Heat capacity rotational 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... 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

Figure 10.10 Internal rotation contribution to the heat capacity of CH3-CCI3 as a function of temperature. Reprinted from K. S. Pitzer. Thermodynamics, McGraw-Hill, Inc., New York, 1995, p. 374. Reproduced with permission of the McGraw-Hill Companies.

One of the more interesting results of these calculations is the contribution to the heat capacity. Figure 10.10 shows the temperature dependence of this contribution to the heat capacity for CH3-CCU as calculated from Pitzer s tabulation with 7r = 5.25 x 10-47 kg m2 and VQ/R — 1493 K. The heat capacity increases initially, reaches a maximum near the value expected for an anharmonic oscillator, but then decreases asymptotically to the value of / expected for a free rotator as kT increases above Vo. The total entropy calculated for this molecule at 286.53 K is 318.86 J K l-mol l, which compares very favorably with the value of 318.94T 0.6 TK-1-mol 1 calculated from Third Law measurements.7... 

Table A4.6 gives the internal rotation contributions to the heat capacity, enthalpy and Gibbs free energy as a function of the rotational barrier V. It is convenient to tabulate the contributions in terms of VjRTagainst 1/rf, where f is the partition function for free rotation [see equation (10.141)]. For details of the calculation, see Section 10.7c.

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... 

Fig. 4 Rotational contribution to the molar heat capacity C for ortho, para and nominal hydrogen. Note that 1 cal/deg-mol = 4.18 J K Lmor1.

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. 

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 ... 

Rotation. Next, consider the rotational contribution to the heat capacity for a molecule of symmetry number a, for which we found in Problem 5.3.11 that qrot = (kBT/hcB) = (8n2kBT Ie/ah2 = 0.0419IeT/a. Thus we get... 

PROBLEM 5.5.1. Calculate the relative ortho and para contributions to the rotational heat capacity for D2, which has two nucleons, both with 1=1 ... 

In Figure 4, A gives a graph of the contribution due to three torsional Debye modes with 6 = 200° K., while B represents the heat capacity contribution of three hindered rotational degrees of freedom with ... 

This assumption is made arbitrarily to fit the maximum in the heat capacity curve. Rotation of four hydrogens is required to get the right heat capacity at the maximum. Curve C of Figure 4 represents the Debye lattice contribution of the... 

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... 

---