Energy distribution, Boltzmann

Control of sonochemical reactions is subject to the same limitation that any thermal process has the Boltzmann energy distribution means that the energy per individual molecule wiU vary widely. One does have easy control, however, over the energetics of cavitation through the parameters of acoustic intensity, temperature, ambient gas, and solvent choice. The thermal conductivity of the ambient gas (eg, a variable He/Ar atmosphere) and the overaU solvent vapor pressure provide easy methods for the experimental control of the peak temperatures generated during the cavitational coUapse. 

The formidable problems that are associated with the interpretation of LP kinetic data for nonstatistical IM reactions can be entirely avoided if the reactions can be studied in the HPL of kinetic behavior. In the HPL, the energy content of the initially formed species, X and Y, in reaction (2) would be very rapidly changed by collisions with the buffer gas so that the altered species, X and Y, would have normal Boltzmann distributions of energy. Furthermore, those Boltzmann energy distributions would be continuously refreshed as the most energetic X and Y within the distributions move forwards or backwards along the reaction coordinate. The interpretation of rate constants measured in the HPL is expected to be relatively straightforward because conventional transition-state theory can then be applied. 

The coefficient D, being proportional to a normalizing coefficient C of the Maxwell-Boltzmann energy distribution W = C exp(- / ), is determined by the parameters of the hat-curved model as... 

Note that this expression differs from the quantum mechanical partition function in Eq. (A.16) it is the high-temperature limit, where hi/i/kBT is small. The Boltzmann energy distribution function, as given by Eq. (A.36), takes the form... 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

Figure 3-9. Boltzmann energy distributions at 10°C (solid line) and 30°C (dashed line). The inset is a continuation of the right-hand portion of the graph with the scale of the abscissa (E) unchanged and that of the ordinate [ ( )] expanded by 104. The difference between the two curves is extremely small, except at high energies although very few molecules are in this "high-energy tail, there are many more such molecules at the higher temperature.

Very few molecules possess extremely high kinetic energies. In fact, the probability that a molecule has the kinetic energy E decreases exponentially as E increases (Fig. 3-9). The precise statement of this is the Boltzmann energy distribution, which describes the frequency with which specific kinetic energies are possessed by molecules at equilibrium at absolute temperature T ... 

Figure 3-10. Schematic representation of the progress of a chemical reaction in the forward direction. The barrier height to be overcome is the activation energy A (analogous to /min for crossing membranes). Only reactant molecules possessing sufficient energy to get over the barrier, whose fraction can be described by a Boltzmann energy distribution (Fig. 3-9), are converted into products.

The Boltzmann energy-distribution law states that the probability that a molecule possesses energy E is proportional to More-elaborate... 

For a system having a Maxwell-Boltzmann energy distribution, current classical theories of ion-molecule interactions predict a collision or capture rate coefficient given by... 

Now we want to determine the relation between temperature and the energy involved in other kinds of molecular motions that depend on molecular structure, not just the translation of the molecule. This relation is provided by the Boltzmann energy distribution, which relies on the quantum description of molecular motions. This section defines the Boltzmann distribution and uses it to describe the vibrational energy of diatomic molecules in a gas at temperature T. 

The Boltzmann energy distribution is one of the most widely used relations in the natural sciences, because it provides a reliable way to interpret experimental results in terms of molecular behavior. You should become skilled in its applications. 

A new Deeper Look section introduces the Boltzmann energy distribution and applies it to determine the relative populations of molecular energy states. 

Figure 2.28 Boltzmann energy distribution and the AC for catalyzed and uncatalyzed processes. More molecules have the energy to cross the lower AG cat barrier than the higher AG uncat barrier.

Following Debye-Huckel, the distribution of ions can be calculated via the Boltzmann energy distribution. The application of this law is based on the concept that the ion cloud represents a space charge which is most dense in the vicinity of the central ion and decreases with growing distance to the central ion. A number of simplifying assumptions concerning the state of ions is made ... 

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