An isothermal equilibrium theory analysis of a simple two-step pressure-swing adsorption (PSA) process utilizing an adsorbate-adsorbent system that exhibits a favorable Langmuir isotherm was carried out. Analytic expressions, either simple or recursive, were obtained that describe process operation and process performance during the approach to periodicity. These expressions are a function of cycle number and various process parameters. A recursive relationship for the dimensionless penetration depth for each cycle was determined and, although no closed form is readily available (and likely does not exist), the recursive relationship is easily applicable in any spreadsheet program. All other expressions were derived as functions of the penetration depth, thus lending a full analysis to the capabilities of a spreadsheet program. The analysis is primarily focused on the case of no breakthrough, because of the fact that breakthrough forces the system prematurely to periodicity and is therefore trivial for the approach analysis. The resulting expressions were used to examine process performance upon the approach to periodicity for several hypothetical systems, and the effects of various parameters on the number of cycles required to reach a periodic, or virtually periodic, state were examined. From an understanding or educational point of view, this analysis clearly shows how the so-called "heel in the bed", i.e., the adsorbate loading remaining in the bed after the end-of-purge step, forms on the very first cycle and continues to increase cycle after cycle until periodicity is attained. This buildup of the heel in the bed is characteristic of all PSA processes, with a slower buildup resulting from a more nonlinear isotherm or a smaller purge-to-feed ratio.