Vibrational heat capacity

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Statistical thermodynamics tells us that Cv is made up of four parts, translational, rotational, vibrational, and electronic. Generally, the last part is zero over the range 0 to 298 K and the first two parts sum to 5/2 R, where R is the gas constant. This leaves us only the vibrational part to worry about. The vibrational contr ibution to the heat capacity is... 

The heat capacity can be computed by examining the vibrational motion of the atoms and rotational degrees of freedom. There is a discontinuous change in heat capacity upon melting. Thus, different algorithms are used for solid-and liquid-phase heat capacities. These algorithms assume different amounts of freedom of motion. 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

The heat capacity is largely determined by the vibration of die metal ion cores, and tlris property is also close to tlrat of tire solid at the melting point. It therefore follows tlrat both the thermal conductivity and the heat capacity will decrease with increasing teirrperamre, due to the decreased electrical conductivity and the increased amplitude of vibration of the ion cores (Figure 10.1). 

On the experimental side, one may expect most progress from thermodynamic measurements designed to elucidate the non-configurational aspects of solution. The determination of the change in heat capacity and the change in thermal expansion coefficient, both as a function of temperature, will aid in the distinction between changes in the harmonic and the anharmonic characteristics of the vibrations. Measurement of the variation of heat capacity and of compressibility with pressure of both pure metals and their solutions should give some information on the... 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

Equation (10.148) does not correctly predict CV.m at low temperatures because it assumes all the atoms are vibrating with the same frequency. In the ideal gas, this is a good assumption, and in the previous section we used an equation similar to (10.148) to calculate the vibrational contribution to the heat capacity of an ideal gas. 

The electronic contribution is generally only a relatively small part of the total heat capacity in solids. In a few compounds like PrfOHE with excited electronic states just a few wavenumbers above the ground state, the Schottky anomaly occurs at such a low temperature that other contributions to the total heat capacity are still small, and hence, the Schottky anomaly shows up. Even in compounds like Eu(OH)i where the excited electronic states are only several hundred wavenumbers above the ground state, the Schottky maximum occurs at temperatures where the total heat capacity curve is dominated by the vibrational modes of the solid, and a peak is not apparent in the measured heat capacity. In compounds where the electronic and lattice heat capacity contributions can be separated, calorimetric measurements of the heat capacity can provide a useful check on the accuracy of spectroscopic measurements of electronic energy levels. 

The graph in Fig. 6.20 shows how Cv for iodine vapor, I2(g), varies with temperature. At very low temperatures Cv>m = jR, but soon rises to jR as molecular rotation takes place. At still higher temperatures, molecular vibrations start to absorb energy and the heat capacity rises toward R. At 298 K, the experimental value is equivalent to 3.4R. 

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

Estimate the molar heat capacity (at constant volume) of sulfur dioxide gas. In addition to translational and rotational motion, there is vibrational motion. Each vibrational degree of freedom contributes R to the molar heat capacity. The temperature needed for the vibrational modes to be accessible can be approximated by 6 = />vvih/, where k is Boltzmann s constant. The vibrational modes have frequencies 3.5 X... 

Use the estimates of molar constant-volume heat capacities given in the text (as multiples of R) to estimate the change in reaction enthalpy of N2(g) + 3 H,(g) —> 2 NH.(g) when the temperature is increased from 300. K to 500. K. Ignore the vibrational contributions to heat capacity. Is the reaction more or less exothermic at the higher temperature ... 

ALL CHANGES IN PHASE involve a release or absorption of calories. One reason for this is that each solid has its own heat capacity. That is, there is a characteristic heat content for each material which depends upon the atoms composing the solid, the nature of the lattice vibrations within it, and its structure. The total heat content, or enthalpy, of each solid is defined by ... 

Spin-state transitions have been studied by the application of numerous physical techniques such as the measurement of magnetic susceptibility, optical and vibrational spectroscopy, the Fe-Mbssbauer effect, EPR, NMR, and EXAFS spectroscopy, the measurement of heat capacity, and others. Most of these studies have been adequately reviewed. The somewhat older surveys [3, 19] cover the complete field of spin-state transitions. Several more recent review articles [20, 21, 22, 23, 24, 25] have been devoted exclusively to spin-state transitions in compounds of iron(II). Two reviews [26, 27] have considered inter alia the available theoretical models of spin-state transitions. Of particular interest is the determination of the X-ray crystal structures of spin transition compounds at two or more temperatures thus approaching the structures of the pure HS and LS electronic isomers. A recent survey [6] concentrates particularly on these studies. 

When the temperature is such that hv kT, neither of the limiting cases described earlier can be used. For many solids, the frequency of lattice vibration is on the order of 1013 Hz, so that the temperature at which the value of the heat capacity deviates substantially from 3R is above 300 to 400 K. For a series of vibrational energy levels that are multiples of some fundamental frequency, the energies are 0, hv, 2hv, 3hi/, etc. For these levels, the populations of the states (n0, nu n2, etc.) will be in the ratio 1 e hl T e Jh,/Ikr e etc. The total number of vibrational states possible for N atoms is 3N... 

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