Molecular value

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

Tokiwa and Ohki (10) have shown that 1) the Ka value of the micellized LDAO is different from the molecular form, viz., 10 2) while the values of Ka are independent of the degree of protonation ( ) below the critical micelle concentration (CMC), they are dependent on 3 at concentrations above the CMC. These authors did not explain why the micellar pKa (5.9) at 3 = 0 is different from the molecular value (pKa = 4.9). 

Corresponding values for F, evaluated by finite differences from the governing equations, are shown in Fig. 3.15. As PCp increases, circulation causes F to rise more rapidly, but the effect is not large Tp for a given F decreases by at most a factor of three as Pep/(1 + k) increases from zero to infinity. In fact, the Kronig-Brink curve in Fig. 3.17 is closely approximated by Eq. (3-80) with p replaced by 2.5 p. Thus circulation causes an effective diffusivity at most 2.5 times the molecular value. For negligible external resistance, the short-time approximation given by Eq. (3-72) becomes... 

For the case of turbulent flow, the basic transport laws have to be changed to account for turbulence, and many models have been developed since the time of Reynolds. He tried to keep the simple form of Eqns. (3.4-1), (3.4-2), and (3.4-3), but with transport coefficients modified with the so-called eddy diffusivities that are added to the molecular values. So, the net effect of turbulence would be to increase the transfer rates. 

In general however only measurements on powdered samples are performed, which yield the average magnetic susceptibility. In this case there is no way for obtaining the principal molecular values and directions. 

If we can expect that the eddy momentum and energy transport will both be increased in the same proportion compared with their molecular values, we might anticipate that heat-transfer coefficients can be calculated by Eq. (5-56) with the ordinary molecular Prandtl number used in the computation. It turns out that the assumption that Pr, = Pr is a good one because heat-transfer calculations based on the fluid-friction analogy match experimental data very well. For this calculation we need experimental values of C/ for turbulent flow. 

The stereochemical interpretation of the alkali-graphite compounds may also be explained in this way (57). If the volume of graphite is deducted from the molecular value of the compound CgMe, the alkali metal increments shown in Table III, column 2, are obtained. Compari-... 

Since these local group susceptibility values are derived under the explicit assumption that all electron motions are localized, the difference between the observed magnetic susceptibility and that calculated from local group values should provide a quantitative measure of electron delocalization and perhaps aromaticity.19 There are also a variety of other applications for the local susceptibilities given in Table 8. For instance, molecular values can be measured in a number of small molecules by molecular beam techniques.1 In many of these molecules an accurate measurement of the susceptibility anisotropy is difficult. Magnetic susceptibilities calculated by the methods of this paper can thus be combined with g values obtained by other methods to extract quad-rupole moments. Alternatively, calculated susceptibility anisotropies... 

The situation is somehow similar to the effective diffusion the presence of many spatial scales below the observational one increases the effective diffusion well above molecular values (Sect. 2.2.2) and may even change the character of diffusion turning it into anomalous (Sect. 2.8.1). Here, the convoluted geometry generated by the flow changes the production term in the chemical kinetics from what is expected in a fast binary reaction occurring at a smooth interface (/3 = 0) to an anomalous kinetics (f3 > 0). 

As the pressure was further increased, the pumping speed abruptly decreased to a value near the free-molecular value [ ]. Unfortunately the study did not extend to higher pressures to determine if this speed would remain constant. Later analyses indicated that the pumping speeds were probably valid, but the reason for the decrease is unknown. 

Gas Flow in Nanochannels, Fig. 2 Poiseuille coefficient versus rarefaction parameter 5 solid lines, kinetic equation solution [11] pointed line, free-molecular value based on Eq. 12 dashed line, Navier-Stokes solution Eq. 16... 

Using typical molecular values of cIab 0.35 nm and fji = 14 amu for room temperature collisions between N2 molecules, one observes the magnitude of a hard sphere collision rate constant to be on the order of 2.6 X 10 cm /molecule s. 

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