A unified framework for van 't Hoff's law: addressing the complexity of osmotic concentration

The original van 't Hoff's law established the theoretical foundation for osmosis but applies only to ideal solutions and membranes. To address real-world complexities (non-ideal solutions, diverse membranes, etc.), multiple variations have emerged over a century. In resolving osmosis-related conceptual issues, our previous work introduced several new fundamental concepts to fill gaps in the study of osmosis and redefined osmotic concentration (OC) as a membrane-dependent, osmosis system-level parameter, not a parameter of any isolated solution. This article examines the multiple factors influencing the initial OC (OC₀) before osmosis occurs and demonstrates that the multiple forms of van 't Hoff's law can be unified using OC₀ into one general form through mathematical reasoning. Building upon this unified framework, we further propose an extended formulation to accommodate more complex osmosis systems. These general forms of van 't Hoff's law overcome the limitations of the original and may be widely applied to real-world dilute solutions and membranes. We also perform an initial validation of our work using measured data in the literature. This work represents a significant theoretical advance in the understanding of osmosis and has potential to impact multiple disciplines that teach and research it, including physics, chemistry, physiology and clinical disciplines.