Binary intermolecular collisions, collision

The exact kinetic theory of dilute gases leads to expressions for transport coefficients in terms of certain quantities called collision integrals, which depend on the dynamics of binary intermolecular collisions. These integrals are defined in this section. 

The ordinary diffusion equations have been presented for the case of a gas in absence of porous medium. However, in a porous medium, whose pores are all wide compared to the mean free path and provided the total pressure gradient is negligible, it is assumed that the fluxes will still satisfy the relationships of Stefan-Maxwell, since intermolecular collisions still dominate over molecule-wall collisions [19]. In the case of diffusion in porous media, the binary diffusivities are usually replaced by effective diffusion coefficients, to yield... 

The first part of this equation originates from intermolecular collisions (molecular diffusion) and the second from collisions between the molecules and the pore walls (e.g. Knudsen diffusion). Equation (9.338) is valid, strictly speaking, only for equimolar diffusion in a binary solution (Na =-Nb). In porous media, the diffusion coefficient calculated from (9.338) should be corrected for porosity and tortuosity according to eq. (9.103). 

Calculating the mean free path requires assumptions about intermolecular collisions. First, the gas is assumed to be a dilute gas, i.e., a gas for which the mean molecular diameter of the molecules is small compared with the mean molecular spacing. This condition is verified with a good precision for all usual gases under standard pressure and temperature conditions. It involves binary collisions between molecules, and the mean free path may be expressed as X = c j = sJ%RTl%j, ratio of the mean thermal velocity c to the collision rate v. The mean thermal velocity depends on the temperature T and on the specific gas cmistant R. 

Collision-induced absorption takes place by /c-body complexes of atoms, with k = 2,3,... Each of the resulting spectral components may perhaps be expected to show a characteristic variation ( Qk) with gas density q. It is, therefore, of interest to consider virial expansions of spectral moments of binary mixtures of monatomic gases, i.e., an expansion of the observed absorption in terms of powers of gas density [314], Van Kranendonk and associates [401, 403, 314] have argued that the virial expansion of the spectral moments is possible, because the induced dipole moments are short-ranged functions of the intermolecular separations, R, which decrease faster than R 3. We label the two components of a monatomic mixture a and b, and the atoms of species a and b are labeled 1, 2, N and 1, 2, N, respectively. A set of fc-body, irreducible dipole functions U 2, Us,..., Uk, is introduced (as in Eqs. 4.46), according to... 

The kinetic theory of dilute gases accounts for collisions between spherical molecules in the presence of an intermolecular potential. Ordinary molecular diffusion coefficients depend linearly on the average kinetic speed of the molecules and the mean free path of the gas. The mean free path is a measure of the average distance traveled by gas molecules between collisions. When the pore diameter is much larger than the mean free path, collisions with other gas molecules are most probable and ordinary molecular diffusion provides the dominant resistance to mass transfer. Within this context, ordinary molecular diffusion coefficients for binary gas mixtures are predicted, with units of cm /s, via the Chapman-Enskog equation (see Bird et al., 2002, p. 526) ... 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

As indicated above, the vibrational predissociation rates of hydrogen bonded complexes tend to be strongly dependent upon the nature of the vibrational mode initially excited. Another example of this is in the HCN-HF complex where excitation of the C-H stretch yields a lifetime of 13.5 ns while the H-F stretch excited state has a lifetime of only 0.058 ns. Once again this strong mode dependence can be rationalized on the basis of the proximity of the particular intramolecular vibration to the intermolecular bond. In view of the fact that the dissociation of a binary complex can be thought of in terms of a half collision, one might expect that this difference would also be observed in the collisional relaxation data available in the literature. Indeed, Smith and co-workers have carried out an extensive study on the vibrational relaxation of HCN by HF, as well as... 

Effective collision cross sections are related to the reduced matrix elements of the linearized collision operator It and incorporate all of the information about the binary molecular interactions, and therefore, about the intermolecular potential. Effective collision cross sections represent the collisional coupling between microscopic tensor polarizations which depend in general upon the reduced peculiar velocity C and the rotational angular momentum j. The meaning of the indices p, p q, q s, s and t, t is the same as already introduced for the basis tensors In the two-flux approach only cross sections of equal rank in velocity (p = p ) and zero rank in angular momentum (q = q = 0) enter die description of the traditional transport properties. Such cross sections are defined by... 

First it is assumed that the viscosity data of all binary mixtures from among the components of a multicomponent mixture are available at the temperature of interest. In this case the prediction of the viscosity of a multicomponent mixture proceeds from an analysis of the viscosity data for each binary mixture. The first step is to estimate the ratio of cross sections A, . It turns out that this ratio is remarkably insensitive to temperature, to the intermolecular pair potential chosen for its evaluation or to the occurrence of inelastic collisions (Maitland et al. 1987 Vesovic et al. 1995). Consequently, it may be estimated from calculations for any reasonable potential model or from the correlations of the extended law of corresponding states discussed elsewhere (see Chapter 11), once a scaling parameter for energy is available. If this parameter is not listed for the system of interest (Maitland et al. 1987), it may itself be estimated with sufficient accuracy using the combination rule... 

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