  Motivated by the possibility of comparing theoretical predictions of Lake
Vostok's composition with future in situ measurements, we investigate the
composition of clathrates that are expected to form in this environment from
the air supplied to the lake by melting ice. In order to establish the best
possible correlation between the lake water composition with that of air
clathrates formed in situ, we use a statistical thermodynamic model based on
the description of the guest-clathrate interaction by a spherically averaged
Kihara potential with a nominal set of potential parameters. We determine the
fugacities of the different volatiles present in the lake by defining a
"pseudo" pure substance dissolved in water owning the average properties of the
mixture and by using the Redlich-Kwong equation of state to mimic its
thermodynamic behavior. Irrespective of the clathrate structure considered in
our model, we find that xenon and krypton are strongly impoverished in the lake
water (a ratio in the 0.04--0.1 range for xenon and a ratio in the
$\sim$0.15--0.3 range for krypton), compared to their atmospheric abundances.
Argon and methane are also found depleted in the Lake Vostok water by factors
in the 0.5--0.95 and 0.3--0.5 ranges respectively, compared to their
atmospheric abundances. On the other hand, the carbone dioxide abundance is
found substantially enriched in the lake water compared to its atmospheric
abundance (by a factor in the 1.6--5 range at 200 residence times). The
comparison of our predictions of the CO$_2$ and CH$_4$ mole fractions in Lake
Vostok with future in situ measurements will allow disentangling between the
possible supply sources.
