Frequency distribution function

Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with a constant fractional rate of surface renewal, v, but an upper limit on surface age equal to the life of the jet, r, show that the surface age frequency distribution function, 4>(t), for this case is given by ... 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

It is possible to demonstrate the relation between v and T giving an arbitrary assumed frequency distribution function f(T,v). This is done in Figure 12 for two temperatures T and T ST according to the function ... 

Therefore, if it is possible to find a frequency distribution function corresponding to experimental results, it would be possible to find a relation using Equation 24 which would correspond to a Maxwell distribution. With this equation the variable 1/8T X Nme/Nm of Figure 11 should be transformed into the variable l/8v X Nme/Nm. If pairs of values using this performance corresponding to the intersection points of the correlation frequencies with the theoretically determined distribution (cf. Figure 5) are gained, the theoretically determined distribution function would be confirmed by experimental results. 

The inner-integral of Equation (2) was numerically integrated using a four-point Gaussian quadrature. The mean bubble length was calculated from the first moment of the frequency distribution function given in Equation (2). 

Figures 14a-14c show SEM micrographs of the original particte population recovered from HIPS and the separated large and small particle sIk fractions. Figure 15 shows frequency distribution functions of these three particle size distributions.

Fig. 13. Frequency distribution functions of actual bubble size (T27). Experimental conditions are the same as in Fig. 12.

Example An aerosol is composed of particles randomly distributed in space with an average concentration Noo particles per unit volume. Let r be the center-to-center distance between two panicles that are nearest neighbors. Values of r vary among the panicle pairs, that is, there is a distribution of r values. Determine the frequency distribution function forr and calculate the average value of r (Chandrasekhar, 1943). 

The inadequacy of the Debye approximation in describing the details of the frequency distribution function in a real solid is well known. This results in noticeable disparities between Debye temperatures derived from the results of different experimental techniques used to elucidate this parameter on the same solid, or over different temperature ranges. Substantial discrepancies may be expected in solids containing two (or more) different atoms in the unit cell. This has been demonstrated by the Debye-Waller factors recorded for the two different Mdssbauer nuclei in the case of Snl4,7 or when the Debye-Waller factor has been compared with the thermal shift results for the same Mdssbauer nucleus in the iron cyanides.8 The possible contribution due to an intrinsic thermal change of the isomer shift may be obscured by an improper assignment of an effective Debye temperature. 

Figure 2. Relative contributions of particulate size classes to total number, surface area, and volume concentration for a hypothetical power-law frequency distribution function, p =3...

The displacement vector of an atom can be decomposed along three coordinate axes. Each of 3 N lattice waves has its frequency Vi and its amplitude Ui. The value of the mean-square amplitude can be calculated if the frequency distribution function gi(v) is known. According to the definition of the mean value of a function, we have... 

Note, that the line width of a single-frequency laser is also strongly related to temporal coherence a narrow fine width means high temporal coherence. The line width can be used to estimate the coherence time, but the conversion depends on the spectral shape, and the relationship between optical bandwidth and temporal coherence is not always simple. In the case of exponential coherence decay, e.g. as encountered for a laser whose performance is limited by noise, the width of the frequency distribution function (ftdl width at half maximum (FWHM)) is... 

The anti-crystal holes should be permeated by the photons obeying the frequency distribution function with an upper limit. This is due to the interface between the anti-crystal hole and the ordered part be able to act as a filter for the photons. The molar photon energy loss of the anti-crystal holes, AUh, due to the cavity radiation from Tb to T/ is given... 

More recent experiments have shown that such a model is an extreme over-simpliflcation. Impurities form local modes or resonance modes under a variety of conditions which depend on relative masses and relative interatomic interactions. A detailed theory has been worked out for the recoil-free fraction of an impurity (Mannheim, 1968). It is expressed in terms of the host lattice phonon frequency distribution function, the ratio of the masses of the host and impurity atoms and the ratio of the force constant between host atoms to that between host and impurity atoms. In Figure 6.4 the experimental results and the theoretical curves for Fe in vanadium are shown. Further details and recent experimental results for this model may be found in the Proceedings of the Indian National Science Academy International Conference on the Applications of the Mossbauer Effect (1982), pp. 619-63. 

In the more general case can be expressed in terms of the phonon frequency distribution function G(Q) which can be measured experimentally... 

This analysis requires the use of the particle size frequency distribution function, defined by... 

The frequency distribution function in equation 18.66 can be substituted into the statistical thermodynamic expressions for the various state functions, and various thermodynamic properties determined for crystals. We are interested in the expression for the heat capacity. It is (omitting the details of the derivation) ... 

Historical Note 1. Einstein s first work on heat capacity dates to 1907 [6]. In that article, Einstein limited his attention to solids formed by equivalent oscillators—the so-called Einstein model. In this case, the frequency distribution function is a Dirac 5 distribution centered on a... 

Table 12.7 Equilibrium distances (A) and interaction energies (kJ mol ) for hydrogen bonds as in Fig. 12.14. In the diagonal entries, the second Une gives the corresponding separation and peak heights in frequency distribution functions (equation 12.1)...

G Gilat, JL Raubenheuner. Accurate numerical method for calculating frequency-distribution functions in solids. Phys Rev 144 390, 1966. 

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