Translational energy, quantization

In eollisional exeitation, translational energy of the projeetile ion is eonverted into mtemal energy. Sinee the exeited states of the ions are quantized, so will the translational energy loss be. Under eonditions of high energy resolution, it is... 

For reactions between atoms, the computation needs to model only the translational energy of impact. For molecular reactions, there are internal energies to be included in the calculation. These internal energies are vibrational and rotational motions, which have quantized energy levels. Even with these corrections included, rate constant calculations tend to lose accuracy as the complexity of the molecular system and reaction mechanism increases. 

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. 

Molecular entropies For a perfect monoatomic gas, there is only translational motion. According to quantum mechanics, the translational energy of molecules in a box is quantized and the size of the quantum is proportional to the reciprocal of the atomic weight. Heavier gases have smaller gaps and the number of states available and degeneracies are greater. 

The other cause, the density effect, is especially important at high densities, where molecules are more or less confined to cells formed by their neighbors. In analogy to the well-known quantum mechanical problem of a particle in a box, the translational energies of such molecules are quantized, and this has an effect on the thermodynamic properties. In 1960 Levelt Sengers and Hurst [3] tried to describe the density quantum effect in term of the Lennard-Jones-Devonshire cell model, and in 1980 Hooper and Nordholm proposed a generalized van der Waals theory [4]. The disadvantage of both approaches is that, in the classical limit, they reduce to rather unsatisfactory equations of state. 

Thus every value of W (provided W > F ) and therefore of X is possible, that is to say, the translational energy of a free electron is not quantized. For the electron in the box there are, however, boundary conditions that must be fulfilled... 

The basis for this separation lies in the fact that electronic transitions occur on a much shorter timescale, and rotational transitions occur on a much longer timescale, than 10 vibrational transitions. The translational energy of the molecule may be ignored in this discussion because it is essentially not quantized. 

This gives a form of quantization for translational energy in one degree of freedom, which does in fact agree with that later formulated on the basis of a more general theory. 

The quantization of translational energy has already been considered. For vibrational systems the equation is found to yield physically admissible solutions only for values of E defined by the relation E = n+ )hv. The successive energy levels differ by hv as required. The lowest value occurs when the integer n is zero, so that E = Jiv. Schrodinger s equation, unlike the quantum rule which it has superseded, predicts the existence of a so-called zero-point energy. The assumption that there is such a thing is in fact required for the explanation of certain phenomena, so that in this respect the new equation possesses an important advantage. 

The influence of the constant term i is connected purely with the entropy. If i becomes larger, the vapom pressure becomes greater also. Scrutiny of the formulae reveals several points of interest. As the mass of the molecules or atoms increases, the vapour-pressure constant increases also. This is because a greater mass leads to a more fine-grained quantization of the translational energy in the gas phase. There are thus relatively more possibilities of existence as... 

Now a given translational energy corresponds to a multiplicity of states, since each rectangular component of the momentum is itself quantized, and numerous values of and can satisfy... 

Figure 1 Triangular contour plots showing the variation of detailed rate constants. Values of vibrational energy (V ) and rotational energy (RO are plotted, ignoring quantization along the rectilinear axes and those of translational energy (T ) are indicated by the dashed diagonal lines. Units are kcal mol (1 kcal mol = 4.18 kJmol ). Panel (a) shows the variation of the detailed rate constants for reaction (22) in the exothermic direction, i.e. kt ( , i) kt ( , J ,T),as determined by i.r. chemiluminescence experiments.

Molecules may absorb or emit energy in three different ways. These include electronic transitions, vibrational transitions, and rotational transitions. Molecules also possess translational energy but it is not observed with present instrumentation. The total quantized energy of a molecule is therefore given by the expression... 

For the purposes of illustration consider a system of four indistinguishable atoms which move about independently in a large container. For simplicity it will be supposed that they are free to move only along the x co-ordinate. Let Cq, e, etc., be the quantized translational energies of the separate atoms, counting upwards from the lowest level q. It will be supposed that these levels are equally spaced thus = = =... 

X It should be noted that discrete energy states occur in most problems, but not all. For example the translational energies of a particle not confined in a container are not quantized. Quantization arises from the requirement that shall be single-valued, etc., together with the boundary conditions of the particular problem. 

Earlier, we discussed the quantized electronic energy levels of an atom (Chapter 7) and of a molecule (Chapter 11). In addition, the kinetic energy levels of a molecule s motions—vibrational, rotational, and translational—are quantized. And now we ll see that the energy state of a whole system of particles is quantized, too. 

This expression shows that the difference decreases as the length L of the box increases and that it becomes zero when the walls are infinitely far apart (Fig. 9.19). Atoms and molecules free to move in laboratory-sized vessels may therefore be treated as though their translational energy is not quantized, because L is so large. The expression also shows that the separation decreases as the mass of the particle increases. Particles of macroscopic mass (like balls and planets and even minute specks of dust) behave as though their translational motion is unquantized. Both these conclusions are true in general ... 

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