Donnan equilibrium

Schematic representation of the Donnan equilibrium across a cell membrane

In physical chemistry, the Donnan equilibrium describes the uneven distribution of dissolved charged particles ( ions ) that occurs when a semi-permeable membrane is permeable to the solvent and some, but not all, of the ions present in the solution . In this case, the ions passing through are distributed on both sides of the membrane in different concentrations . This uneven distribution leads to a potential difference ( Donnan potential or better: Donnan voltage ) and a difference in osmotic pressure ( Donnan pressure ). The extent of the deviation is indicated by the Donnan coefficient .

This effect is named after the chemist Frederick George Donnan , who in 1911 proposed a theory to explain such membrane equilibria. The concentration distribution is therefore referred to as the Donnan equilibrium (also Gibbs-Donnan equilibrium ), the underlying process as the Gibbs-Donnan effect or Donnan's law .

The Donnan effect is particularly important for living cells but also for technical systems.

Basics

The prerequisite for the Gibbs-Donnan effect to occur is the presence of a type of ion that is not allowed through by the semipermeable membrane. This is regularly the case with macromolecules such as soluble proteins or nucleic acids in biological cells . With the appropriate membrane properties, this can also apply to an ion type of a low molecular weight salt . In the case of dissolved sodium chloride (table salt), the diameter of the hydrated Cl ^- ion is considerably larger than that of the smaller Na ⁺ ion . The Donnan effect also occurs when ions cannot diffuse freely due to anchoring at an interface , as is the case with membrane proteins or charged polymer molecules (see ion exchangers ).

An osmotic system is in equilibrium when the chemical potentials of the solvent and the other substances that have passed through are equal on both sides of the membrane. In the presence of charge carriers, the sum of the equivalent concentrations must also be balanced on both sides in the volume (apart from the double layer ) , because the interior of a conductor is (in equilibrium) field-free and therefore free of net charge . If a non-permeating type of ion is now more concentrated on one side of the membrane, this must be compensated for by permeating ions. Their concentration is therefore different in equilibrium on both sides. A potential difference in the range of a few millivolts (mV) is associated with this uneven distribution . If the uneven distribution leads to a difference in the activity of the water, an osmotic pressure arises.

Donnan's coefficient

The (Gibbs) Donnan coefficient r _D , a dimensionless number, indicates for the permeable ions how the concentration ratios change due to the potential difference. Are z. B. on both sides of the membrane (indices _a , _i for outside , inside ) the permeable, monovalent ions K ⁺ ( potassium ) and Cl ^- ( chloride ) are present and inside a macromolecule with z charges P ^{z +} , then Cl ^{- become} inside pulled and K ⁺ pushed outwards:

${\displaystyle r_{\mathrm {D} }=\mathrm {{\frac {[K^{+}]_{a}}{[K^{+}]_{i}}}={\frac {[Cl^{-}]_{i}}{[Cl^{-}]_{a}}}} >1\,,}$

where the equality applies because the potential difference acts equally but in opposite directions on the two ions.

r _D can be calculated using the neutrality conditions

${\displaystyle \mathrm {[K^{+}]_{a}-[Cl^{-}]_{a}} =0\,}$   and

${\displaystyle \mathrm {[K^{+}]_{i}-[Cl^{-}]_{i}+z[P^{z+}]_{i}} =0\,}$

also specify as a function of the concentration of the charged macromolecule:

${\displaystyle r_{\mathrm {D} }={\sqrt {1+\mathrm {\frac {z\,[P^{z+}]_{i}}{[K^{+}]_{i}}} }}\approx 1+\mathrm {\frac {z\,[P^{z+}]_{i}}{2\,[K^{+}]_{i}}} \,,}$

where the approximation applies to dilute solutions. The permeable ions are distributed more unevenly, the higher the concentration and charge number of the macromolecule. The latter is often dependent on the pH value via the degree of dissociation .

Donnan's potential

The relationship between membrane potential and Donnan coefficient results from the Nernst equation :

${\displaystyle \Delta \Phi ={\frac {RT}{F}}\cdot \ln(r_{\mathrm {D} })}$

ΔΦ is the difference in the electrical potential, R the universal gas constant , T the absolute temperature in K and F the Faraday constant .

The following table with examples shows that the presence of a non-permeable ion can lead to large differences in the concentration of the ions that have passed through. At the same time, considerable Donnan potentials can arise, quoted from:

z * [P z + ] i	[K + ] a = [Cl - ] a	[K + ] i	[Cl - ] i	r D	ΔΦ (mV)
0.002	0.0010	0.00041	0.00241	2.44	22.90
	0.0100	0.00905	0.01105	1.10	2.56
	0.1000	0.0990	0.1010	1.01	2.58
0.02	0.0010	0.00005	0.02005	20.05	76.96
	0.0100	0.00414	0.02414	2.41	22.65
	0.1000	0.0905	0.1105	1.10	2.56

(all concentration data in mol · l ⁻¹ , temperature: 298 K). The height of the membrane potential depends on the ratio of the charged macromolecule to the mobile ions. If such ions are present in high concentration, the presence of the immobile ion has only a minor effect.

Osmotic pressure

In equilibrium, the osmotic pressure results from the ratio of the activities of the solvents on both sides of the membrane:

${\displaystyle \Pi ={\frac {RT}{V_{L}^{m}}}\ln {\frac {a^{(1)}(L)}{a^{(2)}(L)}}\!\,}$

Where: the molar volume of the solvent, R  is the universal gas constant and T is the absolute temperature in  K . ${\displaystyle V_{L}^{m}}$

See also

• Goldman equation
• Colloid Osmotic Pressure

literature

• Walter J. Moore, Dieter O. Hummel: Physical chemistry . Walter de Gruyter, Berlin 1986, ISBN 3-11-010979-4 .

Web links

Commons : Gibbs-Donnan effect  - album with pictures, videos and audio files

• Lecture notes biophysics. (PDF; 302 kB) University of Rostock
• Membrane potential . University of Hamburg: Botany online

Individual evidence

1. ↑ Sehon Hannes. Physical chemistry . Herder Freiburg publisher. i. Brsg. 1976, ISBN 3-451-16411-6
2. ↑ ^{a b} Moore Walter J., Hummel Dieter O .: Physical Chemistry , pp. 650f. Walter de Gruyter, Berlin 1986, ISBN 3-11-010979-4 .
3. ^ Charles Tanford : Physical Chemistry of Macromolecules . John Wiley, New York 1961.