Equilibrium vibrational properties of ground state nitrogen up to 35,000 K

Abstract

The Rydberg-Klein-Reese (RKR) potential is extended beyond observed vibrational levels by an inverse sixth power attraction along with a short range repulsion that preserves the slope and approximates the second derivative of the potential at the observational limit. Vibrational energies are obtained to the dissociation limit by digitizing the action integral. For the ground state nitrogen molecule, this results in 64 vibrational states, compared with 58 levels provided by the JANAF series approximation and 68 levels by the Morse approximation. The vibrational energy and specific heat of ground state nitrogen based on this extended RKR potential are calculated to 35,000 K. Variations between models are due primarily to differences in the number of levels rather than to different anharmonicity of upper levels. N the subsonic flight era, fluid properties of the atmosphere were adequately accounted for by ignoring vibrations in the gas and using constant specific heats. As aeronautical science moved on to low Mach number supersonic flight, the excitation of vibrational energy had to be accounted for. The harmonic oscillator model then adequately accounted for variation in internal energy and specific heat up to a few thousand degrees Kelvin. With the advent of space flight, temperatures on the order of 15,000 K were encountered in flow about re-entry vehicles like Gemini and Apollo. Then vibra- tional eigenvalues up to about 4 eV or so were required to evaluate vibrational energy and specific heat accurately. Fortunately spectro- scopists have measured vibrational levels of N2 and O2 accurately to this level, and sixth degree power series were found to fit these mea- surements very well.1 Currently aerospace science contemplates space vehicles returning to Earth from deep space missions, such as the exploration of Mars. Then transient temperatures up to about 35,000 K are encountered in portions of the flow about such ve- hicles, and vibrational eigenvalues beyond those measured, all the way to the dissociation limit, become important. Since there is no guarantee that a power series fit to the lower levels will extrapolate accurately beyond these levels, this paper investigates the vibrational properties of N2 that can be deduced from reasonable extension of the RKR intermolecular potential that has been determined from spectroscopic data. For temperatures up to the characteristic vibrational temperature 0 = fiw/k (3395 K for N2 and 2274 K for O2), the vibrational en- ergy and specific heat of diatomics are provided rather well by the harmonic oscillator model. At much higher temperatures than 0, the model predicts that the vibrational partition function Qv approaches T/0 as a limit, whereas the dimensionless energy (Ev - E^)/RT and the specific heat CV/R both approach unity. Deviations from this simple model occur for two reasons: anharmonic effects in- crease the number of vibrational levels in a given energy range, and truncation of states at the dissociation energy limits the vibrational levels to a finite number, rather than the infinite number allowed in the harmonic oscillator model. It is well known that these effects cause energy and specific heat to be larger than harmonic oscillator values at temperatures larger than 9. However, it seems not always recognized that when T/0 is much greater than unity, the dimension- less vibrational energy and specific heat do not approach a constant limit but go through a maximum and then tail off toward zero. The nitrogen molecule, which is the major component in Earth's atmosphere, will be used to illustrate typical deviations from