Solids, characteristic temperature

Figure 12-11. Thickness dependence of the electron only j(V) characteristics at L=0.22, 0.31, and 0.37 pm. Solid lines have been calculated for an exponential distribution of electron traps of the total density 101 cnTJ and a characteristic temperature T,.= 1500 K (Ref. [41[).

These ideas are best learned by using them yourself. Let us take an example from your own laboratory work. You have observed the heating of a variety of solid materials sulfur, wax, tin, lead, silver chloride, and copper. Each melts at a characteristic temperature. This fact led us to the generalization A solid melts to a liquid at a characteristic temperature. Your confidence in the generalization was bolstered by the additional information (communicated to you, not experienced) that this applies to hundreds of thousands of substances. ... 

Early discussions, by Tammann and by Hedvall, considered the possible existence of a common characteristic (approximate) temperature for a solid at which chemical interactions with other reactants became detectable. For example [111], such a characteristic temperature for CaO, measured in various reactions with CuS04, Co3(P04)2, MgC03, and MnSi03, was found to be 788—838 K. Similarly, the onset of reaction of BaO with the sulphates of Mg, Ca, Sr, Co, Cu, and Zn occurred between 601 and 645 K. In the latter example, it has been shown that the fusion of Ba(OH)2 (an impurity not easily excluded from BaO) could contribute to the initiation of reaction. Eutectic formation during the reactions of BaCl2 with alkali metal sulphates... 

Characteristic temperature intervals of gas - solid body interaction... 

The boiling points of different types of liquids vary widely. They are an important physical characteristic both of liquids and of the many solids that melt to become liquids and then boil at a certain characteristic temperature. 

Figure 22. The configurational entropy Sc per lattice site as calculated from the LCT for a constant pressure, high molar mass (M = 40001) F-S polymer melt as a function of the reduced temperature ST = (T — To)/Tq, defined relative to the ideal glass transition temperature To at which Sc extrapolates to zero. The specific entropy is normalized by its maximum value i = Sc T = Ta), as in Fig. 6. Solid and dashed curves refer to pressures of F = 1 atm (0.101325 MPa) and P = 240 atm (24.3 MPa), respectively. The characteristic temperatures of glass formation, the ideal glass transition temperature To, the glass transition temperature Tg, the crossover temperature Tj, and the Arrhenius temperature Ta are indicated in the figure. The inset presents the LCT estimates for the size z = 1/of the CRR in the same system as a function of the reduced temperature 5Ta = T — TaI/Ta. Solid and dashed curves in the inset correspond to pressures of P = 1 atm (0.101325 MPa) and F = 240 atm (24.3 MPa), respectively. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21350 (2005). Copyright 2005, American Chemical Society.)...

Heating up amorphous solids, we observe an unsteady increase A0th of the thermal expansion coefficient in a characteristic temperature range. This effect is coupled with the formation of the free volume (Fig. 10) and a strong decrease of... 

For solids the matter is not quite so simple, and the more exacting theories of Einstein, Debye, and others show that the atomic heal should be expected to vary with the temperature. According lo Debye, there is a certain characteristic temperature lor each crystalline solid at which its atomic heal should equal 5.67 calories per degree. Einstein s theory expresses this temperature as hv /k. in which h is Planck s constant, k is Bolizmanns constant, and r, is a frequency characteristic of ihe atom in question vibrating in the crystal lattice. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equilibrium, or the three roots (vapor, liquid, solid) characteristic of the triple point. 

Fig. 3. Analytical results of residual electron density as a function of time for various values of characteristic temperature T0. The values of other parameters used are tabulated in Table I. The dashed curves are the analytical results obtained from (15), and the solid lines are those obtained from (20). For reference purposes numerical results are shown by dotted curves.

Fig. 3.25. Experimental and theoretical J-V characteristics of theAko electron only device at higher temperatures. The solid lines are fits of the theoretical prediction to trap model where the current obeys the power law J = V H, while the symbols are the experimental data. From these measurements characteristic temperature (7c) of 2300 K can be derived [54].

Thermal Properties at Low Temperatures For solids, the Debye model developed with the aid of statistical mechanics and quantum theory gives a satisfactory representation of the specific heat with temperature. Procedures for calculating values of d, the Debye characteristic temperature, using either elastic constants, the compressibility, the mmting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ea., Oxford University Press, 1985, pp 24-29). 

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