Statistical distribution over vibrational-rotational states

The results for the Ba + 02 reactions,99 which employed a beam-scattering gas arrangement, show a statistical distribution over vibrational states corresponding to a temperature 2500°K. The distribution over rotational states was also statistical, corresponding to a temperature of only 500°K, but higher gas pressures were required to resolve rotation and some rotational relaxation may have occurred. The results are in accord with the conclusion,... 

Equilibrium Statistical Distribution of Diatomic Molecules over Vibrational-Rotational States... 

Quantum mechanical, classical and statistical probabilities agree, on average, reasonably well with the experimental results [133] shown in Fig. 37 (vibrational distributions of NO were also measured by Harrison et al. [310]). In the experiment a high population of the state n o = 1 is found already 100 cm above its threshold. Moreover, the measured probabilities show some indications of fluctuations. Because of the limited number of data points, the inevitable incoherent averaging over several overall rotational states of NO2 and the averaging over the various possible electronic states of the 0 and NO products, these fluctuations are less pronounced than in the quantum mechanical calculations on a single adiabatic PES and for J = 0. 

If one looks at a molecule such as heptane, for example, one can add all of the appropriate increments and calculate the heat of formation with acceptable accuracy by the method previously described. But there are a few things that are really not proper about that kind of calculation. Heptane in the gas phase at 25°C (where heats of formation are defined) is actually a complicated mixture (a Boltzmann distribution) of a great many conformations, most of which have different enthalpies and entropies. Additionally, each of these conformations is also a Boltzmann distribution over the possible translational, vibrational, and rotational states. The Benson method works adequately for many cases like this because these statistical mechanical terms can be lumped into the increments and averaged out, and they are not explicitly considered. By adjusting the values of the parameters in Eq. (11.1) or (11.2), much of the resulting error of neglecting the statistical mechanics can be canceled out, or at least minimized, in simple cases. But we would like for this scheme to work for more complex cases too. As the system becomes more complicated, errors tend to cancel out less well. So let us go back and approach this problem in a more proper way. 

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