A rigorous derivation of the ideal gas law from energy constraints

Standard introductory courses in thermodynamics typically introduce the ideal gas equation of state as an empirical postulate or a result of kinetic theory. In this work, we show an alternative way of deriving the equation pV∝NT from general postulates. By assuming only the functional independence of internal energy from volume and enthalpy from pressure under isothermal conditions, or, equivalently, the opposite conditions, we show how the ideal gas law emerges as a necessary consequence of these energy-based symmetries. This serves a dual purpose: it acts as a rigorous exercise in applying thermodynamic relations and multivariable calculus, while simultaneously providing an alternative conceptual perspective on the foundational principles of thermodynamics.