An apparatus has been constructed, similar to that of G. Hertz, making use of the difference in the thermal velocities of molecules of different weight to separate the isotopes of a gas. The problems of design and their solution are discussed in detail. Data are given to show the separation and equilibrium time of apparatus of several types, when used to separate mixtures of various initial concentrations of hydrogen and nitrogen, carbon dioxide and nitrogen, and carbon dioxide and argon. Mass spectrometer measurements on the end-products of a separation process performed on the neon isotopes, with an apparatus containing fourteen mercury diffusion pumps, show the separation to be the same as for mixtures of carbon dioxide and argon, for which the mass ratio, 44:40, is the same as the 22:20 ratio of the neon isotopes. Separation processes performed with an apparatus of twenty-four pumps on carbon dioxide-argon mixtures reveal that the separation factor--i. e., the ratio of the relative abundance of the heavy gas at one end of the system to its relative abundance at the other end, when equilibrium has been reached--and the equilibrium time are practically independent of the initial mixing ratio of the components. Hertz' approximate theory of the diffusion apparatus is extended to permit the calculation of the "enrichment factor"--i. e., the ratio of the relative abundance of the heavy gas at the "heavy" end of the system, after equilibrium is reached, to its relative abundance in the gas initially admitted to the system. The effects of impurities, viscosity, pressure gradients in the system, and insufficient pumping speed are discussed quantitatively. Work done on methane, to concentrate C13, and on nitrogen, to concentrate N15, is described. Photographs of the band spectra of C2, CN, and N2 are given which show isotope bends of sufficient intensity to make possible quantitative mass measurements on the rare isotopes C13 and N15. Methane has been produced containing 16% of C13, and nitrogen gas has been produced containing 3.3% of N15, instead of the 1% and 0.3% in the respective normal gases.