The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation

5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation 

There are in principle also energy levels associated with nuclear spins. In the absence of an external magnetic field, these are degenerate and consequently contribute a constant term to the partition function. As nuclear spins do not change during chemical reactions, we will ignore this contribution. 

The assumption that the energy can be written as a sum of terms implies that the partition function can be written as a product of terms. As the enthalpy and entropy contributions involve taking the logarithm of q, the product of q % thus transforms into sums of enthalpy and entropy contributions. 

For each of the partition functions the sum over allowed quantum states runs to infinity. However, since the energies become larger, the partition functions are finite. Let us examine each of the q factors in a little more detail. 

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THE IDEAL GAS, 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. 

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. 

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]. 

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. 

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/6-31G and MP2/6-31G (0.25) constants (see next section), are evaluated widiin the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HO-IG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,... 

Rigid rotor-harmonic oscillator-ideal gas approximation. The AMBER 4.1 free energy values are summarized in Table 7. The entropy term is important and compensates for the interaction energy (enthalpy) term. A similar type of compensation has also been found in the case of DNA base pairs [40]. FI-bonded structure 4 remains the most stable and also HB6 and HBl structures remain as the second and third most stable ones. The following order of stability is however, changed. The H-bonded structure 7 and the T-shaped structure are surprisingly more stable than H-bonded structures 2, 3 and 5. Analyzing veirious... 

Partitia functias. If in addition to the ideal gas assumption above, we suppose that the species can be approximated by a rigid rotor-harmonic oscillator treatment, then the molecular partition function of a species i may be separated into its molecular... 

The thermodynamic properties were computed with the molecular geometry and vibrational frequencies given above assuming an ideal gas at 1 atm pressure and using the harmonic-oscillator rigid-rotor approximation. These properties are given for the range 0-2000°K in the Appendix (Table AI). 

To arrive at K and k, our task is to express the following terms appearing in eqns. (5.15) and 5.16) the partition functions (Q) of reactants, products and of the activated complex, the heat of reaction at absolute zero, AHq, the enthalpy of activation at absolute zero, Hq, and the tunnelling correction factor, P. For an ideal gas the total partition function can be expressed within the rigid-rotor and harmonic oscillator (RRHO) approximation as a product... 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

In the gas phase, it is usually sufhcient to calculate the partition functions and associated thermal corrections to the enthalpy and entropy using the standard textbook formulae [31] for an ideal gas under the harmonic oscillator-rigid rotor approximation, provided one then makes explicit corrections for low-frequency torsional modes. These modes can be treated instead as one-dimensional hindered internal rotations using the torsional eigenvalue summation procedure described in Ref. [32]. Rate and equilibrium constants can then be obtained from the following standard textbook formulae [31] ... 

ZPE and thermal and entropic corrections at the appropriate experimental temperatures can be calculated using the frequencies in conjimction with the standard textbook formulas for the statistical thermodynamics of an ideal gas under the harmonic oscillator/rigid rotor approximation. Equations (4) and (5) relates the rate constant and equilibrium constant with the Gibbs free energy, which can be described in terms of the enthalpy (H) and the entropy (S) in the following equation ... 

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