Theory of Specific Heats

In terms of the assumed frequency distribution and the formula (1.3), we can now at once write down a formula for the specific heat. This is 

The integration in Eq. (3.2) cannot be performed analytically. It is worth while, however, to rewrite the expression in terms of a variable 

It is customary to define a so-called Dobye temperature 0/ by the equation 

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 more extended table will be found in Nerimt, Die UiundluKon des neucn Waunenatzea  

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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 SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

Einstein A (1907) The Planck theory of radiation and the theory of specific heat. Ann Phys 22 180-190... 

In this way the study of spectra gives even more detailed information than the consideration of specific heats. It is found that infra-red rotation and vibration spectra are only given by those molecules, such as HC1, which are composed of a negative and a positive portion. Strictly homo-polar molecules do not absorb in the infra-red, although the theory of specific heats, and the analysis of the visible spectrum, leave no doubt about the existence of... 

BORN, MAX (1882-1970). A German-born British physicist. Max Born studied mathematics and physics and in 1904 became David Hilbert is private assistant for. While at the University of Breslau, he won a competition on the stability of elastic wires and it became the dissertation for his Ph.D. After graduate school, he studied special relativity for a while, then became interested in the physics of crystals. In 1912. he published the Born-Karman theory of specific heats and his work on crystals is a cornerstone of solid-state theory. 

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

The Einstein equation was the first approximation to a quantum theoretical explanation of the variation of specilic heat with temperature. It was later replaced by the Debye theory of specific heat and its modifications. 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

Black also found that it took different but specific amounts of heat to raise the temperature of the same weight of different substances by the same number of degrees he thereby laid the foundations for the theory of specific heat. 

We see from Table IX-3 t.lmt. fnr prantififtlly all the molecules the characteristic tcirme " pompi d to room tempcraturcr so that at all ordinary temperature wo n o the quantum theory of specific heat. We also note that in every case the characteristic temperature for vibra-... 

It was to explain these deviations from the law of Dulong and Petit that Einstein developed his theory of specific heats. He treated the vibrations of the separate atoms by quantum theory, just as we did in Sec. 5, Chap. IX, and derived the formula... 

The actual dependence of pATsp on the temperature is rather complicated because of the dependence of the specific heat Cp on T, which is given by Debye s theory of specific heat for the reacting oxides and corresponding lattice dynamical model for crystalline solids. Simple assumptions regarding the net change in specific heats of the components involved in the dissolution reactions, however, allow one to avoid these complications [3]. 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

Einstein s theory of specific heat leads to the same result. This theory connects the molecular motion in solid bodies with Planck s theory of radiation, and has been confirmed in the main by the experimental researches of Nernst and his collaborators in the last few years. Einstein assumes that the heat motion in solid bodies consists of vibrations of the atoms about a point of equihbrium, as distinct from the translational motion of the molecules which we assume for gases. The energy of these vibrations—and this is the characteristic feature of the theory, and also of Planck s theory of radiation—is always an integral multiple of a quantity of energy e, which, in turn, is the product of a universal constant (. e. a constant independent of the nature of the substance) and the frequency i/ (number of vibrations R,... 

According to this new theory of specific heat, the specific heat of all sohd bodies, and probably also of super-cooled hquids, vanishes at the absolute zero, and increases with the temperature in the manner shown in the curve (Fig. 8). For every substance there is a certain range of temperature in which the specific heat varies rapidly with the temperature. Above this Ann. d. Fhysik. 22, 189 (1908). 

Debye, P. Zur Theorie der spezifischen Warmen. [On the theory of specific heats.] Annalen der Physik 39, 789-839 (1912). English translation in Collected Papers of Peter J. W. Debye, pp. 650-696. Interscience New York (1954). 

We now return to the law of cavity radiation. We have seen in the preceding section that Planck s hypothesis has been brilliantly successful not only for cavity radiation, but also in the theory of specific heats. The latter success furnishes additional strong support for the quantum theory. 

Theories of this kind have been developed by Tetrode,f Sommerfeld, and Keesom, all on the assumption that a gas behaves at low temperatures like a solid and that therefore Debye s well-known theory of specific heats becomes applicable in its essentials. ... 

Not only the structure of atoms but also their combination to form molecules and extended bodies, and the laws of motion of the latter, are governed by the same quantum laws. We may mention, for example, the more precise development of the theory of specific heats of solid bodies already referred to, and further the theory of the band spectra of molecules, which we shall deal with in detail in this book. 

In this chapter we shall discuss those properties of ice crystals which derive essentially from the thermal motions of water molecules within the crystal structure. In broad outline the theory describing these phenomena is simple and well known and leads to simple generalizations like the Debye theory of specific heats. However, because of the structure of the water molecule and, deriving from it, the structure of the ice crystal, such theories in their simple form represent only a first approximation to the observed behaviour. The coefficient of thermal expansion, for example, is negative at low temperatures and the specific heat is only poorly described by a Debye curve. It will be in tracing the reasons for some of these deviations from simple behaviour that most of our interest will lie. 

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