The Donnan equilibrium: I. On the thermodynamic foundation of the Donnan equation ofstate

Abstract

The thermodynamic equilibrium between charged colloids and an electrolyte reservoir isnamed after Frederic Donnan who first published on it one century ago (Donnan 1911 Z.Electrochem. 17 572). One of the intriguing features of the Donnan equilibriumis the ensuing osmotic equation of state which is a nonlinear one, even whenboth colloids and ions obey Van ’t Hoff’s ideal osmotic pressure law. The Donnanequation of state, nevertheless, is internally consistent; we demonstrate it to be arigorous consequence of the phenomenological thermodynamics of a neutral bulksuspension equilibrating with an infinite salt reservoir. Our proof is based on an exactthermodynamic relation between osmotic pressure and salt adsorption which,when applied to ideal ions, does indeed entail the Donnan equation of state. Ourderivation also shows that, contrary to what is often assumed, the Donnan equilibriumdoes not require ideality of the colloids: the Donnan model merely evaluates theosmotic pressure of homogeneously distributed ions, in excess of the pressureexerted by an arbitrary reference fluid of uncharged colloids. We also conclude thatresults from the phenomenological Donnan model coincide with predictions fromstatistical thermodynamics in the limit of weakly charged, point-like colloids.