Origin of the Temperature Dependence of the Ultrasonic “Rotational Relaxation” Time

Abstract

An investigation of the origin of the anomalous temperature dependence of the rotational equilibration time is reported. Systems of H2, D2, and N2 have been studied. In each case the rotational system has been represented as a multilevel ensemble. The transition constants kij for transitions j → i are obtained from a simple, line-of-centers collision model and contain one common adjustable parameter which is fixed by fitting room-temperature ultrasonic velocity dispersion data to the model. Equilibration rates are obtained via numerical solution of the master equation. Velocity dispersion curves are then calculated under the assumptions of gaseous ideality and rotational Boltzmann equilibrium at all times. The individual transition constants kij obtained in the formalism exhibit both positive and negative temperature coefficients depending upon the i–j spacing. Positive coefficients are associated with large rotational spacing while negative values result from small spacings. The calculated ultrasonic velocity dispersion curves correspond to experiment with near quantitative accuracy over a 700°K temperature range. The ultrasonic frequency at which dispersion occurs decreases with increasing temperature, and this effect does not appear to be due to the negative temperature coefficient exhibited by the transition constants for the low-J states. Likewise, the effect is not directly associated with a decrease in collision rate at unit pressure as T increases. In general, it is concluded that the apparent negative temperature coefficient associated with the over-all ultrasonic equilibration process is due to the multilevel character of the system and to the fact that the adjacent level spacing increases linearly with rotational quantum number. It is shown that the equilibration process can not be adequately described by a simple relaxation expression with a single time constant, and the misleading character of the term “rotational relaxation time” is pointed out.