Heat capacity linear molecules

In Eq. (3.22a), Cj i t is the internal heat capacity per molecule of the ith component. M is the total number of terms in the expansions (3.20). In the limit as M approaches infinity, the expansions are presumed to become exact, and the coefficients aj , bj , and are obtained exactly as solutions of an infinite set of linear simultaneous equations. However, the only coefficients actually needed for the calculation of the transport properties are ajoo> bjoo, and Cjqq. To get numerical values of these to a first approximation, the set of equations is truncated by taking only three terms (mn = 00, 10, 01) in Eq. (3.20a), which controls thermal conduction and thermal diffusion, and only one term (mn = 00) in each of Eqs. (3.20b) and (3.20c). 

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 ... 

In each case, CPm has been calculated from Q>m = Cv nl 4- R.) Note that the molar heat capacity increases with molecular complexity. The molar heat capacity of nonlinear molecules is higher than that of linear molecules because nonlinear molecules can rotate about three rather than only two axes (recall Fig. 6.17). 

STRATEGY We expect the temperature to rise more as a result of heating at constant volume than at constant pressure because at constant pressure some of the energy is used to expand the system. Oxygen is a linear molecule and its heat capacities can be... 

CrCl2 molecules have been isolated in solid inert-gas matrices and their i.r. spectrum has been recorded 33—1000 cm The isotopic shifts and i.r. selection rules indicate a linear 10°) structure. The d-d spectrum of gaseous CrClj has been discussed in terms of ligand field theory.The heat capacity of anhydrous CrCl3 in the temperature interval 2—20 K has been determined and the sublimation and decomposition pressures of the compound have been recorded. ... 

The normal vibrations and structural parameters of Sg S, S, and Sjj have been used to calculate several thermodynamic functions of these molecules in the gaseous state. Both the entropy (S°) and the heat capacity (C°) are linear functions of the number of atoms in the ring in this way the corresponding values for Sj, Sg, Sjo and can be estimated by inter- and extrapolation For a recent review of the thermodynamic properties of elemental sulfur see Ref. 

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 ... 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

SOLUTION The molar heat capacities of oxygen, a linear molecule, are... 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

On the basis of initial calorimetric measurements (Gill et al., 1976 Olofsson et al., 1984 Dec and Gill, 1984, 1985), one can represent the enthalpy of transfer of hydrocarbons from the gaseous phase to water by a linear function of temperature in the temperature range 15-35°C. Bearing in mind Kirchhoff s relation between enthalpy and heat capacity change in the reactions, one can conclude that the transfer of nonpolar molecules to water leads to an increase of heat capacity by a value that is independent of temperature in the mentioned temperature range. 

The entropies, heat capacities, and thermodynamic functions of gaseous cyclooctaselenium have been calculated from spectroscopic and structural data for temperatures of up to 3000 K (64). Both the heat capacities and entropies of sulfur rings S (n = 6, 7, 8, 12) at a given temperature depend linearly on the ring size n (65). Therefore, it has been assumed that analogous relationships exist for the cyclic Se molecules, and the following equations have been derived from the data of Se2 and Seg at 298 K (64) ... 

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 adopted heat capacity values are based on the studies by Montgomery (1, 405.79-433.31 K) and West (2, 373-678 K). Liquid sulfur undergoes a second order transition with a maximum reported at 432.02 0.20 K (1 ) and 432.25 0.30 K (2) this has been attributed to the polymerization of Sg molecules (3). We adopt the tabulated heat capacity values of Montgomery ( ) up to 434 K and those of West (2) above 434 K. The heat capacity is assumed to be constant at 7.568 cal mol above 810 K. Below T 388.36 K, the heat capacity values are obtained by linear extrapolation using the slope of the values in the region T to 420 K. The entropy is calculated in a manner similar to that used for the enthalpy of formation. 

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... 

Heat capacity theory permits a correlation with the chemical structure of the repeating unit (U). In the solid state, only vibrational contributions need to be considered (skeletal and group vibrations). For an approximate discussion of the skeletal vibrations, the molecule is considered to be a string of structureless beads of the given formula weight. For linear macromolecules with similar backbones, the geometry and force constants are similar so that intramolecular skeletal vibrations are fixed by the mass of the structureless bead. The inter-molecular vibrations of linear macromolecules have quite low... 

Contributions to the molar heat capacity for a linear-molecule ideal gas... 

It was observed that the values of both the heat capacities and the entropies of the sulphur ring compounds S6(g), 87(g), Sg(g), and 812(g) vary linearly with the number of atoms in the molecule. Even the 82(g) molecule fits the linear relationship if its entropy... 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

The total heat capacity of a molecular substance is the sum of each contribution (Figure 17.13 of the text). When equipanition is valid (when the temperature is well above the characteristic temperature of the mode T 0m) we can estimate the heat capacity by counting the numbers of inodes that are active. In gases, all three translational modes are always active and contribute 7/ to the molar heat capacity. If we denote the number of active rotational modes by (so for most molecules at normal temperatures vjj = 2 for linear molecules, and 3 for nonlinear molecules), then the rotational contribution is 71 / . If the temperature is high enough for vibrational modes to be active the vibrational contribution to the molar heat capacity is u /f. In most cases vy 0. It follows that the total molar heat capacity is... 

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