For nitrous oxide, the first anesthetic for which uptake was measured in humans, Severinghaus noted empirically that a plot of the log of uptake against the log of elapsed time produced a straight line with slope -0.5, suggesting that uptake is proportional to the inverse square root of time. This result is something of a black box model, based on empirical curve fitting without regard to physiology. Some authors (e.g., Lowe) repeatedly returned to this inverse square root of time model as a benchmark while others (e.g., Hendrickx) questioned its validity and demanded the relationship be expressed with a physiologic model whose structure matches the known physiology being modeled. Nevertheless, the fact that authors have repeatedly come back to this inverse square root of time model as a benchmark suggests that it might have some underlying validity which has not previously been recognized. We re-explored this mathematically in an attempt to reveal hitherto undiscovered insights or limitations. In this study, we examined the square root of time model (viewed as a power function) and compared it with multi-compartment models. Further, we explored the stability of this relationship to systematic variation in the power value and also to the superimposition of noise-like perturbations, seeking conditions under which it might not work. Based upon this theoretical analysis, we also speculate on the existence of a physiological compartment with a time constant between that of the vessel-rich group (VRG) and muscle, and what the identity of such a compartment might be.