Rigid-rotor harmonic-oscillator transition states

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. 

Figure 2.1.b. Schematic diagram of the energy levels and vibration-rotation transitions (after Herzberg, 1950). The lower lines (v/ = 0) correspond to the fundamental vibrational state with different rotational states (J// = 0,1, 2,3,4). The first vibrationally excited state (v/ = 1) with related rotational states (J/ = 0 to 4) are shown by the upper lines. The transitions corresponding to the P and R branches are indicated. Transition AJ = 0 is not allowed in the case of the idealized rigid rotor, harmonic oscillator for a linear molecule, but can be observed (Q-branch) in other cases. The notations are from Herzberg (1950). 

The partition function ratios needed for the calculation of the isotope effect on the equilibrium constant K will be calculated, as before, in the harmonic-oscillator-rigid-rotor approximation for both reactants and transition states. One obtains in terms of molecular partition functions q... 

Evaluation of the transition moment for the harmonic oscillator and for the two-particle rigid rotor gives the selection rules Au = 1 and A/ = 1 stated in Sections 4.3 and 6.4. 

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