Binary density operator

B. Binary Density Operator in Three-Particle Collision Approximation— Boltzmann Equation for Nonideal Gases... 

C. Binary Density Operator for Higher Densities—Bound States... 

Thus the binary density operator is given only by the single-particle density operator Fl, which depends on the earlier time t0. Therefore, retardation effects appear. In order to eliminate this time we use the formal solution (1.30) for F, ... 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

Our starting point is the expansion (2.37) for the binary density operator,... 

B. Binary Collision Approximation for the Two-Particle Density Operator— Kinetic Equations for Free Particles and Atoms... 

As in Section II, the initial values F12(t0) and F123(/0) for the binary and the three-particle density operators must be determined. For this purpose we have to generalize the Bogolyubov condition of the weakening of initial correlations given by (2.4) for systems that do not support the formation of bound states. 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

Electrolytic (coukxnetric) hygrometers The quantity of electricity required to carry out a chemical reaction is measured. The principle is based upon Faraday s law of electrolysis. Water is absorbed on to a thin film of dessicant (e.g. P2O5) and electrolysed. The current required for the electrolysis varies according to the amount of water vapour absorbed. The current depends also upon the flowrate. Capable of high precision. Used in the range 1000 to 3000 ppm of water by volume. Somewhat complicated procedure. Recombination of products to water is necessary after electrolysis. Density, pressure and flowrates have to be maintained precisely. Contamination can poison the cell. It is ideal for binary mixtures but is of limited range. Suitable for on-line operation. 

Subsequent to polymer manufacture, it is often necessary to remove dissolved volatiles, such as solvents, untreated monomer, moisture, and impurities from the product. Moreover, volatiles, water, and other components often need to be removed prior to the shaping step. For the dissolved volatiles to be removed, they must diffuse to some melt-vapor interface. This mass-transport operation, called devolatilization, constitutes an important elementary step in polymer processing, and is discussed in Chapter 8. For a detailed discussion of diffusion, the reader is referred to the many texts available on the subject here we will only present the equation of continuity for a binary system of constant density, where a low concentration of a minor component A diffuses through the major component ... 

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