Osmotic pressure
= Osmotic pressure
{title2=$\Pi$}
{wiki}
<Osmotic pressure> is the excess <pressure> needed on a <chemical solution> to maintain osmotic <thermodynamic equilibrium> with pure <solvent> across a <semipermeable membrane>. For a nearly incompressible <solvent>, equality of <chemical potential> gives $\Pi\simeq-(RT/\bar V_w)\ln a_w$. It is an equilibrium property; an observed initial swelling rate is a different measurement.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/osmotic pressure>[Gold Book, osmotic pressure: equilibrium definition, solvent activity formula and dilute ideal limit]{ref}
For a dilute ideal <chemical solution> of retained particles,
$$
\Pi=\frac{nRT}{V}=RTc_{\mathrm{osm}}.
$$
Here $n$ is the <amount of substance> of independently dissolved <solute> particles, measured in moles; $V$ is the final <volume> of <chemical solution>, not the starting <volume> of pure <solvent>. Count the resulting <ions> separately if a <solute> dissociates; do not multiply twice if $n$ already counts them. With $c_{\mathrm{osm}}$ in $\mathrm{mol\,m^{-3}}$ and $R$ in $\mathrm{J\,mol^{-1}\,K^{-1}}$, $RTc_{\mathrm{osm}}$ has units $\mathrm{J\,m^{-3}}=\mathrm{Pa}$. This resembles the <ideal gas law>, although the particles here are dissolved in a liquid.
The assumptions are uniform <temperature>, a dilute ideal <chemical solution>, a nearly incompressible <solvent>, and a <semipermeable membrane> that retains the counted <solutes> while passing <solvent>. Association, incomplete dissociation and interactions change the relation between added material and independent particles; nonideal <thermodynamic activities> invalidate simple particle counting. The formula does not describe a concentrated brine merely because the apparatus contains a membrane.
A short derivation explains why particles are counted. In an ideal mixture, $a_w=x_w=n_w/(n_w+n)$, with $n_w$ the <amount of substance> of <water>. If $n\ll n_w$, then $-\ln x_w=\ln(1+n/n_w)\simeq n/n_w$ and $V\simeq n_w\bar V_w$. Substitution gives $\Pi\simeq RTn/V$. This is a dilute-limit approximation, not a claim that <water> <molecules> stop moving or that dissolved particles are a gas.