A Common Model for Ion Hydration Thermodynamics

Marcus (1987, 1991) presented a model that is applicable to all the thermodynamic functions of hydration. It follows the thought process of dissolution described at the beginning of Sect. 2.2  

Here Y is, for the present purposes, H or 5, Y is the unitary part, describing the hydration process proper. At 25 °C AhydH differs by 2.29 kJ mor from the standard enthalpy Ahyd/f° and Ahyd5 differs by - 18.9 J K mor from the standard entropy Ahyd5 ° per mole of ions. These difference quantities pertain to step 6 in the thought process the relaxation of the fixed points in the ideal gas and solution phases and turning on the full translational degrees of freedom. The numerical values 

The interaction of the solute ion with the water in its surrounding (step 4 in the thought process) is described by the electrostatic terms ATEiifc f) + AFei2(z, r). The first pertains to the first hydration shell of the ion and the second to water outside this shell. For the Gibbs energy of hydration these two terms read  

The second term in the square brackets is the Born expression applicable at distances n + Ari, i.e., beyond the first hydration shell of thickness Ar. The first term describes the electrostatic interaction inside this shell, characterized by a relative permittivity e now, approximated by the square of the refractive index of water at the sodium D line. With the relevant de /rfT and de/tfT values for water at 25 °C, the enthalpy of hydrationEq. (2.28)is AHeh+A//ei2 = -69.5z2[(0.35(Ari/ri)+1.005)/(ri+Ari)] kJ mor The entropy is then A5eu + A5ei2 = —4.06z [(1.48(Ari/ri)+1.00)/(ri + Ari)] J mol The thickness of the first hydration shell, Ar, depends on the number of water molecules, hi, in it, the hydration number. According to the model (Marcus 1987) hi = 0.36 zi /(r/nm), that is, it is proportional to the charge number of the ion and inversely proportional to its radius. The volume occupied by hi water molecules is nhid l6, where cfw = 0.276 nm is the diameter of a water molecule. Hence the volume of the first hydration shell is given by  

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