A statistical mechanical model of a bilayer of dipalmitoyl-3-sn-phosphatidylcholine molecules in equilibrium with an aqueous phase saturated with an n-alkane is presented. A mean-field approach developed in previous work on a solventless bilayer (Gruen, Biochim. Biophys. Acta. 595:161--183, 1980) is extended to allow alkane chains to exist in the hydrophobic core of the membrane. As the alkane chains are chemically similar to the lipid chains, much of the analysis follows directly from the solventless model. Novel features of the present model are the inclusion of (a) a term which models the free energy cost of creating space for alkane conformations, (b) a term which constrains the chains to pack evenly in the hydrophobic region of the membrane, and (c) a term which estimates the free energy of mixing of the lipid and alkane molecules in the plane of the bilayer. On uptake of alkane, the dimensions of the bilayer increase. Allowance is made for an increase in thickness and/or an increase in area per lipid. A thermodynamic framework is established which allows evaluation of the free energy of a bilayer of arbitrary dimensions with a view to predicting the equilibrium structure.