  The Langmuir-McLean isotherm is often interpreted as providing an
approximation to the most probable grain boundary segregation as a function of
the bulk mole solute fraction $x_B$, even though $x_B$ is not an independant
parameter in the free energy minimization on which it is based. In this paper
it is shown that the most probable segregation for a specified $x_B$ differs
from the standard Langmuir-McLean relation. Numerical solution of the derived
equation suggests that two potentially stable interface compositions are
associated with most bulk compositions. One solution represents a state with an
excess of solute along the boundary relative to the bulk, while the other
represents a deficit. The vacancy content ratio between the interface and the
bulk plays a large role in determining the shape of the derived isotherm.
