Angular Boltzmann distribution

In impact theory the result of a collision is described by the probability /(/, /)dJ of finding angular momentum J after the collision, if it was equal to / before. The probability is normalized to 1, i.e. / /(/, /)d/=l. The equilibrium Boltzmann distribution over J is... 

The association rate data determined in this study can be used to make quite a precise binding energy estimate for the aluminum ion-benzene complex. The relation between the association rate constant and the binding energy was made with use of phase space theory (PST) to calculate as a function of E, with a convolution over the Boltzmann distribution of energies and angular momenta of the reactants (see Section VI). PST should be quite a reasonable approximation for... 

In more complex situations, where e.g. rotations and vibrations are involved, there will be squared terms relating to each rotational and each vibrational mode. Each rotational mode involves the square of the angular velocity and contributes one squared term. Each vibrational mode involves a term from the potential energy and a term for the kinetic energy of vibration, and contributes two squared terms if the vibration is harmonic. If there are a total of 2s squared terms, then this is called energy in 2s squared terms. The Maxwell-Boltzmann distribution is correspondingly more complex (Section 4.5.8). 

Now on multiplying Eq. (33) by e [0o = 0(0) denotes a sharp initial value] and averaging over a Maxwell Boltzmann distribution of 0o and 0o (that procedure is denoted by the angular braces) we have a differential recurrence... 

O or equivalently the drag coefficient becomes very large. (The noninertial limit typically corresponds to // 10 s.) In this limit the Maxwell-Boltzmann distribution for the angular velocities has set in so that orientation and angular velocity variables are decoupled from each other as far as the time behavior of the particle orientations is concerned. Thus on setting / = 0 in Eqs. (5.23)-(5.25), we have... 

To determine a rate constant, k (b) and the recombination probability 6 nust be obtained. For particles interacting via a centro-symmetrlc potential of mean force, Eq. 2 can be used to obtain kp(b). For most of the model studies considered in this section, however, Eq. 2 is not strictly valid since the potential of mean force has an angular (0) dependence. Making the reasonable assumption that the relative orientation of the dimer follows a Boltzmann distribution at R = b, an equation similar to Eq. 2 can be derived (34). To determine 6, dynamical trajectories are computed using Eq. (5) or equations derived therefrom, (34) starting at R = b 8 A and with relative orientations selected at random from a Boltzmann distribution. Trajectories were terminated at R > q = 10 A and 6 was determined from the results of 5,000 to 10,000 separate trajectories. The interested reader is referred to reference 34 for more details. The rate constant results are summarized in Table II. 

Here NaBe and Na are the number densities of interacting 8 Be and 4 He nuclei and the angular brackets denote thermal averaging over a Maxwell Boltzmann distribution ip(E). This averaging leads to ... 

Since the above description of experimental data focused almost exclusively on the sticking coefficient, we now present very briefly some data on the analysis of the scattered molecules. The reader is invited to consult a number of excellent reviews that exist on the subject (Barker and Auerbach, 1984 Comsa and David 1985 Rettner 1988) for more details. The angular, velocity, and internal energy distributions are usually measured. These are compared, respectively, to the predictions resulting from the assumption of equilibrium of the gas and surface (1) the angular distribution is proportional to cos 6 (2) the velocity and internal energies are described by a boltzmann distribution. 

The simplest dynamical model of associative desorption is based upon a one-dimensional PES with, of course, a single barrier in the exit channel for desorption (Van Wiligen 1968). (An exit channel barrier for desorption is equivalent to an entrance channel barrier for dissociative chemisorption.) Assuming an equilibrium distribution of adsorbates at the surface temperature T, the model predicts a Boltzmann distribution of velocities with the Z-component of velocity centered around v = (2Fq/M) where Vq is the barrier height. The angular distribution of desorbed molecules is then... 

Various methods of statistical mechanics are applied to the calculation of surface orientation of asymmetric molecules, by introducing an angular dependence to the intermolecular potential function. The Boltzmann distribution can also be used to estimate the orientational distribution of molecules. The pair potential V(r) may be written as V(r, 6) if it depends on the mutual orientation of two anisotropic molecules, and then we can write for the angular distribution of two molecules at a fixed distance, r, apart... 

---