Energy transfer processes and chemical kinetics at high temperatures

E N E R G Y T R A N S F E R P R O C E S S E S A N D CHEMICAL KINETICS AT H I G H T E M P E R A T U R E S SIDNEY W. BENSON Recent theoretical studies of the rates of recombination of atoms in gases are examined and discussed in terms of the experimental evidence available. Unfortunately most of the latter is of such low precision that many of the fine distinctions cannot be made. It does appear, however, that the recombination process must involve deactivation of nascent molecules by consecutive collisions rather than by large energy transfers. The justification for a consecutive collision process is examined in terms of recent theoretical studies of the vibrational energy transfer process and shown to be quite strong. These studies yield two independent evidences supporting the belief that vibrational energy exchange occurs as a 1 quantum process for both ground level and highly excited oscillators. Some recent studies on ionization processes in strong shocks will be examined briefly and mechanisms will be described for electronic excitation as well as ionization. It is shown that very low values for the ionization activation energy must be accompanied by such low Arrhenius A factors as to make the data implausible. Finally some general considerations will be described governing the process of homogeneous catalysis of recombination at high temperatures and it will be shown that the recombination of H atoms can be accelerated by factors of 100-fold or more by the addition of small amounts (~1%) of suitable molecular species.

Introduction--Unimolecular Reactions The unimolecular reaction of a molecular species, involving as it does the apparently isolatable event of spontaneous change, shares some of the mystery of comparable events such as the radioactive decay of metastable atomic nuclei. In principle we know that there are two requirements for such an event, activation and atomic rearrangement. The former must precede the latter since the rearrangement of an initially stable system will require energy. These two processes are characterized by two quite different time scales. The rate of atomic rearrangement within a molecule could be expected to be of the order of atomic frequencies with a lifetime of about 10-la seconds. On the other hand the process whereby a molecule in a dilute chemical system (i.e., gas) gains a large excess of internal energy relative to R T is one of molecular collision. At STP collisions occur at intervals of about 10 -9 sec. As the pressure decreases, this time becomes correspondingly longer (10 -s see at 1 mm Hg). Such values would seem to iml)ly that unimolecular reactions have rates which are controlled by tile rates at which molecules may gain large amounts of internal energy and only

exceptionally by the rates of internal rearrangement of atoms. We find however that this is true only of very simple molecules containing on the order of from 2 to 7 heavy atoms, i.e., 02, Oa, N204, N2Os, cyclopropane, etc. For more complex molecules we find that the rate of unimoleeular change is independent of the rate of energy transfer. To account for such an observation we must invoke an additional process, that of internal energy migration, or the localization of energy within a molecule. If we picture the internal energy of an isolated molecule as consisting of a discrete number of units, let us say s quanta, and that these are distributed among n independent modes of internal motion (oscillators), then the simple probability that some very large number, m of these shall be localized in some specified way within the molecule is of the order of

(s-- m + n - - 1) I s ! / ( s +

n--

1) I ( n - - m) I

--* (1 -- m / s ) n-~ << 1. For very large values of n and s, this becomes such a sufficiently rare event (i.e., small fraction), that we can understand how strongly energized molecules can have lifetimes many powers of 10

760

761

ENERGY TRANSFER PROCESSES

in excess of the times of motion of their component atoms. This very elementary version of the famous Rice-Ramsperger-Kassel Model of uuimolecular reactions conceals another important concept, the transition state. T h a t is, there must exist a configuration of the molecular system such that it demarcates reactant and product species. For computing rate events we must picture some transition structure through which all reactant species pass irreversibly on their way to products. The mathematical alternative to this (which has never been seriously explored) is a cradle-tograve program in which we follow reactant(s) over its entire history to ground state product(s). In the range of temperatures between 1000~ and 5000~ the average energy content of molecules becomes so high that the rate of internal energy migration is sufficiently rapid to make collisional activation the rate-determining process for a unimolecular reaction. Thus even complex molecules like C2H~ or C3Hs will be expected to split into radicals at 2500~ at rates which are governed by second-order collisional a c t i v a t i o n ) Such processes which are termed, "energy-transfer processes" are then typical of the high temperatures of shocked gases. I t is the purpose of the present paper to review some simple energy-transfer processes from the point of view of the foregoing concepts and to examine the present kinetic evidence concerning them.

R e c o m b i n a t i o n of A t o m s - - E x p e r i m e n t a l Studies One of the most important chemical reactions occurring in shocked gases is the dissociation of diatomic molecules. In the subsequent expansion and cooling of the shocked gases the inverse processes of recombination of atoms to form diatomic molecules take a dominant role. Experimental studies of both of these reactions are of relatively recent origin, dating back about three decades. The measurements of recombination have been done only once in a photostationary system by Rabinowitch and Wood? Although potentially the most accurate method for these rate studies, the original studies gave very crude results and have not been repeated. Studies of the recombination at room temperature by the photoflash technique have yielded the most reliable rate constants to date. They are believed accurate to within 10 to 20 per cent. The precision is limited by the accuracy of high speed spectrophotometry and a detailed knowledge of the behavior of metastable species.

Unfortunately no photoflash measurements have been made at sufficiently high temperatures to obtain reliable estimates of the teml)erature coefficient of the recombination rate constant. Instead the high temperature data have all comc from shock tube studies of the rate of dissociation. The recombination rate constants have been deduced from these latter by means of the thermodynamic relation:

Kcq = k,~/kr,

(1)

where Keq is the precisely known equilibrium constant for the dissociation reaction, k~ is the second order dissociation rate constant, and kr is the third order recombination rate constant. The spread in rate constants measured by shock tube is usually within a factor of 2 to 10. The accuracy of shock tube rate constants is at present completely uncertain since there are few independent studies with which to compare them. If we take as an example the shock tube measurements of the rates of recombination 3 of 0 atoms in the range of 4000~176 one finds that the results of 5 different laboratories cover a tenfold range. The values of kr calculated from Eq. (1) are subject to considerable uncertainty since K,q and kd are both very sensitive to temperature. Thus for 02 at 3000~ the critical ratio, E / R T is about 20 so that a 1 per cent error in temperature will lead to a 20 per cent error in either K~q or kd. Here E is the bond dissociation energy. This is not true of kr and it is hoped t h a t future shock tube work will turn to direct measures of k~ rather than kd. One further technique of measurement of k~ at room temperature has been made popular recently and this is the titration technique first reported by Harteck and co-workers2 Once again the accuracy of the method is still uncertain and there have been spreads of factors of 2 and 3 reported by different laboratories. 5 Here surface reactions and impurities can play an important role. In addition it is not quite certain that the metastable species produced in the initial discharge are without effect. The result of the foregoing is that while it is generally believed that kr decreases with increasing temperature, the magnitude of the effect is almost completely uncertain.

Microscopic Reversibility A number of authors have questioned the validity of Eq. (1) for calculating the values of kr and k~. The basis on which these criticisms rest is that in a given experiment, there is a non-Maxwellian distribution of excited species.

762

CHEMICAL

REACTIONS

AND

PHASE

If this energy distribution shifts with the relative concentration of atoms and molecules in the system, then, it is stated, there is no necessary relation between the mechanism far from equilibrium and near equilibrium. Hence there need be no simple relation such as Eq. (1) between a measured k~ (or ka) and its inverse constant. This argument is not correct. The application of microscopic reversibility can be readily justified if the system is in a stationary state, i.e., if there are no important concentrations of reactants tied up in intermediate states. Under these conditions it does not matter how nonMaxwellian the intermediate states; forward and back reactions will proceed through these at equal rates. Until there is some cogent experimental evidence indicating the contrary Eq. (1) may be considered to hold rigorously for kinetic studies.

The CoUisional Deactivation M o d e l for Recombination We have recently proposed a modification 6 of the original Rice model for atom recombination which is the first to account for two anomalous features of this process. The first anomaly has to do with the fact that the Arrhenius factor of the inverse process, dissociation, is from 100 to 1000 times larger than collision frequencies, Z. This is in sharp contrast to the usual values of the Arrhenius factors for "normal" chemical, bimolecular rate constants which are all smaller than collision frequencies. The anomalous value of Ad was explained by Rice ~ by invoking a pre-equilibrium between ground state molecules Xe and a group of highly energized molecules, X2 ('o possessing energy on the average about R T below the dissociation threshold. His dissociation mechanism is a

X2 + M.~-X2 (~) + M v

(2) X2 (n) + M ~-~ 2X + M. c

If k~ > kr which seems reasonable, then the value of ka is given by

ka = k~K ('),

(3)

where K (~) is the equilibrium constant for formation of X2 ('o. Since the entropy of the X2 ('o species is very likely of the order of 8 to 13 eu, in excess of X2, the value of Ad [which is given by A, exp (AS('O/R), with A~ of the order of the

CHANGES

IN

SUPERSONIC

FLOW

collision frequency Z ] can readily exceed Z by two to three orders of magnitude. I t is important to note that the Rice model already implies the transfer of vibrational energy or collision in small increments. Any theory which proposes very large transfer of vibrational energy on collision will immediately lose the pre-equilibrium of X2 with X2 (n) along with the favorable entropy involved. Such theories will therefore not be able to account for the very large values of Aa. The second anomaly which required explanation is the weak decrease of k~ with increasing temperature which we have discussed briefly. If one uses (3) and (1) to calculate kr one finds that kr is not sensitive to temperature. Our modification of the Rice model which gives such an effect is to replace his pre-equilibrium by a stationary state so that k~ and k~ are not necessarily very different. From the point of view of recombination, the model looks like the following:

X + X ~ X2* X~* + M ~ X2 (~) + M X~ (') + M . ~ X,/n-l) + M

X2 (1) + M ~ - X 2 (~ +

M.

(4)

X~ (") is the highest, bound, vibrational state of X2 while X2* is any pair of X atoms whose separation is less than some assignable distance rm. Employing the usual steady state teehnigue which can be readily justified for this system it is possible to derive the following expression for k~:

kr = XZK*G( T).

(5)

Here X is the probability that a collision of X2* with M will lead to deactivation, K* is the equilibrium constant for 2X,~- X2* while G ( T ) is for all practical purposes the inverse of the number of vibrational states of X., within the energy range R T of the dissociation threshold. The physical interpretation of this result [Eq. ( 5 ) ] is that when 2X atoms come within bonding range of each other, a properly phased collision with a third body M can lead to the loss of sufficient energy so that a bound state of X2 will be formed. The probability of such an occurrence per collision is given by the parameter X. I t is also assumed that the amount of energy lost b y Xe* in this collision will be of the order of R T so t h a t one of the highest vibrational states of the species, X2 (~) will result. This highly excited species X2 (~) may however regain this energy in a further collision and be redissoeiated.

ENERGY

TRANSFER

TABLE 1 Some Rate Constants for Atomic Recombination in Argona kr X 10-~(12/mole~-sec) Calc. Temp Atom (~ H

300 600 1000 2000 6000

N

300 600

Case IB 13.5 12.2 10.5 7.3 2.9 1.27 1.12

Case IIB

Obs.

3.8 3.2 2.7 1.8 0.81

9~7(H~=M) -----

2.1 1.6

1.4 --

1000 2000

0.96

1.3

--

0.67

0.79

--

O

300 600 1000 2000 3400

1.14 0.70 0.48 0.28 0.18

2.0 1.08 0.69 0.36 0.21

1.0 ---0.06

Br

300 600 1000 2000

2.32 1.17 0.70 0.34

6.4 2.8 1.6 0.68

3.1 1.2 0.6 0.3

I

300 600 1000 2000

3.20 1.35 0.75 0.34

10.0 3.7 1.9 0.76

2.9 1.2 0.6 --

a Reprinted in part from reference 6. IB is for Morse potential, IIB for Lennard-Jones potential. As the temperature increases, the probability of redissociation increases and thus the total rate of recombination decreases. I n effect the group of highest vibrational states within R T of dissociation constitute an increasing barrier to recombination with increasing temperature. The net temperature dependence of kr from this model approaches T -~ at high temperatures, i.e., temperatures of the order of (h~/k). Table 1 shows a comparison of some values of kr calculated from Eq. (5), assuming k = 1 and neglecting any contribution to recombination of upper excited electronic states. The values of K* used here are calculated with a Sutherland correction to the viscosity diameters. The Sutherland is on the average about 10 to 20 per cent, thus almost negligible within the present uncertainties.

763

PROCESSES

Within the imprecision of current data, Eq. (5) may be said to be in excellent agreement. It leaves unanswered a number of questions. One of the most interesting of these is why the upper electronic states of X2 which are accessible to 2X atoms do not appear to contribute to recombination. The second concerns the correct value for h. The third has to do with the proper potential function to use to describe X2* while the fourth has to do with the justification for small energy transfers occurring on collision. The value of k, is quite sensitive to the potential of interaction between X atoms at large distances. The Morse (Case IB) and the Lennard-Jones (Case IIB) potentials give results differing by factors of from 2 to 5. This is an unfortunate but real result. It is difficult to foresee any simple resolution of this problem other than an investigation of the X - - - X interaction potential at low energies and large distances. On the question of the contribution of upper electronic states (such as IA and 1Z for 02) there is again no simple answer. One would expect contributions from such states, particularly at low temperatures. Their participation would imply values of k of the order of 0.1 to 0.3 rather than unity as assumed. In addition they would be expected to contribute another source of negative temperature coefficient to kr of the order of T -1. Some independent studies of these states would appear to be well worth while. On the question of small energy transfer we shall offer some evidence in the last section. First we shall present some aspects of an independent, homogeneous, catalytic mode for recombination. H o m o g e n e o u s , Catalytic R e c o m b i n a t i o n of O Atoms We have recently considered the general prospects for a homogeneous catalyst s for the recombination of atoms. Such considerations stem from the idea that if a species C forms a strong bond with an atom X and C has many internal degrees of freedom, then the nascent species, CX* formed in a bimolecular collision process may have a sufficiently long life to be able to be deactivated by collision. I n such cases one can reasonably expect the following chain sequence for recombination: 1

c + x~:cx 2

(8) CX -]- X--~ C + X2. 3

764

CHEMICAL REACTIONS AND PHASE CHANGES IN SUPERSONIC FLOW

Under the conditions that k3(X) > ks so that dissociation of CX is negligible, and further that the reaction of C with Xs (reverse of 3) is also negligible compared to the reaction of C with X, the rate of catalytic recombination R~ will be given by k~(C)(X). Comparing this with the termolecular process in which the third body M is Xs we find

R J R , = k~/kt(X) X ( C ) / ( X 2 ) .

(7)

Now since kl may be expected to be of the order of 10 t~ l/mole-see, while kt is about l01~ F/moleS-see, we sec that R~/Rt will be of the order of 200 ( C ) / ( X s ) ( X ) at 2500~ with all concentrations expressed in atmospheres. Thus for (X) ---- 0.001 atm and ( C ) / ( X s ) = 0.01 (i.e., i mole per cent) the catalytic rate can be 2000 times faster than the homogeneous, thermolecular rate. Under these conditions, the efficiency is still considerable when an appreciable fraction of CX redissociate via step 2 and there is appreciable back reaction of X2 with C. The corollary of all this of course is, t h a t for such species C, one must be certain that they are not present as impuritics in amounts of the order of 1 part in 100,000 or they will make an error of 100% in measurements of kt. The above mechanism, if applied to the very important problem of H atom recombination at very high temperatures and low pressures, suggest a number of possible catalysts approaching the above hypothetical efficiency. These are in descending order of utility, CH4, CsH2, NH3, and HsO. A more detailed analysis of these molecules will be found in an independent paper (reference 8). The chief requirements on C for high efficiency are that it contain at least three atoms and that the C - - H bond strength be very large relative to R T. Both of these guarantee a long life for the nascent CX* species. Practical considerations of thc method of utilization of C will dictate other requirements, not lcast of which is that C and CX bc stable with respect to other kinetic processes in the regime and time of utilization. These wc shall not be able to discuss hem. The possibility of finding efficient catalysts rests on the existence of species CX for which the rate of decomposition is controlled by internal energy migration. Under considerations where this is not the case, the decomp<)sition of CX will be controlled by the rate of energy transfcr and the lifetime of CX* may I>c too short to be of interest in the chain cycle, (6).

Ionization Rates in Hot G a s e s At temperatures above 4000~ ionization becomes an important phenomenon in hot gases

and the process of dissociation into ions and electrons and the inverse recombination show many similarities with atom recombination. In recent years studies of the rates of ionization of rare gas atoms in shock tubes 9,t~ have pointed to mctastable states as being the chief precursor to ionization in the absence of impurities with low ionization potentials. The evidence adduced for this is that the activation energy for the rate of ion production has beep considerably below the ionization potential of the rare gas in question. The mechanism for such a process is one of the two following: Case A 1

M + X~.~- X* + M 2

X* + M--+ X+ + e- + M 3

Case B 1

M + X~.-~-X* + M X* + X--~ X~+ + e-. 3t

If we apply steady state methods to X*, the metastable species, then we find for the rates: RateA = J-kskl(M) ( X ) ] / ( k s + k~)

(8)

RateB = [-k3'k,(M) (X)2]/[-k:(M) + k 3 ' ( X ) ] . (9) When (M) = (X) the two rate laws are indistinguishable. Let us considcr the conscquences of the fact that the ionization activation encrgy is very much less than the ionization potential. Let it be first noted that mechanism A is inherently implausible relative to B if only by virtue of the fact that any second higher excited state of X produced by collision with M will dissociate its clcctron so rapidly (10-15 to 10-14 sec) that the product X M + will be cffcctively left behind. For this reason we shall consider only Case B even where the bond energy of X M + is very small. In order for RateB to give an overall activation energy corresponding to step 1, k3/ > ks (assume M = X ) . However ka' is not without activation energy of its own, namely the ionization potential of X* minus the bond dissociation energy of Xs ~. The former is of the order of 4-5 cV while the latter is about 1-2 cv. Thus ks' should have an activation energy of the order of 3 ev. The only way in which the relation ks < ks~ can be satisfied under these conditions is if the Arrhenius A factor for step 2 is much lower than that for step 3 (i.e., A2 << As'). The assignment of an activation energy to

765

E N E R G Y T R A N S F E R PROCESSES

step 2 is not a solution since the activation energy for step 1 would have to be. raised by an identical amount in view of the relation E1 -- E2 = &E* = the excitation energy of X*. But if ks has a very low Arrhenius A factor, so must kl since R in (A2/A1) = AS* ~ 2 (the entropy of excitation of X*). However at the very high temperatures where the rates have been studied (7500~ these inequalities are not very large. Thus an activation energy difference of 3 ev which will contribute a factor of 10-8 (at 2500~ to the ratio of two rate constants, will contribute 10-3 at 5000~ and 10-2 at 7500~ However the A factor for 3', Aa' is already low. The standard entropy change in reaction 3' is about 22 eu so that A J is about 105 smaller than A4p, the A factor for dissociative recombination. The latter however cannot exceed frequencies of about 1012 1/mole-sec so that A J < 107 1/mole-sec. This in turn implies that As < 105 1/mole-sec which is then also the order of magnitude of At. There is no evidence at present to indicate that the Arrhenius factor for the ionization rate constant is so small. An alternative mechanism has been presented by Petschek and Byron for the latter stages of ionization in shocked argon. They skip over the details of the process, assuming only that collisions of "colder" electrons with metastable Ar* are rate controlling. They further assume that Ar* are in equilibrium with electrons and not with Ar, ground state. But this assumes without evidence that the inelastic collision of e-* + Ar--~ At* + e- is not reversible compared to the further excitation of Ar* + e- --~ Ar + + 2e-. This seems indefensible in terms of detailed balancing. 13 I t is my feeling t h a t there is as yet very meager experimental evidence on which to base any detailed mechanism of ionization in shocks. W h a t does appear to be well established is the difficulty of controlling impurity generated ionization.

Vibrational Energy Transfer From the foregoing discussion it is perhaps evident that the inelastic transfer of energy between an internal oscillation in a complex molecule and the translation degrees of freedom is one of the keys to high temperature reaction rates. This problem has been under active investigation since the early 1930's ~4 with results which are in about order of magnitude agreement with laboratory experiments. There are two aspects of this work which have a direct bearing on the problems of chemical kinetics which it is of interest to discuss here. One has to do with the probability of collisional

loss of a vibrational quantum by a highly excited oscillator (such as X2 (') or X2*) while the other has to do with the relative probability of excitation of a lower energy oscillator to very high vibrational levels by an energetic collision. We have recently at the Douglas Research Laboratories completed some computations of the classical, inelastic co-linear collisions of a diatomic molecule (harmonic oscillator) AB with an atom, C) 5 The computations were done on a machine (IBM 7090) and constituted a precise calculation of the mechanical trajectory of the collision over a broad range of the dynamical parameters of the system. The interactions between the colliding species were represented by a Morse (or Lennard-Jones) potential with shallow minimum between atom B of the diatom AB and the atom C. For simplicity the collision was always between B and C. This is pretty much the model which has been examined by most previous investigators. If 1 is taken as the range of the repulsive forces in the Morse potential

(V/Vo = [1 - exp ( - , ~ ( z -

Xo))] 2 - 1)

(10)

with l = 1/2o~, Vo the well depth and X0 the equilibrium separation of B and C, then a hardsphere, impulsive collision could be represented by setting l ~ 0.01 ~ (a = 50 .~-1). These results could then be checked against the exact explicit solution which is available for the impulsive, hard sphere collision.15 Some surprisingly interesting results came from this latter problem. In Fig. l a is shown a space-time diagram of the hard sphere collision. The straight lines through C represent the trajectories of C relative to A-B for different initial separations of C and A-B. The sinusoidal curve through B represents the harmonic trajectory of B relative to the center of mass of the oscillator (assumed fixed). All possible initial separations of C and A - B will be represented by a family of curves parallel to the lines through C. What is most striking here is the fact that no intersection of the two trajectories can occur between the phases of oscillator motion represented by the region between the points P and Q! Between P and R, no collision can occur because B is moving faster than C and away from it while between R and Q the collision is prohibited by the prior collision at the preceding crest (e.g., P'). What this oversimplified model shows is that the collisions tend to he confined to oscillator phases corresponding to expansion which would on the average cause de-excitation. Two extreme cases are represented in Fig. l b and lc. In Fig. l b we see the result of a hard

766

CHEMICAL REACTIONS AND PHASE CHANGES IN SUPERSONIC FLOW

|

02 - At

60

a = 100

t

,,ME

-,

~o

.

-'~Q

-0o

I

-- . - - f

~" 20

XB~AX ~

(a =2.5

PO-1

-40

"" 0

XA MIH PHASE ~

| P1

I/ v

_

| P2

.0 0

"'"

2

4

6

8

ER x 1012

|

FIG. 2. Probability of excitation and de-excitation per collision from the first vibrational level to ground state.

P

I/ V

FIG. 1. Space-time trajectories for colliding hard spheres showing excluded phase angles. sphere collision between a "cold" atom and a highly excited oscillator. This is the situation we envisage as the rate-determining step in atom recombinations. We see that the relative slopes are such as to favor collision near the peak of the oscillator separation where little kinetic energy is present for de-excitation. The effect of anharmonicity is shown in Fig. ld. The result is to weight even more heavily the configurations of very large separation of the oscillator atoms and again make more probable the exchange of small amounts of kinetic energy. We believe that with all the limitations of the oversimplified model, this result added weight to our assumption that vibrational energy exchange takes place in small rather than large increments. The other extreme situation where no oscillator configurations are excluded is shown in Fig. le representing the hard sphere collision of a "hot" atom C with a "cold" oscillator. There is now no longer any absolute restriction on phases but a further consideration appears. This second consideration has to do with the observation that even in the completely classical system, the probability of excitation shows a sharp cutoff with collisional energy. This cutoff energy increases in magnitude with increasing softness of collision and is in fact a very sensitive function of the parameter a [-Eq. (10)J. This is illustrated in Fig. 2 showing the de-excitation probability per collision, P~-0, as a function of collisional energy and range a for a "soft" collision. The P1-0 curves go through a maximum

and fall to zero with increasing energy. The reason for the maximum is that at higher collisional energies, excitation rather than deexcitation predominates (Fig. lc). If the excitation curves are examined (Po-i) it is found that they exhibit a sharp decrease with increasing energy above their own cutoffs. This is due to the fact that with increasing collisional energy the probabilities of 2 , 3, and higher quantum excitation becomes important. However, inspection of the curves shows that the cutoff for these higher order processes does not occur at an energy just h~ higher than the cutoff for the lower quantum jump. On the contrary, there is a considerably higher energy requirement. What this implies is that if we consider the stepwise excitation of an oscillator (originally in the ground state) to any quantum level nhv, the stepwise rate will always be faster than the multiple quantum excitation. The kinetic reason for this arises from the steady state analysis of the system. For a 2-quantum process for example, the stepwise mechanism is 1 M + X { ~ ~ - M + X~(~) 2 3 M -{- X~(z)~.~- M -~ X f f ), 4

(11)

while that for the 2-quantum jump is: M -[- X2(~

M -F X~(2), (12) 2p The ratio of stationary rates for these processes, neglecting for the momement the deactivation rates 4 and 2', is:

Rn/RI~ = k~ka/Ek~'(k~ + ka)].

(18)

If ]ca > ks then Rn/Rz~ "---) kz/kz' while if as is much more likely ks > ks, then Rn/R12

ENERGY TRANSFER PROCESSES

K12k3/kl' where K12 is the equilibrium constant for X~ (1) and has a temperature coefi%ient of precisely h~. I t turns out however t h a t the activation energy of k3 is much less than t h a t for k2 so that process (11) is always faster than process (12). The exception to this is in the hard sphere case where the threshold energy is v e r y close to the enthalpy change for excitation. E v e n in this case, however, there is a small, not inappreciable activation energy. This arises from the circumstance t h a t if a t o m C were to transfer 100 per cent of the relative energy of the collision to vibration, it would be trapped as a motionless particle next to A B . On the next half-cycle of oscillation E would strike C and be de-excited. A very crude analysis of the excess " a c t i v a t i o n " energy required in the classical, haM-sphere case yields a value of a b o u t 14% of the q u a n t u m excitation. T h a t is, if C is to excite A - B by nh~, then the m i n i m u m relative collision energy of the system must be about 1.14 nh~. Substituting any such result in E v . (13) Yields the same conclusions, n a m e l y a more rapid stepwise excitation.

2. RABINOWITCIt,E. and WOOD, W. C.: Trans. Faraday Soc. 32, 907 (1937); J. Chem. Phys. 41, 487 (1936). 3. WILSON, J.: Thesis, Cornell Univ., 1962. 4. HARTECK,P., REEVES, R. R., and MANELLA,G.: J. Chem. Phys. 29, 608 (1958). 5. HARTECK,P. and REEVES, R. R.: Chemical Re-

actions in the Lower and Upper Atmosphere,

6. 7. 8. 9.

10. 11.

12. 13.

ACKNOWLEDGMENTS This work has been supported in part by the Douglas Aircraft Co., Santa Monica, California, and by the National Science Foundation.

14.

REFERENCES 1. BENSON, S. W.: Developments in Mechanics, Vol. 1, p. 554. Plenum Press, 1961. A brief discussion is present here.

767

15.

p. 219. Interscience, 1961. See this for comparisons of recent work. BENSON,S. W. and FUENO,T.: J. Chem. Phys. 36, 1597 (1962). RICE, O. K.: J. Chem. Phys. 9, 258 (1941). BENS0N, S. W.: In press. JOHNSTON, H. L.: Paper presented at Spring Meeting of the American Chemical Society, Abstracts, April 1962. PETSCHEK, H. and BYRON, S.: Ann. Phys. 1, 270 (1957). BIONDI, M. A.: Chemical Reactions in the Lower and Upper Atmosphere, p. 353. Interscience, 1961. Idem. : Phys. Revs. 35, 653 (1951). Values of A2 reported here are about 107 l/mole-sec. Douo^L, A. A. and GOLDSTEIN,L.: Phys. Revs. 109, 615 (1958). This shows that the equipartition of energy between atoms and electrons occurs extremely rapidly via coulombic collisions even at 0.001% ionization. See HERZFELD, I~. F.: Thermodynamics and Physics of Matter. Princeton Univ. Press, 1955. BENSON,S. W., BEREND, G. C., and Wu, J. C.: J. Chem. Phys. 38, 25 (1963); ibid. 37, 1386 (1962).

Discussion PROF. S. H. BAUER (Cornell University): For an audience which consists of a minority of chemical kineticists, it may be worthwhile to underscore that the notation and formalism of chemical kinetics is based on the assumption that each symbol is to be clearly identified with a distinct chemical species, the concentration of which could be measured, at least in principle. I should also remind you that the "rate constants" which chemical kineticists use to express probabilities of transformation of one such specie to others are averaged over many molecular parameters and states. If one is not on guard as to the significance of these hidden parameters, he will easily become confused. The accepted formulation of a unimolecular process is: k2 A +X~-~,A* +X; kl A* ~ transition state ~ products. k0

The "transition state" of the chemist is to the physicist a region of phase space which corresponds to the accumulation of a large amount of internal energy in a few (critical) vibrational modes. When these modes attain large amplitudes, the atoms are brought into a sequence of configurations required for the transformation of the reactant to the product. The "transition state" is that region of phase space which corresponds to the maximum of the potential energy surface along the minimum energy path (saddle point) in which the representative points are moving toward the configurations representative of the products; the correct vector quality is essential. I must admit that I did not have the proper intuitive feeling for Prof. Benson's statement that at sufficiently high temperatures the rate of internal energy migration is sufficiently rapid to make collisional activation the rate-determining process for a unimolecular reaction. However, on the slide he

768

CHEMICAL REACTIONS AND PHASE CHANGES IN SUPERSONIC FLOW

argued its validity b y writing: k3 ~

~[(~

-

~*)/~]--,.

Of course, if he chooses, he m a y introduce this dependence as an essential postulate; and t h e n he will not obtain the usual Arrhenius temperature dependence of k~. Generally, it has been assumed that k3 =

/0 co k(~)D(e)

de,

where k(e)

=

o

for

~ _> e*

and D(~) is the distribution function over states. Then k3 = v exp (-- e*/kt), and the unimolecular limit is maintained. Thus by performing the average over energies rather t h a n over probabilities for reaction [k(e)], Benson reached this rather unexpected conclusion, for Which, as far as I know, there is no evidence. The m a t t e r of notation arises again in the comparison Of O. K. Rice's model for the dissociation of diatoms with t h a t of Benson.

a X2+M~--~X2 (n)+M

b

Prof. Benson treats X2(") as a distinct chemical species for which one may write a meaningful partition function, and thus deduce an equilibrium constant for the reaction, as written. Prof. Rice did not use this formalism because the X2(")'s are nothing b u t X2 molecules in their uppermost vibrational states which have already been included in the partition functions and equilibrium constants in the manner proposed. Now, the rate constant for dissociation m a y be simply written as the product of the normal collision number times the population of molecules in the states within k T of the dissociation limit. kd = Zn* = Zg(Do -- kT) exp [ - - (Do -- k T ) / k T ]

where g(Do - k T ) is the total n u m b e r of states within k T of the dissociation limit. The last term assumes the essential a t t a i n m e n t of a Boltzmann distribution for these states. Benson stated t h a t his estimate of the population was obtained by solving for a steady state condition, with detailed balance among all the upper states. From the material presented it is not evident t h a t his temperature dependence of the pre-exponential term could differ substantially from t h a t of Rice. I strongly suspect t h a t differences by a factor of T 1/2or even T could be introduced by following different averaging paths. For example, a classical t r e a t m e n t of all the vibra-

tional levels will give a different T dependence t h a n one in which the lower levels are quantized. I wish to counter to some extent the pessimism regarding diatom dissociation data as derived from shock tubes. If one considers all the published works, values for selected rate constants do differ b y a factor of 10. However, values which appeared during the past 2-3 years are m u c h more consistent; these range only over factors of 2-3. In particular, the rate constants for 02 have settled down, and as was not admitted b y Prof. Benson, Jack Wilson has shown t h a t at 2800~ his directly measured recombination rate constant checks within 30% the value deduced from Skip Byron's measured dissociation rates. Finally, with respect to the use of hydrocarbons such as CH~ and C2H2 as catalysts for H atom recombination in nonequilibrium nozzle flow, attention is called to the paper by A. Q. Eschenroeder and J. A. Lordi (this volume, p. 241). They reported on a numerical analysis of the effect of introducing such species in the flow, and concluded t h a t the loss in specific impulse due to the increase in average molecular weight is significant. The over-all gain is not dramatic. PROF. S. W. BENSON (University of Southern California): I n answer to Dr. Bauer's first point, t h a t of the estimate of the half-life or rate constant of a critically energized species, this is done b y a method indicated in m y text "Foundations of Chemical Kinetics". 1 W h a t is done is to set g = 9* -~ n k T on the crude assumption t h a t the mean reactive molecule contains n k T excess internal energy beyond the barrier energy. There is considerable evidence for such an estimate. I n his second point Dr. Bauer confuses Rice's t r e a t m e n t a n d our own. Rice assumed t h a t X2 (~) was in equilibrium with the ground states of X2. We show on the contrary t h a t X2 (n) is in a steady state whose concentration is less t h a n equilibrium. F u r t h e r this steady state concentration decreases with increasing temperature about like T-L I t is this latter behavior which is responsible for the negative temperature dependence of the rate cons t a n t for termolecular recombination. I t h i n k it would be best on this score to refer to the original paper. 2 On his last point concerning the rate of homogeneous catalysis I have not made an effort to calculate the performance gain in propulsion due to small amounts of CH4 or C2H2. I did calculate the effects on the recombination rate of CH4, C2H~, NH3, and H20. They are in the order given with CH4 about a factor of 10 more effective than C2H2. In view of the smaller molecular weight of CH4 vs. C2H2 and the greater catalytic efficiency I would be surprised if it did not increase propulsion in the low t h r u s t (low

769

ENERGY TRANSFER PROCESSES chambcr pressure) engines. B u t t h a t is for the engineers to discover. PROF. H. B. PALMER (Pennsylvania State University): Professor Benson has chosen as an example of shock tube kinetic results the case in which I believe the disagreement is the worst known from shock tube studies. In contrast I might cite the four independent studies t h a t have been made of Br2 dissociation, using differing tubes and differing inert gas dilution ratios. As I recall, the scatter in the entire collection of data is approximately a factor of two. As for checks of shock tube results b y other methods, two examples with which I am familiar come to mind. At the last symposium, Deklau and I reported work on nitrosyl chloride decomposition in shock waves. Our scatter was large, b u t by using the kinetic results of Ashmore and of Waddington and Tolman a t low temperatures, we were able to make a rate constant plot over a large temperature range. Dr. Ashmorc has recently sent me a new result for nitrosyl chloride t h a t falls almost perfectly on our best low T-high T line. Finally, in the case of CH~ decomposition, T. J.

Hirt and I have rate constant data, obtained from kinetic studies of carbon film formation, t h a t agree extremely well with the best line through the shock tube data of Skinner and of Glick. PROF.S. W. BENSON: In regard to the problem of the precision of shock tubery raised by ])r. Bauer and Dr. Palmer I must, of course, confess to playing devil's advocate in some measure. I would be very pleased to know t h a t the precision of rate constants from shock tube studies was within a factor of 50% (spread of 2). This is certainly not true of the data with which I am currently acquainted. I think it is very i m p o r t a n t t h a t Shock Tubers point Out and emphasize the precision of their measurements. I hope that, in future work, independent checks of the type mentioned by Dr. Palmer will be available for comparison. REFERENCES 1. BENSON, S. W." Foundations of Chemical Kinetics, McGraw-Hill, 1960. 2. BENSON, S. W. a n d FUENO, T.: J. Chem. Phys. 36, 1597 (1962).