Einstein specific heat

Vibrational Frequency Spectrum of a Continuous Solid.—To find the specific heat, on the quantum theory, we must superpose Einstein specific heat curves for each natural frequency v1y as in Eq. (1.3). Before we can do this, we must find just what frequencies of vibration are allowed. Let us assume that our solid is of rectangular shape, bounded by the surfaces x = 0, x = X, y — 0, y = F, z = 0, z = Z. The frequencies will depend on the shape and size of the solid, but this does not really affect the specific heat, for it is only the low frequencies that art very sensitive to the geometry of the solid. As a first step in investigating the vibrations, let us consider those particular waves that arc propagated along the x axis. 

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... 

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

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. 

Einstein showed that when a reversible reaction is present sound dispersion occurs at low frequency the equilibrium is shifted within the time of oscillation, the effective specific heat is at a maximum, and the speed of sound c0 is at a minimum. At high frequency the oscillations occur so rapidly that the equilibrium has no time to shift (it is frozen ). The corresponding Hugoniot adiabate (FHA) is shown in the figure. Here the effective heat capacity is minimal, the speed of sound c is maximal cx > c0. From consideration of the final state and the theory of shock waves it follows that C>c0. 

The problem of specific heats, treated by Jeans from the classical point of view, as I said above, was discussed by Einstein in the case of solids, with special regard to the discrepancy observed at low temperature between the measured values and those deduced from the theory he had constructed in 1907 by quantizing the mechanical oscillators3 as Planck had quantized the radiation oscillators. 

In September of 1920, Polanyi took a position in Berlin at the Kaiser Wilhelm Institute for Fiber Chemistry, which was housed in the buildings of the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry directed by Fritz Haber. [8] By this time Polanyi had published his doctoral thesis and papers in several areas of thermodynamics, including papers on Nernst s heat theorem and Einstein s quantum theory for specific heats. [9] In the next 13 years, before he was forced to leave Germany in 1933, Polanyi worked in several areas of physical chemistry in Berlin, afterwards heading Manchester s physical chemistry laboratory for fifteen years. In 1948 Polanyi exchanged his professorship in chemistry at Manchester for a chair in social studies, thus formally becoming a philosopher. 

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... 

Our discussion of the specific heat capacity of polymers on the preceding pages has been quite empirical. There are, in fact, few fundamental rules that can be used for the prediction of specific heat capacity. At very low temperatures, the equations of Debye and Einstein may be used. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

The adiabatic compression of saturated vapours was considered by Bruhat, who also calculated the angle between the liquid and vapour phase isochores in the entropy-temperature diagram. Amagat investigated the discontinuity in specific heats where an isothermal cuts the saturation curve. Hausen, from a complicated formula for the specific heat of steam involving two Einstein terms ( 2.IX N), calculated the heat content and entropy of steam. Leduc found the value of n for dry steam in Rankine s equation for adiabatic... 

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,... 

Horning, R. D., and Staudenmann, J.-L. The Debye-Waller factor for polyatomic solids. Relationship between X-ray and specific-heat Debye temperatures. The Debye-Einstein model. Ada Cryst. A44, 136-142 (1988). 

The important fact that the quantum theory explains the extraordinarily small specific heat of electrons at normal temperatures, as is well known, was pointed out by Sommerfeld. Our model likewise explains this fact, though in a somewhat different way. If, as is permissible here, we work with the spins in the same way as we previously did with the free electrons, it follows by what we have said above that we have to deal with a degenerate Bose-Einstein gas, the specific heat of which at low temperatures is, as we know, given by... 

Recalling the definition of the specific heat as the temperature derivative of the average energy (see eqn (3.99)), it is found that for the Einstein solid... 

Fig. 5.12. Plots of the dimensionless average energy and specific heat resulting from the Einstein model. The average energy is scaled by 3NEe, where Ee = ha>E, while the specific heat is reported in units of 3Nk. The temperature is also plotted in units of the Einstein temperature given by Te = hcoE/k.

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