High-frequency partitioning

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

We can adapt the method described for high-frequency perturbation theory to the partitioning setup, which is another way to obtain the result of adiabatic elimination. We consider a partition represented symbolically in matrix form... 

Step 3 Dissect the high-frequency band into small partitions by defining a grid of dyadic square ... 

For reactions where Ri and R2 are both diatomic and the reaction proceeds through a linear transition state, there will be six vibrations in the transition state of which two will be conserved and are likely to have high frequencies, close to those for the vibrations in Ri and Rj, whereas the other four vibrations are transitional and will be low frequency bending modes. The partition functions (q. ns vibtR Rf and q ibtR will all have values close to unity,... 

The general expressions for the energy and the entropy of a diatomic molecule are given. At high frequencies (these are often important) use the quantum mechanical expressions. The temperature dependent enthalpy and entropy depend on the frequencies of the system. The equilibrium constant for dissociation is related to the product and quotient of partition functions. Usually the partition function due to the rotational degrees of motion is much larger than that of the vibrational motion. 

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