The reduction energies of these gas-phase nanodrops can be related to bulk solution-phase reduction energies by the thermodynamic cycle shown in , where Δ

*H*_{solv} is the Born enthalpy associated with solvating a droplet with

*n* water molecules and nominal charge

*z* into bulk solution. A modified Born equation () is used to obtain the free energy of solvation, Δ

*G*_{solv}, of the product and reactant charged nanodrops (5)

where

*ze* is the charge,

*n* is the number of water molecules,

*R*_{s} is the size-exclusion radius of a water molecule,

*∊*_{o} is the vacuum permittivity, and

*∊* is the dielectric constant of the solvent,

*a*_{am} and

*a*_{i} are factors that relate the size of an ammonia molecule and a metal ion, respectively, to that of a water molecule. The differential of the negative of with respect to temperature is used to estimate the entropy of solvation () (6)

and is combined with the free energy of solvation at 298 K to obtain the enthalpy of solvation at this temperature. The enthalpic AEA is omitted in our subsequent analysis and the solution-phase reduction potentials are referenced to a gas-phase electron. Values of the enthalpic AEA have been reported;,, the absolute half-cell potential could be referenced to a solvated electron by including this term.

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