Summary: We construct approximate solutions of the linearized Boltzmann equation for a gas outside of a completely absorbing sphere, a simple model for a liquid droplet growing in a supersaturated vapor. The solutions are linear combinations of two Chapman-Enskog-type solutions, which carry heat and particle currents, and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path. To construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere, we expand the solution in Burnett functions and truncate the resulting system of equations for the expansion coefficients. For one particular truncation prescription, which generalizes Grad’s 13-moment scheme, good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere; the results for small radii extrapolate smoothly toward the known limit of zero radius. We present results for the reaction rate (the particle current arriving at the sphere divided by the density at infinity) and for the density and temperature profiles in the boundary layer. The explicit calculations are carried out for Maxwell molecules, but the method appears to be suitable for more general intermolecular potentials.