TY - JOUR

T1 - The Donnan equilibrium: I. On the thermodynamic foundation of the Donnan equation of state

AU - Philipse, A.P.

AU - Vrij, A.

PY - 2011

Y1 - 2011

N2 - The thermodynamic equilibrium between charged colloids and an electrolyte reservoir is named
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 equilibrium is the ensuing osmotic
equation of state which is a nonlinear one, even when both colloids and ions obey Van ’t Hoff’s
ideal osmotic pressure law. The Donnan equation of state, nevertheless, is internally consistent;
we demonstrate it to be a rigorous consequence of the phenomenological thermodynamics of a
neutral bulk suspension equilibrating with an infinite salt reservoir. Our proof is based on an
exact thermodynamic relation between osmotic pressure and salt adsorption which, when
applied to ideal ions, does indeed entail the Donnan equation of state. Our derivation also shows
that, contrary to what is often assumed, the Donnan equilibrium does not require ideality of the
colloids: the Donnan model merely evaluates the osmotic pressure of homogeneously
distributed ions, in excess of the pressure exerted by an arbitrary reference fluid of uncharged
colloids. We also conclude that results from the phenomenological Donnan model coincide with
predictions from statistical thermodynamics in the limit of weakly charged, point-like colloids.

AB - The thermodynamic equilibrium between charged colloids and an electrolyte reservoir is named
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 equilibrium is the ensuing osmotic
equation of state which is a nonlinear one, even when both colloids and ions obey Van ’t Hoff’s
ideal osmotic pressure law. The Donnan equation of state, nevertheless, is internally consistent;
we demonstrate it to be a rigorous consequence of the phenomenological thermodynamics of a
neutral bulk suspension equilibrating with an infinite salt reservoir. Our proof is based on an
exact thermodynamic relation between osmotic pressure and salt adsorption which, when
applied to ideal ions, does indeed entail the Donnan equation of state. Our derivation also shows
that, contrary to what is often assumed, the Donnan equilibrium does not require ideality of the
colloids: the Donnan model merely evaluates the osmotic pressure of homogeneously
distributed ions, in excess of the pressure exerted by an arbitrary reference fluid of uncharged
colloids. We also conclude that results from the phenomenological Donnan model coincide with
predictions from statistical thermodynamics in the limit of weakly charged, point-like colloids.

U2 - 10.1088/0953-8984/23/19/194106

DO - 10.1088/0953-8984/23/19/194106

M3 - Article

SN - 0956-8984

VL - 23

JO - Journal of Physics: Condensed Matter

JF - Journal of Physics: Condensed Matter

M1 - 194106

ER -