  We measured the binding energy of N$_2$, CO, O$_2$, CH$_4$, and CO$_2$ on
non-porous (compact) amorphous solid water (np-ASW), of N$_2$ and CO on porous
amorphous solid water (p-ASW), and of NH$_3$ on crystalline water ice. We were
able to measure binding energies down to a fraction of 1\% of a layer, thus
making these measurements more appropriate for astrochemistry than the existing
values. We found that CO$_2$ forms clusters on np-ASW surface even at very low
coverages. The binding energies of N$_2$, CO, O$_2$, and CH$_4$ decrease with
coverage in the submonolayer regime. Their values at the low coverage limit are
much higher than what is commonly used in gas-grain models. An empirical
formula was used to describe the coverage dependence of the binding energies.
We used the newly determined binding energy distributions in a simulation of
gas-grain chemistry for cold cloud and hot core models. We found that owing to
the higher value of desorption energy in the sub-monlayer regime a fraction of
all these ices stays much longer and up to higher temperature on the grain
surface compared to the single value energies currently used in the
astrochemical models.
