We describe a simple electrostatic model of hydrated ions [M(H2O)n]+ (n = 3–18, M = Li, Mg, Ca, K) which enables to calculate ion vibration frequency of the ground state. In this model the considered ion with a reduced mass vibrates in quasispherical well formed by the ion-dipole attractive potential and repulsive valence potential, these simplifications allowed to solve one dimensional Schrödinger equation, whilst the calculated ground state was considered as one of triply degenerated state of the three dimensional motion of the ion vs. hydration shell. The reduced partition function ratios were calculated from the vibration frequencies using Urey's (1947) harmonic approximation formula.
The results obtained in this way are in good agreement with those obtained by much more laborious ab initio molecular orbital methods, like SCF Hartree-Fock, DFT, MP2, etc. Moreover, we were able to extend calculations to hydrated Li and K ions surrounded with two shells of water molecules. These results are the first estimations of the upper limit of isotope fractionation in water solutions, which are 99.3‰ for Li and only 2.5‰ for K isotopes.
A.I. Boldyrev J. Simson P. v. R. Schleyer 1993 Ab initio study of the electronic structures of lithium containing diatomic molecules and ions J. Chem. Phys. 99 8793 8804.
T. Chacko D. Cole J. Horita 2001 Equilibrium oxygen, hydrogen and carbon isotope fractionation factors applicable to geologic systems Rev. in Mineralogy & Geochemistry 43 1 81.
S.A. Clough Y. Beers G.P. Klein S. L. Rothman 1973 Dipole moment of water from Stark measurements of H2O, HDO, and D2O J. Chem. Phys. 59 2254 2259.
S. Halas P. Mackiewicz 2003 Radii of di-electron systems: H, He, Li+, Be+, B3+, C4+.a revision of the ionic radius for lithium Ann. Pol. Chem. Soc. 2 909 913.
M. Humayun R.N. Clayton 1995 Precise determination of the isotopic composition of potassium: Application toterrestial rocks and lunar soils Geochim. Cosmochim. Acta 59 2115 2130.
G.G. Malenkov 1962 Geometry of structures consisting of water molecules in hydrated crystals J. Struct. Chem. 3 206 226.
J.R. O'Neil 1986 Theoretical and experimental aspects of isotope fractionation Rev. in Mineralogy 16 1 40.
L. Pauling 1960 The Nature of the Chemical Bond Cornell Univ. Press Ithaca, New York.
J.R. Rustad W.H. Casey Q.-Z. Yin E.J. Bylaska A.R. Felmy S.A. Bogatko V.E. Jackson D.A. Dixon 2010 Isotopic fractionation of Mg2+(aq), Ca2+(aq), and Fe2+(aq) with carbonate minerals Geochim. Cosmochim. Acta 74 6301 6323.
W. Salejda M.H. Tyc M. Just 2002 Algebraic methods for resolving of the Schrödinger equation Polish Scientific Publishers PWN Warsaw.
P.L. Silvestrelli M. Parrinello 1999 Water molecule dipole in gas and in the liquid phase Phys. Rev. Lett. 82 3308 3311.
G. Singer P.A. Rock 1972 Thermodynamics of lithium exchange reactions. III. Electrochemical studies of exchange between isotopic metals and aqueous ions J. Chem. Phys. 12 5556 5561.
P.B. Tomascak 2004 Development in the understanding and application of lithium isotopes in the earth and planetary systems Rev. in Mineralogy & Geochemistry 55 153 195.
Weast, R.C. Ed. 1980: CRC handbook of chemistry and physics, 61st ed. — CRC Press Inc. 1980–1981.
Urey, H.C. 1947: The thermodynamic properties of isotopic substances. — J. Chem. Soc., 1947 (London), pp. 562–581.
K. Yamaji Y. Makita H. Watanabe H. Sonoda H. Kanoh T. Hirotsu O. Kenta 2001 Theoretical estimation of lithium isotopic reduced partition function ratio for lithium ions in aqueous solution J. Phys. Chem. A105 602 613.