Specific Heat-Theory

The theoretical value of the frequency of vibration, depending on the curvature of the cmrve at its minimum, is naturally more uncertain. Calculation shows that the curve gives a frequency of vibration of 5300 cm. S about 20% higher than the value 4360 cm. from experiment. As for the moment of inertia, while it is larger than most of the values from specific heat theories, it is in accord with the larger values which have been found by Richardson and Tanaka from analysis of the hydrogen bands. 

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). 

The temperature factor 2B can be evaluated according to the specific heat theory of Debye-Waller so that... 

In statistical mechanics (e.g. the theory of specific heats of gases) a degree of freedom means an independent mode of absorbing energy by movement of atoms. Thus a mon-... 

The molecular weight (mean relative molecular mass) was obtained by determination of density but, in order to determine that the gas was monatomic and its atomic and molecular weights identical, it was necessary to measure the velocity of sound in the gas and to derive from this the ratio of its specific heats kinetic theory predicts that Cp/C = 1.67 for a monatomic and 1.40 for a diatomic gas. 

The problem has been largely worked at from both sides from the theoretical side the point of view has been almost exclusively that of the kinetic gas theory. It must be kept in mind, however, that it is possible that a purely mechanical theory may not be sufficient to cover the phenomena, as has recently appeared in the case of the specific heats of solids. 

There is, however, a fatal objection to the theory of Boltzmann. At very low temperatures the oscillations will be small, and should conform to the theory. But the atomic heats, instead of approaching the limit 5 955 at low temperatures, diminish very rapidly, and in the case of diamond the specific heat is already inappreciable at the temperature of liquid air. A new point of view is therefore called for, and it is a priori very probable that this will consist of a replacement of the hypothesis of Equipartition of Energy adopted by Boltzmann. This supposition has been verified, and the new law of partition of energy derived... 

At low temperatures the first term only will be significant, and hence the theory leads to the striking result that at very low temperatures the specific heat of a solid is proportional to the cube of the absolute temperature, or its energy to the fourth power of the... 

By using Lipatov s theory, interrelating the abrupt jumps in the specific heat of composites at their respective glass transition temperatures with the values of the extents of these boundary layers, the thickness of the mesophase was accurately calculated. 

This result is independently verified by Dennison (17) who has recently given a satisfactory theory of the specific heat of hydrogen. The observed specific heat as interpreted by Dennison requires that I0 be equal to 0.464 X 10 40 g. cm.2. The very recent measurements by Cornish and Eastman (18) of the specific heat of hydrogen from the velocity of sound are said to agree very well with Dennison s theory if I0 be given the value of 0.475 X 10-40 g. cm.2. 

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

The expression in Eq. (29) can be evaluated numerically for all values of t, and the results for three different waiting times are shown in Fig. 11 for c = 0.1. The value of Tmin = 2.0 ps at E/To = 5.7 x lO", derived from the present theory (also consistent with Goubau and Tait [101]) was used. The results for t = 10 ps demonstrate that, due to a lack of fast relaxing systems at low energies, short-time specific heat measurements can exhibit an apparent gap in the TLS spectrum. Otherwise, it is evident that the power-law asymptotics from Eq. (30) describes well Eq. (29) at the temperatures of a typical experiment. 

Another approach to relating the hardness to atomic parameters is that of Grimvall and Thiessen (1986) in which hardness is related to vibrational energies. Their theory is slightly modified here by using vibrational energy densities instead of the energies themselves. Specific heat data measure the excitation... 

It has also to be remembered that the band model is a theory of the bulk properties of the metal (magnetism, electrical conductivity, specific heat, etc.), whereas chemisorption and catalysis depend upon the formation of bonds between surface metal atoms and the adsorbed species. Hence, modern theories of chemisorption have tended to concentrate on the formation of bonds with localized orbitals on surface metal atoms. Recently, the directional properties of the orbitals emerging at the surface, as discussed by Dowden (102) and Bond (103) on the basis of the Good-enough model, have been used to interpret the chemisorption behavior of different crystal faces (104, 105). A more elaborate theoretical treatment of the chemisorption process by Grimley (106) envisages the formation of a surface compound with localized metal orbitals, and in this case a weak interaction is allowed with the electrons in the metal. 

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

Afterwards, measurements of the specific heat have been carried out in a wider pressure range [29,37-39]. It was thus evident that there is a discontinuity at the transition temperature which, at the melting pressure, gave a AC/C 2, (well above the 1.43 value expected from BCS theory) which decreased as the pressure diminished. 

The experimental data usually give the specific heat at constant pressure cP. Theories usually refer to the specific heat at constant volume cv. The specific heat cP is greater than cv by a factor (1 + jgT), where f5 is the volumetric coefficient of thermal expansion and yG is the so-called Griineisen parameter ... 

The model proposed by Anderson and Phillips gives a phenomenological explanation of the properties of the amorphous materials without supplying a detailed microscopic description [42]. Low-temperature measurements of the specific heat of amorphous solids have however shown that instead of a linear contribution as expected from the TLS theory, the best representation of data is obtained with an overlinear term of the type [43,44] ... 

Even Anderson et al. [39] pointed out that an important consequence of the tunnelling model was the (logarithmic) dependence of the measured specific heat on the time needed for the measurement of c. The latter phenomenon was due to the large energy spread and relaxation time of TLS. In 1978, Black [45], by a critic revision of the tunnelling theory, has been able to explain the time dependence of the low-temperature specific heat. 

David, D.J. Determination of specific heat and heat of fusion by differential thermal analysis. Study of theory and operating parameters, Aural Chem., 36(11) 2162-2166, 1964. 

The theory fails to explain the molar specific heat of metals since the free electrons do not absorb heat as a gas obeying the classical kinetic gas laws. This problem was solved when Sommerfeld (1) applied quantum mechanics to the electron system. 

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