Ab-fine problems

Ab Fine Problems in Physical Chemistry and the Analysis of Adsorption-Desorption Kinetics... 

In spite of the already high, and continuously increasing, power of ab initio methods, there are situations which remain much too complex to be modeled with them. These situations (e.g., the vibrational modes of, or the adsorption-energy distribution on, highly dispersed systems, like powders or colloids) can often be described quite accurately on experimental grounds but are of difficult theoretical description. Because the experimental data (e.g., the specific heat as a function of temperature or the adsorption isotherm) can be written in terms of unknown microscopic quantities (like the vibrational density of states or the adsorption-energy distribution, in the above examples), one can try to extract the inaccessible microscopic information from the accessible macroscopic data. This generates a kind of ab fine problem. 

This chapter is devoted to an anlaysis of a few ab fine problems of physical chemistry, with a special emphasis on the extraction of the activation-energy distribution from experimental desorption kinetics. 

II. AB FINE PROBLEMS AND THEIR ROLE IN PHYSICAL CHEMISTRY... 

In the following, we consider three ab fine problems of much interest in physical chemistry, essentially related to the extraction of the following ... 

The ab fine problem is related to finding the density of states by solving Eq. (5) for gj and g E) when Z T) is known. This problem, an age-old problem of physical chemistry, was first formulated by Bauer in 1939 [5]. 

Einstein himself did not believe seriously in his model in fact, in a second article The Present State of the Problem of Specific Heats presented to the first Solvay Congress in 1911 and later published in the Proceedings, commenting on a proposal of Nemst and Lindemann [7], he wrote, One can therefore suppose that the body behaves as a set of oscillators with different frequencies. [...] A theoretical meaning could only be attributed to a formula in which all the infinite frequency values compare in a sum [8]. However, not only has the Einstein model had a historical and tutorial relevance, but also formula (15) continues to be used for the vibrational specific heat of isolated defects. Nemst and Lindemann had observed that most of the experimentally known specific heats could be adequately represented by taking 8 oo — (o ) + 8 (jo — oo ) for g(a>) [7] in a way, this formula can be considered the first attempt to solve the ab fine problem. 

The ab fine problem associated with Eq. (14) was not considered carefully because, soon after Einstein s proposal, on one side, Debye developed a model for the specifie heat of crystalline solids able to rationalize most experimental data, whereas, on another side. Bom and von Karman posed the bases for the foundation of a dynamic theory, the theory of lattiee dynamics in the harmonic approximation, able to give an adequate solution of the direet problem [i.e., able to determine the fi equency spectrum g co) from atomic properties]. 

The problem of finding which frequency distribution g(co) reproduces the experimental specific heat Cy(T) is therefore of a certain importance. This ab fine problem was first raised and solved by Montroll (in an appendix to a fundamental article otherwise devoted to the direct problem) via a formula involving the function Cy T) for T IR+ [16]. That the experimental errors affect the resulting frequency spectrum significantly was first demonstrated by Lifshitz [17] and later confirmed by Chambers [18]. 

It is rare that any of the above-described overaU isotherms fits the experimental isotherm satisfactorily in the whole physical range. This state of affairs generates the ab fine problem of accounting for the observed behavior in terms of adsorption-energy distribution of the adsorbent. 

The solution of this problem requires a preliminary choice of the local isotherm. For the reasons stated in Section VA, the most convenient choice is the Langmuir isotherm (19). In that case, the ab fine problem is related to the solution of the integral equation... 

Thus, solving the ab fine problem for the adsorption-energy distribution fi-om the overall isotherm is equivalent to finding the energy distribution of a gas of fermions when the average occupation is known as a function of its Fermi energy Q. 

In the previous sections I have tried to provide the reader with consolidated results. He or she may have reached the conclusion that in spite of the ill posedness of the considered ab fine problems, most of the difficulties have been removed and what remains to do is simply to apply the obtained results to practical situations. 

When considering the heat capacity of crystalline solids, there is no ambiguity about the kernel to be used in the ab fine problem it is determined by the Bose-Einstein stastisties. 

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