Boltzmann distribution, modelling

Finally, the probability factor rj, which is taken to be coverage-independent in the model of a homogeneous surface with no lateral interactions between adsorbed particles, will be expressed by means of the Arrhenius formalism based on the Boltzmann distribution, viz. 

Such conformational dependence presents challenges and an opportunity. The challenges he in properly accounting for its consequences. In many cases, exact conformational energetics and populations in a sample may be unknown, and the nature of the sample inlet may sometimes also mean that a Boltzmann distribution cannot be assumed. Introducing this uncertainty into the data modeling process produces some corresponding uncertainty in the theoretical interpretation of data... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

The diffusion coefficient is estimated using Stokes law, D = kTjQmrja, where rj is the viscosity and a the radius of the rotating sphere. This rough model allows a calculation of the correlation function. It turns out that a Boltzmann distribution remains a Boltzmann distribution so that the system of equations (12)... 

The blackbody radiative mechanism proposed here is in fact very reminiscent of the model advanced by Dimbar in which CW infrared laser irradiation can be regarded as a blackbody source raising the effective temperature of the system. In this model, the population of ions achieves a truncated Boltzmann distribution, which resembles that of a normal Boltzmann distribution characteristic of the effective temperature but which abruptly ends near the dissociation energy of the ion of interest because once this energy is exceeded, the species rapidly imdergo... 

The Gouy-Chapman model describes the properties of the diffuse region of the double-layer. This intuitive model assumes that counterions are point charges that obey a Boltzmann distribution, with highest concentration nearest the oppositely charged fiat surface. The polar solvent is assumed to have the same dielectric constant within the diffuse region. The effective surface... 

A very common approximation goes by the name of rotational isomeric state model (RIS). With such a model it is assumed that the molecular population is placed exclusively in a few energy minima, always according to the Boltzmann distribution. The conformations corresponding to the minima are called conformers, or rotational isomers, and are indicated by one or more letters (G, T) for butane three conformers are considered (G, T, G ), for pentane five (TT, G T, TG, G G, G G ). A more detailed examination of pentane reveals the existence of two further wells (fl, = 65 , 2 = 260° and 0, = 100°, 02 = 295°), sometimes indicated by G G and G G , respectively, close to the forbidden G G" conformation, but with a much tower energy (157, 158). 

As in the case of the Helmholtz model, the plane AA will be negative due to the adsorbed R-species. Therefore, the Na+ and CL ions will be distributed nonuni-formly due to electrostatic forces. The concentrations of the ions near the surface can be given by the Boltzmann distribution, at distance x from the plane AA, as... 

If it is assumed that the Curtin-Hammett principle applies, one need only to compare the energies of the minima on the solid and dashed curves to be able to predict the structure of the major product. These curves also allow a direct comparison of Cram s, Cornforth s, Karabatsos s and Felkin s model for 1,2 asymmetric induction. Both Figures show the Felkin transition states lying close to the minima. The Corn-forth transition states (Fig. 3) are more than 4 kcal/mol higher and should contribute little to the formation of the final products assuming a Boltzmann distribution for the transition states, less than one molecule, out of a thousand, goes through them. Similarly, Fig. 4 shows the Cram and Karabatsos transition states to lie more than 2.7 kcal/mol above the Felkin transition states, which means that they account for less than 1% of the total yield. 

The model of electron scattering in high-mobility systems applied in the simulations is rather simplified. Especially, the assumption that the electron velocity is randomized at each scattering to restore the Maxwell-Boltzmann distribution may be an oversimplification. If the dissipation of energy by electron collisions in a real system is less efficient than that assumed in the simulation, the escape probability is expected to further increase. 

---