The gases diffusion equation:- A unified mathematical framework and rigorous proof for molecular diffusion across the vacuum and planetary atmosphere

Classical Kinetic theory describes gaseous effusion in a vaccum via graham's law and atmospheric diffusion via Fick's law as separate regimes. This paper establishes the gaseous diffusion equation, unifying vaccum transit and planetary atmospheric diffusion under a single analytical formulation. By normalising unit transit times against istopically exact molar mass ( M ). We derive two fundamental constant:- the vaccum diffusion constant ( M0 = 2.72702 ) and the planetary atmospheric diffusion constant ( M1 = 7.36664 ). Furthermore, we present rigorous mathematical proofs incorporating the 9.5 × intermolecular mean spacing gap ( 3.44 nm ) , steric collision, cross - section resistance (X%), thermodynamics scaling ( T 1.75 ), and the planetary pressure multiplier ( 1/P ).