## Specific Heats Of Solids- Physics

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• ### Akhilesh K

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Notes On Specific Heats of Solids.

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SPECIFIC HEATS OF SOLIDS: Classical physics fails again Blackbody radiation is not the only familiar phenomenon whose explanation requires quantum statistical mechanics. Another is the way in which the internal energy of a solid varies with temperature.
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Let us consider the molar specific heat of a solid at constant volume, This is the energy that must be added to 1 kmol of the solid, whose volume is held fixed, to raise its temperature by 1 K. However keeping the volume of solid constant, is difficult while varying the temperature. Hence what is observed experimentally is c , heat capacity at constant pressure. The specific heat at constant pressure c is 3 to 5 percent higher than in solids because it includes the work associated with a volume change as well as the change in internal energy.
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The internal energy of a solid resides in the vibrations of its constituent particles, which may be atoms, ions, or molecules; we shall refer to them as atoms here for convenience. These vibrations may be resolved into components along three perpendicular axes, so that we may represent each atom by three independent harmonic oscillators.
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Dulong-Petit law: The classical treatment to heat capacity The first approach to find an expression for heat capacity of solids is via Dulong-Petit Law. It uses the classical treatment as in gases. According to classical laws, a non-interacting particle will be having 1/2kBT energy associated with each degrees of freedom. For a container having No particles, each having six degrees of freedom (three for potential energy & three for kinetic energy), 2 - 3N0kB = 3R [e: R = NokB] -3R - 5.97 kcal / kmol.l<
• 5
Dulong and Petit found that, indeed, cv 3R for most solids at room temperature and above, and hence is known as the Dulong-Petit law in their honor. However, the Dulong-Petit law fails for such light elements as boron, beryllium, and carbon (as diamond), for which cv = 3.34, 3.85, and 1.46 kcal/kmol . K respectively at 200C. Even worse, the specific heats of all solids drop sharply at low temperatures and approach 0 as T approaches 0 K. Figure shows how cv varies with T for several elements. Clearly 7 something is wrong with the analysis. 6 5 2 1 o Lead Aluminum Silicon Carbon (diamond) 200 1000 400 600 800 Absolute temperature, K 1200
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Einstein's Formula: The quantum treatment to heat capacity In 1907 Einstein discerned that the basic flaw in the derivation of Dulong- Petit law, lies in the figure of kT for the average energy per oscillator in a solid. This flaw is the same as that responsible for the incorrect Rayleigh-Jeans formula for blackbody radiation. According to Einstein, the probability f(v) that an oscillator have the frequency v is given by, f(v) = 1/(ehv/kT - 1). Hence the average energy for an oscillator whose frequency of vibration is v is Average energy E = hvf(v) per oscillator hv/kBT -1 and not E = kT.
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The total internal energy of a solid therefore becomes Internal energy of solid and its molar specific heat is Einstein specific heat formula 3N0hv hv/kBT -1 3N0hv hv/kBT -1 2 hv/kBT kBT (e hv/kBT
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since hv/kBT kBT (e hv/kBT kBT At high temperatures, hv kBT, hv hv/kBT kBT kBT hv kBT kBT kBT Energy Transition acantum harmonic osci llator neglecting as kBT kBT which leads to the Dulong-Petit value, as it should. At high temperatures the spacing hv between possible energies is small relative to kBT, so E is effectively continuous and classical physics holds.
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As the temperature decreases, the value of Cv. The reason for the change from classical behavior is that now the spacing between possible energies is becoming large relative to kT, which inhibits the possession of energies above the zero-point energy. The natural frequency for a particular solid can 1 0.8 0.6 0.4 0.2 Heat capacity of an Enstein Solid 0.4 1.2 0.8 1.6 2 be determined by comparing the experimental data points with 2 hv/kBT -1)2 kBT (e hv/kBT The result in the case of aluminum is v = 6.4 x 1012 Hz
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Why is it that the zero-point energy of a harmonic oscillator does not enter this analysis? As we recall, the permitted energies of a harmonic oscillator are (n + h )h, n = 0, 1, 2, The ground state of each oscillator in a solid is therefore Eo = hhv, the zero-point value, and not Eo = 0. But the zero- point energy merely adds a constant, temperature-independent term of Eo = to the molar energy of a solid, and this term vanishes when the partial derivative (DE/öT)v is taken to find cv
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Debye Formula: Correction to the Einstein treatment Whilst the Einstein theory showed why the specific heat decreases at low temperature, it was not entirely satisfactory because it is assumed that all the atoms vibrated with the same frequency and that they were independent of one another. However all the atoms of the substance are coupled together and they should be considered as one complete system. The important points of Debye theory where it differ from Einstein's theory are 'Vibrating atoms are coupled to each other and cannot be treated independent of one another 'All the atoms are not vibrating with same frequency but there exist a distribution of frequency with a cut-off at highest frequency depending on the inter-atomic separation of ions.
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Based on these facts, Debye (1912) proposed a model for heat capacity in which only certain frequencies can be excited and maintained. 3 cph = 9B 0 (5 12 —n4nN0kB 5 3 cph = 1943.7 n Jmol-lK 1 where n is the number of atoms per formula unit, OD is Debye temperature, OD =hv ax/kBT and signifies the temperatures above which all the modes (for each direction there will be three modes, corresponding to the two transverse and one longitudinal vibrations) are excited. Debye theory emphasized on the distribution of frequencies, corresponding to various modes of vibration.
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Most important, the theory put a cut-off to the frequency distribution at highest frequency of vibration corresponds to the nearest inter-atomic distance. This indicates that modes having wavelength smaller than inter- atomic distance is not possible, which is true. o o Frequency Einstein Model max Frequency Debye Model min = ad 'to maximum phonon energy For most of the metals the Debye temperature lies in the range 300 to 400K. Since the Debye temperature is directly proportional to the maximum lattice frequency, a high value of the frequency implies that we are dealing with a lattice which has very strong inter-atomic forces and light atoms. e.g., for diamond
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20 15 04 0.3 Debye Einstein lee 0.6 0.9 TITO 1.5 Comparison of Classical, Einstein & Debye model for heat capacity. Fit of silver specifie heat data to the Debye curve with To - 21S K. Plot of experimental data of heat capacity and its fitting with Debye model.

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