The Donnan ratio, often denoted by $\lambda$ (lambda), is a fundamental concept in Donnan equilibrium, describing the equilibrium distribution of mobile ions across a semipermeable membrane that separates a solution containing fixed, non-diffusible ions from a solution without them. The formula for the Donnan ratio is:
$$ \lambda = \left( \frac{m_j^F}{m_j^S} \right)^{1/z} $$
This ratio indicates the unequal distribution of diffusible ions at equilibrium, driven by the presence of a non-diffusible charged species on one side of the membrane.
Understanding the Donnan Ratio Formula Components
To fully grasp the Donnan ratio, it's essential to understand each variable in the formula:
- $\lambda$ (Lambda): Represents the Donnan ratio itself. It is a dimensionless value that reflects the concentration gradient of mobile ions at equilibrium across the membrane.
- $m_j^F$: This denotes the molar concentration of a specific mobile ion j in the fixed phase. The fixed phase is the compartment (often a gel, membrane, or cell interior) that contains the non-diffusible, charged molecules (e.g., proteins, polyelectrolytes).
- $m_j^S$: This refers to the molar concentration of the same mobile ion j in the solution phase. The solution phase is the compartment (e.g., external solution, interstitial fluid) where there are no fixed, non-diffusible charged molecules.
- $z$: This is the charge (valence) of the mobile ion j. For example, for a sodium ion ($Na^+$), $z = +1$; for a chloride ion ($Cl^-$), $z = -1$; for a calcium ion ($Ca^{2+}$), $z = +2$.
The relationship derived from the Donnan equilibrium dictates that the ratio of the concentrations of any mobile cation ($j^+$) in the fixed phase to the solution phase will be equal to the reciprocal of the ratio for any mobile anion ($k^-$):
$$ \frac{[Cation]_F}{[Cation]_S} = \frac{[Anion]_S}{[Anion]_F} = \lambda $$
This implies that for a given ion, the product of its concentration ratios raised to the power of its charge is constant. Specifically, for an equilibrium where a fixed negative charge is present in one compartment, mobile cations will be at a higher concentration in that compartment, while mobile anions will be at a lower concentration.
Significance and Applications
The Donnan ratio is crucial for understanding various biological and industrial processes, including:
- Cell Biology: Explaining the distribution of ions across cell membranes, contributing to membrane potential and cellular volume regulation.
- Ion Exchange: Fundamental in ion-exchange chromatography and other separation processes where charged fixed groups are involved.
- Membrane Science: Designing and analyzing membrane systems for desalination, wastewater treatment, and selective ion separation.
- Soil Chemistry: Describing ion exchange in soils, where charged clay particles and organic matter act as fixed phases.
Understanding the Donnan ratio allows for predictions regarding ion distribution, osmotic pressure differences, and electrical potentials across charged membranes or phases containing fixed charges.
