Distillation cascades based on differential condensation (DC) and differential vaporization (DV) stages are considered as an alternative to ordinary (equilibrium stage) distillation for the separation of binary mixtures. In the differential condensation-differential vaporization (DCDV) cascades, stage separation factors can be much higher than for cascades based on equilibrium stages, and as a result, significantly fewer stages and, possibly, lower recycle ratios may be required compared with conventional distillation. This paper presents the theory behind the DV and DC stages and explores the use of the two-stage cascade design (a DV reboiler and DC condenser), to recover oxygen for use in hypersonic flight. Part II will consider N-stage cascades that could possibly be used to separate most systems now separated by ordinary distillation.