Study of Isotope Exchange in Molecular Gases in Equilibrium

Abstract:

Urey [1] and Bigeleisen & Mayer [2] used molecular partition function ratios to calculate the equilibrium constant of a chemical reaction to study the isotope exchange process in equilibrium. That time it was not possible to measure the energies of isotopic molecules relative to the completely dissociated energy states of the molecule and precise determination of zero-point energies were impossible [1]. Using harmonic potential approximation, the vibrational frequencies of the molecules were calculated for ascertaining the partition functions at various temperatures. Vibrational frequencies of a large number of molecules are available from spectroscopic data. In recent times, computational chemistry packages can provide frequencies as well as energies of a molecule very accurately.

We use Gaussian09 [3] as the tool to study the abundances of multiply substituted isotopologues as a function of temperature by calculating the partition function ratios and equilibrium constants using both the frequencies and the energies of the molecules with different isotopologues. The vibrational frequencies of isotopologues of a molecule are obtained from Gaussian frequency calculation after optimizing the molecule for its ground state. For the energy approach, ‘Sum of electronic and thermal energies’ inclusive of electronic, translational, rotational, vibrational, and zero-point energies was utilized in the calculation. Thus the molecular energies are obtained with respect to the completely dissociated molecules and Boltzmann factor can be applied. Only the vibrational and zero-point energies that change with the molecular masses contribute to isotopic fractionation.

We obtain vibrational frequencies and ‘Sum of electronic and thermal energies’ from Gaussian09 [3] and evaluated the relative abundances of multiply substituted isotopologues as a function of temperature. Relative abundances calculated using vibrational frequencies is in agreement with the previous estimates of Wang et al., [4] obtained using spectroscopic data. Accuracy of abundances of isotopologues calculated using energies can be improved with improved accuracy of energy values obtained using Gaussian09 [3].

[1] Urey, H.C., Journal of the Chemical Society, 1947(0):p.562-581. [2] Bigeleisen, J., Mayer, M.G., The Journal of Chemical Physics, 1947. 15(5): p. 261-267. [3] Frisch, M.J., et al., Gaussian 09 Rev. A.02.: 2009, Wallingford, CT. [4] Wang, Z., Schauble, E.A. and Eiler, J.M., Geochimica et Cosmochimica Acta, 2004. 68(23): p. 4779-4797.