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Vibrational relaxation of HBR(ν = 1) in D2 between 300 and 140 K
Chemical Physics 122 (1988) 141-146 North-Holland, Amsterdam
VIBRATIONAL RELAXATION OF HBr(v= 1) IN Dz BETWEEN 300 AND 140 K D. PAPADIMITRIOU, H.V. SCHWARZ and B. SCHRAMM Physikalisch-Chemisches Institut der Universitiit, Im Neuenheimer Feld 253, D-6900 Heidelberg, FRG
Received 4 December 1987
Vibrational relaxation of HBr( v= 1) has been studied in mixtures with Dz in the temperature range between 290 and 140 K with the help of laser-induced fluorescence. V-V rates and V-T, R rates could be determined separately.
1. Introduction An enormous amount of data concerning vibrational energy transfer of hydrogen halide molecules was gathered in the last 20 years and the fundamental principles of energy exchange seem to be well understood [ 11. However, a number of major questions still exist, e.g. the influence of attractive intermolecular potentials or the problem of rotational energy in V-T, R processes. In order to get some more detailed insight into these problems we have started a systematic investigation of low-temperature V-T, R relaxation rates of hydrogen halides in different hydrogen isotopes. In collisions of HBr( v= 1) with H2 or HD only V-T, R energy transfer is possible because of energetic reasons. In HBr ( v= 1) + D2 collisions V-V transfer can occur also and laser-induced fluorescence decay curves are complicated in this case. In this paper we want to present our experimental results of V-V and V-T, R relaxation in HBr (v= 1) + D2 mixtures down to temperatures of 140 K.
2. Experimental A laser-induced fluorescence method was used in order to measure vibrational relaxation rates. The experimental arrangement is described elsewhere [ 2,3] and only a brief summary is given here. A chemical HBr laser, tuned to the P ( 5) line of the v= 1+v= 0 transition with pulses of about 1 us half
width and 1 kW peak power, was used to excite HBr in a fluorescence cell that could be cooled down to 140 K. Fluorescence was collected with a lens vertical to the laser beam onto an InSb detector. Stray light could be blocked effectively with a bandpass filter that only transmits light from the R branch. Fluorescence decay curves were stored in a digital storage oscilloscope ( Vuko VKS 22- 16 ) and could be averaged with the help of a Commodore 8032 computer. The gas handling procedure and further experimental details are explained in ref. [ 3 1.
3. Results A typical fluorescence decay curve is shown in fig. 1. It can be interpreted as a sum of exponential decay curves Z=Z, exp( -n,t)
+Z2 exp( -n,t)
.
Separation of the two exponentials is easily done in the following way. First, the decay constant A2 and the intensity factor Z2are determined from the slowly decaying tail of the fluorescence curve. Then Z2 x exp( -&t ) is calculated for all times beginning with t = 0, which is set at the time of the laser pulse, and subtracted from the measured fluorescence curve. The result is a single exponential decay curve, I, X exp ( -1, t ) . In fig. 1 we have drawn the calculated term Zzexp ( -1 2t) and the full double exponential fit in addition to the measured fluorescence curve. Depending on temperature and mole fraction other
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
D. Papadimitriou et al. / Vibrational relaxation of HBr(v= 1) in Dz
142
This model assumes a fast V-V exchange between HBr and Dz until an equilibrium distribution is reached that is given by [ 41
60
50
P2(~=1)1
3 s f" c
(6)
xexp[--h(vD,-vHB,)/kTl
s -30 B : E
xD, =------XHBr
[HBr(u=l)]
(where square brackets on the left-hand side denote partial pressures and x is the mole fraction) and a subsequent slow V-T, R relaxation of HBr (u= 1). Collisions between two excited HBr molecules may be neglected since only a small fraction of HBr was excited by the laser pulse and since all our measurements were performed with high dilution of HBr in
20
10
I
20
60
I
100
I
I
140
#
I
180
I
I
220
I
260
I
I
300
I
I
340
I,,
360
Xrn.9[arbitrary units]
D2.
Fig. 1. Fluorescence decay curve of a mixture of HBr and Dz at temperature T= 184 K, pressure p= 93 Torr, and mole fraction xHBr= 0.00428. The fit of the slowly decaying part and the double exponential fit are drawn together with the experimental c&e.
fluorescence curves may show a more pronounced fast decay or a more pronounced slow decay. Therefore error bounds for the corresponding decay constants are very much dependent on the special shape of the cunres. In order to explain this result theoretically we take into account the following live relaxation channels: he’
HBr(v=l)+HBr(u=O)-
2HBr(u=O),
Al?=2559 cm-’ ;
(1)
HBr(u=l)+D,(v=O) IWer-o2 HBr(v=O)+D2(u=O), AE=2559 cm-’ ;
k,/ k,* = exP [ -h
(VD? - Z’HB~) lkT1
d[HBr(u=
l)]/dt [HBr(u= 11+b,
P2(u=l)l,
d[D,(u=l)]/dt (2)
=a,[HBr(u=l)] with
HBr(v=O)+D,(v=l),
AIT= -431 cm-’ ;
Dz(v=l)+D(v=O)M=2990
(3) kD,
cm-’ ;
2D2 (v=O) ,
Dz(v=l)+HBr(v=O) kD-HB, -‘D,(v=O)+HBr(u=O), AE=2990
cm-’ .
+kHBr[HBr(u=O) I , b, =k,. [HBr(u=O)]
(4)
,
az=kP2(u=O)l,
b2 = (k,, +kD2-HBr) [HBr(u=O)] +kD,
(5)
.
(7)
Therefore only three rate constants, k,, kHBr_Dzand &-HBr , remain to be determined from our experiment. In order to do this we have to solve the relaxation kinetics. Processes ( 1 )- ( 5 ) lead to a system of two coupled differential equations
=-a,
HBr(v=l)+Dz(u=O) G
The self-relaxation constants of processes ( 1) and (4) are already known. kHBrwas measured by laserinduced fluorescence [ 3,5 ] and k,, by the Raman excitation technique [ 61. The two rate constants k, and k,, are related by the principle of detailed balance, giving
P2
(u=O)
and [D2(u=1)]
1,
beingOatt=O.
D. Papadimitriou et al. / Vibrational relaxation ofHBr(v=
If we take the concentrations of HBr (v= 0) and D2 (v= 0) to be constant, the solution of the system is: [HBr(v=l)]=-12_A, + *
2-
1
,
[exp(-~,t)-exp(-~2t)l
,
+ k
+ km )xm
+ kwmr
bHBr
(10)
.
WI
is of the order of 200-500 s-’ Since kobS I + kobs2 Torr- * in our measurements and k,,, is of the order of 1 s-l Torr- ’ in our temperature range [ 61, k,, in eq. ( 10) might be neglected. Indeed, we changed eq. (10) to
Cab)
(kobs,+koi,s2)x& -kb,=(k,+&-b,)
P2(~=1)1 = &
(al +&J/P
= (kc + km-D;1 + (km
aI - AZ exp(-l,t)
exp( -1,t)
kobsl+k,bs2=
143
I) in Dz
+ ( knBr+ k,r + kuz-unr )xuBrx~: with decay constants 1, =(a, +b,)/2+
[(a, -bZ)2/4+a2bl]‘/2,
& = (a, +bz)/2-
[(a, -b2 )2/4+a2bl
(9a) ]‘I2 .
(9b)
Now partial pressures may be replaced by mole fractions in all the equations, leading to the observed decay constants k+,s,=R,/~,
kw=&/p,
being the total pressure. First the values of kobsland kobs2were determined from the experimental fluores-
p
cence curves. The experiments had been performed in the following way. In a storage bulb there was a mixture prepared from HBr and D2 with a certain mole fraction. After a temperature was set in the fluorescence cell, the cell was evacuated and freshly filled from the storage bulb up to pressure p. Then the fluorescence measurements were performed, a new temperature was set and the whole procedure repeated. In this way we took data at several temperatures with the same mixture. We tried to set the temperatures of the measurements close to the temperatures given in the tables. From all the fluorescence curves, kobs, and kobQ were determined and drawn as a function of temperature for each mole fraction. A smooth curve was drawn by hand through the data points. From this curve we read the values for kobsl and kobs2at the exact temperatures that are given in the tables. In addition we get a good feeling for the errors of our measurements from the scatter of the data points around the smooth curve. Addition of kobsland kobslgives:
taking k,, from the literature [ 6 1. A plot of the lefthand side as a function of XHBr/xbZ yields k+ kHBr_D2as intercept and kHBr+ k,, + kDz-HBras slope of the straight line. Since kHBriS Wd known from Selfrelaxation measurements [ 33 1, k,, + kDZ_HBr can be determined from the slope. These values are summarized in table 1. As will be shown in the following, kHBr_Dzcan be determined separately. Therefore k, is available from the intercept (see table 2 below). k,, is much greater than kD2_HBrso that kJ(k,,
+kD2_HBr)XexP(-~v/kT)
according to the principle of detailed balance, eq. (7). The experimental ratio k,/ ( ket+ kD1_HBr) is compared to exp ( - hAv/kT) in table 1 also. There is very good agreement if one takes into account that measurements were performed with very small mole fractions Xu& only. This confirms the reliability of our measurements and their evaluation. k, and kHBr_D1 can be separated in the following way [7]. Since V-V rates are much larger than V-T, R rates the square roots in eq. (9) may be expanded after taking a factor of a2 + b, in front of it, resulting in: km
(fast)
+
k
x e
+
‘T; Dz
+~,xHB~
x e’
(kHB&Br
+kiBr-DzXDz
)
HBr
kXD,
I(
x
= kxD,
k&D,
(kDp~Br&Br
+ k,
XHBr
+kDzXDz
>
) ,
(lla)
D. Pap~~mitr~~~ et al. / ~ibrat~~nfflrelaxation of HBrfu= 1) in D2
144
Table 1 Combined relaxation constants of HBr-D,
k+kner-m
k + b-n,
L
(s-l Torr-‘)
(s-’
290 250 220 190 170 160 150 140
495 * 64 406+ 53 334+43 255+38 205+41 180_+50 1601t45 150+45
4032k605 4866 + 729 585OrtS80 7420+ 1300 8930+2200 9900 r 3500 112~~~0 12900~50~
exp ( - hAv/kT)
k
Tow-‘)
k +koz-Har
(eq. (7))
0.113kO.023 0.076rtO.016 0.052z!zO.O11 0.031 zk0.008 0.020 rt:0.007 0.016 +O.OOS 0.012~0.~6 0.010~0.005
0.118 0.084 0.060 0.038 0.026 0.021 0.016 0.012
XHBr
+km&
1
.
+
+
XD,
I?’
-
+
.’
k&X&
1
-.-!-XHBrxD,
We took kHBrand kD, from the literature [3,6] and k,/k,, from eq. (7), drew the left-hand side of eq. ( 12) as a function of xH&xDz and fitted a straight line with slope kHBrthrough the points. The intercept gives the sum kHBr_D2 f (k/k,. )kDz-HBr.In this way we took into account all the error bounds of the single measurements simultaneously and estimated the error of our V-T, R relaxation rate, given in table 2. Only the sum kHBr_D2 + (k~/k~.)k~-~~~ can be extracted from our measurements reliably. But kDz_HBr is estimated to be smaller than kHBr-D2[ 41 in accord with the energetically similar system HF-Hz [ 8 ] and k/k,, c 0.1. Therefore, (k,/k,. )kD2_HBris neglected compared to kHBr-Dz.This may lead to an additions error of c 10%.
(lib)
As one can see, the slow decay constant kobSzis determined by the self-relaxation constant kHBr and the V-T, R constants kHBr-,,* and kD2_HBr.The V-V relaxation only enters in the ratio k,/k,. that is given by the principle of detailed balance. We have performed measurements with six different mole fractions ranging from xHB,=0.00375 to xH&=O.O978. In order to evaluate measurements with different mole fractions (but at the same temperature) simultaneously we rearranged eq. ( 11b) to
Table 2 V-T, R and V-V relaxation constants of HBr-D2 T
km-m + (S/kc k-mr
k
(K)
(s-’ Tom-‘)
(s-‘Tow’)
290 250 220 190 170 160 150 140
40+ 10 34+s 32+8 29?rs 25rtS 2327 Zi+S 20&S
455f65 370 f 54 300f45 230+40 180f40 160+50 14Of45 130+45
106Py-v(HBr-0,)
34 26 20 14 10 9 7 7
1). Papadimitriou et al. / Vibrational relaxation ofH&(o=
k, is easily obtained by subtracting kHBr_D2from k,+k,,,, (from eq. (10) ). Its accuracy is much better than that of kHBr_Dz,because it is si~i~~antly larger than kHBr-D1and therefore scarcely suffers from any error in kHBr_D2.kc calculated in this way is also
given in table 2. Now all rate constants for vibrational relaxation of HBr (v= 1) in mixtures with Dz are determined except kD2_HBr.Expressions which allow its determination can be derived also, but they contain small differences of large numbers. Therefore the determination of kDz_HBrwould suffer from extremely large errors and no data will be given here. Our room temperature data of k, and kHBr_,,*can be compared with literature data from Hopkins and Chen [4,10 1. They agree very well within their mutual error bounds. From rate constants k, relaxation probabilites Pv_v can be calculated in the usual way, taking into account gas kinetic collision numbers [ 12 ] : Pv_v=k,/2~a2(IFIMkT)“2.
In this formula all numbers, constants, and conversion factors can be collected to give Pv_v =2.29x
10-4keJMTl’a2,
where T is the temperature in IL, M is the reduced mass of the collision partners in g/mol, and k, is the rate constant in s-’ Torr- ‘. 0, the collision diameter, has to be taken in pm. It was calculated as CI= f(%,, + ab2 ) with ouBr= 34 1 pm and or,, = 294.8 pm
Temperature[K] ,
‘Of
“0 g
300280 260 240 liS,i
220
200 I
160 I
160
140
x7-
65_
1 t
if
*
I
0.15
0.16
0.17
0.16
0.18
0.2
1
T-'@[K-$)3]
Fig. 2. V-V energy transfer probability in DZ(v= 1)+HBr(v=O) collisions.
I) in D2
145 Tmpsrature[K]
300260260
240
220
200
160
160
Fig. 3. V-T, R energy transfer probability + D2( v= 0) collisions.
140
in HBr (u= I )
[ 111. Pv_” calculated in this way from k, is given in table 2. Relaxation probabilities for V-V (in the exothermic direction, calculated from Pv_” of table 2 with the help of eq. (7) ) and V-T, R energy transfer are shown in figs. 2 and 3.
4. Discussion
Most of the experimental data of V-V processes were elaborated at room tern~~t~e only. They show a pronounced dependence of the V-V exchange probability from the energy mismatch A.??[ 12,131, but this is not the only important parameter. Recently, it was shown [ 141 that HBr-N2 has a 100 times lower V-V relaxation rate than HBr-CO2 whereas dE is almost equal. Of course that may be due to different relaxation pathways in triatomic COz compared to diatomic N2. Furthermore, the HBr-N2 rate decreases with temperature whereas the HBr-0, rate increases [ 141, Measurements of the temperature dependence of rate constants therefore are crucial in order to test relaxation theories. Our V-V data are compared with HF-Hz [ 16,171, DF-D2 [l&21 ] and HCl-I& [ 18,19,22-241 data in fig. 4. In all cases exothermic
146
D. Papadimitriou et al. / Vibrational relaxation of HBr(v= 1) in Dz
Acknowledgement 40
: .
2
o H,-HF.
AE=ZOlcm-’
. D,-DF.
AE= .33cm-’
Financial support from Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References x
AE=104cm-’
D,-Ha,
[ 1] St.R. Leone, J. Phys. Chem. Ref. Data 11 (1982) 953. [ 21 B. Schramm and H. Rapp, Ber. Bunsenges. Physik. Chem.
I 1Dcl
200
300
400
500
600
700
800
Tmpmture
900
1000 1100
[K]
Fig. 4. V-V energy transfer probabilities. Comparison of literature data of different hydrogen-hydrogen halide systems.
V-V exchange probabilities were calculated from the rates given in the literature using gas kinetic collision diameters [ 111 a(H,) =296 pm, o(Dz) =295 pm, a( HF) = a( DF) = 280 pm (estimated from liquid density and from ref. [ 20]), a(HC1) = 330 pm, and a( HBr ) = 34 1 pm. As one can see, there is a very similar temperature dependence of the V-V relaxation probabilities, suggesting the same relaxation mechanism in all these hydrogen halide-hydrogen systems. Nevertheless, there is no scaling with AZ?. V-T, R exchange probabilities in HBr-D2 are compared to corresponding values in HBr-H, and HBr-HD elsewhere [ 3 1. All these systems show a very similar behaviour and PI_,,( HBr-Hz ) > P,_0 (HBrHD) > P,_,( HBr-D, ) as is expected from SSH theory. But there is no quantitative agreement with calculations using simple V-T or V-R theories. In order to clarify these discrepancies, measurements of the rotational energy distribution of the HBr vibrator and the hydrogen collision partner after the collision would be highly desirable.
84 (1980) 850. [3] 0. Losert, H.-V. Schwatz and B. Schramm, Ber. Bunsenges. Physik. Chem., to be published. [4] H.-L. Chen. J. Chem. Phys. 55 (1971) 5557. [5] P.F. Zittel and C.B. Moore, J. Chem. Phys. 59 (1973) 6636. [ 61 J. Lukasik and J. Ducuing, Chem. Phys. Letters 27 (1974) 203. [ 7 ] H.-L. Chen and C.B. Moore, J. Chem. Phys. 54 ( 1971) 4072. [8] J.F. Bott, J. Chem. Phys. 61 (1974) 2530. [9] J.R. Hancock and W.H. Green, J. Chem. Phys. 57 ( 1972) 4515. [lo] B.M. Hopkins and H.-L. Chen, J. Chem. Phys. 59 (1973) 1495. [ 111 J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). [ 121 J.D. Lambert, Vibrational and rotational relaxation in gases (Clarendon Press, Oxford, 1977). [ 131 J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980). [ 141 B. Seoudi, L. Doyennette, M. Margottin-Maclou and L. Henry, J. Chem. Phys. 72 (1980) 5687. [15] J.F. Bott, J. Chem. Phys. 74 (1981) 2827. [ 161 J.F. Bott and N. Cohen, J. Chem. Phys. 58 (1973) 4539. [ 171 J.F. Bott and R.F. Heidner III, J. Chem. Phys. 72 (1980) 3211. [ 18 ] B.M. Hopkins, H.-L. Chen and R.D. Sharma, J. Chem. Phys. 59 (1973) 5758. [19] J.F.BottandN.Cohen, J.Chem.Phys. 63 (1975) 1518. [20] S. Murad, K.A. Mansour and J.G. Powles, Chem. Phys. Letters 13 1 ( 1986) 98. [21] J.F. Bott, J. Chem. Phys. 60 (1974) 427. [22]H.-L.ChenandC.B.Moore,J.Chem.Phys.54(1971)4080. [23] J.-M. Allee, M. Margottin-Maclou, J. Menard and L.C. Doyennette, Roy. Acad. Sci. Ser. B 279 (1974) 305. [ 241 D. Papadimitriou and B. Schramm, to be published.
VIBRATIONAL RELAXATION OF HBr(v= 1) IN Dz BETWEEN 300 AND 140 K D. PAPADIMITRIOU, H.V. SCHWARZ and B. SCHRAMM Physikalisch-Chemisches Institut der Universitiit, Im Neuenheimer Feld 253, D-6900 Heidelberg, FRG
Received 4 December 1987
Vibrational relaxation of HBr( v= 1) has been studied in mixtures with Dz in the temperature range between 290 and 140 K with the help of laser-induced fluorescence. V-V rates and V-T, R rates could be determined separately.
1. Introduction An enormous amount of data concerning vibrational energy transfer of hydrogen halide molecules was gathered in the last 20 years and the fundamental principles of energy exchange seem to be well understood [ 11. However, a number of major questions still exist, e.g. the influence of attractive intermolecular potentials or the problem of rotational energy in V-T, R processes. In order to get some more detailed insight into these problems we have started a systematic investigation of low-temperature V-T, R relaxation rates of hydrogen halides in different hydrogen isotopes. In collisions of HBr( v= 1) with H2 or HD only V-T, R energy transfer is possible because of energetic reasons. In HBr ( v= 1) + D2 collisions V-V transfer can occur also and laser-induced fluorescence decay curves are complicated in this case. In this paper we want to present our experimental results of V-V and V-T, R relaxation in HBr (v= 1) + D2 mixtures down to temperatures of 140 K.
2. Experimental A laser-induced fluorescence method was used in order to measure vibrational relaxation rates. The experimental arrangement is described elsewhere [ 2,3] and only a brief summary is given here. A chemical HBr laser, tuned to the P ( 5) line of the v= 1+v= 0 transition with pulses of about 1 us half
width and 1 kW peak power, was used to excite HBr in a fluorescence cell that could be cooled down to 140 K. Fluorescence was collected with a lens vertical to the laser beam onto an InSb detector. Stray light could be blocked effectively with a bandpass filter that only transmits light from the R branch. Fluorescence decay curves were stored in a digital storage oscilloscope ( Vuko VKS 22- 16 ) and could be averaged with the help of a Commodore 8032 computer. The gas handling procedure and further experimental details are explained in ref. [ 3 1.
3. Results A typical fluorescence decay curve is shown in fig. 1. It can be interpreted as a sum of exponential decay curves Z=Z, exp( -n,t)
+Z2 exp( -n,t)
.
Separation of the two exponentials is easily done in the following way. First, the decay constant A2 and the intensity factor Z2are determined from the slowly decaying tail of the fluorescence curve. Then Z2 x exp( -&t ) is calculated for all times beginning with t = 0, which is set at the time of the laser pulse, and subtracted from the measured fluorescence curve. The result is a single exponential decay curve, I, X exp ( -1, t ) . In fig. 1 we have drawn the calculated term Zzexp ( -1 2t) and the full double exponential fit in addition to the measured fluorescence curve. Depending on temperature and mole fraction other
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
D. Papadimitriou et al. / Vibrational relaxation of HBr(v= 1) in Dz
142
This model assumes a fast V-V exchange between HBr and Dz until an equilibrium distribution is reached that is given by [ 41
60
50
P2(~=1)1
3 s f" c
(6)
xexp[--h(vD,-vHB,)/kTl
s -30 B : E
xD, =------XHBr
[HBr(u=l)]
(where square brackets on the left-hand side denote partial pressures and x is the mole fraction) and a subsequent slow V-T, R relaxation of HBr (u= 1). Collisions between two excited HBr molecules may be neglected since only a small fraction of HBr was excited by the laser pulse and since all our measurements were performed with high dilution of HBr in
20
10
I
20
60
I
100
I
I
140
#
I
180
I
I
220
I
260
I
I
300
I
I
340
I,,
360
Xrn.9[arbitrary units]
D2.
Fig. 1. Fluorescence decay curve of a mixture of HBr and Dz at temperature T= 184 K, pressure p= 93 Torr, and mole fraction xHBr= 0.00428. The fit of the slowly decaying part and the double exponential fit are drawn together with the experimental c&e.
fluorescence curves may show a more pronounced fast decay or a more pronounced slow decay. Therefore error bounds for the corresponding decay constants are very much dependent on the special shape of the cunres. In order to explain this result theoretically we take into account the following live relaxation channels: he’
HBr(v=l)+HBr(u=O)-
2HBr(u=O),
Al?=2559 cm-’ ;
(1)
HBr(u=l)+D,(v=O) IWer-o2 HBr(v=O)+D2(u=O), AE=2559 cm-’ ;
k,/ k,* = exP [ -h
(VD? - Z’HB~) lkT1
d[HBr(u=
l)]/dt [HBr(u= 11+b,
P2(u=l)l,
d[D,(u=l)]/dt (2)
=a,[HBr(u=l)] with
HBr(v=O)+D,(v=l),
AIT= -431 cm-’ ;
Dz(v=l)+D(v=O)M=2990
(3) kD,
cm-’ ;
2D2 (v=O) ,
Dz(v=l)+HBr(v=O) kD-HB, -‘D,(v=O)+HBr(u=O), AE=2990
cm-’ .
+kHBr[HBr(u=O) I , b, =k,. [HBr(u=O)]
(4)
,
az=kP2(u=O)l,
b2 = (k,, +kD2-HBr) [HBr(u=O)] +kD,
(5)
.
(7)
Therefore only three rate constants, k,, kHBr_Dzand &-HBr , remain to be determined from our experiment. In order to do this we have to solve the relaxation kinetics. Processes ( 1 )- ( 5 ) lead to a system of two coupled differential equations
=-a,
HBr(v=l)+Dz(u=O) G
The self-relaxation constants of processes ( 1) and (4) are already known. kHBrwas measured by laserinduced fluorescence [ 3,5 ] and k,, by the Raman excitation technique [ 61. The two rate constants k, and k,, are related by the principle of detailed balance, giving
P2
(u=O)
and [D2(u=1)]
1,
beingOatt=O.
D. Papadimitriou et al. / Vibrational relaxation ofHBr(v=
If we take the concentrations of HBr (v= 0) and D2 (v= 0) to be constant, the solution of the system is: [HBr(v=l)]=-12_A, + *
2-
1
,
[exp(-~,t)-exp(-~2t)l
,
+ k
+ km )xm
+ kwmr
bHBr
(10)
.
WI
is of the order of 200-500 s-’ Since kobS I + kobs2 Torr- * in our measurements and k,,, is of the order of 1 s-l Torr- ’ in our temperature range [ 61, k,, in eq. ( 10) might be neglected. Indeed, we changed eq. (10) to
Cab)
(kobs,+koi,s2)x& -kb,=(k,+&-b,)
P2(~=1)1 = &
(al +&J/P
= (kc + km-D;1 + (km
aI - AZ exp(-l,t)
exp( -1,t)
kobsl+k,bs2=
143
I) in Dz
+ ( knBr+ k,r + kuz-unr )xuBrx~: with decay constants 1, =(a, +b,)/2+
[(a, -bZ)2/4+a2bl]‘/2,
& = (a, +bz)/2-
[(a, -b2 )2/4+a2bl
(9a) ]‘I2 .
(9b)
Now partial pressures may be replaced by mole fractions in all the equations, leading to the observed decay constants k+,s,=R,/~,
kw=&/p,
being the total pressure. First the values of kobsland kobs2were determined from the experimental fluores-
p
cence curves. The experiments had been performed in the following way. In a storage bulb there was a mixture prepared from HBr and D2 with a certain mole fraction. After a temperature was set in the fluorescence cell, the cell was evacuated and freshly filled from the storage bulb up to pressure p. Then the fluorescence measurements were performed, a new temperature was set and the whole procedure repeated. In this way we took data at several temperatures with the same mixture. We tried to set the temperatures of the measurements close to the temperatures given in the tables. From all the fluorescence curves, kobs, and kobQ were determined and drawn as a function of temperature for each mole fraction. A smooth curve was drawn by hand through the data points. From this curve we read the values for kobsl and kobs2at the exact temperatures that are given in the tables. In addition we get a good feeling for the errors of our measurements from the scatter of the data points around the smooth curve. Addition of kobsland kobslgives:
taking k,, from the literature [ 6 1. A plot of the lefthand side as a function of XHBr/xbZ yields k+ kHBr_D2as intercept and kHBr+ k,, + kDz-HBras slope of the straight line. Since kHBriS Wd known from Selfrelaxation measurements [ 33 1, k,, + kDZ_HBr can be determined from the slope. These values are summarized in table 1. As will be shown in the following, kHBr_Dzcan be determined separately. Therefore k, is available from the intercept (see table 2 below). k,, is much greater than kD2_HBrso that kJ(k,,
+kD2_HBr)XexP(-~v/kT)
according to the principle of detailed balance, eq. (7). The experimental ratio k,/ ( ket+ kD1_HBr) is compared to exp ( - hAv/kT) in table 1 also. There is very good agreement if one takes into account that measurements were performed with very small mole fractions Xu& only. This confirms the reliability of our measurements and their evaluation. k, and kHBr_D1 can be separated in the following way [7]. Since V-V rates are much larger than V-T, R rates the square roots in eq. (9) may be expanded after taking a factor of a2 + b, in front of it, resulting in: km
(fast)
+
k
x e
+
‘T; Dz
+~,xHB~
x e’
(kHB&Br
+kiBr-DzXDz
)
HBr
kXD,
I(
x
= kxD,
k&D,
(kDp~Br&Br
+ k,
XHBr
+kDzXDz
>
) ,
(lla)
D. Pap~~mitr~~~ et al. / ~ibrat~~nfflrelaxation of HBrfu= 1) in D2
144
Table 1 Combined relaxation constants of HBr-D,
k+kner-m
k + b-n,
L
(s-l Torr-‘)
(s-’
290 250 220 190 170 160 150 140
495 * 64 406+ 53 334+43 255+38 205+41 180_+50 1601t45 150+45
4032k605 4866 + 729 585OrtS80 7420+ 1300 8930+2200 9900 r 3500 112~~~0 12900~50~
exp ( - hAv/kT)
k
Tow-‘)
k +koz-Har
(eq. (7))
0.113kO.023 0.076rtO.016 0.052z!zO.O11 0.031 zk0.008 0.020 rt:0.007 0.016 +O.OOS 0.012~0.~6 0.010~0.005
0.118 0.084 0.060 0.038 0.026 0.021 0.016 0.012
XHBr
+km&
1
.
+
+
XD,
I?’
-
+
.’
k&X&
1
-.-!-XHBrxD,
We took kHBrand kD, from the literature [3,6] and k,/k,, from eq. (7), drew the left-hand side of eq. ( 12) as a function of xH&xDz and fitted a straight line with slope kHBrthrough the points. The intercept gives the sum kHBr_D2 f (k/k,. )kDz-HBr.In this way we took into account all the error bounds of the single measurements simultaneously and estimated the error of our V-T, R relaxation rate, given in table 2. Only the sum kHBr_D2 + (k~/k~.)k~-~~~ can be extracted from our measurements reliably. But kDz_HBr is estimated to be smaller than kHBr-D2[ 41 in accord with the energetically similar system HF-Hz [ 8 ] and k/k,, c 0.1. Therefore, (k,/k,. )kD2_HBris neglected compared to kHBr-Dz.This may lead to an additions error of c 10%.
(lib)
As one can see, the slow decay constant kobSzis determined by the self-relaxation constant kHBr and the V-T, R constants kHBr-,,* and kD2_HBr.The V-V relaxation only enters in the ratio k,/k,. that is given by the principle of detailed balance. We have performed measurements with six different mole fractions ranging from xHB,=0.00375 to xH&=O.O978. In order to evaluate measurements with different mole fractions (but at the same temperature) simultaneously we rearranged eq. ( 11b) to
Table 2 V-T, R and V-V relaxation constants of HBr-D2 T
km-m + (S/kc k-mr
k
(K)
(s-’ Tom-‘)
(s-‘Tow’)
290 250 220 190 170 160 150 140
40+ 10 34+s 32+8 29?rs 25rtS 2327 Zi+S 20&S
455f65 370 f 54 300f45 230+40 180f40 160+50 14Of45 130+45
106Py-v(HBr-0,)
34 26 20 14 10 9 7 7
1). Papadimitriou et al. / Vibrational relaxation ofH&(o=
k, is easily obtained by subtracting kHBr_D2from k,+k,,,, (from eq. (10) ). Its accuracy is much better than that of kHBr_Dz,because it is si~i~~antly larger than kHBr-D1and therefore scarcely suffers from any error in kHBr_D2.kc calculated in this way is also
given in table 2. Now all rate constants for vibrational relaxation of HBr (v= 1) in mixtures with Dz are determined except kD2_HBr.Expressions which allow its determination can be derived also, but they contain small differences of large numbers. Therefore the determination of kDz_HBrwould suffer from extremely large errors and no data will be given here. Our room temperature data of k, and kHBr_,,*can be compared with literature data from Hopkins and Chen [4,10 1. They agree very well within their mutual error bounds. From rate constants k, relaxation probabilites Pv_v can be calculated in the usual way, taking into account gas kinetic collision numbers [ 12 ] : Pv_v=k,/2~a2(IFIMkT)“2.
In this formula all numbers, constants, and conversion factors can be collected to give Pv_v =2.29x
10-4keJMTl’a2,
where T is the temperature in IL, M is the reduced mass of the collision partners in g/mol, and k, is the rate constant in s-’ Torr- ‘. 0, the collision diameter, has to be taken in pm. It was calculated as CI= f(%,, + ab2 ) with ouBr= 34 1 pm and or,, = 294.8 pm
Temperature[K] ,
‘Of
“0 g
300280 260 240 liS,i
220
200 I
160 I
160
140
x7-
65_
1 t
if
*
I
0.15
0.16
0.17
0.16
0.18
0.2
1
T-'@[K-$)3]
Fig. 2. V-V energy transfer probability in DZ(v= 1)+HBr(v=O) collisions.
I) in D2
145 Tmpsrature[K]
300260260
240
220
200
160
160
Fig. 3. V-T, R energy transfer probability + D2( v= 0) collisions.
140
in HBr (u= I )
[ 111. Pv_” calculated in this way from k, is given in table 2. Relaxation probabilities for V-V (in the exothermic direction, calculated from Pv_” of table 2 with the help of eq. (7) ) and V-T, R energy transfer are shown in figs. 2 and 3.
4. Discussion
Most of the experimental data of V-V processes were elaborated at room tern~~t~e only. They show a pronounced dependence of the V-V exchange probability from the energy mismatch A.??[ 12,131, but this is not the only important parameter. Recently, it was shown [ 141 that HBr-N2 has a 100 times lower V-V relaxation rate than HBr-CO2 whereas dE is almost equal. Of course that may be due to different relaxation pathways in triatomic COz compared to diatomic N2. Furthermore, the HBr-N2 rate decreases with temperature whereas the HBr-0, rate increases [ 141, Measurements of the temperature dependence of rate constants therefore are crucial in order to test relaxation theories. Our V-V data are compared with HF-Hz [ 16,171, DF-D2 [l&21 ] and HCl-I& [ 18,19,22-241 data in fig. 4. In all cases exothermic
146
D. Papadimitriou et al. / Vibrational relaxation of HBr(v= 1) in Dz
Acknowledgement 40
: .
2
o H,-HF.
AE=ZOlcm-’
. D,-DF.
AE= .33cm-’
Financial support from Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References x
AE=104cm-’
D,-Ha,
[ 1] St.R. Leone, J. Phys. Chem. Ref. Data 11 (1982) 953. [ 21 B. Schramm and H. Rapp, Ber. Bunsenges. Physik. Chem.
I 1Dcl
200
300
400
500
600
700
800
Tmpmture
900
1000 1100
[K]
Fig. 4. V-V energy transfer probabilities. Comparison of literature data of different hydrogen-hydrogen halide systems.
V-V exchange probabilities were calculated from the rates given in the literature using gas kinetic collision diameters [ 111 a(H,) =296 pm, o(Dz) =295 pm, a( HF) = a( DF) = 280 pm (estimated from liquid density and from ref. [ 20]), a(HC1) = 330 pm, and a( HBr ) = 34 1 pm. As one can see, there is a very similar temperature dependence of the V-V relaxation probabilities, suggesting the same relaxation mechanism in all these hydrogen halide-hydrogen systems. Nevertheless, there is no scaling with AZ?. V-T, R exchange probabilities in HBr-D2 are compared to corresponding values in HBr-H, and HBr-HD elsewhere [ 3 1. All these systems show a very similar behaviour and PI_,,( HBr-Hz ) > P,_0 (HBrHD) > P,_,( HBr-D, ) as is expected from SSH theory. But there is no quantitative agreement with calculations using simple V-T or V-R theories. In order to clarify these discrepancies, measurements of the rotational energy distribution of the HBr vibrator and the hydrogen collision partner after the collision would be highly desirable.
84 (1980) 850. [3] 0. Losert, H.-V. Schwatz and B. Schramm, Ber. Bunsenges. Physik. Chem., to be published. [4] H.-L. Chen. J. Chem. Phys. 55 (1971) 5557. [5] P.F. Zittel and C.B. Moore, J. Chem. Phys. 59 (1973) 6636. [ 61 J. Lukasik and J. Ducuing, Chem. Phys. Letters 27 (1974) 203. [ 7 ] H.-L. Chen and C.B. Moore, J. Chem. Phys. 54 ( 1971) 4072. [8] J.F. Bott, J. Chem. Phys. 61 (1974) 2530. [9] J.R. Hancock and W.H. Green, J. Chem. Phys. 57 ( 1972) 4515. [lo] B.M. Hopkins and H.-L. Chen, J. Chem. Phys. 59 (1973) 1495. [ 111 J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). [ 121 J.D. Lambert, Vibrational and rotational relaxation in gases (Clarendon Press, Oxford, 1977). [ 131 J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980). [ 141 B. Seoudi, L. Doyennette, M. Margottin-Maclou and L. Henry, J. Chem. Phys. 72 (1980) 5687. [15] J.F. Bott, J. Chem. Phys. 74 (1981) 2827. [ 161 J.F. Bott and N. Cohen, J. Chem. Phys. 58 (1973) 4539. [ 171 J.F. Bott and R.F. Heidner III, J. Chem. Phys. 72 (1980) 3211. [ 18 ] B.M. Hopkins, H.-L. Chen and R.D. Sharma, J. Chem. Phys. 59 (1973) 5758. [19] J.F.BottandN.Cohen, J.Chem.Phys. 63 (1975) 1518. [20] S. Murad, K.A. Mansour and J.G. Powles, Chem. Phys. Letters 13 1 ( 1986) 98. [21] J.F. Bott, J. Chem. Phys. 60 (1974) 427. [22]H.-L.ChenandC.B.Moore,J.Chem.Phys.54(1971)4080. [23] J.-M. Allee, M. Margottin-Maclou, J. Menard and L.C. Doyennette, Roy. Acad. Sci. Ser. B 279 (1974) 305. [ 241 D. Papadimitriou and B. Schramm, to be published.