Translational energy spacing

Two special cases of the theory illustrate the important features. The first is the relaxation of an ensemble of noninteracting harmonic oscillators in contact with a heat bath, and subject to nearest neighbor transitions in the discrete translational energy space. The master equations which describe the evolution of the ensemble can be written13... 

Figure B2.3.6. CM angle-velocity contour plot for the F + D2 reaction at an incident relative translational energy of 1.82 kcal mol [26], Contours are given at equally spaced intensity intervals. This CM differential cross section was used to generate the calculated laboratory angular distributions given in figure B2.3.4. (By pennission from AIP.)...

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

The displacement of the center of mass of a molecule gives rise to the external or translation energy of the molecule. It corresponds to the energy associated with the movement in three-dimensional space of a particle whose mass is equal to that of die molecule. In each of the three directions in space the problem can be treated as a particle in a box. as presented in Section 5.4.1. It was shown that the energy is given by... 

We see that the size of the quantum of translation energy is around 1020 times smaller than kg T in this example. To a very good approximation we can treat translational energy levels as though they were continuous rather than discretely spaced. 

Let s examine the magnitude of the rotational energy spacing term, as we did before for translation energy, choosing H2 as an example. The intemuclear spacing for H2 is... 

A molecule can move through space along any of three dimensions, so it has three translational degrees of freedom. It follows from the equipartition theorem that the average translational energy of a molecule in a sample at a temperature T is 3 X kT = kT. The molar contribution to the internal energy is therefore NA times this value, or... 

We shall need to know how to evaluate these separated partition functions. The translational energy levels can be derived from the quantum mechanical solution for a particle in a box they are so closely spaced that the partition function can be evaluated in closed form by integration, and has the value... 

These distribution functions show that, at a given temperature, many energy levels will be populated when the energy spacing is small (which is the case for translational... 

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