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 H2, and the recombination of atoms into that state, have been examined in detail. The J = 21 state of H2 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 the individual 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 °K. 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−1, depending upon the 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-H2 and of the 172nd order matrix representing the dissociation of ortho-H2 confirm that the factored model gives an acceptable representation of the dissociation rate of H2 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 at 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, respectively. These calculations predict that there should be a reversal in the isotope effect for both dissociation and recombination of hydrogen and deuterium as follows: (i) with helium as third body, deuterium should dissociate faster than hydrogen at high temperatures, but below about 2000 °K, the dissociation of deuterium will become the slower of the two processes; (ii) with argon as third body, deuterium should recombine faster than hydrogen at high temperatures, but below about 1000 °K, the recombination of deuterium will become the slower of the two processes; (iii) the rate constants for the recombination of hydrogen by hydrogen and for the recombination of deuterium by deuterium will probably cross over near 1000 °K, indicating a need for experiments in this region of temperature.