Thermodynamic function of molecules

METHOD FOR CALCULATION OF THE CONFORMATION OF MINIMUM POTENTIAL-ENERGY AND THERMODYNAMIC FUNCTIONS OF MOLECULES FROM EMPIRICAL VALENCE-FORCE POTENTIALS-APPLICATION TO THE CYCLOPHANES. 

Table 1 Equations for the translational, rigid-rotational, and harmonic oscillator contributions to the thermodynamic functions of molecules in the ideal-gas state and at the standard pressure 101325 Pa...

Boyd, R. H. Method for the calculation of the conformation and minimum potential-energy and thermodynamic functions of molecules from empirical valence-force potentials - application to the cyclophanes, / Chem. Phys. 1968, 49,2574-2583. 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

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. 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

The calculation of the thermodynamic functions of a substance is based upon theuu Boltzmann distribution equation, which predicts the most probable distributionvv of molecules (or atoms) among a set of energy levels. The equation is... 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

The vaporization of solvent molecules from the pure liquid solvent described above should not differ from its vaporization from an infinitely dilute solution of some solute(s) in it, since the vast majority of solvent molecules have other solvent molecules in their surroundings in both cases. As the solute concentration increases in the dilute solution range, it is expected that Raoulf s law will be obeyed, that is, the vapour pressure of the solvent will be proportional to its mole fraction in the solution. If this is indeed the case, the solution is an ideal solution. At appreciable concentrations of the solute this will no longer be the case, due to solute-solute interactions and modified solute-solvent ones. The vapour pressure as well as other thermodynamic functions of the solvent and, of course, of the solute will no longer obey ideal solution laws. The consideration of these effects is beyond the scope of this book. 

Here we have used a rough approximation by calculating the thermodynamic functions of the Ln3Xb molecules from a single set of structural and molecular parameters for the fluorides, chlorides, bromides and iodides each. 

The thermodynamic functions of one mole of compound consisting of No identical molecules in standard state are related to the partition function and its derivatives, thus [44] ... 

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) ... 

If we have a mixture of AT molecules of one gas, N2 of another, and so on, the general phase space will first contain a group of coordinates and momenta for the molecules of the first gas, then a group for the second, and so on. The partition function will then be a product of terms like Eq. (3.5), one for each type of gas. The entropy will be a sum of terms like Eq. (1.14), with n in place of n, and Pt, the partial pressure, in place of P. But this is just the same expression for entropy in a mixture of gases which we have assumed thermodynamically in Eq. (2.7). Thus the results of Sec. 2 regarding the thermodynamic functions of a mixture of gases follow also from statistical mechanics. 

The thermodynamic functions of mixing two substances, a and A, are conveniently expressed in terms of the difference in solubility parameters (5a Sb)- The applicability of this treatment is limited to systems for which the interaction between a and b molecules is the geometric mean of the separate interactions among a molecules and among b molecules. This is rarely true for H bonding liquids. 

In order to calculate the thermodynamic functions of the process described by Eq. (15), it is necessary to known the equilitHium geometry and tl frequencies of the normal vibrational modes of all species involved in the equilibrium process, as well as interaction energy, A . Partition functions, used for relatively strong vdW molecules, were evaluated using the rigid rotor-harmonic oscillator approximation. 

Summary The alkoxy group reactivity in carbofunctional organosilicon amines H2NR Si(OR)3 in hydrolytic and reetherification reactions was studied. Modeling calculations of electronic and molecular parameters and thermodynamic functions of organosilicon amines were performed by computer chemistry methods. The obtained calculated parameters of the molecules agree with experimental kinetic data in terms of alkoxy group reactivity of carbofunctional aminoalkylalkoxysilanes. 

IVICB/GOR] McBride, B. J., Gordon, S., Thermodynamic functions of several triatomic molecules in the ideal gas state, J. Chem. Phys., 35, (1961), 2198-2206. Cited on page 434. 

The thermodynamic functions of the solubilities of many gases in molecular liquids at room temperature have been tabulated [22]. The enthalpy of sorption is negative (exothermic) if the sorption energy exceeds the energy needed to make a hole of molecular size in the polymer or molecular liquid, and positive (endothermic) otherwise. In rough empirical correlations [21], S and AHS are usually related to the boiling temperature, critical temperature, or Lennard-Jones 6-12 potential energy parameter of the gas molecule. AHS can also be modeled atomistically [23-25], and by statistical thermodynamic equation-of-state theories (Section 3.E and Ref. [26]). 

Let us express the osmotic equilibrium condition by starting from the thermodynamic functions of the system. The number N of molecules being fixed, the system may be in various states i, to which the energy and the probability pt are attributed. The system is in equilibrium when the system entropy... 

More extensive and accurate data and additional calculations are necessary to obtain s , e , and from isotherm data over what is required to get the differential energy and entropy from the isosteric equation. The first complete calculation of ss, e and , as well as the differential quantities, has recently been made by Hill, Emmett, and Joyner (95). This paper shows in detail how the methods of this section can be applied in practice. Using heats of immersion, Harkins and Jura (96) made earlier equivalent calculations, but the relationship of their calculated quantities to the thermodynamic functions of the adsorbed molecules was not pointed out until recently by Jura and Hill (92). 

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