Debye temperature limitations

The upper limit of the dimensionless variable Vp, is typically written in tenns of the Debye temperature 9, as... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

The limiting angular vibrational frequency, D, that exists defines the Debye temperature, 0D, as... 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

For the solutions of Exercises 12 through 18, the Debye-Hiickel limiting law is sufficiently accurate, and the numbers in parentheses can be read as either molality or molarity at this level of approximation. The temperature to be used is 25°C. 

For minerals and inorganic samples, low temperature is almost useless to improve structure solution and only marginally relevant to improve the refinement, unless dealing with host-guest materials like zeolites. In facts, for harder materials ambient temperature is already quite comparable and sometimes lower than the Debye temperature. Therefore, resolution is seldom a limitation for structure refinement of minerals at ambient temperature. On the contrary, for macromolecules and especially for proteins, the low temperature significantly increases the number of... 

B is a constant that depends on the properties of the solution, for example, on its dielectric constant, and on the temperature. For water at 25°C, B = 0.51 Ll/2 mol 1/2. The Debye-Hiickel limiting law applies only for solutions of low ionic strength, for example, below 0.01 M for 1 1 electrolytes, such as NaN03, and below 0.001 M for electrolytes of higher charge. 

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... 

Mossbauer s discovery [49] consisted in the fact that when the nuclei of the emitter and the absorber are included in a solid matrix, they vibrate in a crystal lattice [49,54,56], Therefore, owing to the essential quantum character of solid vibrations (see Section 1.4), the atoms located in a solid matrix are limited to a certain collection of quantized lattice vibration energies [54], Consequently, if the recoil energy is smaller than the lowest quantized lattice vibration energy, Ew, then / v = 0D, in which, k is the Boltzmann constant and 0D is the Debye temperature of the solid. In this case, this... 

The Debye temperature can also be obtained from the elastic constants. The measurement of the elastic constants of polycrystalline AIN was used by Slack et al [8] to derive the Debye temperature, giving 0D = 950 K. Therefore, Slack et al have criticised the value of the AIN Debye temperature 0D = 800 + 2 K, derived from the heat capacity measurements by Koshchenko et al [6], as too low. Also, Slack s value differs considerably from Meng s result [7]. Since the cubic dependence T3 approximates the Debye specific heat well in the temperature range below T = 0d/5O [9], it is likely that the upper temperature limit used by Meng is too high and led to error and the difference from the results of Slack et al [8],... 

The factor A in equation (123) is proportional to 1/(D T), as shown on page 150 hence, a further test of this equation is to determine the slope of the plot of log S/So against Vy from Solubility data at different temperatures and in media of different dielectric constants. Such measurements have been made in water at 75° (D = 63.7), in mixtures of water and ethyl alcohol (D = 33.8 to 78.6), in methyl alcohol (D = 30), in acetone (D = 21), and in ethylene chloride D = 10.4). The results have been found in all cases to be in very fair agreement with the requirements of the Debye-Huckel limiting law as may be expected, appreciable discrepancies occur when the saturating salt is of a high valence type, especially in the presence of added ions of high valence. ... 

The mean ionic activity coefficients of hydrochloric acid in water and in dioxane-water mixtures, at an ionic strength of 0.001, together with the corresponding temperatures and dielectric constants are given below. Use the data to test the Debye-Hilckel limiting law, with particular reference to the effect of temperature and dielectric constant. 

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