NearResonant Vibration→Vibration Energy Transfer: CO2(υ3 = 1)+M→CO2(υ1 = 1)+M*+ΔEJohn C. Stephenson and C. Bradley Moore Citation: J. Chem. Phys. 52, 2333 (1970); doi: 10.1063/1.1673309 View online: http://dx.doi.org/10.1063/1.1673309 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v52/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF cHEMICAL PHYSICS VOLUME 52, NUMBER 5 1 MARCH 1970

N ear-Resonant Vibration -7 Vibration Energy Transfer: CO 2(Va = 1) +M -7C0 2(Vl = 1) +M* +L\E

JOHN C. STEPHENSON* AND C. BRADLEY MOOREt

DepartmMlt of Chemistry, University of California, Berkeley, California 94720

(Received 25 September 1969)

The laser-excited vibrational fluorescence technique has been used to determine the rate constants for deactivation of the asymmetric stretching vibration of CO, in collisions with CO2, CH4, C2~, CHaF, CHaCI, CHaBr, CHaI, BCla, and SF •. Rates were determined as a function of temperature in the range 300-800oK. The large deactivation cross sections u for the latter seven molecules decreased as the temperature T in­creased. For the latter six collision partners urx liT. This result is interpreted as a near-resonant vibrational energy transfer process in which three vibrational quantum numbers change as the vibrational energy is shared between the collision partners.

INTRODUCTION

The laser-excited vibrational fluorescence technique has been extensively used to measure rate constants for transfer of vibrational energy from the laser-excited vibration to other vibrations due to molecular col­lisions.1 Most of the published data have been for the asymmetric stretch of CO2, CO2(0001), although rates have also been reported for CH4(OOlO) and CH4(OOOl) ,2 HCl(v= 1) ,a and N20(OOOl).4 In past studies5,6 of vi­bration-to-vibration (V-7V) energy transfer processes in CO2, it has been possible to identify and measure the rate of transfer of a single quantum from one molecule to another, such as

CO2(0001) + N 2(v= 0)-7C02(OOOO) + N 2(v= 1) (1)

or

12C02(0001) + 1aC02 (0000)-712C02 (0000) +laC02(OOOl),

(2) and to observe intramolecular processes7 such as

CO2(0001) + He-7C02(mn I0) + He, (3)

where CO2(mn IO) denotes some (unidentified) state with bending and/or symmetric stretching modes ex­cited. In a single instanceS it has been possible to identify a process in which energy transfer from one molecule to another involved a three-quantum-number change:

CO2(0001) +D20(000)-7C02(OOOO)+D20(020). (4)

In systems of polyatomic molecules there are many possible V -7 V processes in which energy may be shared between two molecules, e.g.,

CO2(OOOl) + CO2(OOOO)-7C02(1000) + CO2(0110). (5)

The rates for this kind of process are very difficult to determine since there are usually many competing paths available, such as

By studying collision partners with vibrational fre­quencies close to the CO2 laser transition, processes like

CO2(0001)+SFr

CO2(1000)+SF6(va= l)+AE= 13 cm-1 (8)

are favored over others by having a small energy dis­crepancy (L\E is the energy difference between vi­brational band centers). In our observations reported here on the deactivation of CO2 (00°l) in pure CO2

and in mixtures with CH4, C2H4, CHaF, CHaCl, CHaBr, CHaI, BCla, and SF&, large rates were found for systems in which a near-resonant process was possible. For the latter seven molecules, the room-temperature de­activation probabilities are 3 or 4 orders of magnitude larger than in COrrare-gas collisions7 [Eq. (3)J, and 4-40 times larger than single-quantum vibrational energy transfer to 14N2 or 15N2 [Eq. (1)J,6 a process with similar L\E.

The temperature dependence of the energy transfer cross sections, as well as their magnitudes, is valuable in determining what part of the intermolecular inter­action is most important in causing energy transfer. Almost all vibrational deactivation rates increase as the temperature T increases,9 a result with which the theoretical arguments of Landau and TelIerlO and later workers are in at least qualitative agreement. However, the temperature dependence of only a few V-7V processes has been measured.ll The only closely reso­nant system for which data are available is N2-C02.12.1a For that case as T increased, the cross section de­creased as 1'-1 from 3000 K to 1000°K.la In this paper rates in the temperature range 30Q-800oK are reported. We find this inverse temperature dependence to be a general property of the near-resonant three-quantum­number-change systems studied.

EXPERIMENTAL

CO2 (000 1) + CO2 ( 00(0) -7C02 ( 1110) + CO2 (0000)

-7C02(0000) +C02(1l10).

Our standard CO2 laser-excited fluorescence appa­(6) ratus has been described previously.s The 4.3-~ fluo­(7) rescence signal was monitored by a Au-Ge (3XlO mm

2333

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2334 J. C. STEPHENSON AND C. B. l\100RE

area) photoconductor, and amplified by a Keithley Model 104 wide-band (25 Hz to 150 MHz) amplifier. The amplifier output was averaged on a Tektronix lSI sampling plug-in and displayed on an X - Y re­corder. The RC time constant of the detector, limited by a lO-K Q bias resistor and the capacitance of the elec­trical feedthroughs, was about 0.25 Jl.sec. Signal decay times were typically 10 Jl.sec. The 2.4-msec radiative lifetime of CO2(000l) makes a negligible contribution to the decay rate.

The variable temperature fluorescence cell was de­signed by W ood. 14 It is similar to the cell of Rosser, Wood, and Gerry,t3 except that our heavy nickel cell has no stainless-steel liner; the Kovar sleeve which holds the sapphire fluorescence viewing window is brazed directly to the cell body. Four iron-constantan thermocouples monitoring the temperature along the cell agreed within about 3% at SOOoK. Agreement was better at lower temperatures. A thermocouple placed against the outside of the sapphire window furnished a lower limit on the temperature of the window itself. Due to radiant heat loss the sapphire will always be at a slightly lower temperature than the surrounding insulated cell. With the cell body at SOOoK, the sap­phire window thermocouple read 730oK; since radiant heat loss increases as P, the temperature difference at lower T was considerably less. The thermocouple was withdrawn from the window during actual experi­ments, since the thermocouple wire itself was a heat drain.

Gases used from Matheson Company, with indicated minimum purity, were CH3CI (99.5%), CH3Br (99.5%), SFs (9S%) , and BCh (99.9%). Methyl iodide was "Eastman Grade" from Eastman Chemical Company. The CH3F was from Columbia Organic Chemical Com­pany, minimum purity 95%. Methane (50 ppm CO2, 75 ppm C2R 6, 5 ppm C3Rs, 20 ppm N2, 5 ppm O2, balance CH4) and CO2 (5 ppm N2, balance CO2) were Matheson Research Grade. The system leak and outgassing rate was :::; 10-4 torr/h. Pressures were read on aU-tube mercury manometer or on a Consolidated Vacuum Corporation triple McCleod gauge.

Our gas handling procedure was to admit a sample from a Pyrex mixing flask to the hot cell, to make the fluorescence decay measurements over a period of about 5 min, and then to pump away the sample. To test for sample decomposition, occasionally the same sample was run at room temperature, at elevated temperature, and then again at room temperature. No difference between the room temperature rates was found after heating any sample to a temperature less than 700oK. No gas decomposition or pressure change during an experiment at any temperature was observed for the data presented.

Several of the deactivating molecules absorb CO2

laser light (e.g., CR3Br, CR3F, BCh, and SF6)' Since our sample cell was inside the laser cavity, these ab­sorption losses prevented the laser from oscillating on

the lines strongly absorbed by the gases. If an added gas had a weak absorpt ion comparable to that of C()~ itself, the interpretation of our dala would not be affected.

The fluorescence decay curves were single exponen­tials. Signal-to-noise ratios ranged from a low of SIN = 20 to over 150; Typically SIN = 70. (The SIN values are the ratios of the maximum signal height to the peak-to-peak value of the high-frequency noise on the X - Y recorder traces.) In the range 700-S20oK, the signal did not decay quite to zero, but to a value 1 %-2% of the initial amplitude. This residual excita­tion due to thermal heating of the gas relaxes on a longer time scale between laser pulses. At a given temperature, the product of the pressure p, times relaxation time r was a constant. Typical pressure range was S-40 torr for the seven molecules which rapidly deactivate CO2 (0001) .

The pr values for a particular sample and tempera­ture are known quite precisely; two times the sample standard deviation divided by the average value of pr is about 5%. Several systematic errors were pos­sible. Since the sapphire window was at a slightly lower temperature than the cell body, there may have been temperature gradients in the gas. For reasons given above, this is important at higher temperatures. At SOOoK the recorded value of T may be 5%-6% too high. That the pr product was constant as p was varied indicates that neither the high-frequency limita­tion (RC=0.25 Jl.sec) nor the low-frequency cutoff of the amplifier (-3 dB at 25 Hz) was so distorting the signal as to cause error. Impurities are unlikely to be important, since the deactivation rates of the sample gases are so large, and the expected impurities (e.g., air, water) are less efficient deactivators than the major components. The reproducibility of the results, as well as the appropriateness of Eq. (9) (see below), is in­dicated by the close agreement of cross sections derived from rates measured for different sample concentrations. For the CO2-added gas results, we feel that the true energy transfer cross sections at room temperature are within ± 7% of our data points or the lines drawn through them, and within ± 13% at the highest tem­peratures. The accuracy of the pure CO2 data ranges from ±2% at room temperature to ±9% at S20oK.

The rate constants k for deactivating CO2(OOOl) were found from the expression

(9)

where kCO,....c02 may be deduced from data given below, XC02 is the mole fraction of CO2, and Xgas is the mole fraction of the added relaxing gas. The energy transfer cross section U vv was found from the expression U vv= k/(vn), where k is in second-l·torr-t, v is the average relative velocity, and n is the number density per torr at the experimental temperature. Probabilities may be defined by dividing u .. by a gas kinetic cross section.

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ENERGY TRANSFER IN CO 2 2335

RESULTS

Measured vibrational relaxation rates are given below for mixtures with CO2 in which nearly resonant three­quantum-number-change processes are possible. These data differ markedly from those for systems in which near-resonant V-tV energy transfer processes are not available. The energy transfer cross sections are much larger than have been found for all nonresonant and many resonant V-tV processes with CO2.1 The cross sections decrease as the temperature is increased from 300 to 800oK, whereas the opposite behavior is ob­served in most previous studies9 of vibrational energy transfer. We believe that this temperature dependence can only be understood in terms of a nearly resonant V -t V energy transfer process.

The most efficient three-quantum deactivators of CO2 are SF6 and BCh. The measured cross sections (Fig. 1) are accurately linear with 1'-1 and are given15

by O"SF6 (A2) = 1480/T(OK) -0.73 and O"BCla (A2) = 590/T(OK) -0.10. The corresponding room-tempera­ture probabilities are 0.06 and 0.029, respectively. The resonant processes involved are Eq. (8) and

CO2 (Oool) + BClr

CO2(loo0)+BC1a(v3= l)+b.E= 7 em-I. (10)

It is unlikely that other resonant processes such as

CO2(OOOl)+BClr

CO2(1PO)+BCh(v4= l)+b.E= 12 cm-1 (11)

are important, since at least one more quantum num­ber must change, since b.E is not decreased, and since the vibrational transition dipoles are weaker.

For the methyl halides (Fig. 2) all of the observed cross sections are between 0.28 to 0.42 A2 at room temperature, corresponding to energy transfer proba­bilities of about 7X 10-a. For the methyl halides the cross sections are again linear with 1'-1 and are givenl5

by: O"CHaF (A2) = 61.3/T(OK) +0.071; O"CHa! (A2) = 89.6/T(OK) +0.091; O"CHaBr (A2) = 107/T(OK) +0.055; and O"CHaCI (A2) = 78.3/T(OK) +0.04. The processes

CO2 (Oool) + CH:;X-t

CO2(loo0)+CHaX(v6= l)+b.E (12)

are closely resonant. The energy difference between band centers is b.E=9, -54,81, and -87 em-I for X= Br, CI, I, and F, respectively. For two cases another transition is nearly resonant:

CO2(Oool) +CHaX-tC02(02°0) +CHaX*+b.E, (13)

where b.E= 16 and 49 cm-I when CHaX*= CHaF(va= 1) and CHaCI(v6= 1), respectively. Since there are no energy levels between 200 and 500 cm-I for the methyl halides, there are no near-resonant processes similar to (11), and all other V-tV processes have b.E's much larger than for (12) and (13).

SFS-C02 4.0 r- • XSF6 '0.0536

• XSF6 '0.0372 ~ 3.0

N

".5 b 2.0

1.0-

T(OK)

800 SOO 500 400 350 300

-

FIG. 1. Cross sections for the deactivation of the asymmetric stretching vibration of CO2 by SF6 and BCla. A sample with X SF• =0.0759 gave <1=4.3 12 at room temperature. For BCla at room temperature, samples with mole fractions XBCla =0.088 and XBc1a =0.127 gave cross sections within 4% of the result shown in the figure for XBc1a =0.0487.

For ethylene the magnitude of the vibrational energy transfer cross section is similar to that of the halo­methanes (Fig. 3), and it likewise decreases with increasing temperature. However, the cross section does not go as 1'-\ but appears to level off in the range 600-S00oK. The most closely resonant process is

CO2 ( Ooo 1) + C~4-t

CO2(1oo0)+C2H4 (V7= 1)+b.E= 12 em-I. (14)

For comparison, especially to the methyl halide and C2H4 data, data on CH4 deactivating CO2 are presented in Fig. 4. The room-temperature cross section is a factor of 20 smaller than those for the methyl halides and ethylene, and the cross section increases in the "normal" way as temperature increases. The processes by which CH4 may deactivate CO2(OOOl) are typical of those which might compete with the resonant proc­esses in the methyl halides and ethylene. These include

CO2(OOOl) +CH4-t

CO2(11IO) + CH4+ b.E = 272 em-I, (15)

CO2(Oool) +CH4-t

CO2(OPO)+CH4(V2= 1)+b.E= 149 em-I, (16)

CO2(Oool) +CH4-t

CO2 (OooO)+CH4 (V2= 1)+b.E=S16 em-I. (17)

That the observed rates are much larger and have the opposite temperature dependence for the molecules with resonant processes available is strong empirical evidence that in the cases studied here the resonant process dominates the observed rates. Similar sorts of comparisons may be made to many room-tempera­ture energy transfer probabilities where resonant proc­esses do not contribute. For example, the probability of deactivation of CO2(Oool) without vibrational ex-

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2336 J. C. STEPHENSON AND C. B. MOORE

T(OK) 800 600 500 400 350 300

0.4

0.3

0.2

O.

I- • XCH3F ~ 0.833 A XCH F ~ 0.460 .... _-1

3 _ --I- • ___

~r

I ------ CH3F-C02

• XCH3I ~ 0.787 _--I-A XCH3I ~0.313 --_-e ,._.----

o 0.4

0.3

0.2

O. I~"""'--- CH3I-C02

~ 0 ~ -;; 0.4 • XCH3Br ~ 0.469

AXCH Br~0.126 ...--3 .Ai

~/ ...--...---

...--t: ...--...--...--0.3

0.2

O. 11---/""-/ CH3Br-C02

2

I

o 0.3

O.

O.

• XCH CI ~ 0.802 o XCH

3CI ~0.430 t--~

AXCH3CI~O.121 ~ .... -3 ~ --~ _-- CH3CI-C02

~--3

FIG. 2. Cross sections for the deactivation of the aysmmetric stretching vibration of CO2 by methyl haJides. XCH,x denotes the mole fraction of methyl halide in the CO ..... ClLlX mixture.

citation of the collision partner is 7X 10-4 for HCI,· 3X 10-4 for HI,s and 1.6X 10-4 for H2.l The probability for deactivation of CO2(O()o1) by CCI4, where a process similar to (11) is possible, is 3.7X1Q-4. These proba­bilities are again much smaller than those observed for the resonant cases in Figs. 1-3. The close resonance

T(OK)

800 600 500 400 350 300 I I I I I

OAI- • C2H4-C02 • ~0.3f--

•• • b

0.2

0.11-

1 I 1

3

FIG. 3. Cross sections for deactivation of the asymmetric stretching vibration of C02 by C2H,. The mole fraction of ethylene in the CO ..... CJ:4 mixture is 0.800. At room temperature a sample with Xc,H,=0.379 gave 0'=0.384 A2.

T (OK)

0.06 800 600 500 400 350 300

• 0.04f--

• N 0« •

b O.02f--

CH4-C02 •

0.01 I

0.10 0.12 0.14 rl/3 (OK)

FIG. 4. Cross sections for the deactivation of the asymmetric stretching vibration of CO2 by CH,. The mole fraction of methane in the CH,-C02 mixture is 0.903.

with a CO2 vibrational tranSItIOn seems to be the important factor, not some other structural feature such as dipole moments or halogen atoms. Thus we assert with some confidence that these large cross sections which decrease as temperature increases are due to near-resonant multiquantum V~V energy transfer.

The data on pure CO2 which were necessary for the analysis of the mixture results above are presented in Fig. 5. The measured rates as a function of tempera­ture have previously been given by Rosser, Wood, and Gerry.IS Their Eq. (3) fits our data within 20% over the range 37S-820oK. We find that k does not give a straight line for T<SOOoK. For comparison to RWG, a fit to our data from Sao-820oK gives loglok (sec-l·ton·-I) = 7.32-35.8]'-1/3.

T (OK)

0.05 800 600 500 400 350 300

1-\ \

0.03

0.02

I-- __

I--

II--0.0

",,- 0.008 oc:(

I--

-;; 0.006 I--

0.004 I--

0.002 I-

0.001 0.10

, 1 I

-

CO2-CO2 -• • \ -

'e -\ \ -

\ ~ -

\ \ . \ \ 1

\ \

I \

0.12 0.14

rl/3 (OK)

FIG. S. Cross sections for the deactivation of the asymmetric stretching vibration in pure CO2•

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ENERGY TRA~SFER IN CO 2 2337

TABLE I. Comparison of calculated dipole-dipole energy transfer cross sections to observed near-resonant three-quantum-number-change processes.

Collision partner

SF,

BCla CRaF

C2B. CHaI CRaBr CRaCI

CO2

Vibration

Jla

117

Jl6

'" ", (0001)

CO2 final state

(1000)

( 1(00) (0200)

( 1(00) ( 1(00) ( 1(00) (1000) (1000) (0200) (1000) (0200)

1'2 tJ.Ea (X 1039

(cm-l ) esu2·cm2)

13 450·

7 16 42f

-87 12 34b 81 4.1i 9 3.0i

-54 1.6i 49

961 1.44k 1064 1.22m

• tJ.E. the energy difference between vibrational band centers. was cal­culated from data in Ref. 32. For SF. the reference is H. Brunet and M. Perez. J. Mol. Spectry. 29, 472 (1969). For BCla the reference is D. F. Wolfe and G. L. Humphrey. J. Mol. Struct. 3, 293 (1969).

b (fR denotes the theoretical cross section for exact resonance. C P. N. Schatz and D. F. Hornig. J. Chern. Phys. 21, 1516 (1953). d K. E. MacCormack and W. G. Schneider. J. Chern. Phys. 19, 849

(1951). • c. J. G. Raw. J. South African Chern. lnst. 8, 21 (1955). f G. M. Barrow and D. C. McKean. Proc. Roy. Soc. (London) A213

27 (1952). c G. A. Miller and R. B. Bernstein, J, Chern. Phys. 63, 710 (1959). hR. C. Golike. I. M. Mills, W. B. Person, and B. L. Crawford. Jr .• J.

DISCUSSION

The unusual magnitude and temperature dependence of the cross sections for deactivation of CO2 (OOOl) by SF 6, BCla, CJI4, and the methyl halides are impossible to understand in terms of the standard approximate theories of vibrational energy transfer. We initially expected that these rates were further examples of vibrational energy transfer due to the vibrating dipole­dipole interaction suggested by Mahan.16 This inter­action is likely to be responsible for the very fast near-resonant single-quantum (two-quantum-number­change) vibrational energy transfer between infrared­active vibrations in triatomic molecules.6 The cases reported here are analogous except that one of the vibrating dipoles is associated with a difference band, C02(O()ol)~C02(1000), instead of with a fundamental transition.

The theory of vibrational energy transfer caused by transition multipole electric moments has recently been discussed in detail by Sharma and Brau,17 Essentially, the rotational line-broadening theory of Andersonl8

and later workersl9 is adapted to the case of collisions causing vibrational energy transfer by substituting the transition moments of the vibrations involved for the permanent molecular electric moments. Cross sections

Molecular diameter T Uexvtl O"theol'ot <TRb

(A) (OK) (A2) (A2) CA2)

5.51d 300 4.3 1.8 3.7 500 2.3 1.1 2.4 700 1.4 0.73 1.8 300 1.85 300 0.28 0.12 0.30 500 0.19 0.071 0.19 800 0.15 0.044 0.11 300 0.28 0.005 0.35

4.07 i 300 0.39 0.23 4.23g 300 0.39 0.053 4.1~ 300 0.42 0.036 4.05g 300 0.30 0.018

300 0.015 4.01

Chern. Phys. 25, 1266 (1956) . i L. W. Flynn and G. Thodos, A.I.Ch.E. (Am. lnst. Chern. Engrs.) J.

8, 362 (1962). i A. D. Dickson, I. M. Mills, and B. L. Crawford. Jr .. J. Chern. Phys.

27, 445 (1957). k S. R. Dray.on and C. Yound, J. Quant. Spectry. Radiative Transfer

7, 993 (1967). This is a re·analysis of the data of T. K. McCubbin. Jr .. R. Darone, and J. Sorrell. Appl. Phys. Letters 8, 118 (1966).

1 H. L. Johnston and K. E. McCloskey. J. Phys. Chern. 44,1038 (1940). m L. D. Gray, J. Quant. Spectry. Radiative Transfer 7, 143 (1967).

Gray indicates that the transition dipole for the 9.4-1' transition is 0.85 times the 10.4-1' transition dipole moment.

are calculated using the impact parameter form of the first Born approximation20 and assuming straight­line constant-velocity trajectories. Three qualitative features of this treatment are noteworthy. First is that for very near-resonant transitions, the cross sec­tions are proportional to T-I. This temperature de­pendence results from the use of the Born approxima­tion with a constant-velocity straight-line trajectory for any form of the perturbing potential. Thus the observation of a cross section inversely proportional to T suggests that the perturbation is effective at distances for which intermolecular potential energy is less than kT, but gives no information on the precise form of the perturbation. Second, the cross sections are proportional to the product of the squares of the transition multipole moments. Finally, the cross sec­tions decrease as M is increased. The form of this dependence is sensitive to the range of the intermolec­ular potential, to the rotational selection rules, to the rotational constants of the collision partners, and to the temperature.

We have carried out calculations of dipole-dipole energy transfer cross sections for comparison to the experimental data. The details are given in the ap­pendix. In Table I are given parameters which enter the calculation: the vibrational transitions involved,

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2338 ]. C. STEPHENSON AND C. B. MOORE

the transition dipole moments, the energy between band centers, and the gas kinetic collision diameters assumed. The dipole derivatives of the vibrations of most common molecules are known from the absolute intensity of the infrared bands21 ; unfortunately the literature on BCh(l'a) is unclear. There are no ad­justable parameters which enter the calculation. The computed theoretical cross sections at several tempera­tures for SF 6 and CHaF are given in Column 9. In Column 10 are cross sections calculated under the simplifying assumption that all transitions are exactly resonant. Such an approximation severely overestimates the probability for a process where more than 10 or 15 cm-1 of energy must go into translation (see Ap­pendix). For CHaF, for which the band-center separa­tion is only 16 cm-t, the approximation is a factor of 2.5 larger than the "exact" calculation. However, the resonance assumption does furnish an upper limit to the cross section due to the dipole-dipole interaction.

The agreement between theory and experiment is good for the molecules with large dipole derivatives, SF6, C2H4, and CHaF.22 Such agreement is excellent for a theory which lacks any variable parameters such as the exponential repulsion parameter or steric factor of Landau-Teller-type theories. Also, contrary to the usual theories, this approach gives the observed T-l temperature dependence of the cross sections. However, for CHaCl, CHaBr, and CHal the theoretical result, even under the drastic assumption of exact resonance, is about 10 times smaller than experiment.

In Table I the molecules are listed in order of de­creasing transition dipole moment. SF 6, and pre­sumably BCh, have transition dipoles and cross sec­tions both an order of magnitude greater than C2H 4

and CHaF. However, as the transition dipole is de­creased by another order of magnitude in going from CHaF and C2H 4 to CHal, CH3Br, and CH3C!, there is no further decrease in the magnitude of the observed cross sections. Clearly there is another mechanism aside from the dipole-dipole interaction responsible for the observed cross sections in CHal, CH3Br, and CH3c!. Furthermore, since the cross sections for CH31, CH3Br, CH3C!, CH3F, and C2H 4 are all of the same order of magnitude, it is not clear that the dipole­dipole process dominates even for C2H4 and CH3F. Thus we may conclude with relative safety that the very large rates for SF 6 and BCh are due to the dipole­dipole interaction, and that the rates for CHal, CH3Br, and CHaC! definitely are not.

The multipole moment theory may be extended beyond the first-order dipole-dipole calculations dis­cussed above by considering higher-order transition moments or by going to higher-order perturbation theory and mixing in permanent moments as well as transition moments. Sharma 17.23 has discussed the effect of including higher transition multipoles (e.g., quad­rupoles and octupoles) in the calculation. The rotational

selection rules for higher multipoIes allow larger changes in the rotational quantum numbers. More of the energy discrepancy may then go into rotation, thus reducil-lg the amount of energy going into translation. Since for a given multipole interaction the cross section increases as the amount of energy going into translation de­creases, the effect of the higher multipoles may be significant for energy exchange processes with large /:;.E. Unfortunately, the error introduced by the straight­line constant-velocity trajectory approximation will be much greater here than in the dipole-dipole case. For the methyl halide-C02 transitions considered here, the CO2 dipole derivative is too small for the dipole (C02)­quadrupole (CH:X) interaction to be important, and the CO2-quadrupole transition is symmetry forbidden. Multipole derivatives higher than quadrupole are not known. Thus, taking higher multipoles into account does not significantly increase the predicted cross sections.

If one takes the multipolar interaction to higher orders of perturbation theory, for instance, the second Born approximation,24 permanent moments as well as transition moments are involved, and a large number of intermediate and additional final states are available, which may increase the predicted cross section. Like higher terms in the multipole expansion, higher orders of perturbation theory will admit transitions with larger rotational quantum number changes, which for large /:;.E will decrease the amount of energy going to translation, thus increasing the probability. Here again the usefulness of the straight-line trajectory Born ap­proximation is open to question. The inclusion of higher multipole moments or higher orders of perturbation may increase the theoretical probabilities above those for the first-order dipole-dipole calculation when /:;.E is large. However, it is unlikely that the calculated resonant cross sections in Column 10 of Table I could be exceeded.

Although the multipolar approach does not explain all of our data, the predictions of other theories are much further from experiment. Theories which consider head-on collisions predict probabilities which increase as the temperature increases; for the resonant case, the probability of energy transfer increases linearly with T. This result was found in the approximate treatments of Schwartz, Slawsky, and Herzfeld25 (ex­ponential repulsive potential, first-order distorted-wave approximation), Rapp and Englander- Golden26 (ex­ponential repulsion, semiclassical perturbation theory), and Thompson27 (Morse potential, first-order dis­torted-wave approximation). The same result was ob­tained in the exact (exponential potential) quantum­mechanical result of Riley and Kuppermann.28 In ad­dition to the incorrect temperature dependence, these theories predict a cross section typically 100-1000 times too small for a near-resonant room-temperature process for which three vibrational quantum numbers

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ENERGY TRANSFER IN CO 2 2339

change. Hence, it appears that a three-dimensional treaimeni29 is necessary if the model calculations are (0 agree even qualitatively with our data.

Thus, rather convincing agreement between first­order dipole-dipole calculations and experiments is found for SF6 and presumably for BCla. The agreement for CHaF and C2H 4 is ambiguous. Neither the dipole­dipole theory nor its simple extensions, nor application of the standard V ~ V theories, provides an explana­tion for the observations on CHaCl, CHaBr, or CHar.

CONCLUSIONS

The results presen ted on the deactivation of CO2 (WI) by C2H 4, CHaF, CHaCl, CHaBr, CHaI, BCla, and SF 6 have shown that intermolecular energy transfer can have high probability in cases where a near-resonant three-quantum-number-change process is possible; room-temperature cross sections range from 0.28 A2 (p= 7X lo-a) for CHaF to 4.3 A2 (p= 6X 10-2) for SF 6. Smaller cross sections were measured for deactiva­tion of CO2(OOOl) in CH4-C02 and CO2-C02 collisions. The cross sections for the latter two molecules increase in the usual way as the temperature increases. For the near-resonant processes reported here the probability decreases as T increases in the range 30G-800oK. For BCla, SF6, and the methyl halides the probability is proportional to T-I. The magnitude and temperature dependence of the cross sections are contrary to the predictions of most vibrational energy transfer theories, except those using the straight-line constant-velocity trajectory Born approximation. In the case of SF 6-C02,

and presumably BCla-C02, the dipole-dipole inter­action is almost certainly responsible for the observed energy transfer. Although agreement of theory and experiment for CHaF and C2H4 is close, this may well be fortuitous. First-order multipolar interactions do not account for the observed vibrational energy transfer between CO2 and CHaCl, CH3Br, or CHar.

ACKNOWLEDGMENTS

We wish to thank Dr. R. E. Wood, who designed the hot cell and helped to construct and maintain the vacuum apparatus, and Mr. D. F. Heller, who helped with the computer programming. Weare grateful to the National Science Foundation and to the Advanced Research Projects Agency of the Department of De­fense (monitored by the U. S. Army Research Office­Durham, Box CM, Duke Station, Durham, North Car­olina 27706, under Contract No. DAHC04 68 C 0044) for financial support. The computer time used in the calculations was donated by the Computer Center of the University of California, Berkeley.

APPENDIX

The rate constant k (cma/second) is given by

k= If 27rbP(v, b)vj(v)dbdv, (AI)

where b is the impact parameter, j(v) is the three­dimensional velocity distribution function, and jl (v, b) is the probability of energy transfer from the initial quantum state to all final states, averaged over initial rotational states. For the case of a linear molecule (Molecule 1) colliding with a symmetric top (Mole­cule 2), for which the vibrational states involved are nondegenerate (e.g., CHaF-C02), the average proba­bility is

P(v, b) = L j(ll)j(J2, K2)4f.J}/1-1/9fiWv2

JIJ2K2

x L C2(J1l 11'10 0)C2(J2 1 J2' I K2 O)j(wb/v). JI'J2'

(A2)

In (A2), j(11) and j(12, K2) are the rotational dis­tribution functions for the initial states, /1-12 and /1-22 are the squares of the transition dipole moments, C(1ll II' I 00) is a Clebsch-Gordan coefficient con­necting the initial (unprimed) and final (primed) states, w is the amount of energy which must go into translation, and j(wb/v) is a resonance function which is unity for zero argument and decreases rapidly with increasing wb/v. An approximate form30 for the reso­nance function is

j(X) = (1+2x+t7rx2+3xa)e-2X• (A3)

For a relative velocity of 4.3X 104 cm/sec and an im­pact parameter of 4.75 A (values corresponding to ~he average velocity at 3000 K of SF 6-C02, and an Impact parameter equal to the Lennard-Jones diam­eter), the resonance function has dropped to 0.5 at w= 12 cm-r, and to 0.01 at w= 26 cm-I. If a transition resulting in w< 10 cm-I is possible, transitions with w> 20 cm-I may be neglected in comparison.

Since the constant-velocity trajectory approximation i~ not appropriate for_ close co~isions, the approxima­tion was made that P(v, b) =P(v, u) for b<u, where u may be taken (somewhat arbitrarily) as the zero of the Lennard-Jones potential. For b>u, P is given by .(A2~. The approximatio12 that for b>u, the probability IS gIven by P(v, b) =P(v, u) (u/b) 4 was tested for about 100 different values of the parameters v u and mass/temperature. We found, following Gray ~nd Van Kranendonk3I and Sharma and Brau,17 that this ap­proximation overestimates by about 20% the result obtained by substituting (A2) into (AI) and nu­merically integrating over band v. The net result of our approximations is to slightly overestimate the theoretical cross section. For the transitions considered here, a probability greater than t is not predicted for any significant values of the variables v or b. Hence "chopping off" the probability at some value less than unity was not an important feature of our cal­culations.

For the spherical top, SF6, the nuclear-spin de­generacy factors are complicated. In our calculations

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2340 J. C. STEPHENSON AND C. B. MOORE

we simply assumed that each J state has a degeneracy of (2J+l)2, which is a good approximation for a molecule with such a large moment of inertia.a2

For the case of CHaX(V6) the transition moment is perpendicular to the top axis, and the rotational transi­tions allowed are M2 = ± 1, 0 and AK2 = ± 1. The asymmetric top C2H4 is a still more complicated case, although the angular integrals involved (direction cosines) are known.aa We have not attempted an exact computer evaluation of the dipole-dipole cross section for these molecules. However, an upper limit on the contribution due to this interaction may be obtained by setting w = 0 for all possible transitions. The proba­bility then becomes6

F(v, b) =4p}JLN91i,2b4v2• (A4)

To account for close collisions we assumed that if F(v, be) =!, then F=4JL12JLN91i,2vW for b'2be and b'2u; F=! for b~be and bc>u; and F=4JL12JLNIi,2v2u4 for bc<u and b<u. Equation (Al) may then be evaluated in closed form. The cross section uv.=k/fJn is then given by

u .. = (7["aCM/RT) 1/2 erf[(2CM/RTu4) 112]. (AS)

In (AS), C = 4JL12JLN9n2, M is the collision reduced mass, R is the gas constant, T is the temperature, u is the Lennard-Jones diameter for the collision pair, and erf(x) is an error function. a4 Equation (AS) is the result given in Column 10 of Table I.

When this resonance approximation is made for CHaF and SF 6, the result is about 2.3 times larger than the "correct" calculation. CHaBr has M= 9 cm-1

(SF6 = 13 cm-1 and CHaF= 16 em-I), so this assumption presumably overestimates the CHaBr-C02 cross sec­tion by a similar amount. Because of the rapid decrease of the resonance function with increasing w, only those transitions with small w contribute. CHaCl and CHaI have large values of AE, so that rotational changes that obey the dipole selection rules and that result in small ware improbable. For these molecules the resonance assumption yields a cross section likely to be a factor of 10-20 too large.

* National Science Foundation Predoctoral Fellow. t Alfred P. Sloan Fellow. 1 C. B. Moore, Acct. Chern. Res. 2,103 (1969), and references

cited therein. 2 J. T. Yardley and C. B. Moore, J. Chem. Phys. 49, 1111

(1968) . 3 H. L. Chen. J. C. Stephenson, and C. B. Moore, Chem. Phys.

Letters 2, 593 (1968). 4 R. D. Bates, Jr., G. W. Flynn, and A. M. Ronn, J. Chem.

Phys.49, 1432 (1968); J. T. Yardley, 49, 2816 (1968). 5 C. B. Moore, R. E. Wood, B. L. Hu, and J. T. Yardley,

J. Chem. Phys. 46,4222 (1967); C. B. Moore, Fluorescence, G. G. Guilbault, Ed. (Marcel Dekker, Inc., New York, 1967), pp. 133-199.

'J. C. Stephenson, R. E. \Vood, and C. B. Moore, J. Chem. Phys. 48, 4790 (1968).

7 J. T. Yardley and C. B. ::\ioore, J. Chem. Phys. 46, 4491 (1967) .

8 D. F. Heller and C. 13. Moore, J. Chem. 1'h\'s. 52, 1005 (1970). -

9 See, for instance, T. L. Cottrell and J. C. McCoubrcy, Molec­ular Energy Transfer in Gases (Butterworths Scientific Publica­tions, Ltd., London, 1961).

10 L. Landau and E. Teller, Physik Z. Sowjetunion 10, 34 (1936). 11 B. Stevens, Collisional Activation in Gases (Pergamon

Press, Ltd., London, 1967), pp. 154-162, and references cited therein; R. L. Taylor, M. Camac, and R. M. Feinberg, Symp. Combust. 11th, Univ. California, Berkeley, 1966, 49 (1967).

12 R. L. Taylor and S. Bitterman, Rev. Mod. Phys. 41, 26 (1969); R. 1.. Taylor and S. Bitterman, J. Chem. Phys. 50, 1720 (1969) .

13 W. A. Rosser, Jr., A. D. Wood, and E. T. Gerry, J. Chem. Phys. 50, 4996 (1969).

14 R. E. Wood, Ph.D. thesis, University of California, Berkeley, Chemistry, 1969.

15 The equations fit the lines on the graphs and represent the cross sections over the temperature range for which we have data. At higher temperatures, deactivation processes which increase with T may dominate, and our equations, if extrapolated to higher T, will then not represent the true deactivation cross sections.

16 B. H. Mahan, J. Chem. Phys. 46, 98 (1967). 17 R. D. Sharma and C. A. Brau, J. Chem. Phys. 50,924 (1969). 18 P. N. Anderson, Phys. Rev. 76, 647 (1949). 19 See especially C. J. Taso and B. Curnutte, J. Quant. Spectry.

Radiative Transfer 2, 41 (1962). 20 D. R. Bates, Atomic and Molecttlar Processes (Academic

Press Inc., New York, 1962), p. 580. 21 A convenient tabulation is found in L. A. Gribov and V. N.

Smirnov, Usp. Fiz. Nauk 75, 527 (1961) [Sov. Phys.-Usp. 4, 919 (1962)].

22 In order for Utheoret/Uexptl (3000K) to be the same for BCb and SF" p.2 for BCla (va) must be 2XlO-39 esu2 .cm2•

23R. D. Sharma, Phys. Rev. 177,102 (1969). 24 D. R. Bates, Quantum Theory (Academic Press Inc., New

York, 1961), Vol. 1, p. 258. 25 R. N. Schwartz, Z. 1. Slawsky, and F. K. Herzfeld, J. Chem.

Phys. 20, 1591 (1952). 26 D. Rapp and P. Englander-Golden, J. Chem. Phys. 40, 573,

5123 (1964). 27 S. L. Thompson, J. Chem. Phys. 49, 3400 (1968). Thompson

considered deactivation of a diatomic, without vibrational energy transfer to the collision partner. The extension of his treatment to energy exchange among vibrations may be made as in Ref. 25. Thompson's Eq. (29) shows that probability is proportional to temperature as the amount of energy going into translation ap­proaches zero.

28 M. E. Riley and A. Kuppermann, Chem. Phys. Letters 1, 537 (1968) .

29 The SSH treatment, Ref. 25, was extended to three dimensions by R. N. Schwartz and K. F. Herzfeld [J. Chern. Phys. 22,767 (1954)]' They concluded that the three-dimensional result was essentially the same as for one dimension. Because of the mathe­matical approximations leading to this conclusion, the predicted temperature dependence is questionable.

30 R. J. Cross, Jr. and R. G. Gordon, J. Chern. Phys. 45, 3571 (1966) .

31 C. G. Gray and J. Van Kranendonk, Can. J. Phys. 44, 2411 (1966) .

32 G. Herzberg, Molecular Spectra and Molecular Structure (D. Van Nostrand, Co., Inc., Princeton, N.}., 1945), Vol. 2, p.41.

33 H. C. Allen, Jr. and P. C. Cross, Molecular Vib-Rotors (John Wiley & Sons, Inc., New York, 1963), pp. 85-112.

34 The error function normalization is taken as in Ii andbook of Chemistry and Physics (Chemical Rubber Pub!. Co., Cleveland, Ohio, 1967), 48th ed., p. A-158.

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