Heat capacity vibrational contribution

Figure 8.16 (a) IR and (b) Raman spectra for the mineral calcite, CaC03. The estimated density of vibrational states is given in (c) while the deconvolution of the total heat capacity into contributions from the acoustic and internal optic modes as well as from the optic continuum is given in (d). 

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. 

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. 

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 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 ... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The vibrational heat capacity is the largest contribution to the total heat capacity and determines to a large extent the entropy. Analytical expressions for the entropy of the models described in the previous section can be derived. The entropy corresponding to the Einstein heat capacity is... 

Pyda and co-workers [49, 60] measured the reversible and irreversible PTT heat capacity, Cp, using adiabatic calorimetry, DSC and temperature-modulated DSC (TMDSC), and compared the experimental Cp values to those calculated from the Tarasov equation by using polymer chain skeletal vibration contributions (Figure 11.7). The measured and calculated heat capacities agreed with each other to within < 3 % standard deviation. The A Cp values for fully crystalline and amorphous PTT are 88.8 and 94J/Kmol, respectively. 

RADICALC Bozzelli, J. W. and Ritter, E. R. Chemical and Physical Processes in Combustion, p. 453. The Combustion Institute, Pittsburgh, PA, 1993. A computer code to calculate entropy and heat capacity contributions to transition states and radical species from changes in vibrational frequencies, barriers, moments of inertia, and internal rotations. 

Aminomethylpyridine (picolylamine) is an important ligand in respect to spin cross-over, [Fe(2-pic)3]Cl2 being the key compound." Fleat capacity measurements on [Fe(2-pic)3]Cl2 EtOH gave values of 6.14kJmol and 50.59 JK moC for the spin eross-over entropy the determined entropy was analyzed into a spin contribution of 13.38, an ethanol orientational effeet of 8.97, and a vibrational contribution of 28.24 JK mol. " This compound exhibits weak cooperativity in the solid state." The heat capacity of [Fe(2-pic)3]Cl2 MeOH is consistent with very weak cooperativity." [Fe(2-pic)3]Br2 EtOH shows a lattice expansion significantly different from that expected in comparison with earlier-established behavior of [Fe(2-pic)3]Cl2 EtOH." ... 

For molecules with three or more atoms, the number of vibrational frequencies is 3n - 5 for a linear molecule and 3n — 6 for a nonlinear molecule. The partition function for all vibration frequencies is the product of all these qt, and the heat capacity for all vibration frequencies is the sum of their individual contributions ... 

The heat capacity of a molecule is equal to the sum of all the contributions from translation, rotation, vibration, and electronic degrees of freedom (table 4.19). 

Let us compute the C of the molecules H2, HCl, and CI2 from 0 to 100 K. Quantum mechanics shows that for the vibration contributions towards heat capacity... 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

The heat capacity at constant volume Cv from the translational and rotational degrees of freedom are determined via Eqs. 8.124 and 8.128, the vibrational contributions to Cv are calculated by Eq. 8.129, and the electronic contribution to Cv is from Eq. 8.123. For an ideal gas, Cp = Cu + R, so Cp=41.418 J/mole/K. The experimental value is Cp=38.693 J/mole/K. Agreement with experiment gets better at higher temperature. At 1000 K, Cp from our calculation is 59.775 J/mole/K, compared to a value of 58.954 from the NIST-JANAF Tables. The difference between theory and experiment is due entirely to our use of the vibrational frequencies obtained from the ab initio results, rather than using the experimental frequencies. 

Evaluate the translational, rotational, and vibrational contributions to the constant volume heat capacity Cv for 0.1 moles of the A127C135 molecule at 900°C and a pressure of 1 mBar. The molecular constants needed are given in the previous problem. 

The translational contribution to the heat capacity ClMrans is found using Eq. 8.124, and the rotational contribution CUjrot by Eq. 8.127 or 8.128. The vibrational contribution C ,vib... 

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