THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in
VOLIJME 61
U. 8.Patent Office)
(0 Copyright,
1957, by the Americsn Chemical Society)
JULY 30, 1957
NUMBER 7
COLLISIONAL ENERGY EXCHANGE I N GASES.l USE OF THE SPEETROPHONE FOR STUDYING RELAXATION PROCESSES IN CARBON DIOXIDE BY MARILYN E. JACOX AND S. H. BAUER Contribution from the Department of Chemistry of Cornell University, Ilhaca, New York Received January $1, 1967
T o start the discussion, we shall call attention to the significant role of tho interaction potential for a colliding molecule pair in affecting translatiori-vibration ( h v ) energy transfer. This will be followed by a rough summary of the selection rules for excitation which are believed to be operating during the transitions induced by relatively gentle collisions and by an enumeration of the specifically chemical problems which arise from the existence of t-a relaxation times. The problems which the theories which have been developed for handling inter-molecular mechanics have not yet solved will be qualitatively analyzed; as an example, we shall discuss the conditions when one should or should not expect several distinct relaxation times for polyatomic molecules. A brief list of the various methods for measuring relaxation times will be given, and finally we shall describe the spectrophone and demonstrate that it is a useful device for measuring relaxation times for specific v-t energy transfers.
The Role of the Interaction Potential.-In contrast with the well formulated and concise methods of intramolecular mechanics, satisfactory procedurefi for handling those problems in intermolecular mechanics which have chemial content have yet to be developed. Differences between the behavior of macroscopic models, molecular systems which obey classical mechanics, and colliding molecules as described by quantum mechanics are significant2J ; regrettably, the desired q.m. solutions are very difficult to obtain. These differences are illustrated by three models for a collision: in (cy) the atoms are represented by hard spheres with specified masses and diameters, and t,he bonds by springs. The forces between the “atoms” in the molecule under observation and those of the incident molecule are very short range, characteristic of a classical, macroscopic view. In (p) the atoms are represented realistically as kernels wrrounded by electron clouds which provide short range atom t o atom bonding within a molecule, and relatively long range repulsive forces between a colliding pair. Intermolecular motions (1) Presented a t the Symposium on “Intermolecular Energy Transfer,“ A.C.S. Meeting, Atlantic City, N. J., September 18, 1956. (2) IC. F. Herzfeld, Section H (p. 646 ff.) in “Thermodynamics and Physics of Matter,” ed. F. D. Rossini, Princeton University Press, Princeton, N. J., 1955. (3) E. Bauer, J . Chem. Phys., 23, 1087 (1955).
and most interatomic motions are treated classically, but one grafts on quantum restrictions on the energy levels permitted for internal states. If one further relaxes the simple harmonic formulation for atomic vibrations and permits weak anharmonicity, he can permit vibrational energy to flow internally between “almost normal” modes. Model (y) requires a strict q.m. description, which not only leads to rather complicated intermolecular potentials and fuzzy collision orbits, but insists that except very near dissociation limits molecules do occupy stationary energy states. Thus, the normal coordinate description of intramolecular motions only approximately locates the levels, but a molecule remains in a specified state until some external perturbation induces a transition to another. This non-adiabatic feature in radiationless transfer of energy (as induced by collisions) introduces considerable complications. -c
The interaction potential ( V as a function of R, the vector separation of their centers of mass) for the extreme model (a)is shown in Fig. 1. When such a collision occurs it is clear that the atom in the molecule which is hit by the incoming member experiences the impact over a time so short that the other atoms do not know that a collision has occurred. Consequently, the collisional energy is
*
MARILYN E. JACOXAND S. H. BAUER
834
Model (CY)
Q
trr
Y
>
1
s
.
,
interaction
I
,\
/+--
strong interaction
of interaction potentials (for specified 5:). Re, separation at minimum of potential energy function; D,depth of well [ = AHO); t:, collective parameter for all internal variables of both molecules; Sl‘, range over which interaction potential is effective. Fig. 1.-Types
Ro, separation a t turn-around point;
directly transferred to the spring. This is the prerequisite for a very high efficiency for energy transfer. Model (0)permits the other extreme-a very gradual potential; as the colliding molecules approach and recede there is continual adjustment of the entire system to the potential generated by each member and the process may be thought of as ‘being completely adiabatic. There is no energy exchange during such a collision because the duration of the collision is much greater than the period of vibration. I n real molecules one assumes a potent(ia1intermediate between the very hard and the very soft extremes. Consider then one collision, as described by the usual variables, E (measures the relative kinetic energy), b (the collision parameterspecifies the initial direction of approach of the colliding molecules). The detailed shape of the potential may be unique for the molecular species involved; it is thus that a chemical dependence on energy transfer is introduced. From Fig. 1 it is clear that the deeper the potential well for attraction the smaller will be their separation (Ro) at the turn-around point and the larger will be
[dV(%;t)/dK]~,. These two parameters essentially determine the inherent transition probability for excitation. the smaller Ro the more intensely disturbed will be the electron distributions around the molecules, and the steeper the potential function the more rapidly will the fields change as a result of the collision. This ca.n be considered as the high frequency components of the perturbing field which enters into the time dependent perturbation calculation for the transition probability. In setting up the quantitative theory for t-v transfer, one may utilize either these two parameters or
,
Vol. 61
two equivalent parameters such as the depth of the well ( A H o o ) and the separation at the minimum point in the potential function (Re). From classical theory one is led to expect that the steeper the slope a t the turn-around point the g r a t e r the transition probability. On the other hand, from general quantum mechanics considerations one may argue that the optimum condition for energy transfer occurs when the “collision time” is in the neighborhood of the period of oscillation for the excited vibrator. This is expressed by
wherein is a measure of the magnitude of the potential over which an effective interaction can occur, Q is the average relative speed over that region, and yo,, is the frequency for this vibrator. When the pair approach each other with a speed greater than the optimum, the probability for energy transfer should decrease since the time spent during collision becomes too small. The collision orbit is essentially but not entirely classical, Because the masses of the colliders are high, their equivalent wave length is considerably less than the range in distance (ARp) over which the potential is effectively operating X/2a
P o 3 > Poi or Poq, but P i q >> Poi
This supports only in part Rice’s theory14 for the dissociation of diatoms. His rate constant
[wherein (giri2) gives the total degeneracy of the levels within kT of the dissociation limit] is based on the assumption that dissociation is limited by the collision rate and the “nearly maintained” equilibrium population of the upper levels. It is implied that the rotation-vibration levels just below and immediately above the dissociation limit differ only in density and degeneracy from the well characterized levels near the ground state. Now, when the rate of reaction becomes sufficiently high so the near equilibrium condition cannot be maintained, it is the rate of vibrational excitation, i.e., the speed with which these upper levels are populated, which becomes the limiting factor. Since the efficiency for excitation increases as the magnitude of the energy jump decreases, the slowest processes may be the excitation of molecules in the lower vibrational levels to the adjacent or nextadjacent one above. The relaxation process for the complete system, covering both t u and t-vdiss, has yet to be quantitatively formulated and (12) A. Eucken and E. Niimann, ibid., B36, 163 (1937). (13) €3. Widom and 8. H. Bauer, J . Chem. Phys., 21, 1670 (1953) (14) 0. K. Rice, cbid., 9 , 258 (1941); 21, 750 (1953).
. *
COLLISIONAL ENERGY EXCHANGE IN GASES
July, 1957
solved; the statistical mechanics for t-v relaxation only has been successfully treated.I5 The classical model is the other extreme; it is assumed that some collisions will be violent enough to induce dissociation in a single step from the low lying states. Supporters of the single-step dissociation mechanism argue that the rough selection rules are not restrictive when very hard collisions are involved. However, a quantitative treatment using the higher terms in eq. 2 has not yet been presented; the troublesome question is how --e
to estimate the relative magnitudes of
cbij
(R) and
4
837 I I
i 3
E, 6. Possibly, it may prove easier to solve the inverse problem, Le., the rate for three body recombination. Unpublished work16 Fig. 2.-Climbing the vibrational ladder us. direct fragappears to have successfully accounted for the ap- mentation; gi is the total vibrational degeneracy which may parent inverse temperature dependence of the re- be due, in pmt, to the ovevlapping of lour lying electronic combination rate constant, as recently reported.” states. This Rice’s theory does not do. There are indicau3 Yi VI tions that dissociation transitions occur from all IR; II IR;I R; (PI vibrational levels with a maximum probability from those of intermediate excitation. Of further chemical interest are the empirical correlations between the observed relaxation times and molecular constants, the most striking being 1 2349 the apparent dependence of the collision number 1388 on the lowest vibrational frequency. Fogg and Lambert18 found two classes of compounds for which, roughly t (010--ooo) T (loo-ooo) t(001--ooo)
+i(R) as functions of
class I Zl0 = exp(0.00S4vmi,) class I1 Zlo = exp(0.0175~~i.)
I n general, molecules in which Vmin is strongly active in the infrared belong to class I, while those in which vmin is inactive belong to class 11. Also, in a rough way, the relative efficiencies for vibrational de-excitation of ethylene increase with increasing flexibility of the hydrocarbon molecules tested.lg Current Problems.-In a polyatomic molecule wherein complex vibrations can occur, as expressed by a normal coordinate analysis, which relaxation times should be considered as of importance for chemical activation? Do they all contribute equally and are all the relaxsLon times equal? If not, is energy transferred from one mode to another? I n a specified mode, do the upper levels have a shorter T than the lower levels? Experimentally, these questions could be answered by undertaking: (1) in pure substances, to measure specific relaxation times, Le., 7’s for selectively excited vibrational states; (2) the above to be (15) K. E. Shuler. THISJOURNAL,61, 849 (1957), and references quoted therein. (16) J. C. Keck, private communication; he uses the statistical mechanics developed for “strong interaction“ by E. Fermi. (17) For I n : D . Britton. el al., J . Chen. Phys., 2 6 , 804 (1956); For Brz: D. Britton and N. Davidson. ibid., 26, 810 (1956): H. B. Palmer and D . F. Hornig, Tech. report No. 3 to the ONR, November 15, 1956 [project NR-357-2751 give for kdIss(Argan) = 1.39 x 1011 T‘/n(D/RT)l.g7 exp(--D/RT) [see, J . Chem. Phys., 26, 98 (1957)); for 0 2 : 8. W. Benson and A. E. Axworthy, April meeting ACS, 1950, Abstract No. 85, Physical and Inorganic Division. (18) P. G . T . Fogg and J. D. Lambert, Proc. R o y . SOC.(London), A232, 537 (1955). (19) W. D . McGrath and A. R . Ubbelohde, ibid., A227, 1 (1954); A. R. Ubbelohde, “5th Symposium on Combustion,” Reinbold Publ. Corp., New York, N. Y., 1955, p. 74 ff.
THEORY
short
long
very long
Fig. 3.--Possibilities for parallel relaxation processes in COZ ~ ( Y I Y Z vg). Characteristics of the transitions are indicated, such a8 the splitting due to Fermi interaction between V I and 2 ~ 2 .
measured as functions of the temperature, and particularly of the pressure to determine whether deexcitation occurs as a single step; (3) in mixtures, to measure the specific relaxation times with “incipient reactive” diluents, to explore the role of the interaction potential; (4) to measure the specific relaxation times for molecules excited to elevated vibrational states, as has recently been attempted.20 The question as to the existence of several relaxation times for a polyatomic molecule is outlined further in the case of carbon dioxide (refer to Fig. 3). If one believes that molecules exist in stationary states there is no way that an internal shift in energy can occur unless the molecule is perturbed either by an electromagnetic field or by a collision. The argument which is usually given that due to anharmonicity the energy fluctuates among the various modes applies only in the classical sense to coupled anharmonic osc,illators, Model (p). In Model (y) when several modes can be excited concurrently, the question arises whether de-excitation follows a parallel route so that molecules with several excited modes get de-excited in parallel and in an independent manner, or whether de-excitation occurs by collisions which shift the energy from one mode with a long relaxation time to another with a shorter T, as indicated by the sequence (20) J. D. McKinley, D. Garvin and M . J. Bo.udart, J . Chem. Phys., 23, 784 (1955); F. J. Lipscomb, R . G . W. Norrish and B. A. Thrush. Proc. Roy. 8 o c . (London),!USA, 455 (1956).
MARILYN E. ,JACOX AND S.H. BAUER
838 (001) +(030)
VoI. 61
+ transl. energy
6. Spectrophone : any selected infrared absorbing transition can be obtained-to see. (9) Principle of the Optic-Acoustic Effect.-In 1880, (020)+(010) --+(000) Bellz6first noticed that when an interrupted light The above appears probable for carbon dioxide. beam was allowed to fall on an enclosed sample of Although in carbon dioxide there are indications matter (a cigar was most effective), audible sounds that more than one relaxation time is involvedz1 were produced, and in 188lZ7he reported the use there is no substantial evidence that they are not of spectral dispersion to study the relative ampliall close to 5 p sec., within a factor of two or three. tudes generated by different regions of the specWere the parallel path operative, all theories pre- trum. He named the instrument the “spectrodict that the relaxation time for the (001) vibration phone.” Bell’s observations may be simply interwould be in the order of seconds. The sequential preted in modern terms. If the cell is filled with an infrared absorbing gas, the incident radiation ~ ) path, however, requires that the measured T ( ~ ~excites the internal degrees of freedom of the moledepart from a strictly reciprocal pressure dependence. This must be tested. To date two dis- cules above their equilibrium population. Equilitinct relaxation times have been unambiguously bration with the translational degrees of freedom produces a slight temperature increase; a t confound in only one gas, dichloromethane.6 stant volume, this generates a slight pressure inGeneral Methods for Measuring t-v Relaxation Times.-In order to answer some of the questions crease. When the radiation is interrupted periodilisted above, we have listed below the variety of cally, conduction of heat from the sample during the methods which are available for measuring relaxa- dark periods produces a lowering of the temperature, tion times in gases. It is instructive to keep in and thus a very slight pressure pulsation is promind that the intervals to be measured are of the duced. The frequencies used by Bell were in the order of microseconds or less and analytical de- audio range. If the infrared radiation absorbed by the vibravices are needed which are sensitive to changes in tional degrees of freedom of the molecules is imthe populations of energy states. mediately converted into translational energy, the 1. Ultrasonic, dispersion and absorptionZ~4J~l* -to see. CPv must be appreciable (ie., only pressure buildup should have a maximum a t the low lying energy states contribute) ; not specific instant the infrared beam is c u t off. In practice, the average number of collisions required for this to a selected mode. 2. Impact Tubezlc~zz:measures A S on rapid energy conversion may be as great as lo4, correcompression-to sec. ; dominated by low lying sponding to a time delay in the order of microseconds in the buildup of the pressure pulse and to a phase energy states; not specific. 3. Density Ratio across a Shockz3-could go shift, between the integrated light intensity pulse down to lo-* sec.; sensitivity limited to lower en- and the resulting pressure fluctuations. However, ergy states; useful for extended temperature stud- not until Slobodskaya’s experiments2ldwas the posies; could be developed for excitation of higher sibility recognized of relating phase lags in the sound pressure pulses to the time delays in collisional enenergy states; not specific. 4. Shock Detachment Distancez4 in front of ergy transfer. Using a mechanical interrupter for determining the phase shift of the sound produced, projectile; not specific. 5. Competitive measurement of collision and Slobodskaya measured the individual relaxation radioactive lifetimes in the infraredz6; specific times for carbon dioxide excited by 15.0,4.2and 2.7 p radiation. All three of the observed relaxation to selected emission band. times were between 1-7 psec. In our measurements (21) fa) K. F. Buschmann and K. Schafer, Z . phusik. Chem., 810, Slobodskaya’s technique has been modified in that a 73 (1941); (b) W. H. Pielemeier, H. L. Saxton and D. Telfair, J . Chem. completely electrical method of phase shift detection Phys., 8 , 106 (1940); (c) S. H. Bauer and M. R. Gustavson, Disc. is used; this permits more precise measurements of Faraday Soc., 17, 69 (1954); (d) P . V. Slobodskaya, Izvest. Akad. smaller time delays. An alternate electronic set-up Nauk. U . S. S. R . Ser. Fiz., 12, 656 (1948). It appears questionable that the acoustic method is sufficiently sensitive to demonstrate the has been developed by J. C. Decius for his studies presence of more than one relaxation time, unless these differ by very of v-t relaxation times in carbon monoxide.28 large factors. Amine and Legvold [J. Chem. Phys., 26, 515 (1957)l Design of the Spectrophone.-The mode of promeasured relaxation times b y sound dispersion in mixtures of haloduction of a reference signal for phase shift detecmethanes. In a mixture of two gases for which the individual relaxation times differ by a faotor of ten, they observed a single relaxation tion and the optical layout for introducing the intertime intermediate between the two. Thus it is clear that even when rupted infrared beam into the spectrophone cell relaxation of two modes of excitation follows a parallel process, the are shown in Fig. 4. The former is provided by a acoustic method is not able to show the presence of the separate regalvanometer bulb in the upper part of the figure, laxation times. to the left of the chopping wheel; the resulting (22) (a) A. Kantrowitz, J . Chem. Phys., 14, 150 (1946); (b) w.C. Griffith, J . AppZ. Phys., 2 1 , 1319 (1950). modulated beam is detected by a 929 phototube. (23) A variety of devices are available for measuring the density Radiation for excitation of the gas sample is proratio across a shock. Typical references are given. Interferometer: E. F. Smiley, E. H. Winkler and 2.I. Slowsky. J . Chem. Phus., 20, 923 (25) D. A . Dows and s. Killingbeck, Cornel1 University dissertation,
.1
(1952): also, ref. 9: Schlieren: E. L. Resler and M. Scheibe, J . Acoust. Soc. A m . , 27, 932 (1955); reflection of light from shock front: W. H. Andersen and D. F. Hornig, J . Chem. Phus., 24, 787 (1956); attenuation of an X-ray beam: G. B. Kistiakowsky and P. H. Kydd, ibid., 23, 271 (1955); attenuation of an electron beam: D. Venable and D. E. Kaplan, J:AppZ. Phus., 26, 639 (1955). (24) R. N . Schwartz and J. Eckerman, Navord Report No. 3904, this method has yet to be developed.
unpublished data (method yet to be developed). (26) A. G. Bell, Proc. Amer. Ass. Adu. Sci., 29, 115 (1880). (27) A. G. Bell, Phil. Mag., 11, 510 (1881). (28) Doctoral dissertations, submitted to the Department of Chemistry, Oregon State College, Corvallis, Oregon: G. C. Turrell, “Determination of the Vibrational Lifetime of CO by the Spectrophone Method,” 1954; Walter Jones, “Vibrational Relaxation Time Studies on CO by the Infrared Spectrophone Method,” 1957.
.L
.
JuIy, 1957
CoLLrSIONAL
ENERGY EXCHANGE IN GASES
vided by two Nernst glowers operated at an estimated temperature of 2330°K. An f/1.0 firstsurface spherical mirror focuses the radiation from the glowers on a rectangular slit. Immediately preceding this slit is a duraluminum wheel which has 85 equally spaced slits cut in its periphery. I t is driven a t 1800 r.p.m. by a synchronous motor so that interruption frequency is 2550 C.P.S. The resulting light intensity pulse is approximately trapezoidal, with equal periods of maximum light intensity and of complete darkness. Three filters (Eastman Kodak Research Laboratories) were used for isolating the 2.7, 4.2 and 15 p bands absorbed by carbon dioxide. These filters, whose transmission curves are shown in Fig. 5 have been designated, respectively, as filters B, A and S. The infrared radiation then passes into the spectrophone cell, details of which are shown in Fig, 6. This cell was machined in a block of brass and was fitted with aluminum inserts. The cell insert converts the main gas chamber in the front part of the cell to approximately a rectangular cone having dimensions close to those of the diverging light beam, permitting maximum absorption of the infrared radiation before reflection of the beam from the walls. The microphone diaphragm, used to detect the small pressure fluctuations, is in a vertical plane parallel to the sodium chloride window and 3.8 cm. from it. Small venting grooves in the inserts, which hold the microphone in place, allow for equalizing the pressure on the two sides of the diaphragm. The small magnitude of the pressure signal (Apes, 10-9 atm.) places stringent requirements on the microphone and associated electronics. For low inherent noise, reasonably high sensitivity and an essentially flat audio-frequency response we selected a vented Capps condenser microphone (Model CM-2003X). Since the effective noise is determined by the first stage of the amplification train, considerable care was taken in constructing a lownoise cathode follower using selected resistors and tube. Figure 7 is a block diagram of the electronics as developed for measuring the phase shift in the microphone signal. The output of the microphone cathode follower is led to the input of a Tektronix Type 122 Preamplifier, which has an amplification factor of a thousand. The amplified signal, in turn, is passed into a band pass filter (Burnell, Model 5-23102), which has a flat voltage-frequency characteristic near 2550 C.P.S. and is down to one-tenth of maximum transmission a t approximately 1300 and 4400 C.P.S. Because the signal is still of the order of millivolts, it is further amplified by a second Tektronix Type 122. The magnitude of the output voltage is read with a Hewlett-Packard Model 300A Harmonic Wave Analyzer tuned to a 60 cycle band pass a t 2550 c . P . s . , ~while ~ its phase shift is determined by comparison with the phototube signal using a Phazor Model 1OOL4 Null Meter.
Spectrophone Light Source Box with Cell in Position
Fig. 4.-Optics
I""
Filter
n
layout for spectrophone.
>1
0
~ ; i
(29) A t t h i s setting, a signal 30 cycles from center is attenuated by 40 DB. The equivalent noise input of the cathode follower is then about 5 X IO-* volt, a n d is within less t h a n a factor of two of t h e theoretioal limit.
839
100
Filter
B
Filter
S
CO2
'2"6
.5
Fig. 5.--Infrared
transmission of filters and active gases.
The phototube output goes to a cathode follower, and after amplification by a factor of one hundred by a Tektronix Type 122 passes into a resistancecapacitance phase shift network, where it is further amplified and filtered. This signal is finally fed
840
MARILYN E. JACOX AND Rubber Insert
X,p:,
Venting Groove
NaCl Windc
Face Plate
Horizontal Section,
Fig. 6.-Second
spectrophone cell.
Fig. 7.-Blu~ic wagrain
01
s. H. B h U E R
Vol. 61
Interpretation of Spectrophone Data.-Stepanov and GirinaO have made a theoretical analysis of the time dependence of the temperature in the spectrophone cell. They assumed that the cell is an infinitely long cylinder and that thermal energy is generated uniformly throughout the volume of the cylinder. This implies that a uniform light intensity strikes the base of the cylinder and neglects the weakening of the intensity of the beam by absorption along the path. Solution of the resulting heat conduction problem shows that when an intermittent beam of infrared radiation shines on a gas sample, the average temperature of that gas rises and then levels off to a steady state. Superimposed on this background temperature rise is a temperature fluctuation during each cycle. However, the maximum temperature in any cycle after the steady state has been reached does not coincide with the end of the light intensity pulse; because of the finite v-t relaxation time the temperature continues to build up for a short interval after the light has been turned off, as is qualitatively illustrated in Fig. 8. With some simplifying approximations, Stepanov and Girin30aobtained
the electronics.
where T D and T L are, respectively, the radial averages of the instantaneous temperatures in the cell (measured with respect to the temperature of the surroundings) during the dark and light periods, e is the length of a dark period or of a light period, 1 is the time measured from the beginning of the dark or light period, p is the pressure-dependent probability for the conversion of vibrational energy into heat by inelastic collisions, and g is the light intensity-dependent probability for the return transition from the excited state to the normal state with release of a light quantum. Further Time.
Fig. 8.-Schematic of the temperature drift in the spectrophone. The steady-state temperature is reached in a time well under one second.
into the reference terminals of the phase null meter. By adjustment of the resistance and capacitance settings of the phase shift network, the phase of the reference signal is varied to give a null, indicating that it is ninety degrees out of phase with the microphone signal. A phase detector operating on the niultipIying and time-averaging principle is particularly advantageous here, as the signal to noise ratio of the microphone amplifiers output is only about 15. Provided that the reference signal is a noise-free sine wave and that the period of the output meter is sufficiently long, the noise and higher harmonics of the microphone signal should average out. The Phazor Model lOOA Null Meter incorporates a multiplying circuit. The product is averaged over about 1 sec., the period of the output meter. The phase sensitivity of our circuit a t 2.5 Kc. is equivalent to '/4 psec., as checked by inserting known R C networks.
and where r is the light intensity-dependent probability for the excitation of molecules, N is the number of (30) B. I. Stepanov and 0. P. Girin, J. E x p . Theor. Phya. USSR., 947 (1950). (30a) As indicated in the text, St,epanov and Girin presented a theoretical analysis of the time dependence of temperature in the spectrophone cell assuming highly idealized boundary conditions. A much more realistic solut,ion for this temperatiire distribution has now buen presented in a doctoral dissertation by Walter D. Jones (ref. 28). For his boundary conditions, Jones not only inserted a zero temperature rise along the circuInference of the cylinder, b u t also a t the front and back plabes; i . e . . he treated a cylinder of finit,e length. He also took into consideration the decrement in intensity of the infrared radiation due to absorption by the gas. He found that the temperature distribution of the gas itself relaxes with a time constant which may be comparable t,o or larger than the a-l relaxation time being measured. t o 3 degree dependent on the m a g n i h d e of the optical absorption coefficient,. It may well be that our observation8 are being masked by this temperature relaxation process, as would be expected for a strongly absorbing gas soch as carbon dioxide. This point is being aotively studied a t present.
ao,
COLLISIONAL ENERGY EXCHAR'GE
July, 1957
molecules per unit volume, h is Planck's constant, Y is the frequency of the incident ra.diation, R is the radius of the cylinder, k is the heat conductivity of the gas, al equals 2.4048, and Cv is the heat capacity a t constant volume for a unit volume of the gas. From the estimated temperature of the Nernst dowers (assumed black body) and the geometry the optical system we estimated the energy available for absorption in the spectrophone cell a t each carbon dioxide band. In turn, we used this to estimate T in eq. l l a . For the 15 u band q is negligible compared to p, which is of the order of 2 X lo6 set.-' (the reciprocal of an assumed relaxation time of 5 X sec.). After making some allowance for the incomplete absorption of radiation by the carbon dioxide sample and for radiation loss in the filters, the 2.7, 4.2 and 15 fi bands have been assigned estimated p1 values of 2.7,0.72 and 0.,078", respectively. For carbon dioxide a t atmospheric pressure, p1 equals approximately 0.67 sec.-l in our apparatus. Further mathematical analysis is necessary to deduce an expression for the pressure fluctuations in the spectrophone cell in terms of the voltage read on the wave analyzer. Corresponding to the removal of higher harmonics by the band pass filters, a Fourier analysis for the coefficients of the two fundamental frequency components is made of expressions 10a and lob. It is necessary to choose a single reference point for the time in both the light and dark periods; this point is the beginning of the light period. For the approximate values of ( p + q ) and p1 previously given and for 0 equal to 2 X sec. (half-period for 2550 c.P.s.), there results
of
IN
GASES
84 1
pressure signal to an equivalent voltage. Ethsor in Table I are the results of these signal estimates, together with the observed values, Eobsd, and with E,,,, the equivalent voltages which would be obtained were it possible for all the energy to go into the fluctuating signal. TABLEI THEORETICAL AND OBSERVED MICROPHONE OUTPUTFOR COz AT ONEATMOSPHERE Absorption band
2.7
Etheor
E3'obsd
Emmsr
(P)
(Irv.)
(PVJ
(IrV.)
2.7 4.2 15 4.2
77 20 2
3.1 4.9 4.0 10.4
303 80 8.5 629
+
+ 15
141
All voltages are root mean square values. Because each of the observed values includes a noise contribution, the true spectrophone sound intensities are somewhat smaller. Omitting the proportionality constant between temperature and voltage, the filtered microphone signal is
s(t)= AI sin -e + B, cos --7rte 7rt
Referred to a pure sine wave input with zero at t = 0, the phase lag due to the gas is +gas = tan-l (&/Al). The phototube signal, too, is filtered to remove all but the fundamental frequency. In addition, it is passed through the phase shifter, which introduces the phase angle 9. Therefore, the phototube signal which reaches the reference terminals of the null meter may be represented by
Since the null meter multiplies the two signals and integrates them with respect to time, a t a null
Jyo S ( t ) P ( t )dt
and
=0
Substitution and simplification leads where Al is the coefficient of the sin wt term and B1 is the coefficient of the cos ut term. The amplitude of the temperature fluctuations is
+
Amplitude = ('/.AI2
21/2 2 7plp10 (13)
1/~B12)1/p
Referring to eq. l l a and I l b , it is seen that the amplitude of the oscillating temperature for a given absorption band is directly proportional to the energy input but independent of the cell dimensions and of the interruption frequency. I n contrast, the average steady-state temperature rise is given by '/z (TI, 4 TD) = '/2 ul, and depends on the square of the radius of the cylinder (determines heat loss), on the energy available per unit volume, and on the chopping frequency. Its magnitude is of the order lo3 times the oscillating component for COz a t 2.5 Kc. The corresponding pressure fluctuation is obtained by multiplying by the gas constant per unit volume. Assuming that the microphone signal is directly proportional to the pressure fluctuation in the cell, from the known sensitivity of the microphone (as given by the manufacturer) we then converted the computed
tan 4 = - ( / ~ , / B I = ) tan
(&as
_ - (-p -+ ale - -2nfT -
j=n/2)
(14)
7r
The chopping frequency is f = '/z0, and r , the relaxation time at the pressure of the experiment, is l / ( p + q ) . From eq. 14 tan
= '/tnfr
(15)
In obtaining the gas relaxation time from the observed phase shift pressure dependence, apparatus phase shifts other than those introduced by the phase shift network must be considered. In addition to the fixed optical and electrical phase shifts, which are completely independent of the pressure in the spectrophone cell, there are phase shifts due to the inertia of the microphone diaphragm and to the time required for the pressure (31) We surmise that the particularly large discrepancy for the 2.7 9 hand is due to our underestimating the extent of absorption by stmospheric COz, and thus to an overestimate of the available energy (we had estimated 4.3 ergs om.-* cycle-'). The absorption Coefficient used was obtained with the usual pressure broadening technique. However, the spectrophone is almost an infinite resolution device and records only the energy available over the narrow line absorption frequencies. This point is currently under investigation.
MARILYN E. JACOXAND S.H. BAUER
+= 3 Run *240
L'
Phase Shift
-
Pressure Dependence for COO
Vol. 61 dapll
- 2&1/P
The term T has been dropped, as it, mcrely denotes the direction of approach to the null. When is not a function of the pressure, this relationshm will be linear in l/p, and r1 is obtained easily from the slope. After insertiiig the appropriate conversion factors, the measured phase shift vcrsus 1OO/p slope is related to the relaxation time by
A 2.7~ 0
4 . 2 ~
c
Experimental Procedure and Observations
Carbon dioxide was chosen for the most intensive study. Because traces of moisture greatly accelerate collisional energy exchange special care was taken t o dry it thoroughly (over P ~ 0 6a t 800 p.s.i. for several months). Other gases were used without further purification. Before filling with the gas to be measured the cell was evacuated to a pressure of 10-6 mm. for at least one hour; it was filled with the gas to be measured, allowed to stand for a half hour, then evacuated and immediately filled again. The measurements consisted of setting the phase shift network for a null for each filter and for each pressure. The cell was fitted with a two-bulb gas buret which allowed a series of measurements to be taken over the range from 1 to 0.05 atm. without removing the cell from its mount for refilling. Figure 9 shows the observed phase shift as a function of the reciprocal of the pressure for a typical carbon dioxide run. The curves deviate considerably from linearity. Nevertheless, it is possible to fit reasonably good straight lines, with different slopes, to the data for the Ahigher and lower pressure extremes. Clearly, there are II I 100 instrumental pressure dependent phase shifts present. To assess the importance of these extraneous phase shifts p(cmHg)Fig. %-Phase shift-pressure dependence for COP (dry). we investigated other gases and gas mixtures known to have very short relaxation times. For ethane, GriffithZ2b reported 3.0 X 10-9 see. while Lambert and R o w l i n ~ o n ~ ~ signal generated in the cell to travel to the micro- obtained a relaxation time of 5.2 X 10-9 see., attributed to 275 cm. infrared-inactive torsion vibration of the phone diaphragm. These phase shifts will be de- the molecule. Our results in the spectral regions transmitted noted, collectively, by C$app. The condition for a by the filters were very similar to those for carbon dioxide. null then becomes It is possible that ethane may undergo a second relaxation of psec. duration since the vibration to which the previously +pan (Pam r / 2 = (P (16). observed mpsec. time has been assigned cannot be excited To show explicitly the pressure dependence by infrared radiation, and, conversely, the vibrations studied (see eq. 6), let r , the gas relaxation time at any in the spectrophone experiments contribute relatively little the specific heat of ethane a t room temperature. Regiven pressure, be replaced by rl/p with p the gas to sults for a mixture of 0.915 mole fraction carbon dioxide pressure in atmospheres and rl the relaxation time with 0.085 mole fraction hydrogen are plotted in Fig. 10. , ~ ~observed rea t one atmosphere, a constant for a given tempera- From the data by Eucken and B e c l ~ e r the laxation time of this mixture should be about 0.3 psec. a t ture. Substitution of (15) in (16) gives one atmosphere. Still another series of measurements using moist carbon dioxide are shown in Fig. 1 1 . The exart degree of saturation with water vapor is unknown. Assuming half saturation of the carbon dioxide with water vapor a t The above equation is not solved readily for the de- room temperature, Eucken and Becker's data would predict sired 71. Various approximations have been tried ; a relaxation time of less than 0.2 psec. Essentially the same the most satisfactory of these under the conditions pressure dependent behavior was obtained in a preliminary series of measurements on nitrous oxide. Studies of mixof our experiment requires the expansion of tan-l tures of ethane with hydrogen and with nitrogen, which have p/27rjr1. For T~ equal to second, 1 ) / 2 ~ j 7 ~considerably different sound velocities, gave no indication is approximately six a t atmospheric pressure and that the sound velocity in the gas sample is imporcant. The conclusion that microphone diaphragm damping 2550 c . P . s . ~ Hence ~ the approximation and hence the inertial phase lag is pressure dependent receives support from fragmentary data on the Western Electric 640AA condenser microphone, which served as the prototype for the Capps microphone. A t about 2.5 Kc. the is appropriate. When this is substituted into (17) former showed 15-20' phase lag a t 1 atm. (air); its sensitivity increased from -48.5 D B to -46.0 DB when the the measured phase shift is atm.36 Since the damping pressure was reduced to about (32) The assumption that p/2rfr1 is much greater than one is also constant depends on the resistance, total mass and stiffness, given by Stepanov and Girin as a condition for the direct reading and since the effective magnitudes of these in turn depend on property of Slobodskaya's instrument. At the interruption frequency the gas enclosed in the microphone cavity as well as on a
+
+
used in our experiments, this approximation becomes relatively poor at the lower pressures. When 71 is 10-6 second, a rather large value, the approximation introduces only a 5% error s t 0.4 atm. but an 18.5% error at 0.2 atrn. If T I is 10-6 sec., the error involved in the approximation is less than 5% a t pressures as low a8 0.05 atm., the approximate phase shift tending to be smaller than the true value.
(33) J. D. Lambert and J. S. Rowlinson, Proc. Roy. Soc. (London), 8204, 424 (1950). (34) A. Eucken and R. Becker, 2. physilc. Chem., B27, 235 (1934). (35) A. L. DiMattia and F. M. Wiener, J . Acoust. S o c . Am., 18, 341 (1946); F. W. Wiener, ibid., 20, 707 (1948).
c
July, 1957
COLLISIONAL E N E R G Y
diaphragm, some pressure dependence of the inertial phase lag is to be expected. Spectrophone cell design and circuitry are now in process of revision to permit measurement a t each pressure of this part of the phase increment. At a recent iiieetina of tho An~ericnnPhysical Society, Grcens1,nu and Blackman presented new data on relaxation tinics in carbon dioxide as studied by means of strong shock waves ( B d . A m . Phus. Sac., Series 11, 2, 217, Aid. RAO, 1957). They report the appearance of two distinct relaxation times, one of 2 Msec. a t tiOOOI
