July, 1957 k'N. =
k'
NON-EQUILIBRIUM 1
nk1Vn-i =
5 m k' k nk E Nn-l k'
=
+k' nk
nkNnO
RATEPROCESSES
849
where Nnois the equilibrium population of n. If, as now assumed, the energy supply is rate determining, IC' > nk and the rate is nkN2
STUDIES I N NON-EQUILIBRIUM RATE PROCESSES. 11. THE RELAXATION OF VIBRATIONAL NON-EQUILIBRIUM DISTRIBUTIONS IN CHEMICAL REACTIONS AND SHOCK WAVES1,' BY KURTE. SHULER Contributionfrom the National Bureau of Standards, Vashington, D. C. Received J a n u a r y 81, 1967
In some recent papers with R. J. Rubin and E. W. PvIontroll a theoretical treatment has been developed for the radiative and collisional relaxation of a system of harmonic oscillators prepared initially in a vibrational non-equilibrium distribution. In this paper a discussion is given of the application of these theoretical studies to experimental results on relaxation processes in chemical kinetics and in shock waves. It is pointed out that the analysis of relaxation data in terms of half-iives loses its meaning when more than two energy levels are involved in the relaxation process. To obtain information on the efficiency of intermolecular energy transfer in these multilevel systems it is necessary to follow in detail the time behavior of the population in several, and preferably all of the populated energy levels. As an example, the experimental data on the relaxation of vibrationally excited 0 2 produced by a secondary reaction in the flash photolysis of C l 0 ~ is discussed in detail. A short discussion is given of the analysis of shock wave data on vibrational relaxation. The concept and the prescription of the "vibrational temperature" introduced by Bethe and Teller in this analysis are shown to be valid from the exact solution of the relaxation equations.
1. Introduction I n the preceding paper of this series3 and in previous papers with R. J. Rubin4 we have discussed the radiative and collisional relaxation of a system of harmonic oscillators contained in a constant temperature heat-bath and prepared initially in a vibrational non-equilibrium distribution. I n the paper with M ~ n t r o l l the , ~ exact solution for the relaxation equation has been given and expressions have been derived for the relaxation of initial Boltzmann distributions, Poisson (distributions and &function distributions as well as for the relaxation of the moments of the distributions, Using the latter results, explicit formulas were derived for the relaxation of the internal encergy of the system of oscillators and for the time (dependence of the dispersion of the distributions. ;Since the collisional transitlion probability, Le., ithe probability per collision (or per unit time) that a n oscillator will exchange its vibrational energy with the translational energy of the heat-bath molecules, is a parameter of the relaxation equation it is evident that an analysis of the relaxation process will yield data on the efficiency of the intermolecular vibrational/translational energy transfer. Conversely, a knowledge of this efficiency permits the calculation of the time scale of relaxation. This can be of use in planning for the experimental techniques to be employed in the study of the relaxation process. I n this paper the results derived previously will 1 e applied to a study of the relaxation of vibrational non-equilibrium distributions in chemical (1) This work was supported by the U. S. Atomic Energy Commission. (2) Presented a t the Symposium on Intermolecular Energy T r a d e r , American Chemical Society, Atlantic City, Sept. 18, 1956. (3) E. W.Montroll and K. E. Shuler, J . Chem. Phya., 26,454 (1957). (4) R. J. Rubin and K. E. Shuler, ibid., 26, 59, 68 (1956); 26, 137 (1957).
reactions and shock waves. Recent experimental work in chemical kinetics has shown quite definitely that the products of various exothermic chemical reactions are formed in a non-Boltzmann vibrational distribution. Some examples of these types of specific vibrational excitations of product species are furnished by the reaction H 03 +OH* + 02 (1) where the OH is apparently formed predominantly in the 9th vibrational level of the 21Ti ground electronic state,6the reactions
+
0
0
+ NO2 +NO + c10 +
+ ClOZ
----f
02*
02*
(11 A) (11 B)
where the 02,in the 32-gground electronic state, has been observed with appreciable concentration up to the eighth vibrational level6 and the reaction C+C+hZ+C2*+M
(111)
discussed by Herzberg7 where the Czappears to be formed in the 6th vibrational level of the electronically excited TIg state.* After a general discussion of the ana,lysis of relaxation data for multilevel systems in Section 2, the data of Lipscomb, Norrish and Thrush6 on the relaxation of vibrationally excited O2 produced in reactions I1 will be considered in some detail in Section 3. The efficiency of intermolecular energy transfer, in particular that between the vibrational and translational degrees of freedom, can also be determined from an analysis of vibrational relaxation be(5) J. D. McKinley, D. Carvin and M. J. Boudert, ibid., 23, 784 (1955). (6) F. J. Lipscomb, R. G . W. Norrish and B. A. Thrush, Proc. R o y . Soe. (London),233A, 455 (195G). (7) G. Hersberg, Astroph. J., 89, 290 (1939). (8) I t is outside the scope of this paper to discuss the mechanism
and the energetics of the specific vibrational excitation of the product molecules in these reactions. This problem will be considered in a later communication.
KURTE. SHULER
850
hind shock waves. Studies of this type have been carried out by Smiley and Winkler9 for C1, and C02, by Blackmanlo for 0 2 and Nz, and by Griffith, Brickl and Blacknianll for CHI, CC12F2,C 0 2 and NzO. I n Section 4 of this paper we shall discuss the applicability of our relaxation theory t o the analysis of relaxation data obtained from shock wave experiments. 2. Analysis of Relaxation Data A search of the literature, and in particular of
the papers quoted above16-’ shows that there is in general not sufficient data for a detailed application of the relaxation equations developed in references 3 and 4. This is due in large part to experimental difficulties encountered in the spectroscopy and the time resolution of rapid transient phenomena which frequently involve unstable chemical species. There is every reason to believe that this situation will improve in the future. Another reason for the unsuitability of much of the “chemical” relaxation data, as distinguished from shock wave data, is the method of presentation in terms of “half-lives.” For relaxation studies, half-lives are meaningful quantities if, and only if, the relaxation is a simple, one term, exponential process in time for which there is then a simple relation between the half-life time t l / , and the collisional transition probability kij
t i / * = In 2/kij
Vol. 61
This is always the case for radiative excitation and deactivation. I n the case of collisional excitation and deactivation a first-order process will be found when the species undergoing the excitation and relaxation process are contained in a large excess of (chemically) inert gas or solvent so that binary collision between the excited species can be neglected in formulating the relaxation equations. The relaxation equations in references 3 and 4 are examples of this type of linear relaxation equation. The solutions of eq. 2 are w
m xn
=
(3)
q.,kexkl
k=O
In eq. 3 the hk are the roots of the secular equation lA - XI1
=
0
(4)
where A = lanjl is a square matrix of order m and I is the unit matrix of the same order. The expansion coefficients q n k are determined by the X k and the initial condition. It can be shown13that for closed linear systems, i.e., linear systems with conservation of mass, such as given by eq. 2 the characteristic roots hk of eq. 4 are negative (or, in the case of complex roots, have negative real parts) and that a t least one of the roots h k is a zero root. For a two level system with m = 2, the solutions of eq. 2 are thus of the form XO = qoo qolexlt (5a) XI
(1)
=
910
+ +
(5b)
qlleX*t
which is independent of the initial concentration of where qoo and qlo are independent of t and where X1 the relaxing species. A relationship of this form and hz are negative roots. It is evident that even (eq. 1) is met with in the relaxation processes which for a two level model, the time decay of the exhave been studied most frequently in the past such cited state (designated by subscript 1, for instance) as the quenching of electronically excited species is not of a purely exponential form. Under certain and vibrational relaxation in sound waves. In conditions which would depend upon the physical both of these processes one is dealing essentially parameters of the relaxation system it may be valid with a two level system where only the lower level to neglect the terms qoo and qlo in eq. 5a and 5b. is significantly populated.12 It is only for such A half-life determination for such a two level model systems that a simple, one term, exponential re- could then yield meaningful parameters for the analysis of the relaxation process. laxation behavior mill be found. It is evident from the development presented The relaxation of a discrete level system with a linear relaxation process is governed by the set of above that in a m-level model there will be a t least first-order linear differential equations where xn(t) (m - 1) exponential terms of the form exp(Xkt). In this case, it is no longer possible to make a meaningful analysis of the relaxation in terms of a “half-life” nor is it possible to specify a single ren = 0, 1, . . . m laxation time for the system or even for each level of the system.14 An example in point is furnished is the fraction of species in level n, and where the by the relaxation of a system of harmonic oscilconstants anj are related to the collisional and lators from an initial Poisson distribution by inradiative transition probabilities for transitions elastic collisions with heat-bath molecules and by between levels n and j. By a linear relaxation proc- radiative transitions. The general relaxation equaess we understand a relaxation mechanism which is tions for such a system of harnionic oscillators of first order with respect to the concentration of (with an infinite set of energy levels) are3v4 the species whose relaxation is being investigated. (9) E. F. Smiley and E. H. Winkler, J . Chem. Phys., 22, 2018 (1954). (10) V. H. Blackman, Teclinical Report 11-20, Dopt. of Physics, Princeton University, May, 1955. (11) W. Griffitli,D. Brickl and V. Blackman, Phys. Rev., 102, 1209 (1956). (12) I n the quenching of electronically excited species one is usually dealing with only two energy levels (ground state and first excited state) since the higher lying electronic levels are energetically too high to have a significant popiilation. I n the case of sound waves (and ultrasonics in general) one is dealing with low energy excitation sources in which the energy input into the system of oscillators is not sufficient, in general, to excite more than the first vibrational level.
9 ) = (ne-Oz,-, dt K
-
[n
+ ( n + l ) e - ~ ] * . , ~+
+
(n 0,1, . . . . a 2,(0) = e - w / n ! n
l)Z,,+lJ
(Fa)
=
(Gb)
(13) For a detailed discussion see, e.g., T. Z. Hearon, BulE. Math. Biophya., 15, 121 (1953). (14) Depending upon the density of the roots XQ and after a sufficiently long timet, all the exponential terms may he nealigihly small compared to t h a t for the smallest root. Under these conditions the very late stages of the yelaxation of a multilevel systeni inay be of a simple exponential form.
NON-EQUILIBRIUM RATEPROCESSES
July, 1957
where x,(t) represents the fraction of oscillators in the vibrational energy level n a t time t , K is related to the efficiency of the collisional and radiative energy exchange between the oscillators and the heat bath, 6’ = hv/kT with Y equal to the frequency of the harmonic oscillator and T equal to the temperature of the heat bath and where a = 5 is the mean value of the level number n in the initial distribution. The solution of this set of relaxation equations is3
85 1
inert gas (Nz)which acts as a constant temperature heat-bath. Since O2 is homonuclear it will lose its excess vibrational energy solely by collisions, which brings about some simplification in the calculations. The departure of this reaction system from the model used for our theoretical study is twofold: the possibility that the vibrationally excited 0 2 is deactivated by collisions with the reaction intermediate C10 (or NO) in the energy exchange reaction 02*
+ c10
----f 0 2
+ c10
(V)
and the fact that O2 is not a harmonic oscillator. The first point, as will be shown below, readily can be taken care of by a proper modification of the where relaxation equations. The anharmonicity of the O2 will of course introduce some error into the calT = ~ t (l e-0) (7a) culations based on our harmonic oscillator model. is the dimensionless time and where L,[Z] is the It is believed that for the relatively lorn lying enLaguerre polynomial of order n. An inspection of ergy levels considered here ( Y ” = 8) this error is eq. 7 shows very clearly that the time dependence not very large; its magnitude can, however, not of X, is much too complicated to permit an anal- be gaged until the relaxation theory has been exysis of the relaxation in terms of half-lives or in tended to a system of anharmonic oscillators. terms of a single relaxation time. Work on this problem is now in progress.14a The analysis presented above applies primarily Since the data obtained by Lipscomb, Norrish to the relaxation of the population in the various and Thrush are presented in terms of “half-lives” discrete energy levels of a molecular system. of the population in one energy level (VI’ = 6) it is Spectroscopic studies of transient phenomena not possible to make a direct comparison between yield data which pertain to the concentration of the their data and our relaxation theory. Instead, it emitting or absorbing species in their various inter- will be necessary to employ a rather oblique apnal energy levels as a function of time. It is one proach. Most of the work of Norrish and his coof the theses of this paper that a meaningful anal- workers was carried out with ClO? and we will ysis of the relaxation of such multilevel systems therefore restrict our attention to these data. Recan be carried out only for data which give the action IIb is exothermic by 61 kcal. and since the concentration of the relaxing species in several highest excited vibrational level observed for Oz, (more than one or two) of the significantly popu- which was VI‘ = 8, is only 34 kcaL6 above v” = 0, lated energy levels as a function of time over a there is certainly sufficient energy in reaction IIB to sufficiently long time interval to permit the deter- account for the observed excitation. In the abmination of the mathematical form of the relaxa- sence of any information about the initial distribution curve. tion of the vibrationally excited 02, we will assume, 3. The Relaxation of Vibrationally Excited Oxygen admittedly without any particular justification, The kinetic study which looks most promising that all the O2 molecules are produced initially in for the analysis of the relaxation behavior of ex- the 8th vibrational level. We are then dealing cited molecular systems and for the concomitant with an init,ial &function distribution such that 1 for n = 8 determination of the efficiency of intermolecular 0 for all other n energy transfer is the work recently initiated by Lipscomb, Norrish and Thrush.6 These investiga- The solution of the relaxation eq. Ga for this initial tors in studying the flash photolysis of chlorine distribution is3 dioxide and nitrogen dioxide in a great excess of in- l)m+n ert gas (Nz) found that oxygen molecules were X , ( t ) = (1 - ee)em@ (e-7 ( e - . - eo) e - r - e@ formed with up to eight vibrational quanta in their F ( - n , -?It, I ; u2) (9) electronic ground state. From a study of the resinh 012 action kinetics they concluded that these excited u=sinh +/2 oxygen molecules had their origin in reactions 11, i.e. where F is the hypergeometric function, T is the dimensionless time given by eq. 7a, 0 has been de0 + NO2 +NO + 0 2 * fined in connection with eq. 6, and where m is the 0 +- CIOZ +CIO + 0 2 * initially populated level, i e . , m = 8 in this case. subsequent to the primary dissociation steps The formula given for x,(t) in eq. 9 and 9a is not NO2 + hr +NO + 0 very convenient for making calculations. It can (1V-Q c102 hv ----f c10 -+ 0 (IVB) be verified readily by a few simple algebraic operThis system forms a good example of the model (14e) NOTE ADDED I N PRooF.-Recently completed oa!culations by studied in references 3 and 4. -4diatomic species Beesley, Montroll, Rubin and Shuler have shown that the anharrnonicity acts only as a small pert,urbation on the relaxation behavior of a ( 0 2 ) whose concentration in the total volume of gas of oscillators. The difference in the fractional level population is quite small is excited to an initial vibrational system znis of the order of the anharmonicity. This work will be published non-equilibrium distribution in a large excess of shortly.
+
852
KURTE. SHULER I.o
Vol. 61
I .c
@ @ c
7’0 7=0.01 7’0.1
8 8 8
D 7-0.5 E 7 = 1.0
0.8
F
7
0.8
= 2.0
e = 7.840 D
0.6
F Xn.
0.4
0.4
0.2
0.2
4
12
8
n. Fig. 1.-The fraction znof 02 molecules in the vibrational levels n a t various times 7 for the relaxation of an initial 6function distribution with all the 0 2 initially in level n = v” = 8. The value of 0 = 7.840 corresponds to a temperature T = 288°K. This plot gives the transient distributions of vibrationally excited 02.
ations and from elementary properties of the hypergeometric series that eq. 9 can be written as (1
Zn(7)
= -
- e@)em@pm m! n! - e@) i = o ( n c - i ) ! ( n - i ) ! ( i ! ) * p“-”p - a)$
(e-.
2
(10)
where
and
Equation 10 converges rapidly for all T and is suitable for making calculations.14b For 0 2 (?Z2,-),one finds, using the data given by Herzberg,15 that hv/k = 2258°K. With the large excess of NZemployed in the flash photolysis work on ClOz, the “heat-bath” Nz can be assumed to be a t room temperature throughout the duration of the experiment. We have chosen T = 288°K. since some calculations have been published
-
(14b) NOTEADDED I N PROOF.-It may be noted t h a t for 0 + m (which for purposes of calculation is equivalent to 0 > 15) eq. 9 reduces to the simple form
T I , or in our notation T > To. The relaxation of the vibrational temperature for this case is shown in curves A and C of Fig. 1 in reference 3. To check how well the theory agrees with experiment one could transform the usual data of density us. shock strength or density US. distance behind the shock wave into T v i b us. time and compare these data with the temperature relaxation curve calculated from eq. 23. It should be pointed out again that the above theoretical results have been derived for a system of harmonic oscillators. Work is now in progress to extend these calculations to a system of anharmonic oscillators. The question as to whether it is possible to carry over this result of an exactly defined vibrational relaxation temperature to the anharmonic oscillator case must await the completion of this study. The use of state variables in shock wave studies obviates the need for detailed information about tJhe level population in the analysis of the data. To
Vol. 61
obtain some detailed information about the relaxation of vibrational non-equilibrium distributions and, in particular, about the efficiency of intermolecular energy exchange involving molecules in the higher vibrational levels it is, however, desirable to have such information. This suggests that some thought should be given to the instrumentation of shock tubes with spectroscopic equipment. At high shock strength, where higher vibrational energy levels will be excited, such a study might yield some interesting data on vibrat,ional and dissociation energy lag and on the efficiency of intermolecular energy transfer. Acknowledgments.-I have benefited greatly from the opportunity to discuss the contents of this paper with E. W. Montroll. I also wish to acknowledge some helpful correspondence with Professor R. G. w. Norrish of Cambridge University in regard to his work on vibrationally excited oxygen. I am indebt,ed to Mrs. J. Kimrey for her assistance with the calculations presented in Section 3.
ENERGY EXCHANGE I N SHOCK WAVES BY DONALD F.HORNIG* Metcalf Chemical Laboratories, Brown University, Providence 12,R. I . Received January 50, 19.57
Shock waves provide a means to change temperature, flow velocity and other variables in gases in as little as 10 to 15 collisions. Studies in the shock front region show that translational equilibration occurs in about the distance predicted by the Navier-Stokes equation, a few mean free paths. Rotational equilibration seems to require from 1-5 collisions, except in hydrogen, where 300:ollisions are required. Shock waves have also been used to study vibrational equilibration a t Because gases can be heated to high temperatures in a few collisions, they are also a useful tool temperatures up to 6000 in the study of chemical reactions and have been applied to the kinetics of dissociation reactions.
.
It has been pointed out already in this symposium that shock waves have become an important new tool in the study of vibrational relaxation in gases. They have, in fact, become useful in the study of all kinds of energy exchange as well as in the study of fast chemical reactions. In each case the basic reason is tha,t they afford a means to change the pressure and temperature in a gas over a wide range in a very short time. The formation of a shock wave readily can be understood in terms of a hypothetical experiment in which we imagine a tube of gas with a piston inserted a t one end. If the piston is slightly accelerated, the gas ahead of it is slightly compressed, and the disturbance propa.gates down the tube a t the velocity of sound. If the piston is again accelerated, another pressure pulse moves down the tube but a t a higher velocity than the first. The reason is that although the second disturbance also moves a t sonic velocity, it is moving in a gas which is itself in motion a t the first piston velocity and which has been slightly heated by the first compression. Therefore, it gains on the first disturbance and eventually catches up with it. If the piston is accelerated to a finite velocity, each successive disturbance catches up with the preceding disturbances so that down the tube a discontinuity in pressure and temperature is formed, behind *
Department of Chemistry, Prinrrton University, Princeton, N. J.
which the compressed and heated gas moves a t the velocity of the piston. The shock tube affords an easy experimental means of achieving this result.'-3 The gas under investigation is confined in one section of a tube, a t a pressure usually ranging from a fraction of a mm. to a few atmospheres. This section is serarated by a suitable diaphragm from a high pressure section filled with a gas, preferably a light gas such as H 2 or He, a t a pressure normally between 1 and 100 atmospheres. When the diaphragm is broken, either by spontaneous bursting or vhen pricked with a needle, the high pressure ga.s expands into the low pressure gas, acting as the piston in the preceding discussion and setting up a shock wave in the low pressure gas. In order to understand the properties of a shock wave which make it useful for the study of energy exchange it is convenient to consider it in a coordinate system in which the shock wave is at rest and the gas is flowing by. This situation is illustrated in Fig. l. The gas enters at the left at the velocity of propagation of the shock wave, M times the velocity sound, c, in the initial gars and (1) W. Payman and W. Shepherd, Proc. Rou. Soe. (London), 8 1 8 6 , 293 (1946). ( 2 ) W. Bleakney, D. I
