Application of the quantum theory

The postulates of the older theory may be summarized as follows if n be the number of degrees of freedom in the molecule, then we have simply 

According to this a monatomic gas, with its three degrees of freedom for linear motion, must have the molecular heat R this is, of course, true, and is always rightly reckoned as one of the finest results of the kinetic theory. Diatomic rigid molecules have five, triatomic rigid molecules six degrees of freedom, since in the first case there are two, in the last case three, more degrees of freedom of rotation. As the above table shows, there occurs in many cases a remarkable approximation to the values 

Finally, there are possible also vibrations of the atoms within the molecule, which are apparently to be treated just like the vibrations of atoms in the crystal. These explain the occurrence of molecular heats higher than R for diatomic, or than for polyatomic gases. 

On closer examination, however, several doubtful points arise in particular, the older theory has difficulty in explaining the gradual rise of specific heat at higher temperatures, which is always observed in practice. I was able to show (51) in 1911 that application of the quantum theory not only overcomes these difficulties, but may lead to totally new points of view. I pointed out also at the Solvay Congress (1911) that even the conceptions with which the kinetic theory supplies us for a monatomic gas cannot be completely satisfactory, and that quite a different state of affairs must exist, particularly at very low temperatures (cf. also Nemst, 47 and 66). 

With the aid of the quantum theory we may make the following general picture of the behaviour of the specific heats of gases. 

---

The application of the quantum theory predicts that there can be no radiation produced of higher frequency than that given by the Einstein quantum equation for the maximum energy that an electron can have... 

In his interesting book, The Conceptual Development of Quantum Mechanics, Jammer15 reports that in 1908, A. Haas submitted his dissertation on quantum theory to the University of Vienna. It was probably the first application of quantum theory to the calculation of energy levels. However, his contribution was rejected since the application of the quantum theory, then considered a part of the theory of heat, to spectroscopy was considered ridiculous. 

Most of these types of spectroscopy are of use in chemistry. The levels probed by low-energy photons are sensitive to the detailed structure of molecules, and can be used to help identify compounds that have been newly synthesized. Visible and UV spectroscopy is important in the study of chemical bonding. X-ray spectra are characteristic of particular atoms, and are important in some methods of chemical analysis. The application of the quantum theory to the appropriate types of energy levels is essential in all these applications. A few relevant examples will be given in later chapters. 

The hydrogen atom is the most important chemical application of the quantum theory. The solutions of SchrSdinger s equation—known as atomic orbitals—form the basis of our understanding, not only of atomic structure, but also of chemical bonding in molecules and solids. 

All these theories are now only of historical interest because developments in spectroscopic techniques and the application of the quantum theory and wave mechanics are throwing an entirely new light upon our understanding of the causes of colour. In the first place, spectroscopy has shown that all organic compounds, whether they contain chromophores or not, absorb radiation. The fact that some are coloured is purely fortuitous because it so happens that their strong absorption bands lie within the narrow range of radiation to which the human eye is sensitive. Colour, therefore, is only a special aspect of a general phenomenon. 

C. F. Matta. Applications of the Quantum Theory of Atoms in Molecules to Chemical and Biochemical Problems Ph.D. Thesis (McMaster University, Hamilton, Canada, 2002) (http // chem. utor onto. ca/ cmatta/). 

One of the most important applications of the quantum theory of molecules in condensed media is the prediction of their chemical reactivity. It is often necessary to predict the rate or equilibrium constant of a chemical reaction in different organic and inorganic media. However, in most cases the theoretical calculations have been limited to the aqueous solutions. Therefore, it would be important to study the applicability of the solvation theories in different dielectric media. An appropriate reaction for such study is the tautomeric equilibrium between acetylacetone (1) and its enol form (2)... 

Application to Photochemical Side-reactions.—Einstein has pointed out that the application of the quantum theory to photochemical processes renders probable the relation... 

A very important application of the quantum theory to the energy content of a substance at very high pressures is due to M. Pol nyi, who arrives by it at the result that at very high pressures the specific heats must always converge towards zero. 

Rutherford s theory was rescued and placed on a firmer basis by Bohr, a Dane working with Rutherford in Manchester at the time. In 1913 Bohr suggested the application of the quantum theory of Max Planck to the electron. 

Hitherto wc have applied the quantum theory only to mechanical systems whose motion may be calculated by separation of the variables. We proceed now to deal in a general manner with the question of when it is possible to introduce the angle and action variables wk and Jfc so admirably suited to the application of the quantum theory. For this purpose it is necessary, in the first place, to fix the J s by suitable postulates so that only integral linear transformations with the determinant 1 are possible for it is only in such cases that the quantum conditions (1) Jk=nkh... 

This formula enshrines many important and remarkable results. In the first place it gives a maximum value of at a given frequency, as required by experiment, and as was inexplicable without the application of the quantum theory. This explanation is the historic triumph of Planck by which the quantum theory was founded. The account which has just been given of the matter differs, however, considerably from the original one and rests largely upon later considerations, which in their turn depended historically upon the initial discovery. 

In order to understand how this is possible, it is first necessary to know something about the general nature and structure of band spectra. At one time they appeared to be of a quite unintelligible complexity, which, however, the applications of the quantum theory and of wave mechanics have shown to depend upon combinations of relatively simple elements. 

In the application of the quantum theory to the simplest example of a diatomic molecule, the important new factor is the existence in the molecule of an axis defining a specific direction. An atom possesses no such axis. There exists therefore for the molecule a quantum number A which measures the number of units of angular momentum in the component of the electronic orbital motion projected along the axis joining the nuclei. According as A = 0, 1, 2,..., the state is called S, II, A,..., by analogy with the atomic states 8, P, D,..., which are determined by the values of I (p. 199). 

---