Donnan Equilibrium Calculator

The Donnan equilibrium (also called the Gibbs-Donnan effect) describes the unequal distribution of diffusible ions across a semipermeable membrane when a non-diffusible charged species (such as a protein, polyelectrolyte, or colloidal particle) is present on one side. This phenomenon, first described by Frederick G. Donnan in 1911, is fundamental to understanding ion distribution in biological systems, the behavior of polyelectrolyte solutions, and the operation of membrane-based separation processes.

Consider a membrane permeable to small ions (like Na+ and Cl-) but impermeable to a charged macromolecule (like a protein with charge z). When the macromolecule is placed on one side of the membrane with a salt (like NaCl) present on both sides, the ions redistribute themselves until the electrochemical potential of each permeable ion is equal on both sides. The result is an unequal distribution: the side containing the macromolecule has a higher total ion concentration, creating an osmotic pressure difference and an electrical potential difference across the membrane.

This calculator quantifies the Donnan equilibrium by computing the ion concentrations on both sides of the membrane, the Donnan ratio, the Donnan potential, and the osmotic pressure difference. It supports polyelectrolytes with charges of plus or minus 1, 2, or 3, and accounts for temperature effects. Understanding the Donnan equilibrium is essential for designing dialysis systems, understanding cell membrane potentials, and predicting the behavior of polyelectrolyte solutions in industrial and biomedical applications.

The Donnan equilibrium is described by two fundamental equations that must be satisfied simultaneously: the Donnan equilibrium condition and the electroneutrality condition. Together, these equations determine the distribution of ions across the membrane.

The Donnan equilibrium condition states that the product of the permeable ion concentrations on one side equals the product on the other side: [Na+]in x [Cl-]in = [Na+]out x [Cl-]out = Cs squared, where Cs is the external salt concentration. This condition ensures that the electrochemical potential of each permeable ion is equal on both sides at equilibrium.

The electroneutrality condition requires that the total positive charge equals the total negative charge on each side of the membrane. Inside the membrane: [Na+]in = [Cl-]in + |z| x Cp, where z is the charge of the polyelectrolyte and Cp is its concentration. This equation accounts for the charge contributed by the non-diffusible polyelectrolyte.

Solving these two equations simultaneously (using the quadratic formula) yields the internal ion concentrations, from which the Donnan ratio, Donnan potential, and osmotic pressure can be calculated. The Donnan ratio is r = [Na+]in / [Na+]out, the Donnan potential is Delta psi = (RT/F) x ln(r), and the osmotic pressure is calculated from the total ion concentration difference using the van't Hoff equation.

The Donnan equilibrium plays a crucial role in many biological processes, from maintaining cell volume to generating nerve impulses. Understanding this phenomenon is essential for physiology, medicine, and biotechnology.

Cell membrane potential: The Donnan equilibrium contributes to the resting membrane potential of cells. Proteins and other charged macromolecules inside cells cannot cross the plasma membrane, creating an unequal ion distribution. This generates a potential difference (typically -70 mV for neurons) that is essential for nerve impulse transmission and muscle contraction.

Cell volume regulation: The Donnan effect creates an osmotic pressure that tends to draw water into cells. Cells must actively pump ions (primarily Na+ out and K+ in via the Na+/K+-ATPase pump) to counteract this effect and prevent swelling. Failure of these pumps leads to cell swelling and lysis, which occurs during ischemia (loss of blood flow) and in certain diseases.

Dialysis and kidney function: The principles of Donnan equilibrium govern the operation of hemodialysis machines used to treat patients with kidney failure. The dialysis membrane allows small ions and waste products to pass through while retaining large proteins. The unequal distribution of ions across the membrane must be carefully managed to maintain proper electrolyte balance.

Polyelectrolyte solutions: Synthetic polyelectrolytes (like DNA, proteins, and charged polymers) in solution exhibit Donnan effects that affect their osmotic pressure, viscosity, and interactions with small ions. These effects are important in drug delivery, water treatment, and the formulation of personal care products.

This calculator computes the ion distribution across a semipermeable membrane in the presence of a charged polyelectrolyte.

1. Enter the temperature (K): Standard biological temperature is 310 K (37 degrees C). Room temperature is 298.15 K (25 degrees C).
2. Enter the polyelectrolyte concentration (M): This is the molar concentration of the charged macromolecule on the inside of the membrane.
3. Select the polyelectrolyte charge: Choose the charge per monomer unit. Common values are -1 for proteins at physiological pH, -2 or -3 for DNA and RNA.
4. Enter the external salt concentration (M): This is the concentration of the 1:1 electrolyte (like NaCl) present on both sides of the membrane initially.
5. Read the results: The calculator displays the ion concentrations inside and outside the membrane, the Donnan ratio, the Donnan potential, and the osmotic pressure difference.

The Donnan equilibrium has practical applications in medicine, biotechnology, water treatment, and materials science.

Hemodialysis: In kidney dialysis, a semipermeable membrane separates blood from dialysate. The Donnan equilibrium determines how ions redistribute across the membrane. Dialysate composition is carefully formulated to achieve the desired final blood electrolyte concentrations, accounting for the Donnan effect. The protein concentration in blood creates a Donnan potential that affects the final ion distribution.

Drug delivery: Polyelectrolyte hydrogels used in controlled drug release swell due to the Donnan osmotic pressure. The degree of swelling depends on the charge density of the polymer and the ionic strength of the surrounding medium. Understanding and controlling the Donnan effect allows engineers to design drug delivery systems with predictable release kinetics.

Water treatment: Ion exchange resins operate on principles related to the Donnan equilibrium. The charged functional groups on the resin attract counterions from solution while excluding coions. This selective ion uptake is the basis for water softening, deionization, and purification processes.

Food science: The Donnan effect influences the texture and moisture retention of food gels and proteins. In meat processing, the water-holding capacity of muscle proteins is affected by the Donnan equilibrium, which determines how much water is retained during cooking and processing.