Hybridization 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... 

The attachment coefficient, 3, corresponding to the hybrid theory can be shown to be (Fuchs, 1964)... 

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). 

The average attachment coefficient for the hybrid theory is of the form ... 

A four point (n = 4) approximation is adequate since 3 does not change much when 5 and 6 point approximations are used It is apparent that if Og = 1 (a monodisperse aerosol) equation (14) reduces to the hybrid theory with the CMD in place of the diameter. 

Figure 5 illustrates the effect of the geometric standard deviation on the attachment coefficient using the hybrid theory. 

Figure 5. Attachment coefficient vs CMD using the hybrid theory for various particle distributions.

The most complete theory for aerosol coagulation is that of Fuchs (1964). Since the attachment of radon progeny to aerosols can be considered as the coagulation of radon progeny (small diameter particle) to aerosols (large diameter particle), it is reasonable to use Fuchs theory to describe this process. The hybrid theory is an approximation to Fuchs theory and thus can be used to describe the attachment of radon progeny to aerosols over the entire aerosol size spectrum. 

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. 

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 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. 

Figure 5 shows the variation of the hybrid theory with CMD for various Og. It is obvious that assuming an aerosol to be mono-disperse when it is in fact polydisperse leads to an underestimation of the attachment coefficient, leading in turn to large errors in calculation of theoretical unattached fraction. 

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. 

Calculation of the attachment coefficient is required for theoretical prediction of the unattached fraction of radon progeny. The hybrid theory, which is a form of Fuchs theory with certain justifiable assumptions, can be used to describe attachment to aerosols under all conditions of Og and CMD. 

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. 

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. 

Utilize the concepts of molecular orbital and hybridization theories to explain multiple bonds, bond angle, diamagnetism, and paramagnetism. 

As discussed in Chapter 1, the development of PES showed that the spectra could be simply interpreted if one assumed that electrons occupy delocalized molecular orbitals (25,26). In contrast, VB theory, which uses localized bond orbitals (LBOs), seems completely useless for interpretation of PES. Additionally, since VB theory describes equivalent electron pairs that occupy LBOs, the experimental PES results seem to be in discord with this theory. An iconic example of this failure of VB theory is the PES of methane that displays two different ionization peaks. These peaks correspond to the a and t2 MOs, but not to the four equivalent C—H LBOs in Pauling s hybridization theory. 

In this section, we first discuss the bonding in two linear triatomic molecules BeH2 with only a bonds and C02 with both a and n bonds. Then we go on to treat other polyatomic molecules with the hybridization theory. Next we discuss the derivation of a self-consistent set of covalent radii for the atoms. Finally, we study the bonding and reactivity of conjugated polyenes by applying Hiickel molecular orbital theory. 

If we prefer to describe the bonding of a polyatomic molecule using localized two-center, two-electron (2c-2e) bonds, we can turn to the hybridization theory, which is an integral part of the valence bond method. In this model, for AX systems, we linearly combine the atomic orbitals on atom A in such a way that the resultant combinations (called hybrid orbitals) point toward the X atoms. For our BeH2 molecule in hand, two equivalent, colinear hybrid orbitals are constructed from the 2s and 2pz orbitals on Be, which can overlap with the two Is hydrogen orbitals to form two Be-H single bonds. (The 2p and 2py... 

In the next chapter, we will present various chemical applications of group theory, including molecular orbital and hybridization theories, spectroscopic selection rules, and molecular vibrations. Before proceeding to these topics, we first need to introduce the character tables of symmetry groups. It should be emphasized that the following treatment is in no way mathematically rigorous. Rather, the presentation is example- and application-oriented. 

There are several reasons for this unsatisfactory state of affairs. Most important is perhars the different conceptual demands on theories of chemistry and physics respectively. In this instance there has been no effort to re-interpret mathematical quantum theory to satisfy the needs of chemistry. The physical, or Copenhagen, interpretation, which is essentially an ensemble theory, is simply not able to handle the individual elementary units needed to formulate a successful theory of chemical cohesion and interaction. Computational dexterity without some mechanistic basis does not constitute a theory. Equally unfortunate has been the dogmatic insistence of theoretical chemists to drag their outdated phenomenological notions into the formulation of a hybrid theory, neither classical nor quantum even to the point of discarding... 

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