Distribution of energy

FIGURE 3 5 Distribution of energies (a) The number of molecules with energy greater than fact st temper ature T, is shown as the darker green shaded area (b) At some higher tempera ture T2 the curve is flatter and more molecules have energies in excess of fact... 

Central to the categorization of plasmas are electron temperature and electron density. Electrons have a distribution of energies, so it is useful to assume a MaxweUian distribution, in terms of electron energy, E, such that... 

The distribution of energy under idealized conditions is given by Planck s equation in which the energy radiated into a hemisphere in W/cm in... 

The spectral distribution of energy flux from a black body is expressed by Planck s law ... 

FIG. 5-11 Distribution of energy in the spectrum of a Llackbody To convert microns to micrometers, multiply by unity To convert ergs per square centimeter-second-micron-K to watts per square meter, per meter, per K, multiply by 10 . ... 

Heterogeneity Adsorbents and ion exchangers can be physically and chemically heterogeneous. Although exceptions exist, solutes generally compete for the same sites. Models for adsorbent heterogeneity have been developed for both discrete and continuous distributions of energies [Ross and Olivier, On Physical Adsorption, Interscience, New York, 1964 Jaroniec and Madey, Rudzinsld and Everett, gen. refs.]. 

We assume, as a matter of definition, that a raised weight represents a distribution of energy most useful in this respect, as it merely requires to fall in order to perform, on an appropriately connected mechanism, an amount of work equal to the whole of the potential energy of the system composed of the earth and weight in the initial configuration. 

Any distribution of energy which can be completely converted into the potential energy of a raised weight is called available energy. If a part only of the given distribution of energy is so convertible, we speak of this as the available part, and the rest as the unavailable part of the total quantity of energy. 

The partition function turns out to be a very useful quantity in our calculations, and it is important that we understand its properties. As we said earlier, the name comes from the realization that r is a measure of the distribution of energy among excited states, as can be seen by writing r... 

The interaction energy of the valence electron with the two atomic 3d electrons, with parallel spins, is accordingly —0.67 ev, and the width of the energy band that would be occupied by uncoupled valence electrons is 1.34 ev. The number of orbitals in this band can be calculated from the equation for the distribution of energy levels for an electron in a box. The number of levels per atom is... 

Ng, K. and Yianneskis, M. (1999) Observations on the distribution of energy dissipation in stirred vessels. Eluid Mixing 6 Symposium, 1999, Bradford. 

In all of the above equations, is assumed to be constant and uniform throughout the flow field. In most items of bioprocess equipment, however, there is a spatial distribution of energy dissipation. The definition of an average or a maximum energy dissipation rate is notoriously difficult in the case of bioprocess equipment such as high pressure homogenisers, centrifuges, pumps and microfiltration units which all have complex flow fields. 

Kinetics on the level of individual molecules is often referred to as reaction dynamics. Subtle details are taken into account, such as the effect of the orientation of molecules in a collision that may result in a reaction, and the distribution of energy over a molecule s various degrees of freedom. This is the fundamental level of study needed if we want to link reactivity to quantum mechanics, which is really what rules the game at this fundamental level. This is the domain of molecular beam experiments, laser spectroscopy, ah initio theoretical chemistry and transition state theory. It is at this level that we can learn what determines whether a chemical reaction is feasible. 

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

Discuss the basic assumption underlying the Boltzmann distribution of energies for an ensemble of molecules. 

S.4. Guidelines for scale-up of semibatch reactors for fast homogeneous reactions in the absence of data on chemical kinetics and on the distribution of energy dissipation in the reaction zone... 

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