KARLF. HERZFELD A N D VIRGINIAGRIFFING
844
TABLE I1 SUMMARY OF EXPERIMENTAL RELAXATION TIMES Low pressure
High pressure
5 1
Composition Run no.
237 240 252 277 275 284 257
COz(dry) COz(dry) COz(dry) COz(moist) COz HZ N20 CZHB
+
A
+ (4.05.3 p )
1.7 1.2 0.7 .8 .2 .9
..
(#Bid.)
B S (2.4- (74 . 0 ~ )17 P )
1.5 3.0 1.7 1.4 1.4
(psk’c.)
A B S (4.- (2.4(75.3 p 4 . 0 ~ 17 ~1
1 . 4 7.7 2.1 6 . 6 0 . 8 7.2 1.3 7.0 0.9 5.7 . . .7 5.8 1 . 5 1.3 13.5
9.1 8.7 9.2 6.6 6.8
8.2 7.4 8.3 6.8 6.3 . . 7.0 9.9 10.3
Ref.
(21d) COz(condition?) 4 . 0 7 . 0 1 . 6 (21a,b) COz(very dry) 6.0
For both the bending and the asymmetric stretching bands of our sample of carbon dioxide T~ is between 1.0 and 1.5 psec. while for the 2.7 p combination bands r1 is about 2.1 psec. There are no indications of values of T~ for carbon dioxide greater than about second, although theory for the parallel de-excitation mechanism predicts much longer relaxation times for the more energetic vibrational modes. This argues strongly for a sequential mechanism (as per eq. 9), and suggests that the measured pressure dependence for TOO^
1701.
61
may not be strictly reciprocal. The low pressure relaxation time for the asymmetric stretching vibration of carbon dioxide with added hydrogen is decreased considerably beyond experimental error, suggesting that hydrogen selectively de-excites the asymmetric stretching vibration in carbon dioxide. The low pressure relaxation times of the moist carbon dioxide sample do not differ markedly from those of the dry samples. Within experimental error, ethane and nitrous oxide have each yielded only one relaxation time. The average value of T I for nitrous oxide, 0.8 ysec, compares well with the value 1.0 psec. reported by Eucken and Numann.I2 Although residual instrumental effects cannot be ruled out, the observed T~ for ethane, 1.3 psec., may correspond to a relaxation process not previously observed. Acknowledgments.-The authors thank sincerely Professors True McLean and Henry S. McGaughan (Electrical Engineering, Cornell University) for their extensive aid in designing the electronic circuitry. We also acknowledge helpful discussions with Dr. D. C. Hoesterey (Bell Telephone Labs.) regarding microphone characteristics. A grant from the Research Corporation permitted us to purchase many of the basic units. One of us (M.E.J.) is indebted to the General Electric Company and du Pont for fellowships. This work was also supported in part by the ONR.
ULTRASONICS AND THE TRANSFER OF ENERGY BY COLLISIONS BY KARL F. HERZFELD AND VIRGINIAGRIFFING Departments of Physics and Chemistry, Catholic University of America, Washington, D. C. Received January $1, 1967
The theoretical work on the connection between energy transfer in collision and ultrasonics started due to the interest of F. 0. Rice in gas kinetics. The sound velocity in an ideal gas depends on the specific heat, and if the transfer of energy to vibrational degrees of freedom is too slow, the effective specific heat in an ultrasonic wave is smaller than the static one, so that dispersion and absorption result. The relaxation times in gases a t room temperature and I atm. vary from 3 X 10-8 second for oxygen to 0.7 X 10-8 second for hexane, and decrease with increasing temperature. It has been possible to develop a quantum theory for the calculation of these times for simpler molecules, the results of which agree fairly well with experiment. However, even in the most rapid cases it takes 50-100 collisions to remove a quantum from the lowest excited vibrational state, which seems 1/5 to 1/10 as fast as the results of reaction rate measurements indicate.
1. Introduction I n 1927 when F. 0. Rice and K. F. H. were both at Johns Hopkins University, F. 0. Rice had been interested for several years in the mechanism by which the energy necessary for monomolecular endothermic reactions is transferred, and in the rate with which this happens. Simultaneously there was interest at Hopkins in ultrasonic waves, on the part of R. W. Wood and J. C. Hubbard. F. 0. Rice then suggested that ultrasonic waves might provide a means to measure this rate of energy transfer. Out of this remark grew the joint paper in which this was first put into definite, albeit primitive form. I n this report we will discuss in sequence: How does ultrasonics allow a determination of the rate of energy transfer?; the results of such measurements; the status of the quantum theoretical (11 K. F. Herzfeld and F, 0. Rice, Phya. Rev,, 31, 691 (1928).
calculation of relaxation time; the relation of these results to reaction kinetics. We will restrict ourselves to ideal gases. 2. Ultrasonics and Intermolecular Energy Transfer When a gas is compressed adiabatically, the pressure increases not only because the density increases, but also because the work done in compressing the gas becomes thermal energy and therefore the temperature goes up. By how much the temperature goes up depends on the size of the reservoir into which the thermal energy can go, ie., on the specific heat. The larger the specific heat, the smaller the temperature increase and the lower the pressure increase. The sound velocity depends on the rigidity or, more precisely,, the ratio of pressure increase to density increase. There follows the well known
formula
. I
ULTRASONICS AND TRANSFER OF ENERGY BY COLLISIONS
July, 1957
which, for an ideal gas, can be written v2=
@ +RE )RT
The qualitative discussion given above concerns the term R/C,. If C, is large compared to R, the gas behaves as if the compression were isothermal. If one compresses a gas slowly, the increase in thermal energy appears,directly in the translational degrees of freedom (increase in velocity of the molecules reflected from a piston moving to meet them). Only after the increase of the translational energy can energy leak from these into the internal degrees of freedom, to re-establish equilibrium between translational and internal degrees of freedom, In ideal gases such an exchange can only occur during collisions (we restrict ourselves to uncharged molecules). Assume now that the establishment of equilibrium between internal and external degrees of freedom has a relaxation time r. If the change in internal temperature is slow compared to r , there is time to establish an equilibrium, and the whole static C, is effective. If, however, the rate of temperature change is increased, in particular if in a periodic change-as in a sound wave-the frequency is increased, there is less and less time for the energy to leak from the external into the internal degrees of freedom and, therefore, the effective specific heat decreases. One finds (C")df
=
i w7
c,o - C' 1 + iw7
(3)
where C,o is the static specific heat, and C' the internal specific heat with relaxation time r. The calculation of the frequency dependence of V is now a matter of algebra, combining (2) and (3). Since the energy transfer can occur only during collision, and since the time between collisions is proportional to l / p , while what happens during one collision is independent of the number of collisions, it follows that r is proportional to l / p ; the ratio 2, introduced in Dr. Bauer's talk 7/70011
(4)
is independent of the pressure. 2 is roughly the number of collisions necessary to decrease the deviation from equilibrium by a factor l/e. As a consequence of the preceding, the sound velocity depends on frequency and pressure only in the combination u / p (as long as the gas remains ideal). This is experimentally important, in so far as it is easier to get a continuous range of pressures than of frequencies. We next discuss C'. On principle the internal degrees of freedom are those of rotation and vibration. It is found that the former have in general a much shorter relaxation time than the latter, Except for Hz, HD, DZand OH it takes only a few collisions to achieve equilibrium between rotation and translation (see the following paper of D. Hornig). Accordingly, we will assume that C' contains only vibrational degrees of freedom while the rota-
-
+
451 KC
o
92 Kc 9 Kc
I
='-0
I.o
845
I
I
I
2.0
4.0
3.0 log,,(f/p).
Fig. 1.-Dispersion of sound in CSZ according to Richards and Reid. The theoretical limiting velocities are indicated. The full line is the theoretical curve according to single relaxation theory.
tion is always in equilibrium with translation. Of course in the range of larger w / p values than those of importance in vibrational relaxation, this would not be true any more. Therefore we assume for linear molecules C , , = C,o
- C'
for non-linear molecules C,,
= CvO
=
R
- C' = 3R
Figure 1 shows an older measurement (by Richards and Reid2) of dispersion in CS2. If one assumes that the whole vibrational specific heat is involved in the relaxation, the limiting velocities a t low and high frequency can be calculated a priori and are plotted in the figure. The good agreement is a strong argument for the theory. In this case the only unknown quantity to be determined is the relaxation time. This is related to the frequency f " of the mid point by 7 = - -1 G O (f")-' (5) 27r
co, -
C'
Figure 2 shows absorption measurements3 for
cs2.
Since from (2) and (3) the velocity is a complex quantity, this implies absorption. Physically, this results from a phase difference between pressure and density. Figure 2 shows the absorption per wave length as function of u / p . The fully drawn curve is again that theoretically predicted. Figure 3 gives4 the dispersion in a more complicated molecule, CCL. The limiting velocities are again indicated, as calculated a priori from specific heats, and the good agreement at small and large values of w / p is a proof for the correctness of the assumed mechanism. (2) W. T. Richards and J. A. Reid, J . Chem. Phys., 2, 193 (1936).
(3) F. A. Angona, J . Acousl. Soc. Am., %a, 1116 (1953). (4) A. Busala, D. Sette and J. C. Hubbard, J . Chem. Phys., 23, 787 (10.55).
846
F. HERZFELD AND VIRGINIAGRIFFING
KARL
Vol. 61 TABLE I1 Clr
T , "C.
15 (7) 34000 180000
Author Ze,, Ztheor
Clz-He
727
1100
(8)
(8)
20 (7)
550 2100
200 780
1800
Nr0
T, "C. Author Z,,,
"ool
Zthoor
40
Substance
60 80 100 Ilp
ZOO 400 [Kclofrn).
I
€00
1000
2000
Fig. 2.-Absorption of sound per wave length in CS, according to Angona. The full line represents the theoretical curve according to single relaxation theory.
"C. Rel. time in sec.
z
Author Substance
0 "C. Rel. time in 10-8 sec.
I
142
z '
I
I
J
I, I
r
134
I
i
' " - - + I "
I
V,= 134.84 two 23.3
,
Fig. 3.-Dispersion of sound in CCla according t,o Busala, Sette and Hubbard. The theoretical limiting velocities are indicated. The full line is the theoretical curve according to single relaxation theory.
The vibrational specific heat of 0 2 a t room temperature is only 0.031R; with special methods5 it is possible to measure the dispersion even here, although y increases only from 1.3952 to 1.4000. 3. Results Tables I and I1 show 2 as function of temperature for 02,C12 and NzO although for the higher temperatures in O2 and Clz the results have been found not by ultrasonic measurements, but with the shock tube. TABLE I 0 2
T,"C. Author ZeXP
Zthear
200 (9)
400 (9) 2100 520
3300 1500
cos
CSI
25
25 (3) 8000 1800
(10)
9600 8800
TABLE I11
,150
20
0 (9) 8400 11000
DO0
15 (5) 2 . 1 X lo' 8 9 X lo7
1100 (6) 3GOOO
24000
lG80 (6) 12000 7100
2640 (6) 1800 2100
Tables I11 and IV show relaxation times and 2 values for a number of organic vapors a t room temperature. One notes that 2 is spread over a wide range, from 60 to 2 X 10'. There are not many other physical coiistants of matter with such a wide range (elec(5) H. and L. Knoetzel, Ann. Physik, 2 , 393 (1948).
Author Substance "C. Rel. time in sec. Author
CZHS C3H8 C6H14 Ethane %-Propane n-Butane %-Hexane 20 20 20 35 0.3-0.5 50-30 (11, 12)
nk and the rate is nkN2
STUDIES I N NON-EQUILIBRIUM RATE PROCESSES. 11. THE RELAXATION OF VIBRATIONAL NON-EQUILIBRIUM DISTRIBUTIONS IN CHEMICAL REACTIONS AND SHOCK WAVES1,' BY KURTE. SHULER Contributionfrom the National Bureau of Standards, Vashington, D. C. Received J a n u a r y 81, 1967
In some recent papers with R. J. Rubin and E. W. PvIontroll a theoretical treatment has been developed for the radiative and collisional relaxation of a system of harmonic oscillators prepared initially in a vibrational non-equilibrium distribution. In this paper a discussion is given of the application of these theoretical studies to experimental results on relaxation processes in chemical kinetics and in shock waves. It is pointed out that the analysis of relaxation data in terms of half-iives loses its meaning when more than two energy levels are involved in the relaxation process. To obtain information on the efficiency of intermolecular energy transfer in these multilevel systems it is necessary to follow in detail the time behavior of the population in several, and preferably all of the populated energy levels. As an example, the experimental data on the is discussed in detail. relaxation of vibrationally excited 0 2 produced by a secondary reaction in the flash photolysis of C l 0 ~ A short discussion is given of the analysis of shock wave data on vibrational relaxation. The concept and the prescription of the "vibrational temperature" introduced by Bethe and Teller in this analysis are shown to be valid from the exact solution of the relaxation equations.
1. Introduction I n the preceding paper of this series3 and in previous papers with R. J. Rubin4 we have discussed the radiative and collisional relaxation of a system of harmonic oscillators contained in a constant temperature heat-bath and prepared initially in a vibrational non-equilibrium distribution. I n the paper with M ~ n t r o l l the , ~ exact solution for the relaxation equation has been given and expressions have been derived for the relaxation of initial Boltzmann distributions, Poisson (distributions and &function distributions as well as for the relaxation of the moments of the distributions, Using the latter results, explicit formulas were derived for the relaxation of the internal encergy of the system of oscillators and for the time (dependence of the dispersion of the distributions. ;Since the collisional transitlion probability, Le., ithe probability per collision (or per unit time) that a n oscillator will exchange its vibrational energy with the translational energy of the heat-bath molecules, is a parameter of the relaxation equation it is evident that an analysis of the relaxation process will yield data on the efficiency of the intermolecular vibrational/translational energy transfer. Conversely, a knowledge of this efficiency permits the calculation of the time scale of relaxation. This can be of use in planning for the experimental techniques to be employed in the study of the relaxation process. I n this paper the results derived previously will 1 e applied to a study of the relaxation of vibrational non-equilibrium distributions in chemical (1) This work was supported by the U. S. Atomic Energy Commission. (2) Presented a t the Symposium on Intermolecular Energy T r a d e r , American Chemical Society, Atlantic City, Sept. 18, 1956. (3) E. W.Montroll and K. E. Shuler, J . Chem. Phya., 26,454 (1957). (4) R. J. Rubin and K. E. Shuler, ibid., 26, 59, 68 (1956); 26, 137 (1957).
reactions and shock waves. Recent experimental work in chemical kinetics has shown quite definitely that the products of various exothermic chemical reactions are formed in a non-Boltzmann vibrational distribution. Some examples of these types of specific vibrational excitations of product species are furnished by the reaction H 03 +OH* + 02 (1) where the OH is apparently formed predominantly in the 9th vibrational level of the 21Ti ground electronic state,6the reactions
+
0
0
+ NO2 +NO + c10 +
+ ClOZ
----f
02*
02*
(11 A) (11 B)
where the 02,in the 32-gground electronic state, has been observed with appreciable concentration up to the eighth vibrational level6 and the reaction C+C+M+C2*+M
(111)
discussed by Herzberg7 where the Czappears to be formed in the 6th vibrational level of the electronically excited TIg state.* After a general discussion of the ana,lysis of relaxation data for multilevel systems in Section 2, the data of Lipscomb, Norrish and Thrush6 on the relaxation of vibrationally excited O2 produced in reactions I1 will be considered in some detail in Section 3. The efficiency of intermolecular energy transfer, in particular that between the vibrational and translational degrees of freedom, can also be determined from an analysis of vibrational relaxation be(5) J. D. McKinley, D. Carvin and M. J. Boudert, ibid., 23, 784 (1955). (6) F. J. Lipscomb, R. G . W. Norrish and B. A. Thrush, Proc. R o y . Soe. (London),233A, 455 (195G). (7) G. Hersberg, Astroph. J., 89, 290 (1939). (8) I t is outside the scope of this paper to discuss the mechanism
and the energetics of the specific vibrational excitation of the product molecules in these reactions. This problem will be considered in a later communication.
