Heat capacity translational contribution

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... 

The molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

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... 

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). 

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... 

FIGURE 6.18 The variation of the molar heat capacity of iodine vapor at constant volume. Translation always contributes rotation contributes except at very low temperature, and vibrations of the molecule contribute at very high temperatures. When the molecules dissociate, the heat capacity becomes very large, but then settles down to a value characteristic of 2 mol I atoms undergoing only translational motion. 

The thermodynamic properties of a substance in the state of ideal gas are calculated as the sums of contributions from translation and rotation of a molecule as a whole, vibrations and internal rotation in the molecule, and electronic excitation. For example, for entropy and heat capacity the following equations hold ... 

Heat capacities of polyatomic molecules can be explained by the same arguments. As discussed in Chapter 3, bond-stretching vibrational frequencies can be over 100 THz. At room temperature k T heat capacity (which explains why most diatomics give cv % 5R/2, the heat capacity from translation and rotation alone). Polyatomic molecules typically have some very low-frequency vibrations, which do contribute to the heat capacity at room temperature, and some high-frequency vibrations which do not. 

The combined translational, rotational and vibrational contributions to the molar heat capacity, heat content, free energy and entropy for 1,3,4-thiadiazoles are available between 50 and 2000 K. They are derived from the principal moments of inertia and the vapor-phase fundamental vibration frequencies (68SA(A)36l). 

As in Section 5.3, we assume that there are additive translational, electronic, vibrational, and rotational contributions to the heat capacity ... 

Vibrational Partition Function/ The thermodynamic quantities for an ideal gas can usually be expressed as a sum of translational, rotational, and vibrational contributions (see Exp. 3). We shall consider here the heat capacity at constant volume. At room temperature and above, the translational and rotational contributions to are constants that are independent of temperature. For HCl and DCl (diatomic and thus linear molecules), the molar quantities are... 

The crystal data compilation of Donnay and Ondlk ( ) tabulated both ZrCl and ZrBr as cubic structures. Thus, the adopted heat capacity values are estimated so as to parallel those for ZrCl. The heat capacity values below 300 K are calculated by summing contributions due to hindered translations, librations, and internal vibrations of the crystal. The parameters used in the calculations are determined by a correlation with corresponding parameters for ZrCl (6) and a consideration of the sublimation data for ZrBr (6). The high temperature heat capacities are obtained graphically. 

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... 

ISe. Classical Calculation of Heat Capacities.— For a diatomic molecule two types of rotation are possible, as seen above, contributing RT per mole to the energy. Since there are two atoms in the molecule, i.e., n is 2, there is only one mode of vibration, and the vibrational energy should be RT per mole. If the diatomic molecules rotate, but the atoms do not vibrate, the total energy content E will be the sum of the translational and rotational energies, i.e., RT + RT = RT, per mole hence,... 

If the expression for the translational partition function is inserted into equation (16.8), it is readily found, since tt, m, fc, h and V are all constant, that the translational contribution Et to the energy, in excess of the zero-point value, is equal to %RT per mole, which is precisely the classical value. The corresponding molar heat capacity at constant volume is thus f P. As stated earlier, therefore, translational energy may be treated as essentially classical in behavior, since the quantum theory leads to the s ame results as does the classical treatment. Nevertheless, the partition function derived above [[equation (16.16) [] is of the greatest importance in connection with other thermodynamic properties, as w ill be seen in Chapter IX. 

The normal heat capacity of atomic chlorine, due to translational energy only, is Rj i.e., 2.98 cal. deg.-i g. atom-i, d so the electronic contribution would be appreciable even at 300° K. 

At 300 K, the translational and rotational contributions to the heat capacity will be classical, i.e., f and R, respectively, making a total of or 4.97 cal. deg. mole . If the vibrational contribution 1.12 is added, the total is 6.09 cal. deg. mole . (The experimental value which is not very accurate, is close to this result some difference is to be expected, in any case, because the calculations given above are based on ideal behavior of the gas. The necessary corrections can be made by means of a suitable equation of state, 21d.)... 

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