The falling liquid film has become a popular means of transferring heat and mass from a vapor to a binary liquid, especially in gas-fired heat pump systems. Ideally, the required amount of heat and mass transfer can be accomplished by using a simple cylindrical tube; however, increasingly stringent size and weight requirements for the machine prohibit use of the simple cylindrical surface, and other more complex surfaces with higher absorption capacities have been sought. In this article, absorption of a single component and condensation of a binary mixture on an axially fluted tube is considered. The solution to the problem hinges on the energy equation, although the entire energy transfer process is mass-transfer-limited. Significant mass transfer is limited to a thin layer near the liquid-vapor interface. Solutions to the energy equation are obtained for both the conduction- and convection-dominated regimes. In the latter, significant heat transfer occurs within a thin layer near the liquid-vapor interface which contains the mass transfer layer; this "boundary layer" structure does not appear to have been recognized in previous work in this area. Using the present results, the capacity of a given tube may be predicted as a function of governing geometrical and physical parameters. The principal objective of this work is to develop the theoretical tools from which computations may be carried out during a design process. The theoretical results may be applied to mixtures typical of application in the absorption heat pump industry.