This dissertation explores the impacts of configurational free volume, introduced through triptycene units, on the transport properties of condensable vapors and gases in thermally rearranged polybenzoxazole-based polymers as well as their polyhydroxyimide precursors which are not thermally treated. Through a series of in-depth studies, we elucidate the fundamental mechanisms governing solubility, diffusivity, and selectivity in these materials, offering insights into their potential applications in molecular separations. Chapter 1 presents pioneering findings on how triptycene-containing polybenzoxazoles (TPBOs) mediate condensable vapor transport, demonstrating size-controlled (entropically driven) sorption and diffusion, quite unlike conventional glassy polymers. This configurational free volume facilitates the tuning of sorption and diffusion selectivity, potentially unlocking routes to de-bottleneck limitations in current state-of-the-art membrane performance. Chapter 2 extends this investigation to light gases (N2, CH4, and CO2), proposing a mechanism for molecule transport in configurational free volume. By varying triptycene content, we analyze the effects (and lack thereof) on the components of CO2/CH4 selectivity, uncovering that configurational free volume exclusively regulates light gas diffusion selectivity in the Langmuir mode without affecting the other components of selectivity. Chapter 3 addresses the critical issue of data reliability in solubility measurements, proposing a standardized methodology for estimating uncertainty in sorption and adsorption measurements. This study lays out a framework for understanding factors that contribute to measurement error, and comprehensively addresses accurate comparison between solubility measurements within a single lab, as well as across laboratories worldwide, which is critical in the previous two chapters. Chapter 5 investigates the swelling behavior of TPBOs, revealing that configurational free volume and thermal rearrangement collectively and synergistically enhance swelling resistance, a crucial factor for membrane stability and performance in real-world applications. It also brings in techniques from Chapter 3 and data science to demonstrate a novel dilatometry analysis method. This comprehensive study not only lays out an advancement in our understanding of configurational free volume, but also provides measurement and error analysis standardizations in the membrane field. The work may be of practical use to researchers seeking to improve the rational design of membranes and the design of extremely accurate apparatuses geared towards membrane and polymer sciences, such as the sorption and dilation apparatuses discussed herein.