vibrational relaxation time measurements in ch4 and ch4-rare gas mixtures

Physica 86A (1977) 490-512 O North-Holland Publishing Co.

VIBRATIONAL RELAXATION TIME MEASUREMENTS IN CH4 AND CH4-RARE GAS MIXTURES

M. H. DE V A S C O N C E L O S and A. E. DE VRIES*

Centro de Fisica Molecular des Universidades de Lisboa I.S,T., Lisboa, Portugal

Received 26 August, 1976

With a spectrophone vibrational relaxation times in CH4 and in mixtures of CH, with rare gases were measured. Both the amplitude and the phase method were used. The two infrared active modes of CH, (v4 and v3) were investigated separately. The relaxation times, at one atmosphere, after exciting the lowest mode v,, were found to be: ~'(CHa-CH,) = 1.65/xs; r (CH, -He) - 1.97 ~s; r (CH, -Ne) = 8.6 t~s; r ( C H 4 - A r ) = 12 ~ s and r ( C H 4 - K r ) ~ 60 tzs. From these values one may infer that vibrational-rotational (V-R) energy transfer is the dominant relaxation mechanism. By exciting the higher mode the first step in the deactivation of v~ was found to be a V-V transfer to

the lowest modes v,, v2.

I. Introduction

From the many methods to measure vibrational relaxation times, the optic-acoustic way and the laser fluorescence techniques are distinct because they allow to study separately the relaxation of different vibrational modes. Also the deexcitation scheme from a higher energy mode to the ground state can be deduced by these methods.

Read [1] gives a survey of most measurements obtained through the optic-acoustic method, which were done before 1968.

Spectrophone phase-lag measurements on CH4 were performed by Cottreil and McCoubrey [2] who found 1 . 6 ~ s a t m by exciting the v3 mode. The relaxation from the v4 mode was only investigated till 30 Torr because of low signal; within the experimental errors it agreed with the results for the v3 mode.

Yardley et al. [3,4] measured relaxation times for the removal of energy from v3 and for the appearance in v4 with a laser technique. Their results for methane and mixtures of methane with several rare gases will be compared with ours in section 5.

* FOM-Insti tuut voor Atoom- en Molecuulfysica, Kruislaan 407, Amsterdam/Wgm., The

Netherlands.

490

VIBRATIONAL RELAXATION TIME MEASUREMENTS 491

The spectrophone method has been described in a previous paper [5]. We therefore give only a very short description:

Consider an infrared active gas. If radiation of the right wavelength passes through, the molecules will absorb the energy directly as vibrational quanta. The excited molecules will loose the excess of vibrational energy by radiation or by' inelastic collisions. In the latter case one has V-T or V - R - T energy transfer and thus an increase in temperature. If the radiation is chopped, at frequencies between 150 and 2000 cps, and the gas is enclosed in a constant volume a sound is obtained. By use of filters it is possible to excite only one vibrational mode and to obtain information about the relaxation steps.

This can be done in two ways: a) Amplitude-frequency response method: By observing the sound wave

amplitude as a function of frequency. b) Phase method: By measuring the phase lag between the absorbed

radiation and the sound produced.

2. Experimental

Fig. i shows a block diagram of the experimental set-up. The spectrophone [5,6] is a gas cell made of a block of brass with a

cylindrical internal cavity of approximately 2 cm 3. On one side there is a NaC1 window. On the other side a Bruei & Kjaer ½" microphone is mounted, which has a sensitivity of 1 mV//z bar.

The spectrophone is connected with the filling system through a needle valve. Another valve is present for the connect ion with the vacuum system,

FILL

MIRROR

N . . . . .

f ..........

MIRROR "~r

MANOMETER

SPECTROPHONE

i i

' i

PUMP

M I CAT"ODE 1 i °LL°WER

I ! DPGfTAL ] PRE WAVE VOLTMETER k AMPLIFIER - - " ANALYZER - -

E J

J t I POWER FREQUENCY PHASE I SUPPLY - - - - METER METER

~._ l Fig. I. Block diagram of the experimental set-up. D- Chopping disc; F- Filter; M - Microphone;

N- Nernst filament.

492 M.H. DE V A S C O N C E L O S A N D A.E. D E VRIES

~E o

.~ 3000

2000

1000

/

/// i \ " ,, : \

/ :

J , 7.7 p

• .

3

41

~o

CH~

Fig. 2. Vibrational levels of CH, below 3100cm ' with the range of filters used indicated at the right-hand side. * means infrared active mode.

which allows pressures of about 2 x 10 6Torr. The filling sys tem can be evacua ted separa te ly by ano ther set of vacuum pumps. The pressure in both vacuum sys tems can be measured by a Penn ing and a L.K.B. gauge. The spec t rophone as well as the filling sys tem may be baked out to 150°C.

A Bourdon manomete r al lows pressure measu remen t s in the spec t rophone f rom 0 to 220 Torr with an accuracy of 0.25 Torr.

The infrared source is a Nerns t f i lament which can work at 2000 K. Two spherical mirrors and a gold plated cyl inder focus the radiat ion on the gas-cell window.

Several filters may be put into the radiat ion path in order to isolate the vibrat ional mode to be exci ted: a 3.3---0.5/x band pass filter to isolate the u3 mode (see fig. 2); a 7.7 - 1.1 t~ band pass filter and a long wave pass filter with the edge at 5 # for the v4 mode.

The radiat ion is chopped by a disc with 25 aper tures turned by a P .A.R. chopper . This makes it possible to have chopping f requencies be tween 150 and 2000 cps with a stabil i ty bet ter than 0.5% per hour. The chopper gives a re fe rence signal which is used in the phase method to synchron ize the wave-ana lyser .

Both the microphone power supply and the pre-amplifier are bat tery dr iven in order to have as little 50 cps noise as possible. The preamplif ier has also a low f r equency and a high f r equency roll-off which a t tenuates f requenc ies lower than 10 cps and higher than 10 kcps.

The wave ana lyser is working as a synchron ized amplifier with a Q of 50 having a t ransmiss ion of one at the resonan t f r equency and a re jec t ion larger than 80 dB at f requencies far f rom resonance . The gain can be changed in ten cal ibrated steps from 1 to 104.

The digital phase meter has an accuracy of 0.2 ° at the f r equency used. The gas-cell and the filling sys tem are mounted on a cement block of 500 kg


which is suppor ted by six shock breakers in order to isolate the mic rophone from vibrat ions of the surroundings.

Measurements are pe r fo rmed at t empera tures close to 300 K.

3. Re laxat ion equat ions for a two-s tate and a three-state mode l

3.1. T w o - s t a t e m o d e l

For a two-state model the opt ic-acoust ic effect was already studied by Gorel ik [7] in 1946. Later Cottrel l [2] ex tended this s tudy as fol lows:

Cons ider a two-s ta te gas charac ter ized by two temperatures . The transla- t ional t empera ture To and the vibrat ional t empera tu re T~, which can be defined by:

Nt /No = exp ( - hv /kT t ) , (1)

where No and Nt are the numbers of molecules in the ground and excited states and v the vibrat ion wave number.

If the chopped radia t ion energy is of the form

H = H ° + H ' exp (io~t) (2)

one can write the conse rva t ion of energy equat ions:

(" dT0 ~ . o - ~ = CJ, o( T, - To),

c dTt ~'-d-t- = CLfto( To - T,) + H ° + H '

(3)

exp (kot), (4)

Co and C~ are the heat capaci ty of t ransla t ion plus rota t ion and of vibrat ion, respect ively . The coll ision f r equency for deexci ta t ion of one molecule is deno ted by f~0-

The energy flow to the env i ronmen t and the spon taneous emission of radiat ion have been neglected, as both t imes involved are much larger than v ib ra t ion- t rans la t ion re laxat ion times.

Assuming solut ions of the form:

T0 = T~ + T~ exp (icot), (5)

T~ = T o + T'~ exp (io~t), (6)

one gets:

H'f to (7) T;I =

ia~(Co + Ct)flo - to:Co

494 M.H. DE VASCONCELOS AND A.E. DE VRIES

and for the osc i l l a to ry par t of the p res su re :

and

pH' exp 04 ' )

P " T°o~CT[ 1 + (w~'Jp)Z(Co/CT) 2] 1t2 ( 8 )

oJ~'lo Co* tg (rr/2 - ~ ) - - - (9)

p CT

C-r = Co + C~ is the total hea t capac i ty , p is the p r e s s u r e (in a tm) and

r~o = Plf, o (10)

is the re laxa t ion t ime r e d u c e d to o n e a t m o s p h e r e . Eq. (8) is u sed in the a m p l i t u d e - f r e q u e n c y r e s p o n s e m e t h o d and eq. (9) in

the p h a s e m e t h o d .

3.2. Three-state model

For a t h r e e - s t a t e m o d e l the re are t w o poss ib i l i t ies : exc i t ing the h igher or the m i d d l e s ta te (see fig. 3).

a) A t h r e e - s t a t e t r e a t m e n t , exciting the higher mode, was first t r ied in 1966 [8, 9] for a t w o - s t e p r e l axa t ion of the t ype

t~, f.o h v 3 ~' nhv~ -~ t r ans la t ion ,

fi is the col l i s ion f r e q u e n c y for v ib ra t ion e n e r g y t r ans fe r f r o m sta te i to s ta te ].

\ \ \~3m

V 2

I i I I I f r e e It f~o I f40 I I I I

Fig. 3. Two three-state models.

* If ~o is such that one may neglect acoustical relaxation.

V I B R A T I O N A L R E L A X A T I O N T I M E M E A S U R E M E N T S 495

According to the last more correct treatments [10--12] eqs. (8) and (9) must be replaced by:

pH' 1 + exp (i~b)

P'= T°o°JCr[ ( ~ _ ~ ) 2 ] ½ [ 1 + 1 + \(°Jr'°C°~2] }~Aexp( idp)p C-r/ J (11)

with

,:arctg(o, ) (o.°0% arc tg \ ~ /

o r

tg (7r12 - ¢) =

+ y) ,oop-' + ,Co E [ ( Co) Co 2 ] - 2 ' (I2)

l + ° j 2 Y - E 'r"°r3" + Y E r ' ° ] P

where

y = (v3- nvm)lv3. (13)

These equations apply if C3 "~ C~ and CT-~ Co+ Cm (see appendix). In the derivation of these equations the energy flow to the environment is

neglected, as well as the spontaneous emission of radiation. In the case of CH4 both relaxation processes (one step and two-step) can be

observed. C H 4 has four fundamental modes (see fig. 2): a triply degenerated bending

mode at [13] v4 = 1306 cm -t, which is infrared active; a doubly degenerated mode [14] v2 = 1533cm-1; a symmetric stretching [15] v~=2917cm -1 and another infrared active triply degenerated mode [15] v3 = 3019 cm -I, which is an asymmetric stretching mode.

The vibrational heat capacities of the lower modes of C H 4 are, at 300 K: (74 = 0.444cal /mole. K and C2 = 0.137 cal /mole. K, higher modes having negligible heat capacities. The total heat capacity, at constant volume, is CT = 6.45 cal/mole - K.

In the case of exciting the v4 mode of CH4 the two-state model may be app l i ed -equa t ions (8) and (9). However, because of easy V-V transitions v2 and v4 are strongly coupled and state l should be considered as one vibrational group including v 4 and v2. Experiment seems to confirm the theoretical prediction of quick v4-~v2 transitions [3, 16].

In describing the v3 deexcitation the three-state model has to be u s e d - equations (11) and (12).

496 M.H. DE V A S C O N C E L O S AND A.E. DE VRIES

b) A t h r e e - s t a t e m o d e l exc i t ing the l o w e r m o d e , is n o w c o n s i d e r e d : If the e n e r g y t r ans fe r 124---*-~122 is m u c h q u i c k e r than any t r ans fe r to

t r ans la t ion , the r e l axa t ion s c h e m e af te r exc i t a t ion of the 124 m o d e of CH4 can be d e s c r i b e d by the fo l l owing c o n s e r v a t i o n of e n e r g y equa t i ons :

T 4 124 C 4 - - = - C4f4o( T4 - To) + --(C4f42 + C j24 ) ( T2 - T4) + Ho + H ' e ~'~ (1 4)

at v2

C20T2 C2f2o(T2-- T , O - - ( C j 4 : + C f f z 4 ) ( T 2 - T4), (15) at

a To = C4f40(7"4 - To) + C?f,o( T~ - To) + 1 - (C4f42 + Cff,4)( T~ - T4). Co (~--7- - - . "

(16)

Also here the e n e r g y f low to the e n v i r o n m e n t and the s p o n t a n e o u s e m i s s i o n of r ad ia t ion h a v e b e e n neg l ec t ed , fii is the col l i s ion f r e q u e n c y for t r ans i t ions f r o m sta te i to s ta te j, per mo lecu le . O t h e r s y m b o l s w e r e a l r eady m e n t i o n e d .

A s s u m i n g so lu t ions of t he f o r m T -- T ° + T' exp (iwt), i = 0, 2, 4 and so lv ing

for T;~ yields:

wi th

U' T; . . . . (17)

iwCod

n = f [ z u f + g(1 + z u x ) + i z ( f - Yg)l

and

(17a)

d = f [ z u f + g( ! + zux)](1 + s + sz)

+ i [zf( l + s + u + uzs ) + g(1 + z x ) ] - ¢o2z, (176)

w h e r e

C2 [2o f = f~, prg ' , z C4 f~,'

V4 124 C4 x = - - , y = 1 - - - and s = - -

V 2 122 Co

g = z (f24 + f42),

(no te tha t x + y = 1).

If f24 > w, f24 > f ~ (or "r24 "~ r4o) is a s s u m e d (i.e., g -> f, g > a)) one can d rop t e rms in f z , ~of, o0 2 in (17a) and (17b). This resu l t s in:

H ' 1 - io~p ~'r~B .

Tc'~ ~ iooCoD I + iwp-~'r4oF (18)


with

B - - - z Y D = l + s + sz, F = l + zx (19) 1 + zux" (1 + zux)D"

Writing p ' = p T ~ / T ° in the polar form Ae ~ we obtain for the ampli tude

p H '

A -~ T~,~CoD(1 + to2r~p-2F2) ~ (20)

and for the phase angle

d~ = arc tg (top-1~-40B) -½"rr - arc tg (~op-1740F),

tg (17r - d~) = ~°P-1¢4°(B + F) ~ ~op_lr4o(B + F). (21) 1 - ~oZp-2r~oBF

For the exper imenta l condi t ions , toP < 3 x 105, one may neglect both ~o2p 2r~B2 and o~2p-2T~oBF with respect to unity.

Compar ing eqs. (20) and (21) with the results for the one-s tep relaxat ion (8) and (9) one expects the measu red re laxat ion t ime, rt0, found by applying the one-s tep formulas to the v4 m o d e to be larger than the ' true' relaxat ion t ime ~'4o of this mode.

For the ampl i tude m e t h o d

( Co/ CT)'rjo = ~'4oF, (22)

where , as a l ready stated, the exci ted state l is to be cons ide red as a group including v4 and v2. For the phase me thod

(Co /CT)7" Io ~- r4o(B + F). (23)

For CH4 z = 0.309, y = 0.148, s = 0.076, while u and v =- f J f 2 4 = (g/f24- z) are calculated to be about 0.2 (u = 0.19, v = 0.22). With these values: B = 0.0435, F = 1.092. With Co/CT = 0.910 this gives ~'40 = 0.83"rt0 for the ampli tude me thod and r40 ~ 0.80r~0 for the phase method.

Due to the approximat ions made these conclus ions are h o w e v e r to be taken quali tat ively: In CH4 the re laxat ion t ime "r4o, f rom the v4 mode , is shor ter than the measured re laxat ion t ime ~'~0 cor responding to the v4, v2 group. Because of the vicinity of v2 (and of v4~--~v2 transi t ions fo l lowed by v2 ~ T as well as v 4 ~ T) the re laxat ion f rom i: 4 suffers a delay.


4 . R e s u l t s

We obta ined re laxat ion t imes mainly by the ampl i tude- f requency response m e t h o d - s e e eqs. (8) and (11). The sound ampli tude was measured as a funct ion of f r equency in the range of 180 to 2000 cps, at several pressures . This was done for pure CH4 and for mixtures with inert gases; for each mixture several concent ra t ions were studied.

The phase m e t h o d - ( 9 ) and ( 1 2 ) - w a s also used for pure CH4. The phase lag b e t w e e n sound and light signals was measured as a funct ion of pressure f rom 320 to 8 Torr; several f r equenc ies were used.

4.1. Calibration gas

Because of the non-l inear f r equency response of the mic rophone in the gas cell and of heat conduc t ion to the walls a calibrating gas has to be used. This calibrating gas must have an acoust ic behaviour similar to that of the gas being studied, but a relaxat ion t ime, re, not measurable with the spec t rophone .

For this cal ibrat ion 7% of e thane was added to the gas being invest igated [17]. According to Yardley et al. [4] ~'(CH4-C2H6) = 6.4 × 10 -3 ~,s • atm, which

gives for the calibrating gas mixture, 7% C2H6 + 93% CH4, a relaxation t ime of re = 0.09 ~s" atm. Recent ly , Hue tz et al. [18] showed that the relaxat ion t ime for the mixture CH4-CzH6 has a non-l inear behaviour with concent ra t ion . According to their impact tube m e a s u r e m e n t s our calibrating mixture has: ~-~ = 0.15 t~s • atm.

When using the ampl i tude me thod even this larger value of r,. is small enough, because the de te rmin ing factor is 2 2 2 ~'cP ~ 1, which remains true for the exper imenta l f requenc ies and pressures .

For the phase method , however , one must take this new result into account . The measured quant i ty is $ - 4 ' ~ and for the one-s tep relaxat ion one may write:

tg (7rt2 - 4~ - 7r12 + 4~c) = - tg (& - &c)

( to /p) (Co/Cr)(r - re) z

1 + (to/p)2(Co/CT)2r~ %" (24)

For the two-s tep relaxation:

tg (~r/2 - $ ) - tg (~-/2 - 4'c) - = - ( 2 5 )

tg (~b $~) 1 + tg (~r/2 - 4') tg (-rr/2 - t~c)"

To obtain t g ( ~ r / 2 - ~ ) one uses the two-step formula, eq. (12), while the one-s tep formula, eq. (9), may be used for tg (7r/2 - 4~¢). Indeed for the mixture CH4-CzH 6 all V - V t ransfers are done through small and very quick steps so that only the last, V - T step has to be taken into account .

V I B R A T I O N A L R E L A X A T I O N TIME M E A S U R E M E N T S 499

All gases used contained therefore less than 2 ppm of hydrocarbons other than C H 4 , the purity of CH4 being t> 99.9995% and that of the noble gases as follows" He i> 99.995%, Ne I> 99.99%, Ar t> 99.995% and Kr >i 99.95%.

4.2. Pure C H 4

4.2 .1 . The v4 mode

We obtained relaxation times mainly by the amplitude m e t h o d - s e e eq. (8). Fig. 4 gives two typical curves of log A[ versus log f for this method. Through the experimental points least-squares fits were drawn according to

eq. (8). The lines presented correspond to 1.61/zs • atm for P = 15 Torr and to 1.59/zs • atm for P = 12 Torr. These relaxation times refer to the vibrational group including v4 and v2.

The complete results are given in table I. The measurements were per-

Iog[A f)

t J 120

P:lSlorr 115 ,

i

105i " ~

-. ~ - i

2'1 213 25 2"/ 219 3.1 • t o g f

Fig. 4 Log (Af) versus log f. Experimental points and best fit theoretical curves for two different pressures of CH4: P = 15Torr gives ~ - = l . 6 1 ~ s . a t m and P = 1 2 T o r r gives ~-= 1.59/Ls-atm.

TABLE I Relaxation time of the (v,, u2) group, by the amplitude method

P (in Torr) 20 20 16 15 15 I5 14 12 12 12

r (in

~zs • arm) 1.63 1.66 1 . 6 1 1.62 1.63 1 . 6 1 1.64 1.59 1.66 1.69

500 M.H. DE V A S C O N C E L O S A N D A.E. DE V R I E S

fo rmed at pressures be tween 20 Torr and 12 Torr. The 7.7/x band pass filter was used as well as the 5/z long wave pass filter. The Belt effect, which is the sound genera ted by chopped radiat ion hitt ing the walls, however , had no influence on the results .

The average value and s tandard devia t ion are:

~' t0(CH4) -- 1.63 -+ 0.03 ~s • atm.

For the phase method eq. (24) has been applied. Fig. 5 shows a plot of phase- lag versus the inverse of pressure for two

different chopping f requencies . The curves are least squares best fits to the exper iment points according to eq. (24) with rc = 0.15 /xS-a tm and Co/CT =

0.91. Measu remen t s done via this method gave an average resul t of 7%(CH4)=

1 .67-- -0 .15/xs-a tm in agreement with the result via the ampli tude method a l though with a higher s tandard deviat ion.

0

15

10

f= 320 c.p,s.

f-- 380 c.p,s.

I I t I I I l I I

0.05 0.10

,. p-~(torr -I)

Fig. 5. Phase - l ag versus the inverse of p ressure for the re laxa t ion of the (v4,v=) group. The curve with ~ o = 2 ~ × 3 8 0 s ' l gives r = l . 6 8 / x s - a t m and the curve with ~ o = 2 7 r × 3 2 0 s - ' gives ~-=

1.70 txs " atm.


4.2.2. T h e v3 m o d e

The different possible ways for deac t iva t ion of the v3 mode are so many, that one is forced to schemat ize the re laxat ion paths. Referr ing to fig. 2 such a schemat iza t ion cer ta in ly is possible , leading to only three different ways .

The first possibi l i ty , is a direct V - T transi t ion, which can be descr ibed by the two-s ta te formulas (8) and (9). Apply ing this two-s ta te scheme to the exper imenta l resul ts ob ta ined with the ampl i tude method, a value of r = 1.41 p.s . atm is found, i.e. a value smaller than the value of 1 . 6 3 / z s - a t m for the v4, v2 group, which is ex t remely unlikely.

The other two possibi l i t ies involve both V - T and V - V transi t ions. Again referr ing to fig. 2 the included levels could be divided into two groups: the group 2v2, vt, v2+/J4, 2v4, and the group v z, u4, a l ready discussed in the preceding sect ion. Assuming now that V - V t rans i t ions be tween the members of one group are very quick, as well as t rans i t ions be tween the two groups, one may assign a vibrat ional t empera tu re T m and a vibrat ional specific heat Cm to the whole sys tem including both energy groups. One may decide from the measu remen t s via which group the v3 level decays to the ground state, as the amount of t rans la t ional energy involved in each process is quite different.

If the deexci ta t ion path goes via the members of the upper group, the v2+ v4, v~, 2v2 and 2 u 4 levels are quickly popula ted and one again could descr ibe the re laxat ion by the one-s tep f o r m u l a s - for negligible y and "r3m, the two-s tep equat ions (I1), (12) reduce to the one-step equat ions (8), (9). The value obta ined for -r, in this way, should agree with the one obta ined for the v4 exci ta t ion, but as a l ready stated this is not the case, so one is enti t led to conclude that this pa th is not possible. For example a decay v3 ~ 2v4~ T has a value of y = 0.14 which, a l though different f rom zero, is small compared to values of y when re laxat ion goes via the lower group, and indeed negligible as far as eqs. (11) and (12) are concerned.

For the last possible way of d e a c t i v a t i o n - r e l a x a t i o n via the v4, v: g r o u p - the three-s tep formula should be used. There are now two possible ap- proaches : consider ing that "rm0 is known the values of rs,~ can be ca lcula ted using a value of y = 0.57 or y = 0.49 for the step to e i ther v4 or v z. The results of these ca lcula t ions are tabula ted in table II, for measu remen t s via the ampl i tude method.

W h e n using the phase method the measured quan t i ty is (~b-~c) . For the two-s tep re laxat ion tg (~b - 4~c) is given by eq. (25) with tg (~r/2 - ~b) of eq. (12) and tg (~r/2 - 4~c) of eq. (9). For negligible y and ~'3m one can use the one-s tep equat ion (8) which gives "r~0 = 1.37/xs • atm, again a value smaller than -r~0 = 1.62 tzs • atm. Fol lowing the same reasoning as for the ampl i tude method one gets the values of rsm which are given in table III.

Tables II and III show that the deac t iva t ion scheme goes via the v4,u2 group with a V - V re laxat ion t ime '/'31 ~ 0.4 ~ s • atm.

502 M.H. DE V A S C O N C E L O S A N D A.E. DE VRIES

4-

i

)==(

._=

r=

A

1

I

t"q

/ ~_~ I ~ ~ ~,


T A B L E I I I

r3,,, f r o m eq. (12), v ia ~'4 and v2, as a f u n c t i o n of f r e q u e n c y in pu re CH, . P h a s e m e t h o d .

F r e q u e n c y y ~ (in and ~ c p s l

d e c a y ' ~ t y p e ~ .

0.57 v3 ~ v,

0.49 v~ -~ v2

480 430 430 430 430 380 380 380

M e a n S t a n d a r d

va lue dev ia t .

o f ~3m

0.5 0.4 0.4 0.5 0.4 0.5 0.8 0.5 0.5 0,1

0.4 0.3 0.2 0.3 0.3 0.4 0.7 0.4 0.4 0.1

4.3. CH4-rare gas mixtures

4.3.1. Lowest mode For mixtures of methane with atoms CH4-X, the relaxation time is given

by:

1 x l - x + (26)

r(mixt) r ( C H 4 ) , r ( C H 4 - X ) '

where x is the mole fraction of CH4 and z ( C H : X ) is the relaxation time of CH4 by collisions with X.

For these mixtures only the amplitude method was used. In figs. 6, 7, 8 and 9 one can see plots of r-~(mixt) versus x for He, Ne, Ar

and Kr, respectively. From these plots one gets straight lines, by a least-

1/'[(Mixt) l 0,7-

0.5--

0,3-

°-1 t I " - - ~ - - - ] . . . . .

0.1

C H~ ÷ H, ~7~

F ....... 7 .... T ....... [ ........ T . . . . . . F - - l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5 1,00

= Mol .e f r a c t i o n o f C H 4

Fig. 6. r - l ( m i x t ) ve r sus mole f r a c t i o n o f C H , fo r C H , - H e mix tu r e s . T h e m e a s u r e d po in t s lie in

b e t w e e n the s ta rs .

504 M.H. D E V A S C O N C E L O S A N D A.E. D E V R I E S

i - . -

1/ ' [ (Mixt) !

0 2 - -

C H4 ÷ N~

0 . 5 4

0.3

7.7)2

T 1 T ' i T 0.1 0.5 1.00

~'- Mote fract ion of CH 4

Fig. 7. ~- ' (mixt] ve r sus mole f r ac t ion of C H , for C H , - N e mix tu res . The m e a s u r e d points lie in

b e t w e e n the s tars .

1/ ' [ (Mix t )

T 0.7,

05

03~

[]1

CH4+Ar 7 7/~

T 1 1 ~ 0~5 ~ ~ r r 0.1 1,00

= Mole f ract ion of CH 4

Fig. 8. ~- ' (mixt ) ve r sus mole fraction of CH. for CH~-Ar mixtures. The measured points lie in

between the s tars .

squares fitting to eq. (26), such that the intersection with x = 0 gives "r(CH4- X). Points drawn in the figures are extreme values. For each concentrat ion at least three measurements were performed at different pressures.

Table IV gives a summary of our results for the relaxation t imes of the u4,v2 group, "q0(CH4-X), together with the most recent results of other authors.

V I B R A T I O N A L R E L A X A T I O N T I M E M E A S U R E M E N T S 50'

. . . . . . . . . . . . . . . . . . .

I

v't¢Mixt) C H~ + K~ z~,

0.5--

-1

(3,3

I

/ - - V - P ! ~ r ' I - ~ T T

0.1 0.5 1.00

- - " " - M o l e f r a c t i o n of CH 4

Fig. 9. ~- ' (mixt) versus mole f rac t ion o f C H , for C H , - K r mixtures . The m e a s u r e d points lie in b e t w e e n the stars.

TABLE IV Relaxat ion t imes of the (v,, v~) g roup of C H , - X , at room tempera tu re

Gas r,o(CH4-X) in /~s. a tm or

mixture This w o r k Ref. 18 Ref. 19, 20 Ref. 4

C H , - C H , 1.65---0.1 "~ 1 .65±0 .02 1.56---0.01 1.9_+0.1 C H 4 - H e 1.97___0.1 b~ 3 . 2 ± 0 . 1 - 2.3_+ 0.2

C H , - N e 8.6_+2 b~ 19___ 1 11.2 11 ± 1 C H , - A r 12.1 ---4 b~ ~ 50 16.6 20 ___ 2 C H , - K r ~ 60 ¢~ - 19.9 29 ___ 3

Me thod S p e c t r o p h o n e I m p a c t tube Ul t r a sounds F luo re scence

"~ This value and the s tandard devia t ion refer to m e a s u r e m e n t s done using both exper imenta l me thods .

b~ M a x i m u m possible er ror abou t the fitted straight line (see figs. 6, 7, and 8). ~ It is impossible to evaluate an e r ror in this region of ~" ' (mixt) (see fig, 9).

4.3.2. The ~3 mode

When excit ing the asymmetric stretching mode of C H 4 in mixtures with atoms, one can obtain relaxation times for each CH4 concentration by means of eq. (11), with

y(mixt) = Xf3 m (C H4-CH4)y (CH4) + ( 1 - x)f3,,, (CH4-X)Y (CH4-X) (27)

Xf3 m ( C H 4 - C H 4 ) + ( 1--x)fsm ( C H 4 - X )

506 M.H. D E V A S C O N C E L O S A N D A.E. DE V R I E S

and

C0(mixt) xC0(CH4) + ( 1 - x)C(X)

Cv(mixt) XCT(CH4) + (1 - x)C(X) ' (28)

where /3,,(CH4-CH4) is the number of t ransi t ions 9 3 ~ 9 m per second per molecule due to CH4-CH4 collisions, while f3,,(CH4-X) refers to CH4-X collisions; y(CH4-CH4) and y ( C H 4 - X ) - s e e eq. ( 1 3 ) - i n d i c a t e to which state the deexci ta t ion of ~,~ takes place due to CH4-CH4 or to CH4-X collisions, respect ive ly ; C ( X ) is the specific heat of the rare gas, X.

Fol lowing the same approach as for pure CH4 and cons ider ing a deexci tat ion way for which the one-s tep relaxation equat ion can be u s e d - negligible y and ~'3,, - one obtains values of z(mixt) for several pressures of each concent ra t ion . With these values straight lines, similar to those of figs. 6 to 9, can be drawn. From these plots the fol lowing results are obtained:

~-(CH4-He) = 1.76±0.1/ . ts • atm r (CH4-Ne) = 10.5 __.2/zs • atm

"r(CH4-Ar) = 18_+ 6/xs - atm r (CH4-Kr) ~ 70/J.s. atm.

C o m p a r e d with the results of table IV, obtained by excit ing the bending mode , one can see that for Ne, Ar and Kr a deexci ta t ion via the upper group (2uz, u~, 9: + 94, 294) is possible. In that case, however , the V - V relaxat ion t imes cannot be neglected. On the o ther hand, for He one has, like for pure CH4, that a deexci ta t ion via the upper group is not possible.

So, also in collisions with noble gas atoms a two-step relaxation must be cons idered . In the case of He the two-s tep relaxat ion is only possible via the lower group (94, 92), while for the heavier noble gas a toms also deact ivat ion via the upper group is to be cons idered .

In the same way as for pure C H 4 , the main relaxation t imes for V-V transit ions are calculated for different in termedia te steps. The results of these calculat ions are p resen ted in table V. The first two rows are connec ted with relaxation via the lower u4,9~ group, the o ther two rows with the upper group.

TABLE V Mean va lues of V - V r e l a x a t i o n t imes fo r C H 4 - r a r e gas co l l i s ions , e v a l u a t e d for

the m e a s u r e m e n t s wi th l o w e r CH4 c o n c e n t r a t i o n .

N o b l e l gas ]

/ He Ne Ar Kr 1

d e c a y t y p e

0.57 u~ ~ u, 1.0 --- 0.2 4.1 -+ 0.9 6.5 --- 0.4 4.7 --- 0.4

0.49 u~ -~ ~'2 0.8 -+ 0.2 3.5 -+ I. ! 6.0 -+ 0.4 3.0 -+ 0.6

0.14 u ~ 2 u 4 - 1 . 5 - + 1.7 4.4-+0.7

0.07 u 3 ~ u 2 + u 4 - 1.6-+ 1.9 4.3---0.7


5. Discussion

5.1. The v4 mode Our re laxat ion t ime resul ts for the / /4,//2 group of CH4 and CH4-noble gas

mixtures are summar ized in table IV together with the resul ts of o ther authors . Reasonable ag reement is found be tween our re laxat ion t imes and those obta ined by means of the u l t rasound method and the f luorescence method, excep t for the CH4-Kr mixture. One should bear in mind, however , that the higher ~-(CH4-X) is compared with ~-(CH4-CH4) , the less accura te ly the re laxat ion t ime for m e t h a n e - r a r e gas a tom coll isions can be d e t e r m i n e d - s e e formula (26). The de te rmina t ion by means of the impact tube method seems to give values too high for CH4-rare gas mixtures .

In order to compare the exper imenta l resul ts with the theoret ical calcula t ions it is bet ter to use probabil i t ies . The re laxat ion t imes are ve ry near ly related to the energy t ransfe r probabi l i t ies as fol lows [2]:

Pl0 = Z-Jr'01[1 -- exp ( -- huffkT)] t, (29)

Pt0 is the probabi l i ty of energy t ransfer f rom vibrat ional s tate 1 to the ground state and Z is the coll ision f r e q u e n c y

27 = 4pr2( cr[21xk T)~, (30)

where r0 is the dis tance of c losest approach; the values of r0 used in the calcula t ions were those of ref. [2l].

F rom the re laxat ion t imes tabula ted in table IV and cor responding to the u4,u2 group one obtains then the fol lowing exper imenta l probabil i t ies of energy t ransfe r f rom this group:

Pt0(CH4-He) = 4.6 × 10 5

PI0(CH4-CH4) = 6.0 x 10 -5,

PI0(CH4-Ne) = 1.6 × 10 -5,

P)o(CH4-Ar) = 1.1 × 10 -5,

PI0(CH4-Kr) = 2.0 x 10 -6.

Consider ing the resul ts of the ca lcula t ions done in sect ion 3.b to obtain the probabil i t ies of energy t ransfe r f rom the v4 mode alone one has to mult iply the above probabil i t ies by a fac tor of approx ima te ly 1.2.

Both sets of exper imenta l probabil i t ies are plot ted in fig. 10 as a func t ion of 1/3 /x

The theoret ical probabil i t ies , on the other hand, can be der ived f rom two different models:

For the V - T model the probabi l i ty of energy t ransfer f rom vibrat ional state

508 M.H. DE V A S C O N C E L O S A N D A.E. DE VRIES

10 ~ I

T,oi . . . .

10 -5

10 6

10-:

i \

\ \ \\\\

|He \

2 4 5 6 7

] 1 I [ I I F I I

1.5 2.0 2.5

-- t2/~

Fig. 10. Experimental probabilities, P (CH, -X) , v e r s u s ~ j l J . 3 : • energy transfer from the v,,u~ group; [] energy transfer from the u4 mode, according to approximate formula (22). Lines

represent theoretical calculations of pv_T for several values of a (in :~--L).

i to vibrat ional state j is, according to the S.S.H.- theory [22]:

Psi = Z , , ~ A ( M ) - ~ e x p ~ x v e x p 2kT , - , ( v + u ) J

which can be approx imate ly solved in such a way that the the V-T probabi l i ty depends on the reduced mass, ~, and on the paramete r for an exponent ia l , a, as fol lows:

v v 2{t~]'" [ 3 {16r4~,2t~'~"31 Pis - / x \~-T] exp - ~ \ ~ ] j , (32)

A(M) is a mass factor , in this case equal to 5.67 x l023 g ~, • is the depth of the

V I B R A T I O N A L R E L A X A T I O N TIME M E A S U R E M E N T S 509

intermolecular potential well, v and u are the relative velocities before and after collision, Z0 is a steric factor to account for non-collinear collisions and v = u i - uj.

Using the V-T model with T = 298 K, Z0 = 1/3 [22] and for the values of e / K from ref. 21 the computed values of P~0 -x, from eq. (31) are drawn in fig.

i/3 10, for several values of a, and as a function of /x For the V - R - T model energy transfer f rom vibration is more probably

occurring to rotation than to translation and R-T transitions are very quick as compared to V - R transitions. A simple V - R theory [23] allows calculation of the probability of energy transfer.

/2~r\~1 210~ 3 (01~1/3 + /9 pV-R= Z 0 ~ _ ) m mad --~ ~ (0.~_),16 exp [ - ~ \ - ~ - / ~--~],

where

(33)

0 = hv/k , O~ = 16zr4uEI[ot~kd ~,

I is the moment of inertia, d the distance from the peripheral atom to the axis of rotation, ma the mass of the peripheral atom and m the mass of the vibrating molecule.

Using the V - R - T model there is no dependence of P~0 -R on mass. The values of a which can explain the experimental p robabi l i t i es -cor -

responding to energy transfer from the g)4,/)2 g r o u p - a r e presented in the upper rows of table VI for both models. Z 0 = 1/3 was used for the V-T model (see also fig. 10) and Z0 = 1/5 [23] for the V - R - T model (except for C H 4 - C H 4

collisions where Z0 = 2/5 to account for rotational energy of both collision partners). The last two rows of table VI show values of a from other types of experiments. Molecular beam values [24], [24] were obtained by matching the given r A potential to an exponential one, in the indicated range of r.

TABLE VI The exponential factor, a, in A 1, for CH, collisions

CH4 He Ne Ar Kr

V-T

model 5.1 3.3 5.0 6.3 6,6

V - R - T model 4.0 4,2 3.8 3.6 3.1

Molecular beams 5.6 [24} 4.0 [24] - 3.1 [25] -

Transport properties 5.1 [22] -


It seems possible to conclude that V - T t ransfe r is unable to explain the exper imenta l results. F rom molecular beam exper iments [24, 25] as well as f rom Herz fe ld ' s ca lcula t ions [22], a is expec ted to decrease with increas ing mass of the noble gas coll ision partner . The V - T model, however , together wi th our exper imenta l resul ts leads to an t~ dependence on ~ which is in the wrong direct ion, to an ot (CH4-He) which is too low and to an a (CH4-Ar) which is too high.

By apply ing V - R calcula t ions to our resul ts one is able to predic t the right dependence of a on ~ and to obtain good agreement with molecular -beam resul ts for CH4-He.

If the theoret ica l predic t ions of the two-s tep t rea tment are correct the re laxat ion t ime of the v4 mode, r40 would be around 1.2 t imes smaller than the re laxat ion of the v4,v2 group, ~'t0, which has been cons idered th roughout this paper. This means that the probabi l i ty of energy t ransfe r f rom the v 4 mode is larger than that f rom the v4,v2 g r o u p - f o r example P 4 o ( C H 4 - C H 4 ) ~ 7 × 10 -5 instead of P~0 = 6.0 x 10 -5. It also means that the set of values of ~ one obtains f rom the exper imenta l probabil i t ies , P40, are smaller than those cons idered in table VI and obta ined f rom P t 0 - f o r ins tance ot(CH-CH4) f rom the V - T model, 5.2 ~ - l ins tead of 5.1 ~-1. Anyhow, a shor ter re laxat ion time as well as smaller values of a are to be expec ted when consider ing the v4 mode alone.

5.2. The v3 mode

As was d iscussed in sect ion 4, the exper imenta l resul ts on the re laxat ion of the v3 mode, in pure me thane and m e t h a n e - h e l i u m mixtures , only can be expla ined if a two-s tep mechan i sm is assumed, the first step being f rom v3 to the v4,v2 group and the second one a V - T relaxat ion. This is somewha t surprising, as V - V t rans i t ions to 2v2,vl, /:2 + V4 and 2v4 all are a lmost resonan t and one would expec t these to take place very fast. For mixtures with other noble gases all deac t iva t ion schemes seemed a priori possible. Howeve r , f rom computed results on the basis of eq. ( 1 1 ) - s e e table V - o n e conc ludes that also for m e t h a n e - k r y p t o n coll is ions the deac t iva t ion f rom v3 to the upper group is not possible, while for neon and argon mixtures no definit ive choice can be made.

F rom table V it is also clear that all resul ts are cons is ten t wi th our proposed mechan i sm.

The exper imenta l resul ts of Cottrel l [2] and those of Yard ley and Moore [3] for CH4-CH4 coll is ions seem to cont radic t our exper imenta l results.

Cottrei l could not dis t inguish, with the spec t rophone phase method, be tween the measured re laxat ion t imes af ter exci ta t ion of v4 or of v~. This was due to the fact that measu remen t s could not be done below 30 torr.

Yard ley and Moore [3] using a laser to exci te the v3 vibrat ion measured re laxat ion t imes for V - V energy t ransfe r which are of the order of some ns. F rom these m e a s u r e m e n t s they conclude that the re laxat ion scheme goes via very quick V - V t ransfer to the higher group. It does not seem possible to


reconci le their and our conclus ions . F rom our measuremen t s it is clear that even if r3m, in CH4, is negligible the de-exci ta t ion of v3 goes via the lower v4,v2 group. Indeed for r3,~ = 0 eq. (11) leads to a value of y = 0.44, which is close to the value for the lower group.

Acknowledgements

We would like to thank P ro fe s so r Dr. J. Los , Dr. C.A. ten Seldam and Dr. F, Cannemei j e r for their valuable comments . We are also grateful to Dr. M. J. Ferre i ra dos Santos for her help in computa t iona l problems and to Mr. J. Morais for per forming part of the measurements .

Appendix

The energy conse rva t ion equat ions , for a two-step re laxat ion of the type hv3 ~ nhvm ~ t ransla t ion, are:

dT~ C3

d t = H ° + H,e i~,t _ C 3 f 3 m ( T 3 - T i n ) ,

d T ~

Cm dt = + xC~f3m(T~ - T , . ) - C~f,~o(Tm - To),

dTo Co-: - - = + y C f f 3 ~ ( T 3 - T~) + C J ~ o ( T m - To),

d t

where T3, T,, and To are, respect ively , the tempera tures which charac ter ize the vibrat ional states v3, vm and the ground state; the infra-red radiat ion is of the form H = H ° + H ' exp (itot); f3m and fro0 are the coll ision f requencies for energy t ransfer f rom vibrat ional state 3 to state m and f rom m to zero, respec t ive ly ; C O is the heat capaci ty of t ransla t ion plus rotat ion, C3 and C~ the vibrat ional heat capaci t ies of states 3 and m, respect ive ly ; x = v,Jv3;

y = l - x . These equat ions neglect the spon taneous emission f rom state 3 as well as

t ransfer of energy to the env i ronment . With s- -Cm/Co, z = C J C m and assuming solut ions of the form To =

T~ + T) exp (itot) (] = 3, m, 0) we get

n r /1 T ~ - m

Coi~ d

with

n = f3m(fmO + iwy)

512 M.H. DE VASCONCELOS AND A,E. DE VRIES

a n d

d = - w 2 + iw[fsm(1 + zx) +/,~0(1 + s)] + fmof3m(1 + S + SZ)

w i t h z "~ 1 o n e g e t s e q s . (11) and (12) w i t h Ca- = Co + Cm.

R e f e r e n c e s

1) A.W. Read, Advances Mol. Relaxation Processes 1 (1%7-68) 257. 2) T . L Cottrell and J,C. McCoubrey, Molecular Energy Transfer in Gases (Butterworths,

London, I%1). 3) J.T. Yardley and C.B. Moore, J. Chem. Phys. 49 (1%8) 1111. 4) J.T. Yardley, M.N. Fertig and C,B. Moore, J. Chem. Phys. 52 (1970) 1450. 5) F. Cannemeijer, M.H. de Vasconcelos and A,E. de Vries, Physica 53 (1971) 77. 6) F. Cannemeijer, Thesis, University of Amsterdam (1973). 7) G. Gorelik, Dokl. Akad. Nauk. SSSR 54 (1946) 779. 8) B.J. Lavercombe, Thesis, University of London (1%6). 9) MH. Vasconcelos, internal report: Scriptie, FOM4nstituut voor Atoom- en Molecuulfysica,

Amsterdam (1%8). 10) M. Huetz-Aubert, P. Chevalier and R. Tripodi, J. Chem. Phys. 54 (1971) 2289. 11) R, Tripodi and W.G. Vincenti, J. Chem. Phys. 55 (1971) 2207. 12) F. Cannemeyer and A,E. de Vries, Physica 70 (1973) 135. 13) G. Herzberg, Molecular Spectra and Molecular Structure II (D. van Nostrand, Princeton, 1945). 14) J. Herranz, J. Morcillo and A. Gomez, J. Mol. Spectr. 19 (1%6) 266. 15) T. Shimanouchi, Tables of Molecular Vibrational Frequencies-Consolidated Volume I (Unit.

St. Departm. of Commerce, 1972). 16) W.M. Madigosky, J. Chem. Phys. 39 (1%3) 2704. 17) M.H, Vasconcelos, F. Cannemeijer and A.E. de Vries, Chem. Phys. Lett. 12 (1971) 154. 18) M. Huetz, R. Lenormand and H. Manceau, Advances in Molecular Relaxation Processes 6

(1974) 153. 19) J,G. Parker and R.H. Swope, J. Chem, Phys. 43 (1%5) 4427. 20) J.G. Parker and D.N. Ritke, J. Ac. Soc. 51 (1972) 169. 21) J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids (John

Wiley, New York, 1954). 22) K.F. Herzfeld and T.A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Acad.

Press, New York, 1959). 23) C.B. Moore, J. Chem. Phys. 43 (1%5) 2979. 24) I. Amdur, M.S. Longmire and E.A. Mason, J. Chem. Phys. 35 (1%1) 895. 25) I. Amdur and E.A. Mason, J. Chem. Phys. 41 (1%4} 2695.

vibrational relaxation time measurements in ch4 and ch4-rare gas mixtures

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