Numerical Calculation of Vibrational Relaxation and Dissociation for a Quantum Anharmonic Oscillator

Abstract

The vibrational relaxation and dissociation of gaseous H₂ highly diluted in an argon heat bath have been studied numerically at temperatures between 2500 and 15 000°K. Collisional transition probabilities between the vibrational levels of H₂ were calculated by the method of Schwartz, Slawsky, and Herzfeld in a form modified for a Morse oscillator. The master equation was solved by numerical integration to give the time dependence of the vibrational level populations and of dissociated molecules in a simulated shock‐wave experiment. The absolute values of the calculated rate coefficients for dissociation are in good agreement with experiment, but the predicted vibrational relaxation times are about a factor of 35 shorter than those determined experimentally. At all temperatures, the upper vibrational levels in the dissociating gas are substantially depleted below their equilibrium populations. The Arrhenius activation energy for dissociation is 102.5 kcal mole⁻¹, slightly lower than the dissociation energy of 103.2 kcal but higher than the experimentally determined activation energies by about 10 kcal. An incubation period of about 1 to 1.5 vibrational relaxation times precedes the establishment of approximately steady dissociation. The computed incubation behavior is compared with that predicted from the diffusion theory model of Brau, Keck, and Carrier. The vibrational relaxation behavior of this anharmonic oscillator molecule has been examined. In the absence of dissociation, the relaxation time as defined by the Bethe–Teller equation differs from its SHO value by less than 10%, and is found to show variations of less than 10% with time. The present results are also used to examine some approximations made in theories of coupled vibration and dissociation.