Rotational relaxation of HFL. H. Sentman Citation: The Journal of Chemical Physics 67, 966 (1977); doi: 10.1063/1.434923 View online: http://dx.doi.org/10.1063/1.434923 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/67/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rotational relaxation of HF in a freejet expansion of dilute HF–He mixtures: Information content on statetostaterate constants J. Chem. Phys. 78, 4486 (1983); 10.1063/1.445341 Rotational relaxation rates in HF and Ar–HF from the direct inversion of pressure broadened linewidths J. Chem. Phys. 75, 4927 (1981); 10.1063/1.441932 Rotational relaxation studies of HF using ir double resonance J. Chem. Phys. 65, 2732 (1976); 10.1063/1.433417 Vib–rotational energy distributions and relaxation processes in pulsed HF chemical lasers J. Chem. Phys. 65, 1711 (1976); 10.1063/1.433316 Vibrational relaxation of HF in Ar J. Chem. Phys. 58, 3454 (1973); 10.1063/1.1679675

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Rotational relaxation of HF L. H. Sentman

Aeronautical and Astronautical Engineering, University of Illinois, Urbana, I/linois 61801 and BelI Aerospace Textron, Buffalo, New York 14240 (Received 10 August 1976)

A rotational nonequilibrium model of Hinchen's HF rotational relaxation experiment has been used to determine the two constants in the Polanyi and Woodall rotational transition probability. Agreement between Polanyi and Woodall's and Hinchen's results is demonstrated.

INTRODUCTION

The rotational relaxation of the hydrogen halides was studied by Polanyi and Woodall l in connection with their measurement of the detailed rate constants for the pumping reactions upon which the hydrogen halide la­sers are based. Z They showed that the observed double peak rotational relaxation of the hydrogen halides is fully accounted for by a transition probability of the form l

P~ .. = Cl exp(- CzAE/kT) , (1)

where Cl and Cz are constants that are independent of the vibrational level v and AE = E J - E J'" The constant Cz does depend upon the I A J MAX I that is included in the model. From this data, values for Cl and Cz that give an upper bound on P~" were determined. 1 These re­sults have been used in the study of the effect of rota­tional nonequilibrium on cw chemical laser perfor­mance, 3

Recently, in an ir double resonance experiment, Hinchen4 has measured the pressure dependence of the relaxation time for the transfer of population from v=l, J=3tov=1, J=4, 5,6, and7inHF. Thecol­lis ion partner in these experiments was HF. The rate constants deduced from this data are global rate con­stants for all the transitions that can populate a given rotational level. For this data to be useful in describ­ing the rotational nonequilibrium processes that occur in chemical lasers, it must be related to the transition probability P~'" This paper describes the method for determining the constants Cl and Cz in the Polanyi and Woodall (PW) model from Hinchen's data and presents the values of these constants when the collision partner is HF itself.

966 The Journal of Chemical Physics, Vol. 67, No.3, 1 August 1977

THE EXPERIMENT AND MODEL

In Hinchen's experiment, HF molecules in v = 0 were pumped to v = 1, J" 3 by the absorption of radiation from a pulsed laser operating on a Single P l (4) transi­tion. A cw probe laser operating on a single pz(J) tran­sition was used to monitor the population in the Jth ro­tational level of V" 1. The pressure and temperature in the HF gas cell were constant. The only species in the cell was HF. Before the pumping pulse, the nv have a Boltzmann distribution over v. At the end of the pump­ing pulse, iiz. J has a Boltzmann distribution over J, nl. J has a Boltzmann distribution over J except for a spike at J = 3, and no. J has a Boltzmann distribution over J except for a hole at J = 4. The subsequent transfer of population from nl J is a result of rotational relaxation and collisional acti~ation/deactivation, Since the system was initially in equilibrium at 295 "K (no» ill » nz), the relaxation was modeled by following nl. J and iiz ... whose populations change due to rotation­al relaxation

(2)

J' _ _ kJ _

nz. J' + nm- nz. J + nm (3) k~,

and collisional deactivation

- - kg,J';l,J - -nZ.J,+nm• nl,J+nm , (4)

kl, J;2, JI

where nm = L:v L:J nv. J = constant =~ n in = no + nl + nz is the total mole mass ratio of HF in the cell. Since p and Tare constants, the relaxation is governed by the time depen­dent species equations

(6)

C0l>yright © 1977 American Institute of Physics

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L. H. Sentman: Rotational relaxation of HF 967

10.0

1.0

C1

0.1

0.1 1.0 10.0

where unbarred mole-mass ratios are dimensionless (nv , ,,= nv , ,,/nln)' the detailed rate constants have been expressed in terms of the transition probabilities for deactivation, and the relations between the detailed and global rate constants are developed in Ref. 3. 5 The reference time is defined as t' = [pnm(k,+ kb)r l

, where p=PW/RT, W is the molecular weight (nln= l/W) and R is the universal gas constant per mole. These equa­tions were solved for J m = 20 (42 simultaneous equations) subject to the initial conditions

(7)

100

FIG. 1. Ct versus T for several values of C2; P = O. 04 Torr. The ve rtical lines represent the experi­mental values of T for nt,4

and n t ,6' The intersections of the vertical lines with the Ct versus T lines give the values of Ct required to give the experimental value of T

for the indicated value of C2•

-C2 = O. 71885; - - - C2 =0.359425; -,-,-C2 = O. 071885.

where Anl,3 is the increase in population caused by the pumping pulse. 6 The most recently recommended val­ue' for the rate constant for the collisional deactivation reaction HF(2)+ HF - HF(l)+HF is k,= 7. 5xl014 T- 1

cc/mole sec. At 295 "K, the values for the constants in the corresponding detailed rate constant are Al = 402, A2 = 0 [Eq. (49) in Ref. 3].

RESULTS AND DISCUSSION

Since there are two constants to be determined in Eq. (1), two independent measurements at fixed pres­sure are required. The relaxation times for nl,4 and n1,6 at a fixed pressure (the time for nl," to reach with­in e- l of its maximum value) provide the requisite data, Fig. 2 of Ref. 4. The values of C1 and C2 were deter-

8 r-------r------,r------,-------.------~------_,------_r------,

6

C1 4

2

o L-------~ ______ ~ ____ ~~ ____ ~ ______ ~ ______ _L ______ _L ______ ~

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

J. Chern. Phys., Vol. 67, No.3, 1 August 1977

FIG. 2. Cj that gives the experimental T versus C2

for nj,4 and nt,6 for p = 0.04 Torr. The intersection of these two curves gives the set of values for Cj and C2 that give the experimental value of T for both nt,4 and nj,s' From the figure, C j = 2.45 and C2 = O. 359.

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968 L. H. Sentman: Rotational relaxation of H F

6 ,-------,-------,-------~------_r------_.------_.------_.--_.

5

4 • 3

2

FIG. 3. Inverse relaxation time versus HF pressure for n t ,4 and n l ,6; -theoreti­cal prediction using C t = 2.45 and Cz = 0.359; • n t ,4 data and _ nt,6 data from Fig. 2 of Ref. 4.

o ~ ______ L_ ______ J-______ ~ ______ _L ______ _L ______ ~ ______ ~ __ ~

o 20 40 60 80

H F Pressure (m Torr)

mined as follows: For a set of values for Cl and Ca, Eqs. (5) and (6) were solved numerically and the relax­ation times Tn1 ,4 and Tnl,S were determined from a plot of nl," versus T = tf. For fixed Ca, Cl was varied and a plot of Cl versus the resulting relaxation times was constructed for nl ,4 and nl,s' These calculations were repeated for several values of Ca. The results are shown in Fig. 1. The intersection of the vertical dashed lines representing the experimental T'S with the lines of Cl versus T for fixed Cz give the values of Cl

required to give the experimental T for that value of C2•

From these intersections for each nl ,,,, a curve of Cl

versus C2 that gives the experimental T'S can be con­structed (Fig. 2). The intersection of the curves for nl ,4 and nl ,6 then gives the unique set of values for Cl

and Cz that will give the experimental T for both n l ,4

and ~,6' From Fig. 2, these values for the rotational relaxation of HF by HF are Cl = 2. 45 and C2 = O. 359.

With these values for Cl and C2 , Eqs. (5) and (6) were solved for p = O. 072 and 0.122 Torr. The result­ing relaxation times for nl ,4 and nl,6 are shown in Fig. 3 together with the experimental data from Fig. 2 of Ref. 4. This comparison shows that the determined values for Cl and Cz reproduce the experimentally ob­served pressure dependence of the relaxation times for nl ,4 and nl ,6' Table I shows that there is good agree­ment between the experimental and predicted slopes of the inverse relaxation time versus pressure curves for nl,4 thru n1.7 using the determined values of Cl and C2•

To ascertain if Polanyi and Woodall's observed double peak rotational relaxation is compatible with the deter­mined values for Cl and Cz, Eqs. (5) and (6) were solved using Polanyi and Woodall's measured nl," and nz J distributions2

(a) as the initial conditions. Since th~se measurements were performed in an excess of

100 120 140

H2, the calculations were carried out for two cases; (a) H2 as the collision partner and (b) HF as the colli­sion partner. With Hz as the principal collision part­ner, the subsequent rotational relaxation occurred via the double peak mode, in agreement with Polanyi and Woodall's experiment. However, with HF as the prin­cipal colliSion partner, the double peak relaxation was not obtained unless the vibrational relaxation was neg­ligible. In the H2 case, since ~,;-H2'" lO-z ~,;-HF, the vibrational relaxation was slow enough compared to the rotational relaxation that it did not playa role. These results indicate that double peak rotational relaxation would only be observed in an experiment in which the HF vibrational relaxation is arrested (as in Refs. 1 and 2).

Finally, it is interesting to compare the values of Cl

and C2 determined from Hinchen's experiment with those obtained from Polyanyi and Woodall's data. As given in Ref. 3, ciw = O. 08027, Crw = O. 71885. Hinchen8 has reported kHF-H/kHF-HF = 0.1. Since ~F-H/kHF-HF = (ZP)HF-H2/(ZP)HF-HF = 2. 6973 P HF- H/ PHF- HF '" 2. 6973 (0.08027)/2.45 = O. 0884, the values of Cl and C2 obtained from Polanyi and Woodall's data

TABLE 1. Experimental and predicted slopes of the r-I versus p curves for C1 =2.45, C2 =0.359.

Experimental Predicted

nl,4 4x 107 4.65xI07

nt,5 2.2x107 3.29xI07

nt,6 1. 6x107 2.27xI07

nl,7 1.2x107 1.52xI07

J. Chern. Phys., Vol. 67, No.3, 1 August 1977

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L. H. Sentman: Rotational relaxation of HF 969

(their experiment was performed in an excess of Ha) are in agreement with Hinchen's results.

lJ. C. Polanyi and K. B. Woodall, J. Chern. Phys. 56, 1563 (1972).

2(a) J. C. Polanyi and K. B. Woodall, J. Chern. Phys. 57, 1574 (1972); (b) K. G. Anlauf, P. J. Kuntz, D. H. Maylotte, P. D. Pacey, and J. C. Polanyi, Discuss. Faraday Soc. 44, 183 (1967); (c) J. C. Polanyi and J. J. Sloan, J. Chern. Phys. 57, 4988 (1972); (d) K. G. Anlauf, D. S. Horne, R. G. MacDonald, J. C. Polanyi, and K. B. Woodall, J. Chern. Phys. 57, 1561 (1972).

3L. H. Sentman, J. Chern. Phys. 62, 3523 (1975): 4J . J. Hinchen, Appl. Phys. Lett. 27, 672 (1975). 5The factor G £ = (gU/gL) exp[- (Eu-E L)/kT] enters from the

detailed balance relation that has been used to express the detailed rate constants in terms of deactivation transition

probabilities. E u, EL are the rotational energies for a rigid rotator; gu, gL are the statistical weights of the upper and lower states. The nondimensional rate constants are given by Kc=Zc/(k,+k b), KCl ~Kcexp(-hcw/kT), Ky=Z/(k,+k b),

where Zj = <JIN A (8rrkT / ~j)1I2, k, and kb are the global forward and reverse rate constants for the collisional deactivation reaction, and the rate constants are related to the transition probabilities by k=ZP. The expressions for the transition probabilities are given in Ref. 3.

6Calculations showed that the time for the popUlation of a given J level to reach e- l of its final value is independent of the magnitude of M t ,3' Hence, any reasonable estimate can be used for M l ,3'

7N. Cohen (private communication), Aerospace Corporation, El Segundo, CA, May 1976.

8J . J. Hinchen, Tri-Service Chemical Laser Symposium, Kirtland AFB, New Mexico, 19-21 February 1975.

J. Chern. Phys., Vol. 67, No.3, 1 August 1977

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