  Hydrophobic hydration at metal/water interfaces actively contributes to the
energetics of electrochemical reactions, e.g. CO$_2$ and N$_2$ reduction, where
small hydrophobic molecules are involved. In this work, constant applied
potential molecular dynamics is employed to study hydrophobic hydration at a
gold/water interface. We propose an extension of the Lum-Chandler-Weeks (LCW)
theory to describe the free energy of hydrophobic hydration at the interface as
a function of solute size and applied voltage. Based on this model we are able
to predict the free energy cost of cavity formation at the interface directly
from the free energy cost in the bulk plus an interface-dependent correction
term. The interfacial water network contributes significantly to the free
energy yielding a preference for outer-sphere adsorption at the gold surface
for ideal hydrophobes. We predict an accumulation of small hydrophobic solutes
of sizes comparable to CO or N$_2$, while the free energy cost to hydrate
larger hydrophobes, above 2.5 Angstrom radius, is shown to be greater at the
interface than in the bulk. Interestingly, the transition from the volume
dominated to the surface dominated regimes predicted by the LCW theory in the
bulk is also found to take place for hydrophobes at the Au/water interface, but
occurs at smaller cavity radii. By applying the extended LCW theory to a simple
model addition reaction, we illustrate some implications of our findings for
electrochemical reactions.
