Deactivation of Vibration by Collision in CO, BY KARL F. HERZFELD* Dept. of Physics, Catholic University of America, Washington 17, D.C. Received 16th January, 1962 COZ has one degenerate bending vibration, one totally symmetric and one asymmetric valence- bond vibration. There are a number of different ways by which these can be deactivated by collision. The rates are calculated in the range from 300 to 2500°K and compared with experiments. There seem to exist a contradiction between the results of different experiments. 1. THE PROBLEM The carbon dioxide molecule has three different characteristic modes of vibra- tion : 1 (i) the bending vibration, doubly degenerate, which has a characteristic temperature 8 = Rv/k = 960°K. This is conventionally called v2; (ii) the totally symmetric valence bond vibration, in which the C atom remains at rest, with 81 = 1920; (iii) the asymmetric valence bond vibration, in which C vibrates against the two oxygen atoms, with 03 = 3380.We ignore in the following the Fermi resonance, due to the near coincidence of 282 and 81. The problem considered in the following is the deactivation of these vibrations by collisions, as function of the temperature. This deactivation may occur directly, or by complex collisions.2 For convenience, we measure also the kinetic energy in temperatures, so that a kinetic energy increase by 500°K means an increase of the translational energy of a molecule by 500 k. The following is the list of possible deactivation processes, which will be designated subsequently by Roman numbers. v,; e2 = 960 v , ; el = 1920 direct : direct : or 0, = 960-kinetic energy 960; or O1 = 19204920; hv, + kinetic energy ’ hv, + kinetic energy ’ &+& +960 kinetic energy ; 1920+960+960 ; resonance: 81 +202 ; lt920+2 x 960 ; v,; e, = 3380 direct : , 3380-3380; hv, -+ kinetic energy &-’&+kinetic energy ; 3380+960 - 2420 ; O3 +el + kinetic energy ; 3380+ 1920 -I- 1460 ; 03+282 +kinetic energy ; 3380+2 x 960+ 1460 ; O3 +& + O2 + kinetic energy ; 33804 1920 + 960 + 500 ; 03+302 + kinetic energy ; 3380+3 x 960 + 500.of these processes have been calculated in this paper for the temperature range 300 to 2500°K. Recent measurements are now available. Additional interest comes from the hypothesis of Laidler et aZ.,3 that energy transfer does not easily occur between symmetric and antisymmetric vibration.Although it must * aided by Office of Naval Research. 22K. F. HERZFBLD 23 be emphasized that we only consider energy transfer during a collision, it will be interesting to see whether a similar rule is valid here. It would make processes VII and IX impossible.* 2. THE METHOD The methods to calculate the rates are given in the book of Herzfeld and Litovitz 4 and will not be repeated herz. Intermediate data used in the calculation are given in appendix B. However, several remarks must be made. For the rate, we write in accordance with the book 4 rate = (ZZJ , where zc is the time between collisions, defined as q = 1*271p~, 2 is independent of pressure for an ideal gas and is approximately the number of collisions needed for deactivation.Unfortunately, numerical errors have crept into table 66-5 of the book which are corrected here. The formulae contain a steric factor 20. The present author had deduced an expression for that factor,s which he now believes unjustified because, in the deduction, the interaction potential was divided into one part responsible for elastic scattering and one part responsible for inelastic scattering. This is now believed to be wrong; instead, one should use the averaged potential for elastic scattering. If the methods given in the paper 5 are then used, one finds the original value 20 = 3 for direct deactivation of longitudinal vibrations, and 20 = 3/2 for direct deactivation of the doubly degenerate bending vibration. For the complex deactivations, we write 2 0 = 320/3 leaving 2 0 undetermined theoretically.The equations given by Schwartz 2 apply directly to non-degenerate modes of a molecule hit by a different molecule. Witteman6 has pointed out that in the collision between two C02 molecules, and because of the degeneracy of the bending vibration, many more possibilities exist and that, therefore, the rate of a definite complex collision in which the mode which is excited is specified has to be multiplied by a number > 1 to get the total rate, or 2 divided by that number to get the correct 2. For example, in process 111, OpO2+kinetic energy, four modes 0 2 are available for excitation. The details are given in appendix A. Witteman makes definite assumptions about the 20 for each possibility.Instead, an average 20 is left open for each complex collision. 3. RESULTS Table 1 lists the specific heats (in units R) for the different modes. These have been taken without any refinements like Fermi resonance, from the Planck-Einstein formula, using tables in the Mayer-Mayer book? TABLE 1 VALUES OF c/R T, "K = 300 600 1000 1500 2000 2500 92 0.9072 1.6230 1.8514 1 -93 30 1.9618 1.9756 01 0.0688 0.4536 0.7422 0.8742 0.9257 0.9522 0.1 146 0.4168 0.6666 0.7930 0.8609 9 3 * It should be added that other processes than those enumerated above, I to X, are possible at higher temperatures, in which more than a single excited mode is deactivated, e.g., 01+02+2760 kinetic energy. Such processes have been neglected since their probability is very small.4 RELAXATION IN c02 Table 2 shows for the complex collisions those values of 2 which one gets if the mode to be deactivated is in quantum state 1, all the other modes in quantum state zero and only jumps by unity occur.Table 3 shows the calculated 2 for all processes considered here; for the complex collisions, the values of table 2 have been divided by the “ weights ” for which the formulae are given in appendix A. TABLE 2 z FOR A COMPLEX COLLISION IN WHICH THERE IS A TRANSITION 1+0 AND TRANSITIONS O-tl IN DEFINITE MODES T. OK = process I11 IV VI VII VIII IX X 300 1-lox 106 7400 1 . 8 ~ 1012 1 . 6 ~ 1010 1 - 6 ~ 106 1 . 8 ~ 109 2 . 3 ~ 107 600 1000 1500 2000 2500 68,000 9000 2600 1100 420 5600 2900 2100 1600 1360 1 . 4 ~ 106 4 . 6 ~ 107 3 . 7 ~ 106 585,000 180,000 74,000 4 .6 ~ 108 3.5X 107 5 . 7 ~ 106 1.71 x 106 726,000 284,000 92,200 39,200 22,900 15,600 4.3 x 106 1.4 x 106 618,000 383,000 252,000 TABLE 3 EFFECTIVE (TO BE MULTIPLIED BY UNKNOWN zo/3 EXCEPT FOR I, II AND V) T, OK = 300 600 1000 1500 2000 2500 process I (960) I1 (1920) I11 (1920) IV (1920) V (3380) VI (3380) VII (3380) VIII (3380) IX (3380) X (3380) 16,300 1 . 4 9 ~ 108 265,000 490 4-4x 1011 8 . 8 ~ 108 193,000 392,000 4.7 x 1013 1 . o ~ 109 1160 220 95,200 14,200 1600 270 94 2 . 2 ~ 107 1 . 6 ~ 106 2-1 x 107 981,000 27,300 6280 41,900 6670 73 37 4100 430 170 38 28 297,000 66,400 101,000 21,700 1870 810 1400 510 24 1700 43 8-8 1 . 6 ~ 106 104,OOO 26,500 3340 310 310 It is evident from table 3 that the sequence IV (resonance) is most efficient for the deactivation of 01 (provided there is no extraordinarily large ZO), with sequence I11 approaching it in efficiency above 2000°K. For the deactivation of 03, process IX is most efficient (again with the proviso about ZO), but X is comparable over the whole temperature range, so that, if 20 = 3 for both, the combination of processes IX and X gives the results shown in table 4.The Z values for IX and X at 2500”, as given in tables 2, 3 and 4, may be too large, due to the mathematical approximations made. 4. COMPARISON WITH EXPERIMENTS At 300”K, the most direct comparison is with the “spectrophone” in which a particular vibration is excited by absorption of infra-red radiation, and the timeK. F. HERZFELD 25 delay in the pressure increase is measured. Unfortunately, the experiment is very difficult.Slobodskaya,8 who did it first, found 2 = 14,000 for the bending vibra- tion and 63,000 for the asymmetric valence bond vibration. The first agrees very well with the value in table 3, the second is about half the number in table 4. In any case, a large 20 is not compatible with this. Bauer and Jacox’s measurements 9 vary by a factor 2 within themselves (at low pressure). The average 2 for 02 agrees with that of Slobodskaya, but the 2 for 0 3 is equally low. 0 3 cannot be detected i n ultrasonic measurements. Henderson and Klose 10 find (at 323.7”K) a single absorption peak ; however, its height, i.e., the value of 2C2 + C1 calculated from it, is a few percent too high. No explanation of this excess is known. However, 2 for IV cannot be larger than 2 for I.Dr. Wittleman will discuss his unpublished experimental results, which are in fair agreement with those of Winkler and Smiley.11 The latter do not deviate too much from the values of the direct process I according to table 3. The usual shock-tube methods do not work below 2- 300 ; perhaps the procedure of Hornig 12 might be successful in such cases. In the range 2000-250OoK, Greenspan and Blackman 13 give z = 3.5 x 10-6 sec or 2-7000. They interpret this as relaxa- tion of 01 ; this would, however, imply that process IV has a 20 of about 1000 making its room temperature 2 about 160,000 which is in complete disagreement with Henderson and Klose’s result of good adjustment between 01 and 02. Hurle and Gaydon 14 also have made shock-tube experiments around 2500°K.They find relaxation times of 35 and 54 x 10-6 sec, which can be considerably shortened by impurities, particularly water vapour ; so it seems reasonable to assume that Green- span and Blackman’s values have been shortened in this way. Hurle and Gaydon ascribe their relaxation to 03 and think it probable that the asymmetric valence bond vibration cannot easily exchange energy with the symmetric one. This would eliminate processes VII and IX. If we take 45 x 10-6 sec as an average value, one gets 2-77,000, and one must also deny the effectiveness of VIII and X and leave VI as the most probable sequence. This, however, contradicts the spectrophone results at 300°K. We are, therefore, confronted with a contradiction : the spectrophonic experi- ments seem to prove that at room temperature there is no particular difficulty of energy exchange between the asymmetric valence bond vibration and the sym- metric longitudinal vibration plus one bending vibration during a collision (IX and X).The shock-tube experiments seem to prove that, even in a collision, Laidler’s rule holds, namely, that there is no energy exchange between 01 and 03 and in addition that 03 cannot exchange energy with more than one bending mode. APPENDIX A Consider as an extreme example process 0, 03+302+kinetic energy. Let us first assume low temperatures so that all the bending vibrations are in the ground state. Then the following possibilities exist. (a) The three quanta go into different modes. These are 4 possibilities, since any one of the modes may be unaffected.(b) Two quanta go into one mode, the third quantum into a different mode. There are four different choices for the two quantum modes, and for each, three different choices for the additional quantum, or 12 choices; in addition, however, the two quantum jump has a factor 2 ; therefore (b) makes a contribution of 24. (c) All three quanta g o into the same mode, of which there are four. However, the jump from 0 to 3 gives a factor 6, so one has a contribution of 24. The total factor is 4+ 24+ 24 = 52. At higher temperatures, this is further increased because some molecules are already excited, and the corresponding transitions have higher probabilities. For26 RELAXATION IN COz example, if the average state of the oscillators were o = 2, (a) would contribute 4x 3, (b) 12x 12x 3, (c) 4 x 60, or a total factor of 684.Use the expressions C n j = 1, C j n j = e-eJT(l - e-e/T)-l, C j2n = e-e/T(l+ e-eJT)(l - e-'/T)-2, C j 3 n j = e - e / T ( 1 + 4 e - e / T + e - 2 e / T ) ( l -e-@iT)-3, where nj is the fraction of the molecules in state j. Abbreviate Then one has the following " weights ". (111) 4 f d - l (IV) One starts with This leads to 6[CO'+1)nj12+4CO'+1)0'+2)nj. (IV) 14f3,l ; (VI) 4 f 2 P ; (VII) 2flf3l ; (IX) 8flf,fi-1 - (VIII) 14f,2f,-l ; (X) One starts with 4[c(j + l)njI3 + 12[c(j + 1)(j +2)nj][C(j + l ) n j ] + 4x0 + l)(j + 2)(j + 3)np This leads to 52fZfT' . It can be seen that the way in which the different f appear in the " weights " is deter- mined by the type of process, after the fashion of the chemical mass-action law, but I have been unable to give a general proof for this.APPENDIX B It is assumed that the interaction energy between two C02 molecules obeys the Lennard- Jones law 4E[(912-($7. For C02, ro = 3-996A, &/k = 190°K. For the purposes of our calculation, the Lennard- Jones law is replaced by - E + Ho exp (- r/l). The length I is determined so that for the most important kinetic energy of the molecules, Em, the Lennard-Jones function and the exponential function have the same slope. One now defines an energy 8' (measured in units of temperatures) by Eere v is the vibrational frequency, 6 the effective mass in absolute units per noIea.de, M the effective mass per mole (for the collision of two CO2 molecules: this is 22).InK . F. HERZFBLD 27 the last equation I is in A. In a complex collision, 8 is the energy transferred to translational energy. One then has for the energy of the most efficient molecules at temperatute T : Em = +k(O')"T'. The values used in the calculation are shown in table B. T, OK = Ern18 Z in A e'x 10-5 Ern/& ZinA e'x 10-5 Emf& Z in A 8' x 10-5 Ern18 I in A 1~x10-5 Ern/€ IinA e'x 10-5 Ern18 I in A O'X 10-5 300 10-30 0.2005 6-67 1 16.58 27.793 0.2049 24.39 88.580 19.36 44.522 0-2079 0.2059 13.72 0.2030 15-763 6-566 0.1963 1 -727 TABLE B 600 1000 1500 2000 1 AND 111 16.56 23.58 32.55 37.73 0.2047 0.2076 0.2098 0.2113 6.926 7.109 7.240 7.365 I1 37.70 0.2109 29-400 60.19 29.905 0.2127 V VI VII AND VIfl 2207 3 1.23 41-14 49-98 16.375 16.720 16.986 17.130 0.2069 0.2091 0.2108 0.21 17 Ix AND x 10-58 15.04 19.83 24.17 0.2005 0.2041 0.2060 0,2074 1.808 1.868 1 -9025 1.936 2500 44.15 0-2141 7.555 73-32 34.594 0.2135 102.5 0.2146 94.380 81.78 48.02 0-2138 58.13 17.257 0.2124 28-12 0.2087 1.953 1 Herzberg, Infra-red and Raman Spectra (van Nostrand Co., N.Y., 3rd edn., 1945). 2 Schwartz, Slawsky and Herzfeld, J. Chem. Physics, 1952, 20, 1591. 3 Gil and Laidler, Proc. Roy. SOC. A, 1959, 250, 121 ; 1959, 251, 66. 4 Herzfeld and Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New 5 Herzfeld, 2. Physik, 1959, 156, 265. 6 Witteman, J. Chem. Physics, 1961, 35, 1 . 7 Mayer and Mayer, Statistical Mechanics (Chapman and Hall, London, 1940). 8 Slobodskaya, Izvest. Akad. Nauk S.S.S.R. (ser. Fiz.), 1948, 12, 656. 9 Bauer and Jacox, J. Physic. Chem., 1957, 61, 833. York, 1959). 10 Henderson and Klose, J. Acoust. SOC. Amer., 1959, 31,29. 11 Smiley and Winkler, J. Chem. Physics, 1954, 22, 2018. 12 Hansen and Hornig, J Chem. Physics, 1960, 33, 913. 13 Greenspan and Blackman, Bull. Amer. Physic. Soc., 1957,II, 2, RA9. 14 Hurle and Gaydon, Nature, 1959, 184, 1858.