As a member of the ubiquitous ammonium transporter/methylamine permease/Rhesus (Amt/MEP/Rh) family of membrane protein channels, the 50 kDa Rhesus channel (Rh50) has been implicated in ammonia (NH$_3$) and, more recently, also in carbon dioxide (CO$_2$) transport. Here we present molecular dynamics simulations of spontaneous full permeation events of ammonia and carbon dioxide across Rh50 from Nitrosomonas europaea. The simulations show that Rh50 is functional in its crystallographic conformation, without the requirement for a major conformational change or the action of a protein partner. To assess the physiological relevance of NH$_3$ and CO$_2$ permeation across Rh50, we have computed potentials of mean force (PMFs) and permeabilities for NH$_3$ and CO$_2$ flux across Rh50 and compare them to permeation through a wide range of lipid membranes, either composed of pure lipids or composed of lipids plus an increasing cholesterol content. According to the PMFs, Rh50 is expected to enhance NH$_3$ flux across dense membranes, such as membranes with a substantial cholesterol content. Although cholesterol reduces the intrinsic CO$_2$ permeability of lipid membranes, the CO$_2$ permeabilities of all membranes studied here are too high to allow significant Rh50-mediated CO$_2$ flux. The increased barrier in the PMF for water permeation across Rh50 shows that Rh50 discriminates 40-fold between water and NH$_3$. Thus, Rh50 channels complement aquaporins, allowing the cell to regulate water and NH$_3$ flux independently. The PMFs for methylamine and NH$_3$ are virtually identical, suggesting that methylamine provides an excellent model for NH$_3$ in functional experiments.