Heat capacity frequency dependence

The frequency dependence of e and e" and their magnitudes control the extent to which a substance is able to couple with the microwave radiation and therefore are fundamental parameters for interpreting the dielectric heating phenomenon. Although tan 8 is a helpful parameter for comparing the heating rates of a series of dielectrics with similar physical and chemical characteristics, for more complex mixtures expressions, which take into account the complexity of the electric field pattern, the heat capacity of the compound and the density, have been proposed. 

For a monatomic gas, where the heat capacity involves only translational energy, V is independent of sound oscillation frequency (except at ultra-high frequencies, where a classical visco-thermal dispersion sets in). For a relaxing polyatomic gas this is no longer so. At sound frequencies, where the period of the oscillation becomes comparable with the relaxation time for one of the forms of internal energy, the internal temperature lags behind the translational temperature throughout the compression-rarefaction cycle, and the effective values of CT and V in equation (3) become frequency dependent. This phenomenon occurs at medium ultrasonic frequencies, and is known as ultrasonic dispersion. It is accompanied by... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Figure 18. Frequency dependence of the heat capacity of the modes at temperature 28 K (solid line), 128 K (dotted line), 228 K (dash-dot line).

We shall limit ourselves to the case where the mean energy of each molecule is equal to the sum of the energies of translation, of rotation and of vibration. The heat capacity at constant volume (c/. 10.5) will also be composed of three terms arising from these three kinds of motion. The contribution from the translational motion is f R per mole, and that from rotation is JR or fR depending upon whether the molecule is linear or not. This last statement is only exact if the rotational motion may be treated by classical, as opposed to quantum, mechanics. This is a good approximation even at low temperatures except for very light molecules such as Hg and HD. Finally the contribution from vibration of the atoms in a molecule relative to one another is the sum of the contributions from the various modes of vibration. Each mode of vibration is characterized by a fundamental frequency vj which is independent of the temperature. It is convenient to relate the fundamental frequency to a characteristic temperature (0j) defined by... 

We shall calculate here the coefficient of thermal conductivity and thermal diffusivity for myoglobin. We then need to calculate for each mode of myoglobin the heat capacity with Eq. (15) and to estimate the frequency-dependent energy diffusion coefficient, D(go). We are assuming that the heat capacity, calculated per unit volume of protein, is a function only of its vibrational energy. To estimate the volume of myoglobin, we assume for simplicity that the protein is a sphere with radius 17 A [111]. 

In the second part of the chapter, we have examined the spread of vibrational energy through coordinate space in systems that are large on the molecular scale—in particular, clusters of hundreds of water molecules and proteins—and computed thermal transport coefficients for these systems. The coefficient of thermal conductivity is given by the product of the heat capacity per unit volume and the energy diffusion coefficient summed over all vibrational modes. For the water clusters, the frequency-dependent energy diffusion coefficient was... 

The dynamic problem of vibrational spectroscopy must be solved to find the normal coordinates as linear combinations of the basis Bloch functions, together with the amplitudes and frequencies of these normal vibrations. These depend on k, and therefore the problem must be solved for a number of k-points to ensure an adequate sampling of the Brillouin zone. Vibrational frequencies spread in k-space, just as the Bloch treatment of electronic energy gave a dispersion of electronic energies in k-space. The number of vibrational levels whose energy lies between E and fc +d E is called the vibrational density of states. Vibrational contributions to the heat capacity and to the crystal entropy can be calculated by appropriate integrations over the vibrational density of states, just like molecular heat capacities and entropies are obtained by summation over molecular vibration frequencies. 

K the value of RT is approximately 2.5 kJ mol"1,) However Aevib is usually much greater than kT and under these circumstances vibrations do not contribute significantly to heat capacity (Fig. 9.7). Thus for nitrogen at room temperature the rotational contribution is approximately iiTand the vibrational contribution almost zero. Thus Cv 5/2)1 and y w 7/5 = 1.40 as opposed to the classical prediction of 1,29. For iodine the vibrational spacings are closer (Table 9.1) and we would predict y 1.29 in accord with the classical value. If the temperature is varied the heat capacity of a diatomic or polyatomic gas may show steps as the contributions from rotations and vibrations rise as the energy separations become comparable to kT, The positions of the steps depend on the moments of inertia and the vibrational frequencies of the molecules. 

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

Both the real and imaginary parts of the dynamic heat capacity are finite at the critical point and become larger with decreasing frequency. The dependence on frequency is extremely small at the critical point, as Eqs. (106), (109), and (110) show. The dependence will be even smaller in the immediate neighborhood of the critical point, since the multiplying factor in Eq. (106)... 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

The value of the heat capacity in Eq. (29.51) does not depend on any assumption about the frequencies in the solid. This should be the value of the heat capacity of any monatomic solid if the temperature is sufficiently high. If we deal with one mole of the solid, then N = Na, and N k = R. For one mole, = 3R k 25 J/K mol. This result is the law of Dulong and Petit, recognized for a century and a half. 

We have explored some of the simpler aspects of statistical thermodynamics, a very powerful theoretical tool. If the energy levels of the molecules composing the system can be obtained by solution of the Schrodinger equation, the partition function can be calculated then any thermodynamic property can be evaluated. One of the great virtues of statistical thermodynamics is its ability to reveal general laws. For example, we reached the conclusion that all monatomic solids should have the same heat capacity at high temperatures. Restrictions on the laws are made apparent f or example, the heat capacity of a monatomic solid at low temperatures depends on what is assumed about the frequencies in the solid. 

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