The oxygen‐barrier properties of amorphous copolyesters based on ethylene terephthalate with 10%, or less, of an acid comonomer were examined. Comonomer units were isophthalate, phthalate, 1,5‐naphthalate, 1,8‐naphthalate, 2,6‐naphthalate, 1,8‐anthracenate, 2,6‐anthracenate, and 2,7‐pyrenate. Even 2.5 mol % comonomer significantly affected the permeability. Linear comonomers decreased the permeability. In contrast, small amounts of a kinked comonomer increased the permeability. However, increasing the amount of kinked comonomer further (gradually) decreased permeability P below that of polyethylene terephthalate. Generally, comonomer affected solubility S less than diffusivity D; therefore, changes in P reflected primarily changes in D. The solubility and diffusivity depended on copolymer composition in accordance with static and dynamic free‐volume concepts of gas permeability in glassy polymers. The solubility correlated with the amount of free volume, as determined by the glass‐transition temperature. This study also explored the relationship of dynamic free volume, which determines D, to thermally accessible segmental motions of the polymer chain. The effect of comonomer on D correlated with the intensity of the gauche component of the subambient γ relaxation. Changes in the fraction of gauche‐glycol conformations resulting from copolymerization were confirmed by Fourier transform infrared spectroscopy. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1900–1910, 2001