Equation Debye

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

This expression is known as the Debye equation. It is therefore obvious that if ttg and p. were to be additive properties then it would be possible to calculate the dielectric constant from a knowledge of molecular structure. 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

Starting with the Debye equation and using a number of simplifying assumptions, various authors contributed to the development of a general equation for molecular electron diffraction intensities, in the final form given by Bartell and Kuchitsu4. For every atom pair in a polyatomic molecule,... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

Of the ethers, rate constants for es reactions are available for tetrahydrofuran (THF). Since the neutralization reaction, THF+ + es, is very fast, only fast reactions with specific rates 10u-1012 M s"1 can be studied (see Matheson, 1975, Table XXXII). Bockrath and Dorfman (1973) compared the observed rate of the reaction es + Na+ in THF, 8 x 1011 M 1s 1, with that calculated from the Debye equation, <3 x 1011 M-1s-1. Although the reaction radius is not well known, the authors note on a spectroscopic basis that Na+ and es are strongly coupled in THF Thus, the reaction of a solute with (Na+, es) in THF is much slower, sometimes by an order of magnitude, than the corresponding reaction with es only. Reaction with pyrene is an example. 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

The integration indicated by Equation (11.21) then is carried out in two steps. From approximately 20 K up, graphical or numerical methods can be used (see Appendix A). However, below 20 K, few data are available. Therefore, it is customary to rely on the Debye equation in this region. 

Use of Debye Equation at Very Low Temperatures. Generally, it is assumed that the Debye equation expresses the behavior of the heat capacity adequately below about 20 K [9]. This relationship [Equation (4.68)],... 

Putnam and Boerio-Gates [19] have measured the heat capacity of pure, crystalline sucrose from 4.99 K to above 298.15 K. Their smoothed results up to 298.15 K are shown in Table 11.7. Use the Debye equation and numerical integration of the experimental data to calculate at 298.15 K. 

The dipole moment is a fundamental property of a molecule (or any dipole unit) in which two opposite charges are separated by a distance . This entity is commonly measured in debye units (symbolized by D), equal to 3.33564 X 10 coulomb-meters, in SI units). Since the net dipole moment of a molecule is equal to the vectorial sum of the individual bond moments, the dipole moment provides valuable information on the structure and electrical properties of that molecule. The dipole moment can be determined by use of the Debye equation for total polarization. Examples of dipole moments (in the gas phase) are water (1.854 D), ammonia (1.471 D), nitromethane (3.46 D), imidazole (3.8 D), toluene (0.375 D), and pyrimidine (2.334 D). Even symmetrical molecules will have a small, but measurable dipole moment, due to centrifugal distortion effects. Methane " for example, has a value of about 5.4 X 10 D. 

In the liquid phase, observed electron-ion recombination rate constants kr in a variety of nonpolar media were, as shown in Fig. 15, in good agreement with the values of k, calculated from the reduced Debye equation,... 

The Debye equations (9.42) are particularly important in interpreting the large dielectric functions of polar liquids one example is water, the most common liquid on our planet. In Fig. 9.15 measured values of the dielectric functions of water at microwave frequencies are compared with the Debye theory. The parameters tod, e0v, and r were chosen to give the best fit to the experimental data r = 0.8 X 10 -11 sec follows immediately from the frequency at which e" is a maximum e0d — e0v is 2e"ax. 

Permanent dipole/induced dipole interaction (Debye equation)... 

This is the Debye equation (subscript D) in Table 10.1 that describes the attraction between a... 

As we have already noted, all molecules display the dispersion component of attraction since all are polarizable and that is the only requirement for the London interaction. Not only is the dispersion component the most ubiquitous of the attractions, but it is also the most important in almost all cases. Only in the case of highly polar molecules such as water is the dipole-dipole interaction greater than the dispersion component. Likewise, the mixed interaction described by the Debye equation is generally the smallest of the three. 

On the other hand, if the rate constant for the quenching step exceeds that expected for a diffusion-controlled process, a modification of the parameters in the Debye equation is indicated. Either the diffusion coefficient D as given by the Stokes-Einstein equation is not applicable because the bulk viscosity is different from the microviscosity experienced, by the quencher (e.g. quenching of aromatic hydrocarbons by O, in paraffin solvents) or the encounter radius RAb is much greater than the gas-kinetic collision radius. In the latter case a long-range quenching... 

Liquids with low viscosity or large 3 (high density or efficient momentum transfer across the boundary layer) have a rotational diffusion coefficient close to that of the Debye equation [220], eqn. (110). For viscous liquids, the rotational diffusion coefficient tends to saturate to a viscosity-independent value. Tanabe [235] has found perdeuterobenzene rotational diffusion to be well described by the Hynes et al. theory [221, 222]. 

Samson and Deutch [258] and Hess [259a] have also discussed the reaction of anisotropic molecules, though only Hess considered rotational relaxation effects. No studies have used the experimentally measured values of rotational relaxation times, which may be 1.5—10 times faster than the Debye equation, eqn. (108), predicts. The theory of Sole and Stockmayer [256] will underestimate the rate of chemical reactions when rotational relaxation is faster than they assumed. 

In the method of trial the intensity I of scattered radiation at any angle d is given (for a solid containing two different types of atom) by the Debye equation... 

The diffracted intensity from a molecule in the gas phase is expressed by the Debye Equation [21] ... 

The Debye Equation for Dielectric Constant.—The Debye equation for the dielectric constant of a gas whose molecules have a permanent electric moment no is... 

If each normal mode is equally weighted in the total response, equations identical in form to (2.18) and (2.19) also hold for the cubic array since the connection between x and v obtains regardless of dimensionality. Here, g(v) — g3 (v) may be approximated by the familiar Debye equation ... 

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