For more than a century, monolayer adsorptions in which
adsorbate
molecules and adsorbing sites behave ideally have been successfully
described by Langmuir’s adsorption isotherm. For example, the
amount of adsorbed material, as a function of concentration of the
material which is not adsorbed, obeys Langmuir’s equation.
In this paper, we argue that this relation is valid only for macroscopic
systems. However, when particle numbers of adsorbate molecules and/or
adsorbing sites are small, Langmuir’s model fails to describe
the chemical equilibrium of the system. This is because the kinetics
of forming, or the probability of observing, occupied sites arises
from two-body interactions, and as such, ought to include cross-correlations
between particle numbers of the adsorbate and adsorbing sites. The
effect of these correlations, as reflected by deviations in predicting
composition when correlations are ignored, increases with decreasing
particle numbers and becomes substantial when only few adsorbate molecules,
or adsorbing sites, are present in the system. In addition, any change
that augments the fraction of occupied sites at equilibrium (e.g.,
smaller volume, lower temperature, or stronger adsorption energy)
further increases the discrepancy between observed properties of small
systems and those predicted by Langmuir’s theory. In contrast,
for large systems, these cross-correlations become negligible, and
therefore when expressing properties involving two-body processes,
it is possible to consider independently the concentration of each
component. By applying statistical mechanics concepts, we derive a
general expression of the equilibrium constant for adsorption. It
is also demonstrated that in ensembles in which total numbers of particles
are fixed, the magnitudes of fluctuations in particle numbers alone
can predict the average chemical composition of the system. Moreover,
an alternative adsorption equation, predicting the average fraction
of occupied sites from the value of the equilibrium constant, is proposed.
All derived relations were tested against results obtained by Monte
Carlo simulations.