We have calculated the free energies of aqueous solvation of the hydrogen isotopes H, D, and Mu quantum mechanically, using path integral methods, to understand the effect of equilibrium solvation on the rate constants for the addition of hydrogen atom isotopes to benzene in aqueous solution. Within a classical mechanical model, the free energy of solvation of the hydrogenic solute is independent of its isotopic mass. However, when treated quantum mechanically, these solvation free energies can vary with solute isotopic mass. Our calculated Helmholtz free energies of solvation for the H and D isotopes are nearly the same, 2.53 +/- 0.31 and 2.44 +/- 0.29 kcal/mol, respectively, indicating that quantum effects on the solvation energetics are small for these systems. Similarly, our calculated Gibbs free energies of solvation for H and D are 3.23 +/- 0.32 and 3.29 +/- 0.35, respectively. The calculated free energy of solvation of H is in qualitative agreement with the experimental estimate. We have also calculated the Helmholtz and Gibbs free energies of solvation at 300 K for the light hydrogenic isotope muonium (Mu). (Muonium is a positive muon electron pair that behaves chemically like hydrogen but has one-ninth the mass.) The calculated Helmholtz and Gibbs free energies of solvation for Mu are much higher, about 3.83 +/- 0.71 and 4.23 +/- 1.23 kcal/mol, respectively. These values are used to evaluate the effect of equilibrium solvation on kinetic isotope effects for the addition reaction of hydrogen atom isotopes to benzene. If classical mechanics is used for this reaction, the change in the free energy of activation upon solvation is independent of the mass of the hydrogen atom isotope. Using our calculated free energy of solvation data, we conclude that the rate enhancement in solution compared to gas phase for hydrogen and deuterium atom addition to benzene is well described by equilibrium solvation. This argument does not apply to muonium, indicating that details of the solvent dynamics will be important for the reaction of muonium with benzene in aqueous solution.
