Debye function table

Debye gave tables are also graph for P(q) as a function of size factor . Experimentally,... 

When integrated numerically, the function (3.4) proves to be not unlike an Einstein specific heat curve, except at low temperatures. To facilitate calculations with the Debye function, we give in Table XIV-1... 

A selection of values of the Debye function are tabulated in table 12.3. As T increases D Tj0) tends to unity, so that at temperatures above the characteristic temperature we have a theoretical justification of the empirical rule of Dulong and Petit, namely that 3R for atomic solids. On the other hand at low temperatures D Tj ) tends to zero, and for T/ <0T we have the simple approximate formula... 

Three pairs of C,T data were acquired at temperatures near 80 and 300 K, respectively, from which specific heat values appropriate to 80.0 K and 300.0 K were computed in the manner outlined in Table II. Consider the T = 80K data from Ciattice = C-A-yT, values of the Debye function (DF= Ciattice/3R) and consequently 6d,t were obtained, using standard DF vs. Bd/T tables. By a reversal of that procedure, Ciattice and subsequently C at 80.0 K were deduced. In a like manner, the room-temperature data were corrected to a uniform 300.0 K. 

Table VI illustrates in detail how the low-temperature parameters A, y, and D.LT can be used with the tabulated Debye function to obtain the specific heat values for Inconel X750 (HIP)-STDA at 80 and 300 K, respectively.

Table VI. Debye-Function-Assisted Extrapolation of Low-Temperature Specific Heat Data into the Intermediate-Temperature Regime for Inconel X750 (HIP)...

In Fig. 2.42 results from the ATHAS laboratory on group IV chalcogenides are listed [18]. The crystals of these compounds form a link between strict layer stractures whose heat capacities should be approximated with a two-dimensional Debye function, and crystals of NaCl stracture with equally strong bonds in all three directions of space and, thus, should be approximated by a three-dimensional Debye function. As expected, the heat capacities correspond to the structures. The dashes in the table indicate that no reasonable fit could be obtained for the experimental data to the given Debye function. For GeSe both approaches were possible, but the two-dimensional Debye function represents the heat capacity better. For SnS and SnSe, the temperature range for data fit was somewhat too narrow to yield a clear answer. 

II-Ultrasonic Relaxation. The ultrasonic results for NaSCN, NaNOj, and NaN02 DMA solutions are reported in Table I in the form of A and B parameters (Eq I). Fig. 4 reports representative functions for the NaN02 solutions at 25 C. The solid lines are the computed Debye functions (2). 

Tables of the value of the Debye function D are available. A software package such as Mathematica can easily carry out the evaluation. The appropriate value of d for a given crystal is chosen by fitting heat capacity data to Eq. (28.2-24).

If a table of the Debye function is available find the ratio of your result to the Debye result for several... 

The optical vibrations have also incteased in number by the introduction of the second CHg-group. A look at Tables III. 11 and III. 12 shows that there should only be little effect of the tical vibrations below 150° K. A try to fit a one (or three) dimensional Debye function (Eqs. 11.146 or 11.161) to the data of Table III.15 between 10 and 150° K failed. Neither leads to a constant 0j-temperature as is possible in case of polyethylene, polypropylene and also polystyrene. Of the 10 low frequency vibrations, an average of only 6 seem to be excited at 150° K, indicating that in comparison with polyethylene their average frequencies lie somewhat higher. They must also be higher than comparable frequencies in polypropylene. From the chemical structure of polyisobutylene one would like to conclude that this relative decrease in heat capacity is caused by steric hindrance of the two methyl groups bound to the same carbon atom. 

The important thing to notice about the Debye function is that for a given substance, the lattice heat capacity is dependent only on a mathematical function of the ratio of the absolute temperature to the characteristic Debye temperature. This mathematical function applies for all materials, with 0 varying from material to material. Selected values of 9 are given in Table 3.3. 

Table 11.10 H2O dipole moment (Debye) as a function of theory (valence correlation only), experimental value is 1.847 D...

Table A4.7 The Debye thermodynamic functions expressed in terms of 8D/T... 

In the tables we find i Ag+/Ag = 0.7996 V and 2 Agci/Ag = 0.2223 V. From the above it is clear that primarily the silver-silver chloride electrode functions as a pAg electrode, i.e., it measures oAg+ at an ionic strength above 0.01 (cf., extended Debye-Hiickel expressions) the calculation of [Ag+ ] becomes more difficult, and even more so for [Cl ], where the solubility product value is also involved. 

Table 4.3. Water dimer properties the interaction energy (Ei t) in kcal/mol, the intermolecular distance (R00) in A, and the dipole moment p. in Debye, calculated using the B88/P86 exchange-correlation functional and different basis sets.

Boyer et al. [20] have measured the heat capacity of crystalline adenine, a compound of biologic importance, with high precision, from about 7 K to over 300 K, and calculated the standard entropy of adenine. Table 11.8 contains a sampling of their data over the range from 7.404 K to 298.15 K. Use those data to calculate the standard entropy of adenine at 298.15 K, which assume the Debye relationship for Cp. The value for 298.15 K is calculated by the authors from a function fitted to the original data. 

We have applied Pitzer s equations at T = 298.15 K, but they are not limited to that temperature and can be applied at any temperature where the coefficients are known.k Table I8.l (and Table A7.1 of Appendix 7) gives the Debye-Hiickel coefficients AA, Ah, and Aj as a function of temperature, but the coefficients specific to the electrolyte are tabulated in Appendix 7 only at T = 298.15 K. The usual solution to this problem is to express the coefficients as... 

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

Earlier, when discussing historical development, we mentioned that different workers have used different equations to describe the Debye-Hiickel constant (A, Eq. 2.35) as a function of temperature. For example, at 0°C, the value of this constant is 0.3781, 0.3764, and 0.3767 kg1/2 mol-1/2 for the FREZCHEM, Archer and Wang (1990), and Pitzer (1991) models, respectively. At NaCl = 5 m and 0 °C, the calculated mean activity coefficients using these three parameters evaluated with the FREZCHEM model are 0.7957, 0.7995, and 0.7988, respectively. The largest discrepancy is 0.48%, which is within the range of model errors for activity coefficients (Table 3.5). 

Table B.l lists all the chemical reactions and their temperature dependence. Table B.2 lists the Debye-Hiickel constants A,p and Av) as a function of temperature and pressure. Table B.3 lists the numerical arrays used for calculating unsymmetrical interactions (Equations 2.62 and 2.66). Table B.4 lists binary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.5 lists ternary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.6 lists binary and ternary Pitzer-equation parameters for soluble gases as a function of temperature. Table B.7 lists equations used to estimate the molar volume of liquid water and water ice as a function of temperature at 1.01 bar pressure and their compressibilities. Table B.8 lists equations for the molar volume and the compressibilities of soluble ions and gases as a function of temperature. Table B.9 lists the molar volumes of solid phases. Table B.10 lists volumetric Pitzer-equation parameters for ion interactions as a function of temperature. Table B.ll lists pressure-dependent coefficients for volumetric Pitzer-equation parameters. Table B.12 lists parameters used to estimate gas fugacities using the Duan et al. (1992b) model.

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