Donnan equilibrium

Donnan equilibrium. It is a thermodynamic equilibrium established between two ionic solutions separated by a semipermeable membrane that is permeable to small ions but impermeable to at least one charged species. This phenomenon, first described by Frederick George Donnan, explains the uneven distribution of diffusible ions across such a membrane, leading to a stable electric potential difference. The equilibrium is fundamental to understanding cellular bioelectricity and has wide-ranging applications in industrial processes.

Definition and basic principles

The Donnan equilibrium describes the final distribution of small, permeable ions between two compartments when a large, impermeable charged molecule is present on one side. This impermeable species, often a colloidal particle or a charged polymer like a protein within a cell, restricts the movement of its counterions, creating an electrochemical imbalance. The principle is a direct consequence of the requirement for both **electroneutrality** in each bulk phase and the equality of **electrochemical potential** for each diffusible ion across the semipermeable membrane. A classic demonstration involves a membrane separating a solution of sodium chloride from one containing a sodium salt of a non-diffusible polyelectrolyte. At equilibrium, the concentrations of the small ions satisfy a specific ratio, known as the **Donnan ratio**, and a membrane potential, akin to a Nernst potential, is established. This potential is a key component of the resting membrane potential in excitable cells like neurons.

Derivation and mathematical formulation

The mathematical treatment arises from applying the conditions of equilibrium. For a simple system with a monovalent salt like KCl and an impermeable anion **R⁻** on one side, the electrochemical potential for each permeable ion (e.g., **K⁺** and **Cl⁻**) must be equal on both sides of the membrane. Denoting the two compartments as **I** (inside, with **R⁻**) and **II** (outside), and using concentrations as approximations for activities, the equilibrium condition leads to the relation: [K⁺]_I * [Cl⁻]_I = [K⁺]_II * [Cl⁻]_II. Combined with the electroneutrality conditions ([K⁺]_I = [Cl⁻]_I + [R⁻] and [K⁺]_II = [Cl⁻]_II), this yields the Donnan ratio: [K⁺]_I / [K⁺]_II = [Cl⁻]_II / [Cl⁻]_I = r. The associated **Donnan potential**, **ψ**, across the membrane is given by an equation similar to the Nernst equation: ψ = (RT/zF) ln(r), where **R** is the gas constant, **T** is temperature, **F** is the Faraday constant, and **z** is the ion's valence. This formalism was rigorously developed within the framework of thermodynamics by Donnan and his contemporaries.

Biological and physiological significance

The Donnan equilibrium is critically important in biophysics and physiology, where it is often referred to as the **Gibbs–Donnan effect**. It is a primary mechanism contributing to the resting potential across cell membranes, particularly in conjunction with ion pumps like the Na⁺/K⁺-ATPase. The presence of impermeable intracellular anions, such as proteins and organic phosphates, creates a Donnan distribution that makes the cell interior negatively charged relative to the extracellular fluid. This effect influences ion and water distribution, contributing to **osmotic pressure** and cell volume regulation. It is essential for understanding fluid balance in blood capillaries, as described by the Starling forces, and the function of cartilage and connective tissue. The principle also underlies electrophysiological measurements using instruments like the Voltage clamp.

Applications in chemistry and engineering

Beyond biology, the Donnan equilibrium principle is exploited in various chemical and engineering contexts. In colloid chemistry, it explains the behavior of ion-exchange resins and polyelectrolyte solutions. The equilibrium governs the partitioning of ions in **Donnan dialysis**, a separation technique used for water softening or metal recovery, leveraging membranes like those made from Nafion. It is fundamental to the operation of **membrane electrodes** and certain types of fuel cells. In food science, the principle affects salt distribution in products like cheese. Industrial processes involving electrodialysis and the design of reverse osmosis systems must account for Donnan effects to predict ion transport and efficiency accurately, influencing work at institutions like MIT and Caltech.

Limitations and assumptions

The classical Donnan equilibrium model relies on several simplifying assumptions that limit its direct applicability to complex real systems. It typically assumes ideal solutions, treating ions as point charges without considering **ion-ion interactions** or **activity coefficients**, which deviate significantly in concentrated solutions like seawater or cytoplasm. The model often assumes a perfectly semipermeable membrane with no specific interactions, ignoring phenomena like **membrane surface charge** or **ion channel** selectivity. It does not account for active transport processes mediated by proteins like the Na⁺/K⁺-ATPase, which are dominant in living cells. Furthermore, the equilibrium is a steady-state description and does not address the kinetics of ion establishment or the dynamic changes occurring in systems like synapses during an action potential. More advanced models incorporate elements from Poisson–Boltzmann equation theory or non-equilibrium thermodynamics.

Category:Biophysics Category:Electrochemistry Category:Membrane technology