Rigid rotor harmonic oscillator approximation

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

A2 Corrections to the Rigid Rotor Harmonic Oscillator Approximation in the Calculation of Equilibrium Constants... 

Figure 8. Spectra of CO calculated from the rigid rotor-harmonic oscillator approximation. The top spectrum is CO at 298 K. The bottom is CO at 20 K. This reduction in lines will be very important for simplifying the analysis in larger, more complicated species. 

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. 

It follows from the preceding discussion that the equilibrium constant for complex formation evaluated using the rigid rotor-harmonic oscillator approximation, with molecular constants derived from ab initio SCF calculations with a medium basis set (of DZ quality), is not very accurate. Comparison of the AG° values calculated using extended and medium basis sets indicates that the major uncertainty in AG is derived from AH . TASP is not as dependent on the basis set used. Furthermore, it is evident that the entropy term plays an extremely important rote in complex formation neglecting it may result not only in quantitative, but even in qualitative failure. 

The thermodynamic functions were estimated from those in the present table for HgS(g) (6 ) by adding those for DgS(g) and subtracting those for HgS(g), where both the added and subtracted functions were generated using the rigid-rotor harmonic oscillator approximation. In this calculation the molecular constants for DgS were taken from reference (2). 

The thermodynamic functions were taken from the JANAF table for H2S(g) dated Dec. 31, 1965 (1 ). These in turn were taken from Gordon (8 ) except below 298 K were they were calculated by the rigid-rotor, harmonic-oscillator approximation. Gordon had calculated from 298 K to 6000 K by a method which takes into account second-order corrections for vibrational anharmonicity, vibration-rotation interaction, and centrifugal stretching. The spectroscopic constants used were taken from Allen and Plyler (9). 

For an isolated molecule in the rigid rotor, harmonic oscillator approximation, the (quantum) energy states are sufficiently regular to allow an explicit construction of the partition function, as discussed in Chapter 12. For a collection of many particles the... 

The thermodynamic characteristics of monohydration of monoatomic cations are listed in Table 12. The AE values for all the complexes were obtained using extended basis sets for details see Ref For the majority of the complexes studied, the calculated thermodynamic values, based on the rigid rotor-harmonic oscillator approximation, can be compared with the corresponding experimental characteris-... 

IR-frequencies were determined by Christe et al. [72CHR/SCH] and, together with molecular constant data, they computed thermodynamic properties of SeFsC g) using the rigid-rotor, harmonic-oscillator approximation. The following temperature dependence... 

Using the rigid-rotor harmonic-oscillator approximation on the basis of molecular constants and the enthalpies of formation, the thermodynamic functions C°p, S°, — G° —H°o)/T, H° — H°o, and the properties of formation Af<7°, and log K°(to 1500 K in the ideal gas state at a pressure of 1 bar, were calculated at 298.15 K and are given in Table 9 <1992MI121, 1995MI1351>. Unfortunately, no experimental or theoretical data are available for comparison. From the equation log i = 30.25 - 3.38 x /p t, derived from known reactivities (log k) and ionization potential (fpot) of cyclohexane, cyclohexanone, 1,4-cyclohexadiene, cyclohexene, 1,4-dioxane, and piperidine, the ionization potential of 2,4,6-trimethyl-l,3,5-trioxane was calculated to be 8.95 eV <1987DOK1411>. 

THE IDEAL GAS, RIGID-ROTOR HARMONIC-OSCILLATOR APPROXIMATION... 

---