NernstEinsteinBeziehung

The Nernst-Einstein-Beziehung is a principle in physical chemistry that connects the diffusion of charged species in a solution to their electrical mobility, and thus to ionic conductivity. It combines ideas from Nernst’s description of diffusion and Einstein’s theory of Brownian motion and is widely used in electrolyte theory to relate transport properties to thermodynamic variables. In German-language literature it is commonly referred to as die Nernst-Einstein-Beziehung or die Nernst-Einstein-Gleichung.

The relation has both a microscopic form and a macroscopic implication. For a given ion i with charge q_i, diffusion coefficient D_i, and mobility μ_i in a solvent at temperature T, the Einstein relation is D_i = μ_i k_B T / q_i, where k_B is the Boltzmann constant. For electrolytes, a closely related expression connects the limiting molar conductivity Λ_i∞ of ion i to its diffusion coefficient: Λ_i∞ = (F^2 z_i^2 D_i) / (R T), where z_i is the ion’s charge number, F is the Faraday constant, and R is the gas constant. In a solution containing multiple ions, the limiting molar conductivity is the sum of the individual ion contributions: Λ∞ = Σ_i Λ_i∞.

Applications and limitations are central to its use. The Nernst-Einstein-Beziehung holds for ideal, very dilute solutions where ion–ion correlations and solvent structure are negligible. At higher concentrations, deviations arise due to ion pairing, finite ion size, and solvent–ion interactions, so the relation provides a useful but approximate link between diffusion and conductivity. It remains a fundamental tool for interpreting transport measurements, estimating diffusion coefficients from conductivity data, and understanding electrolyte performance in batteries and electrochemical cells.