Solvation theory to provide a molecular interpretation of the hydrophobic entropy loss of noble-gas hydration

Abstract

An equation for the chemical potential of a dilute aqueous solution of noble gases is derived in terms of energies, force and torque magnitudes, and solute and water coordination numbers, quantities which are all measured from an equilibrium molecular dynamics simulation. Also derived are equations for the Gibbs free energy, enthalpy and entropy of hydration for the Henry's law process, the Ostwald process, and a third proposed process going from an arbitrary concentration in the gas phase to the equivalent mole fraction in aqueous solution which has simpler expressions for the enthalpy and entropy changes. Good agreement with experimental hydration free energies is obtained in the TIP4P and SPC/E water models although the solute's force field appears to affect the enthalpies and entropies obtained. In contrast to other methods, the approach gives a complete breakdown of the entropy for every degree of freedom and makes possible a direct structural interpretation of the well-known entropy loss accompanying the hydrophobic hydration of small non-polar molecules under ambient conditions. The noble-gas solutes experience only a small reduction in their vibrational entropy, with larger solutes experiencing a greater loss. The vibrational and librational entropy components of water actually increase but only marginally, negating any idea of water confinement. The term that contributes the most to the hydrophobic entropy loss is found to be water's orientational term which quantifies the number of orientational minima per water molecule and how many ways the whole hydrogen-bond network can form. These findings help resolve contradictory deductions from experiments that water structure around non-polar solutes is similar to bulk water in some ways but different in others. That the entropy loss lies in water's rotational entropy contrasts with other claims that it largely lies in water's translational entropy, but this apparent discrepancy arises because of different coordinate definitions and reference frames used to define the entropy terms.

Molecular interpretation of Trouton’s and Hildebrand’s rules for the entropy of vaporization of a liquid

“…To account for this, we evaluate DS ;pp b for half a mole of dimer, ignoring its rotational term. Equation (12) strictly, the entropy and enthalpy changes to a fixed gas-phase concentration are not what would be measured experimentally but would be lower by k B and k B T, respectively [45]. We also note that neither entropy change depends on v l as advocated by Pitzer [12], consistent with conclusions made elsewhere based on analyses of experimental data [52,53,13]; only if one were to consider vaporization at constant volume in the liquid phase would a v l dependence arise.…”

“…The ideal-gas partition function of N of molecules with three rotational degrees of freedom at pressure p and temperature T is [45,44] …”

“…where we assume that translational and rotational motion of a molecule may be approximated by a collection of mean-field harmonic potentials [40][41][42][43][44][45][46], u 0 l is the energy minimum of the mean-field potential, v l is the volume per molecule, X rot l is the number of orientational minima, and F i l and s i l are half the force and torque magnitudes along and about the molecule's principal axes measured in a molecular dynamics simulation. The halving avoids double-counting of pairwise forces or equivalently shifts the forces into the reference frame of zero net force [40,41].…”

“…In the case of water, we have a theory to calculate X rot l based on the number of ways that a water molecule can form hydrogen bonds with its neighbors [43,45,44,46] X rot…”

“…More recently a number of computational approaches have been put forward to evaluate the free-volume term which have the advantage that the liquid environment with its diverse range of configurations is modeled more accurately. These methods may be based on Monte Carlo acceptance ratios [38,39], forces [40][41][42][43][44][45][46], velocity [47,48], displacements [49], or normal mode analysis of clusters [50]. The purpose of this work is to test our approach to calculate the entropy of liquids based on forces [40][41][42][43][44][45][46] by comparing the formulation with other theories, predicting the entropy of vaporization for a range of common liquids, comparing with experiment, examining how the free volumes depend on the liquid, and providing a rationale for Trouton's and Hildebrand's rules.…”