Kinetic energy microscopic

One of the primary goals of current research in the area of tribology is to understand how it is that the kinetic energy of a sliding object is converted into internal energy. These dissipation mechanisms detennine the rate of energy flow from macroscopic motion into the microscopic modes of the system. Numerous mechanisms can be... 

The recipe (5.58) is even more sensitive to the high-frequency dependence of kjj than similar criterion (5.53), which was used before averaging over kinetic energy of collisions E. It is a much better test for validity of microscopic rate constant calculation than the line width s j-dependence, which was checked in Fig. 5.6. Comparison of experimental and theoretical data on ZR for the Ar-N2 system presented in [191] is shown in Fig. 5.7. The maximum value Zr = 22 corresponding to point 3 at 300 K is determined from the rate constants obtained in [220],... 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

In a hypothetical system for modeling kinetics, the microscopic cells must be open systems. It is useful to consider entropy as a fluxlike quantity capable of flowing from one part of a system to another, just like energy, mass, and charge. Entropy flux, denoted by Js, is related to the heat flux. An expression that relates Js to measurable fluxes is derived below. Mass, charge, and energy are conserved quantities and additional restrictions on the flux of conserved quantities apply. However, entropy is not conserved—it can be created or destroyed locally. The consequences of entropy production are developed below. 

The mechanism by which equilibrium is attained can only be visualized in terms of microscopic theories. In the kinetic sense, equilibrium is reached in a gas when collisions among molecules redistribute the velocilies lor kinetic energies) of each molecule until a Maxwellian distribution is reached for the whole bulk. In the case of the trend toward equilibrium for two solid bodies brought into physical contact, we visualize the transfer of energy by means of free electrons and phonons (lattice vibrations). 

Figure 20. Microscopic cross section as function of kinetic energy for formation of HeH+ and H+ from reaction of H2+ (t>-0) with helium.85 ...

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