Heat Einstein theory

Einstein Theory of Low-Temperature Heat Capacity of Solids [2], When we consider the heat capacity of solids, we realize that they consist of vibrating atoms or molecules. Their vibrations are quantized, of course, and have the nice name of phonons. Einstein considered a single vibration of an oscillator, along with its partition function ... 

Further work on similar types of cells has been carried out, in which not only is use made of the Nernst Theorem but likewise of the Einstein theory of atomic heat of solids (as modified by Nernst and Lmdemann) This will be taken up after we have discussed Planck s Quantum Theory of radiation and Einstein s application of it to the heat capacity of solids (Vol. Ill)... 

Einstein temperature (0 ) - In the Einstein theory of the heat capacity of a crystalline solid, 0 = hvik, where h is Planck s constant, k is the Boltzmann constant, and v is the vibrational frequency of the crystal. 

Eventually, the answer was found by Albert Einstein and the Polish physicist Marian Smoluchowski (1872-1917), then a professor at the University of Lviv. The title of one of Einstein s papers on the theory of Brownian motion is rather telling On the motion of particles suspended in resting water which is required by the molecular-kinetic theory of heat . Einstein and Smoluchowski considered chaotic thermal motion of molecules and showed that it explains it all a Brownian particle is fidgeting because it is pushed by a crowd of molecules in random directions. In other words, you can say that Brownian particles are themselves engaged in chaotic thermal motion. Nowadays, science does not make much distinction between the phrases Brownian motion and thermal motion — the only difference lies back in history. The Einstein-Smoluchowski theory was confirmed by beautiful and subtle experiments by Jean Perrin (1870-1942). This was a long awaited, clear and straightforward proof that all substances are made of atoms and molecules. ... 

FIGURE18.4 The Einstein theory of heat capacity of crystals agrees reasonably well with experimental measurements. 

Using the Einstein theory, calculate the molar heat capacity of a diamond crystal at 298.15 K, at 500.0 K, and at 1320 K. Compare each value with 3R. The experimental value at 298.15 K is 6.113 JK-lmol-1. 

In this chapter, we have discussed the structure of solids and liquids. Many solids are crystalline, with molecular units arranged in a regular three-dimensional lattice. There are two principal theories for the vibrations of lattices of atoms, the Einstein and the Debye theories. In the Einstein theory the normal vibrational modes of the lattice are assumed to vibrate with the same frequency. In the Debye model, the normal modes of the lattice are assumed to vibrate with the same distribution of frequencies as would a structureless solid. In each theory, the formula for the heat capacity of the solid lattice conforms to the law of Dulong and Petit at high enough temperature. 

Einstein theory of specific heat Atheorypresented by Albert Einstein that proposes that the specific heat of solids is a corrsequence of the vibratiorrs of the atoms in a lattice structure. The theory correctly shows that vrfren the temperature tends towards absolute zero, the specific heat of solids tends to zero. 

One can see that the vibrational frequency no longer contributes to Eq. (3.9). The simple Einstein theory provides a qualitative and surprisingly often a quantitative picture of the temperature behaviour of the specific heat. As required Cvib tends to zero as zero Kelvin is approached but exponentially, that is more steeply 3NkB than the experimentally well-established T law requires. The assumption of a single and temperature-independent frequency is obviously too great an approximation. 

Einstein (f,) remarked that this point of view can be carried over to the theory of the energy content of a solid body if we suppose that the positive ions of Drude s theory ( 198) may be looked upon as the vibrating resonators, and the seat of the heat content of the body (Korperwarme). He calculated the expression ... 

According to Joule s law ( 9), the molecular heat of a compound is the sum of the atomic heats of its components, and since this holds good even when the atomic heats are irregular, i.e., not equal to 6 4, it seems that the heat content of a solid resides in its atoms, and not in the molecular complexes as such. This agrees with Einstein s theory. Hence the molecular heat of a compound should be calculable by means of the formula ... 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

In 1928, the English scientist and idealist Sir James Jean revived the old heat death argument, augmented with elements from Einstein s relativity theory since matter and energy are equivalents, he claimed, the universe must finally end up in the complete conversion of matter into energy ... 

Einstein A. (1905) The motion of small particles suspended in static liquids required by the molecular kinetic theory of heat. Ann. Phys. 17, 549-560. 

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

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

For the history of Brownian motion, see G.L. de Haas-Lorentz, Thesis (Leiden 1912) [German transl. Die Brownsche Bewegung (Vieweg, Braunschweig 1913)] the Notes in A. Einstein, Investigations on the Theory of Brownian Movement (A.D. Cowper transl., R. Furth ed. with notes, Methuen, London 1926 Dover Publications, New York 1956) S.G. Brush, The Kind of Motion We Call Heat (North-Holland, Amsterdam 1976) Vol. 2, ch. 15. 

In a series of papers published from 1905 to 1908, Einstein successfully incorporated the suspended particles into the molecular-kinetic theory of heat. He treated the suspended particles as being in every way identical to the suspending molecules except for the vast difference of their size. Tie set forth several relationships that were capable of experimental verification and he invited experimentalists to solve the problem. 

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. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

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. 

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