The charge-density distribution and the respective plasma potential distribution in an electrodeless low-pressure rf plasma as it is employed in electron-gas secondary-neutral mass spectrometry (SNMS) were determined theoretically by a comparatively simple plasma model. A second-order finite element program combines a Maxwell-Boltzmann like electron density distribution with a collective drift motion of plasma ions directed from the center of the discharge to the wall of the plasma chamber and calculates a self-consistent solution of the space-charge problem by iteratively solving the appropriate Poisson equation. A comparison with the experimentally determined spatial distributions of the electron density and the plasma potential for two different geometries of the plasma chamber shows good agreement with respect to the theoretical calculations. A plasma chamber optimized via this plasma modeling was incorporated into a SNMS instrument. The new chamber's performance was studied with respect to previously used geometries. A distinct enhancement of the intensities of postionized sputtered species was observed and ascribed to a plasma-optical focusing effect; Full exploitation of this beneficial feature appears to be partially limited by a concurrent increase of space-charge accumulation in the vicinity close to the plasma chamber's exit aperture.