Garfinkle, M. J. Chem. Phys. 1983, 79, 2779. Glassman, I . “Combustion”; Academic Press: New York, 1977. Gcddard, W. A., 111 Science 1985, 227, 917. Hampton, R. F., Ed. J. M y s . Chem. Ref. Data 1973, 2 , 267. Happel, J. Catal. Rev. 1972, 6 . 221. Herbo, CI. Bull. SOC. Chim. Be@. 1942, 51, 44. Hjortkjaer, Jes ”Rhodium Complex Catalyzed Reactions”; Polyteknisk Vorlag: Lyngby, Denmark, 1982; Voi. 2. Horiuti, J. J. Res. Inst. Catal., Hokkaido Unlv. 1953, 2 , 87. Horiutl, J. Ann. N.Y. Acad. Sci. 1973, 273, 5. Horiuti. J.; Nakamura, T. Adv. Catal. Re/. Sub]. 17, 1. Johnston, H. S. “Gas Phase Reaction Kinetics of Neutral Oxygen Species”; U.S. Government Printing Office: Washington, D.C., 1968; NSRDS-NBS 20. p 12.
Kembali, C. Discuss. F a r a h y SOC. 1066, 41, 190. Kondrat’ev, V. N. “Rate Constants of Gas Phase Reactions”; Nauka: Moscow, 1970. Madix, R. J. “Chemistry and Physics of Solid Surfaces”; Vanseiow, R., Ed.; CRC Press: Boca Raton, FL, 1979; Vol. 2, p 63. Mars, P.; Schoten, J. J. F.; Zwietering, P. “The Mechanism of Heterogeneous Catalysis”; de Boer, J. H., et ai., Eds.; Elsevier: Amsterdam, 1960; p 66. North, A. M. “The Kinetics of Free Radical Polymerization”; Pergamon Press: Oxford, 1966. Ozaki, A.; Aika, K. “Catalysis, Science and Technology”; Anderson, J. R., Boudart, M., Eds.; Springer-Veriag: Heidelberg, 1981; Voi. 1, p 87. Prigogine, I.; Outer, P.; Herbo, CI. J. Phys. Colloid Chem. 1948, 52, 321. Satterfieid, C. N.; Roberts, G. W. AIChE J. 1988, 14, 159. Sinfelt, J. H.; Hurwitz, H.; Shuiman, R. A. J. Phys. Chem. 1960, 64, 892. Temkin, M. I . I n t . Chem. Eng. 1971, I f , 709. Temkin, M. I.; Pyzhev, V. Acts Physicochim. URSS 1940, 12, 217. Ti”, B. Proc. I n t . Congr. Catal., 8th, 1984 1985 1, 7. Vannice, A. “Catalysis, Science and Technology”; Anderson, J. R., Boudart, M., Eds.; Springer-Veriag: Heidelberg, 1982; Voi. 3, p 139. Van Tiggekn, A. “Oxidations et Combustions”; Technip: Paris, 1968; Voi. 11, Chapter 12. Wagner, C. Adv. Catal. Re/. Subj. 1970, 21, 323.
Received f o r review June 21, 1985 Accepted October 17, 1985
This paper was presented at the National Meeting of the American Chemical Society, Miami, April 1985.
GENERAL ARTICLES Cubic Chain-of-Rotators Equation of State Hwayong Kim,+ Ho-Mu Lln, and Kwang-Chu Chao’ School of Chemical Engineering, Purdue lJnivers&
West Lafayette, Indiana 47907
A cubic equation of the perturbation type is developed to express pressure as being made up of contributions due to repulsive, rotational, and attractive forces. Use of the equation requires p,, T,, and w of a substance to be known. Calculated pvT, vapor pressure, and enthalpy are compared with data and with the Soave equation and the Peng-Robinson equation for a variety of substances over wide ranges of temperature and pressure. The equation is extended to mixtures by using van der Waals one-fluid mixing rules for the equation parameters. Gas-liquid equillbria of fluid mixtures are calculated for low-pressure symmetric mixtures as well as for highpressure asymmetric mixtures of a heavy solvent with a light gas such as hydrogen, methane, carbon dioxide, and nitrogen. Calculated pvT of mixtures is illustrated with two binary systems for gas and liquid states%p to the critical point.
Introduction Equations of state are useful for the calculation of fluid thermodynamic properties. This usefulness has prompted a continual development of new equations. The perturbation type of approach is noteworthy for being productive of some very useful equations of which the cubic equations have received much attention. Equations such as Redlich-Kwong (1949), Soave (1972), and Peng-Robinson (1976) are in wide use due to their simplicity and generality combined with reasonable accuracy. AU of these equations contain van der Waals excluded-volume expression for the repulsive pressure RT/(u - b). The excluded-volume concept is valid for a dilute gas but breaks down at high densities such as those of liquids (Vera and Prausnitz, 1972; Gubbins, 1973; Abbott, 1979; Henderson, 1979). In place of van der Waals form, Carnahan and Starling (1969, 1972) obtained an expression for the repulsive pressure based on molecular dynamics calculations. Donohue and Prausnitz (1978) developed a perturbation ‘Department of Chemical Engineering, University of Delaware, Newark. DE. 0196-4313/86/1025-0075$01.50/0
equation of state to include Carnahan and Starling’s repulsive pressure and, additionally, to account for the rotational motion of polyatomic molecules in terms of equivalent translational degrees of freedom. Chien et al. (1983) obtained an expression for the rotational pressure from Boublik’s (1975) equation for hard dumbbell molecules. The chain-of-rotators (COR) equation of state that Chien et al. developed using the Carnahan and Starling repulsive pressure expression and their new rotational pressure expression appears to be accurate, but complex. In this work we develop a cubic equation as a simplified form of the COR equation of Chien et al. in order to provide the computational ease of cubic equations while retaining the structure of the COR equation. The repulsive and rotational pressure contributions are simulated with simpler functions. The attractive pressure expression is also simplified. The equation is then fitted to the vapor pressure and saturated liquid density at subcritical temperatures and the pu isotherms at supercritical temperatures. The CCOR equation requires the three constants T,,p,, and o for the substance to be described. These constants are known for a large number of substances, in contrast 0 1986 American Chemical Society
