The Rigid Rotor Harmonic Oscillator Approximation

This simplification was already introduced in Chapter 2. In the rigid rotor approximation there is no rotational-vibrational interaction. The molecular Schrodinger 

4 Isotope Effects on Equilibrium Constants of Chemical Reactions 

The last term refers to the 3N — 6 (or 3N — 5) vibrations. Corresponding to each of the terms in Equation 4.70 are sets of quantum numbers (e.g. translational quantum numbers, rotational quantum numbers, etc.) which are independent of each other. From this point it is quite straightforward to show that the partition function can be factored into a product of partition functions corresponding to translation, rotation, etc. 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

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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>. 

Thermodynamic functions of PHJ [51] in the ideal gas state, calculated in the rigid rotor-harmonic oscillator approximation, are presented in Table 19. They are based on the fundamental frequencies derived from the vibrational spectra of phosphonium halides (see above) and the calculated bond length [10] r=1.382 A [50]. 

The rigid-rotor harmonic oscillator approximation provides an adequate description of molecular properties. 

Many scientists in the fields of thermodynamics and computational software use the rigid rotor-harmonic oscillator approximation or other shortcuts due to the relatively small contribution of the internal rotations to the whole enthalpy and entropy values. 

We have adopted a vialue of AHS(CF2,g,298) = -44.6 kcal/mol from the data of Modica and LeGraff (16,17) and of Carlson (19). This yields values of the equilibrium constant for reaction t ) with in a factor of two of those calculated from the data of Farber et (21), which is certainly within the accuracy of both the experiment and the limits of the rigid-rotor, harmonic oscillator approximation at 2000 to 2500 K ( ). The physical and thermochemical data selected here are sumnarized in Table II and the ideal gas thermodynamic functions calculated to 1500 K from these data are summarized in Table III. 

The quantum number v can equal 0,1,2,..., and the quantum number J can also equal 0,1,2,. We refer to the approximation of Eq. (22.2-29) as the rigid rotor-harmonic oscillator approximation. 

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