Hybridization kinetics theory

When the radius of an aerosol particle, r, is of the order of the mean free path, i, of gas molecules, neither the diffusion nor the kinetic theory can be considered to be strictly valid. Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the transition region, r, which, for very small particles, reduce to the gas kinetic theory, and, for large particles, reduce to the classical diffusion theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows 1. the diffusion theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + i centred on the aerosol particle and 2. kinetic theory predicts the attachment of radon progeny to the particle based on a uniform concentration of radon atoms corresponding to the concentration at a radius of r + L... 

This approximation may be considered to be an alternative to the hybrid theory. The value of di can be found by equating the attachment coefficients for the diffusion and kinetic theories (d x = 8D/v). 

Figures 3 and 4 show the variation of the average attachment coefficient with CMD. It can be seen that for particles of CMD less than 0.06 ym and Og = 2 the kinetic theory predicts an attachment coefficient similar to the hybrid theory, whereas for CMD greater than about 1 ym the diffusion theory and the hybrid theory give approximately the same results. For a more polydisperse aerosol (Og = 3) the kinetic theory deviates from the hybrid theory even at a CMD = 0.01 ym. The diffusion theory is accurate for a CMD greater than about 0.6 ym. These results are easily explained since as the aerosol becomes more polydisperse, there are more large diameter particles (CMD >0.3 ym) which attach according to the diffusion theory. In contrast, the kinetic theory becomes more inaccurate as the aerosol becomes more polydisperse.

The variation of attachment coefficient with Og for CMD =0.2 ym and 0.3 ym is shown in Figures 6 and 7. Again it is apparent that the kinetic theory or diffusion theory are correct only at certain CMD and og. Neither is applicable under all circumstances. It is also evident that the kinetic-diffusion theory is a good approximation to the hybrid theory under all circumstances. 

Unattached fractions of RaA (at t = °°) for two mine aerosols and for a typical room aerosol are shown in Table III. It is usually assumed that the attachment of radon progeny to aerosols of CMD < 0.1 ym follows the kinetic theory. In Table III it is apparent that the hybrid and kinetic theories predict similar unattached fractions for monodisperse aerosols. However, for more polydisperse aerosols, the kinetic theory predicts lower unattached fractions than the diffusion theory and thus the diffusion theory is the more appropriate theory to use. It is also evident that the kinetic-diffusion approximation predicts unattached fractions similar to those predicted by the hybrid theory in all cases. 

The kinetic theory and the diffusion theory may be used for certain aerosol distributions. However, these two theories begin to deviate from the hybrid theory as the aerosol polydispersity increases. Use of either the kinetic theory or the diffusion theory may therefore result in large errors. 

Smith, G.P. (1976). The significance of hybridization kinetic experiments for theories of antibody diversity. Cold Spring Harbor Symp. Quant. Biol. 41, 863-875. 

In his valence bond theory (VB), L. Pauling extended the idea of electron-pair donation by considering the orbitals of the metal which would be needed to accommodate them, and the stereochemical consequences of their hybridization (1931-3). He was thereby able to account for much that was known in the 1930s about the stereochemistry and kinetic behaviour of complexes, and demonstrated the diagnostic value of measuring their magnetic properties. Unfortunately the theory offers no satisfactory explanation of spectroscopic properties and so was... 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

Figures 3 and 4 show the variation of the attachment coefficient with count median diameter for the diffusion, kinetic, hybrid and kinetic-diffusion theory for geometric standard deviations of 2 and 3 respectively.

If it is accepted that the hybrid theory is the most complete theory for attachment of radon progeny to aerosols, the magnitude of the error involved in using exclusively either the kinetic or the diffusion theory can be seen from Figs. 3-7 and Table III. 

The kinetic-diffusion approximation predicts an attachment coefficient similar to the hybrid theory for all CMDs and for both Og m 2 and 3 (Figs. 3 and 4). The advantage of this theory is that the average attachment coefficient can be calculated from an analytical solution numerical techniques are not required. 

Although unattached fractions predicted using the kinetic and diffusion theory for high aerosol concentrations, such as mine atmospheres, are comparable, the same cannot be said for unattached fractions predicted at low aerosol concentrations, such as indoor air. For low aerosol concentrations, neither the kinetic nor the diffusion theory predicts unattached fractions close to those predicted by the hybrid theory. Exclusive use of either of these two theories results in large errors. 

Although the hybrid theory is the most correct theory to use in the prediction of unattached fractions, the error in using the kinetic-diffusion theory in place of the hybrid is small. The kinetic-diffusion theory has the advantage that the solution is in analytical form and thus is more convenient to use than the hybrid theory, which must be solved numerically. 

Y. Zhao and D. G. Truhlar, Hybrid Meta Density Functional Theory Methods for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions The MPW1B95 and MPWB1K Models and Comparative Assessments for Hydrogen Bonding and van der Waals Interactions, J. Phys. Chem. A 108 (2004), 6908. 

Zhao Y, Truhlar DG (2004) Hybrid meta density functional theory methods for thermochemistry, thermochemical kinetics, and noncovalent interactions The MPW1B95 and MPWB1K models and comparative assessments for hydrogen bonding and van der Waals interactions, J Phys Chem A, 108 6908-6918... 

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