What osmotic pressure is, physically
Put a solution on one side of a membrane that passes solvent but not solute, put pure solvent on the other, and solvent flows into the solution. Osmotic pressure is the mechanical pressure you would have to apply to the solution to bring that flow to a stop. It is not a pressure the solution exerts on its container; it is the pressure needed to hold back osmosis.
The driving force is entropy, not attraction. The solvent's chemical potential is lower in the solution than in the pure liquid, simply because the solvent is diluted there, and matter flows down a chemical-potential gradient. Applying pressure to the solution raises the solvent's chemical potential in it, and at exactly π the two sides balance.
Like freezing point depression and boiling point elevation, this is a colligative property: it depends on how many particles are dissolved, not what they are. A 0.1 M solution of glucose and a 0.05 M solution of sodium chloride generate the same pressure, because NaCl produces two ions per formula unit. That is why the useful concentration unit here is osmolarity — moles of particles per litre — rather than the molarity of the compound you weighed out.
The magnitudes are large, and that is what makes the property useful. A 0.15 M sodium chloride solution — ordinary physiological saline — generates about 7.5 atm at 25 °C, roughly 110 psi. That is why a red blood cell placed in pure water bursts, and why desalinating seawater takes a high-pressure pump.
The van 't Hoff equation and why it looks like the ideal gas law
π = iMRT. Rearranged as πV = inRT, it is the ideal gas law with the number of dissolved particles standing in for the number of gas molecules. That resemblance is not a coincidence and it is not a derivation shortcut — both laws describe the entropy of a dilute, non-interacting population of particles distributed through a volume. Van 't Hoff won the first Nobel Prize in Chemistry in 1901 partly for establishing exactly this parallel.
M is molarity, moles per litre of solution, in contrast to the other colligative properties which use molality. Osmotic pressure is measured at essentially constant temperature, so the temperature-dependence of volume that forces molality on freezing-point work does not arise here. If your data is in molality you need a density to convert; the molarity calculator handles it.
i is the dissociation factor, and the product i·M is the osmolarity. Use 1 for glucose, urea, sucrose and proteins; 2 for NaCl and KCl; 3 for CaCl₂, MgCl₂ and Na₂SO₄. For a water analysis reported as total dissolved solids, treating everything as sodium chloride with i = 2 is the standard first approximation.
T is absolute temperature, so the pressure is proportional to it in kelvin, not in degrees Celsius. Warming a solution from 5 °C to 35 °C raises the osmotic pressure by 308.15 ÷ 278.15 = 1.108, or about 11%. Reverse-osmosis plants feel this directly: winter feed water needs less pressure to overcome osmosis but more to push through a membrane whose permeability has fallen, and the two effects do not cancel.
R is 0.0820574 L·atm/(mol·K) if you want atmospheres, or 0.083145 L·bar/(mol·K) for bar. This calculator uses the first and converts.
Worked example: physiological saline and seawater
1. Isotonic saline at body temperature. Normal saline is 0.9% w/v NaCl — 9.0 g per litre, molar mass 58.44 g/mol — at 37 °C.
- Molarity: 9.0 ÷ 58.44 = 0.15400 mol/L.
- Osmolarity: 2 × 0.15400 = 0.30801 Osm/L, or 308 mOsm/L.
- Absolute temperature: 37 + 273.15 = 310.15 K.
- RT: 0.0820574 × 310.15 = 25.450 L·atm/mol.
- Pressure: 0.30801 × 25.450 = 7.839 atm, which is 7.943 bar or 115.2 psi.
That 308 mOsm/L is the whole point of the formulation: human plasma osmolality sits between 275 and 295 mOsm/kg, so 0.9% saline is close enough to be infused without lysing or shrinking cells. Compare it with 5% dextrose: 50 ÷ 180.16 = 0.2775 mol/L with i = 1, giving 0.2775 Osm/L — also near-isotonic, by a completely different route.
2. Seawater, and why the ideal answer is too high. Take 35,000 mg/L of total dissolved solids at 25 °C and treat it all as sodium chloride.
- Molarity: 35.0 g/L ÷ 58.44 = 0.59890 mol/L.
- Osmolarity: 2 × 0.59890 = 1.19781 Osm/L.
- RT at 298.15 K: 0.0820574 × 298.15 = 24.4654 L·atm/mol.
- Pressure: 1.19781 × 24.4654 = 29.31 atm = 430.6 psi.
Seawater's measured osmotic pressure is nearer 26–27 atm. The gap is the osmotic coefficient: at this ionic strength seawater's is about 0.90, and 29.31 × 0.90 = 26.4 atm, which lands where the measurement does. This is the single most important correction to make before you trust an ideal number for design work.
How to read the result
For tonicity work, compare osmolarity, not pressure. A fluid is isotonic with blood if its osmolarity is near 290 mOsm/L; hypotonic below, hypertonic above. The pressure figure is the same information in different units, but clinical practice, labels and osmometers all speak in mOsm. Note also the distinction between osmolarity (per litre of solution) and osmolality (per kilogram of water): in dilute aqueous fluids they differ by under 2%, but in a protein-rich or lipaemic plasma sample they diverge, which is why laboratories measure osmolality by freezing-point depression rather than calculating it.
For reverse osmosis, this number is the floor, not the answer. The applied pressure must exceed the osmotic pressure of the concentrate at the membrane wall, which is higher than the feed on two counts: the feed is concentrated as permeate is removed, and a polarised boundary layer sits against the membrane at a higher concentration still. On top of that you need net driving pressure to actually produce flux. A brackish plant treating 2,000 mg/L feed faces an ideal osmotic pressure of only about 1.7 atm, yet runs at 10–15 bar; a seawater plant faces 27 atm and runs at 55–70 bar. The osmotic pressure is real but it is rarely more than half the applied pressure.
Sanity-check the scale. For a fully dissociating salt at 25 °C, each 100 mg/L of NaCl generates about 1.2 psi by this equation — 0.1 g/L ÷ 58.44 × 2 × 24.4654 = 0.0837 atm = 1.23 psi. Water-treatment practitioners often use a rule of about 1 psi per 100 ppm, roughly a fifth lower, and that gap is exactly the osmotic-coefficient and mixed-ion correction described above.
Osmotic pressure of familiar solutions at 25 °C
| Solution | Osmolarity (Osm/L) | π (atm) | π (bar) | π (psi) |
|---|---|---|---|---|
| Brackish feed, 2,000 mg/L | 0.0684 | 1.67 | 1.70 | 24.6 |
| 0.100 M sucrose | 0.100 | 2.45 | 2.48 | 36.0 |
| 5% dextrose (50 g/L) | 0.2775 | 6.79 | 6.88 | 99.8 |
| Human plasma, 290 mOsm/L | 0.290 | 7.09 | 7.19 | 104.3 |
| 0.9% saline (9,000 mg/L NaCl) | 0.3080 | 7.53 | 7.63 | 110.7 |
| Seawater, 35,000 mg/L TDS | 1.1978 | 29.31 | 29.70 | 430.6 |
The seawater row is an ideal figure; the measured value is about 10% lower because seawater's osmotic coefficient is close to 0.90. Plasma osmolality reference range 275–295 mOsm/kg is a clinical figure, not a calculated one.
Osmolarity is not tonicity
Osmolarity counts every dissolved particle. Tonicity counts only the particles a cell membrane cannot cross. Urea contributes to osmolarity but crosses cell membranes freely, so a 300 mOsm/L urea solution is iso-osmolar with plasma yet hypotonic — cells placed in it swell and lyse. The same applies to a 5% dextrose infusion once the glucose is metabolised: iso-osmolar in the bag, effectively free water in the patient.
This calculator returns osmolarity and the pressure that follows from it. Deciding whether a fluid is safe for a given membrane requires knowing which of those particles the membrane actually retains, and that is a clinical or engineering judgement, not an arithmetic one. Nothing here is medical advice.
Assumptions and limits of this calculation
- Ideal, dilute solution. The equation is exact only in the limit of infinite dilution. Multiply by a measured osmotic coefficient above roughly 0.1 mol/L if accuracy matters.
- Complete dissociation at the entered factor. Real electrolytes show ion pairing, so an effective i below the ideal integer is common at moderate concentration.
- A perfectly semi-permeable membrane. Real membranes reject solutes incompletely; the effective pressure difference is the ideal one times the reflection coefficient, which is below 1 for every real membrane.
- One solute, or a surrogate for many. A mixed water analysis treated as NaCl is an approximation. For accurate multi-ion work use a speciation model such as Pitzer or a commercial projection program.
- No concentration polarisation. In a working membrane element the wall concentration exceeds the bulk concentration, so the pressure that must be overcome is higher than this figure.
- Osmolarity, not osmolality. This returns particles per litre of solution. Clinical osmometers report particles per kilogram of water; the two differ by more than 2% in samples with high protein or lipid content.
Where osmotic pressure is used, and what to use instead
Membrane processes. Reverse osmosis, nanofiltration and forward osmosis are all designed around this number, because it sets the thermodynamic floor on the energy of separation. Pushing one cubic metre of permeate out of seawater against 27 atm costs at least 27 × 101,325 J = 2.74 MJ, or 0.76 kWh — and that is the limit as recovery approaches zero. Recovering a real fraction of the feed concentrates what remains and raises the requirement; working plants consume several times the floor. Any claim of desalination below the osmotic limit is a claim to have broken thermodynamics.
Molar mass of macromolecules. Osmometry is by far the most sensitive of the colligative methods. A 1 g/L solution of a 100,000 g/mol polymer is only 10⁻⁵ molar, which would shift a freezing point by 0.0000186 °C — unmeasurable — yet generates 10⁻⁵ × 24.4654 = 2.45 × 10⁻⁴ atm, which is 0.186 mmHg and well within reach of a differential membrane osmometer. Osmometry returns the number-average molar mass specifically, because it counts molecules.
Physiology and food. Cell volume regulation, plant turgor and root water uptake are osmotic. So is the preservation of food in brine and syrup: raise the external osmolarity far enough and microbial cells lose water and cannot grow.
When the ideal equation is not enough, the alternatives are measurement and modelling. A vapour-pressure or freezing-point osmometer gives you the real osmolality directly; a Pitzer or OLI model gives activity coefficients for concentrated mixed electrolytes. For the neighbouring calculations on this site, see molarity, ppm concentration, and Raoult's law, which is the vapour-pressure face of the same thermodynamics.