Mixed matrix membranes capable of separating carbon dioxide from mixtures including carbon dioxide and methane, and processes for purifying methane using the membranes, are disclosed. The membranes are preferably polymer membranes that include discrete carbon-based molecular sieve particles with sizes of between about 0.5 microns to about 5.0 microns. The particles are formed by pyrolyzing a precursor polymer in the form of a powder or film. The pyrolyzed polymer is then ideally milled to desired small size particles. The preferred ratio of particles to polymer is about 0.25 to about 1.0 by volume. A preferred method for preparing the mixed matrix membrane is by dispersing the particles in a solvent, adding a small quantity of the desired polymer or “sizing agent” to “size” or “prime” the particles, adding a polymer, casting a film of the polymer solution, and evaporating the solvent to form a mixed matrix membrane film. The mixed matrix membrane film permits passage of carbon dioxide and methane, but at different permeation rates, such that the ratio of the relative permeation rates of carbon dioxide to methane is larger through the mixed matrix membrane film than through the original polymer. The mixed matrix membrane is preferably in the form of a dense film or a hollow fiber. A mixture containing carbon dioxide and methane can be enriched in methane by selective passage of carbon dioxide over methane in a gas-phase process through the membrane.