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If a membrane passes small ions but confines charged macromolecules or fixed groups to one side, permeant ions redistribute until their electrochemical potentials balance. The two sides can then have unequal ion concentrations and an electrical potential difference.
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A mechanism-focused enrichment guide for curious students, parents and teachers. Worked numbers are explicitly hypothetical; research links identify supporting scientific studies. This is not an examination-syllabus claim or a laboratory operating procedure.
Inside this guide
Chapters 1–3 · Understand the mechanism
1. Start with one ion that cannot cross
2. Keep electroneutrality and electrochemical balance together
Chapters 4–5 · Work through the model
Chapters 6–8 · Evaluate the evidence
6. Understand co-ion exclusion and salt dependence
Imagine two solutions separated by a membrane permeable to small monovalent ions. One side also contains fixed negative groups that cannot cross. Those charges alter how the mobile cations and anions partition.
Equilibrium does not require equal concentration of each ion on both sides. It requires equality of each permeant species’ electrochemical potential, combining chemical and electrical contributions.
The resulting unequal distribution is called Gibbs–Donnan or Donnan equilibrium. It is an ideal framework, not a claim that every biological membrane has reached passive equilibrium.
CHAPTER 2 OF 8 · Understand the mechanism
2. Keep electroneutrality and electrochemical balance together
Away from very thin interfacial charge layers, each bulk solution is approximately electrically neutral. On the fixed-charge side, extra mobile counterions help balance the immobile charge.
For ideal monovalent salt, equality of electrochemical potentials leads to c+inc−in = c+outc−out. If the outside reservoir contains salt concentration cs, the product is cs².
Electroneutrality inside with fixed negative charge concentration X gives c+in − c−in = X. Both relations are needed; using only attraction of positive ions misses the coupled exclusion of co-ions.
Let the external reservoir contain cs = 0.100 M monovalent salt and let X = 0.200 M fixed negative charge inside. Solving the product and difference equations gives c+in ≈ 0.241 M and c−in ≈ 0.0414 M.
Their difference is 0.200 M and their product is approximately 0.0100 M², satisfying both ideal constraints. Counterions are enriched and co-ions depleted, but neither permeant ion is generally driven to exactly zero.
These concentrations use activities approximated by concentrations and assume ideal monovalent ions, a large external reservoir and a known fixed-charge density. Multivalent ions and non-ideal solutions require activity coefficients and a more complete model.
| Region | Cation concentration | Anion concentration |
|---|---|---|
| Outside reservoir | 0.100 M | 0.100 M |
| Fixed-charge side | 0.241 M | 0.0414 M |
| Inside difference | 0.200 M | — |
For a monovalent cation, one sign convention gives Δψ = ψin − ψout = (RT/F) ln(c+out/c+in). With the invented concentrations at 298 K, Δψ is about −22.6 mV.
The negative sign is consistent with the fixed-negative-charge side being at lower electrical potential in this convention. Reversing which side is subtracted reverses the reported sign.
Research directly examining Donnan potential shows why electrical-potential measurements and ion partitioning must be interpreted with interfacial structure in mind. The bulk ideal equation is a model boundary, not every detail of a real membrane.
CHAPTER 5 OF 8 · Work through the model
5. Connect ion partitioning to osmotic pressure
The fixed-charge side in the invented example contains more total mobile ions: about 0.283 M versus 0.200 M outside. In an ideal van ’t Hoff picture, that difference contributes to an osmotic-pressure tendency.
Water movement, membrane stress and hydrostatic pressure can oppose it. Equilibrium of a real swelling material may therefore require mechanical as well as chemical and electrical balance.
Do not turn this into a medical diagnosis. Living cells use pumps, channels, metabolism and structural constraints, so their state is not described by passive Donnan equilibrium alone.
CHAPTER 6 OF 8 · Evaluate the evidence
6. Understand co-ion exclusion and salt dependence
At lower external salt relative to fixed charge, co-ion exclusion can be strong and the Donnan potential magnitude larger. At high salt, screening and the reservoir’s abundant ions reduce the relative partitioning difference.
Ion-exchange membranes use fixed charges to favour counterions over co-ions, but selectivity is not perfect. Pore size, hydration, ion valence and non-ideal interactions matter.
A study of ideal ion-exchange membrane effectiveness discusses Donnan partitioning while examining separation limits. “Ideal” in a model name should not be mistaken for perfect performance in every device.
CHAPTER 7 OF 8 · Evaluate the evidence
7. Test the model without hiding assumptions
Measure concentrations or activities on both sides after sufficient equilibration and record temperature, membrane selectivity and solution volume. A finite outside compartment may change concentration rather than remain a fixed reservoir.
Check whether fixed groups change their ionisation with pH. If X varies during the experiment, treating it as a constant can create an apparently failed model.
A semipermeable-membrane model includes finite thickness, surface charge and correlations. Escalating model complexity is justified when measured deviations exceed uncertainty and simpler assumptions are demonstrably broken.
Must permeant ion concentrations become equal across the membrane? No. Their electrochemical potentials balance, allowing concentration differences alongside an electrical potential.
Why are counterions enriched? They help maintain bulk electroneutrality around fixed charge while satisfying partition equilibrium.
What does the ideal calculation omit? Activities, multivalent-ion correlations, finite reservoirs, variable fixed charge and mechanical constraints. Continue with Physical Chemistry for chemical potential, membranes and osmotic balance.