Generalized Steady-State Equation for Multi-Species Ion Transport Across A Dual-Compartment System

The Nernst and Goldman-Hodgkin-Katz (GHK) equations are the standard closed-form expressions for electrochemical equilibrium/steady-state across membranes but they rely on the assumption of dilute ion concentrations and do not account for ionic size or excluded volume effects. Here, a generalized steady-state equation is derived from the Fundamental Thermodynamic Relation and using a combinatorial approach for Gibbs Entropy that explicitly accounts for finite volume. The resulting closed-form expression is valid for n ion species of arbitrary charge, size, and quantity. It reduces into the Nernst equation assuming dilute solution, and the Boltzmann ratio under perfect saturation. Unlike the Poisson-Nernst-Planck equations and known simulations such as the Lattice Gas Model, the derived expressions remains analytically tractable. The expression does not require curve-fitting or experimental values either. Although motivated by biological membrane systems, the framework is general and applicable to any system involving two coupled ionic compartments, including electrochemical cells and microfluidic devices.