The Master Equation for the Dissociation of a Dilute Diatomic Gas. VIII. The Rotational Contribution to Dissociation and Recombination

Abstract

The dissociation of the J = 21 state of H,, and the recombination of atoms into that state, have been examined in detail. The J = 21 state of H, has two quasi-bound levels, one long-lived and the other short-lived, but the rate constants for dissociation or recombination involving this state are almost completely independent of the tunnelling rates into and out of the quasi-bound levels, and are in fact determined by bottleneck effects occurring lower down the vibrational ladder. Direct integration of the relaxation equations shows that, either excluding or including tunnelling, the dissociation and recombination rate constants obey the rate-quotient law, and that in the latter case the lowest eigenvalue of the relaxation matrix properly reflects the pressure dependence of the dissociation rate constant. Less extensive examination of the dissociation properties of other rotational states indicates that these conclusions are general, except that there is no strong bottleneck effect for very high rotational states (J > 30).It is shown that if full rotational equilibration is assumed, the sum, weighted over all J, of theindividual dissociation rate constants leads to an overall dissociation rate constant which is much too high, suggesting strongly that rotational equilibration cannot occur amongst the very high J states.A factored form of the master equation is then examined in which either only T-V or only T-R processes take place, over the temperature range 1500-5000 OK. It is found that in this approximation the upper rotational states are very strongly depleted, and that the Arrhenius temperature coefficients of the dissociation rate constants are between 92 and 94 kcal mol-', depending upon :he choice of rotational transition probabilities. The calculation suggests that one contributory cause of "low activation energies" in dissociation reactions is strong rotational depopulation of the very high rotational states, and its importance in relation to other possible causes is discussed.The smallest eigenvalues of the 177th order matrix representing the dissociation of para-H, and of the 172nd order matrix representing the dissociation of ortho-H, confirm that the factored model gives an acceptable representation of the dissociation rate of H z in this temperature range; hence the conclusions of the factored model in respect of strong rotational depopulation are probably valid. Finally, it is shown that the second smallest eigenvalue of the full relaxation matrix changes by a factor of three a t 1500 "K or by a factor of ten at 5000 "K when only rotational transition probabilities are varied, thus identifying the relaxation which immediately precedes the dissociation reaction in a shock wave as a T-VR rather than a T-V relaxation.An exploratory series of calculations for deuterium was carried out for the range of temperatures 700-5000 "K, using the latter model which includes full coupling between rotation, vibration, and dissociation, i.e., using matrices of order 348 and 355 for ortho-and para-deuterium, respect...

The Master Equation for the Dissociation of a Dilute Diatomic Gas IX. Rotation–Vibration Relaxation Times

“…Thus, although the relaxation is still essentially rotational, this increasing complexity brings with it a lengthening of the relaxation time, in our case to somewhere of the order of 5 x s. It is precisely the beginning of this behavior that has been observed recently in shock-wave measurements of the rotational relaxation times for hydrogen (12); that our predicted effect is more severe than the observed one is almost certainly attributable to imperfections in our assumed set of transition probabilities (1). At this temperature also, it is possible to identify theoretically a vibrational relaxation time, but its associated instantaneous heat capacity is negligible and the relaxation would not be observable in any coarsegrained experiment.…”

“…the lowest 49 levels, dn-, was most sensitive to the transition probability for v = 0, J = 5 H u = 0, J = 7, and if the lowest 50 levels were used the key transition remained v = 0, but in which a few of the outlying points lie on branches, each only connected to the main network by a single line passing through all the points on the branch: under these conditions the overall relaxation time will be determined mainly by the characteristic time of the slowest relaxation of any of the b r a n c h e~.~ This idea was supported by the fact that when, using the 44-member set, the top-most J-sublevels for each v were connected to each other with higher transition probabilities (-10-per collision), dl,-, became most sensitive to the v = 0, J = 5 +-+ v = 0, J = 7 transition probability with the AJ-restricted set of probabilities VIII(16) also, but not to the same extent as with the probabilities VIII(16a) which are solely dependent on the energy gap. Thus, one can regard a dissociating diatomic molecule (as we have suggested previously (1,8)) as being analogous to a densely interconnected network, and there is only one branch, which in this case is thesequenceoflevelsv = 0 , J = 1;u = 0 , J = 3; v = 0, J = 5; and v = 0, J = 7 which can then connect eitherto v = 0, J = 9 or v = 1, J = 1. In the energy-dependent case, the latter transition is easy, and so the relaxation time is principally determined by the J = 5 to 7 transition of v = 0, with J = 3 to 5 of v = 0 being of lesser importance: in the AJ-dependent case however, the sideways transition from v = 0, J = 7 to v = 1, J = 1 is much less favorable.…”

On the interpretation of measured rotational and vibrational relaxation times. I. Sound dispersion experiments

“…16a of ref. 8. With these probabilities, almost all entries in Table 1 remain unchanged, but the last two rows of probabilities become about the same as rows 1 and 3 respectively; there are also similar enhancements in probability wherever two levels of disparate J and v are close together in energy.…”

“…16 of ref. 8; a representative selection of these transition probabilities is given in Table 1 to illustrate their general behavior, which will be discussed in more detail later in the paper. The diagram is made u p as follows: the leftmost box gives a representation of the initial population distribution (population displayed horizontally) for the relaxation under consideration, and the rightmost box gives an equivalent representation of the Boltzmann distribution for the temperature of the experi- These are essentially what we would characterize as 'rotational' and 'vibrational' relaxations, respectively.…”

“…It has been suggested recently by several authors (2,3,8,20,21) clearly in the context of acoustic measurements, such transitions need not be considered at low temperatures, but very recent experiments suggest that they are not very important in laserexcited relaxations at low temperatures either (22). We have investigated the form the normal modes would have if such processes were strongly allowed, using eq.…”