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

Abstract

The Nernst and Goldman-Hodgkin-Katz (GHK) equations are the standard closed-form expressions for electrochemical equilibrium/steady-state across membranes but both 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 first principles, combining the fundamental work equation with a combinatorial entropy formulation of ionic microstates that explicitly accounts for finite volume. The resulting 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 saturation, for one species. Unlike the Poisson-Nernst-Planck equations and known simulations such as the Lattice Gas Model, the derived expressions remains analytically tractable. 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. The derivation assumes constant volume, pressure, and temperature, with no net external energy input.