= Osmosis concentration-pressure example
Take two reservoirs separated by a <semipermeable membrane> that passes <water> and retains one neutral <solute>. Stipulate $T=300\,\mathrm K$, $c_A=10\,\mathrm{mol\,m^{-3}}$ and $c_B=30\,\mathrm{mol\,m^{-3}}$, equivalent to $10$ and $30\,\mathrm{mmol\,L^{-1}}$. Treat both <chemical solutions> as dilute and ideal. These are invented model conditions, not measurements or recommended biological conditions.
For <pressures> measured at the same height beside the membrane, define $\Delta P=P_B-P_A$. The model predicts
$$
\Psi_A-\Psi_B=RT(c_B-c_A)-\Delta P,\qquad
RT(c_B-c_A)\simeq49.89\,\mathrm{kPa}.
$$
At $\Delta P=0$, net <water> transfer favors A to B. At $\Delta P\simeq49.89\,\mathrm{kPa}$, the opposing <pressure> balances the <osmotic pressure> difference, giving zero net transfer with unequal <concentrations>. At $\Delta P=75\,\mathrm{kPa}$, the sign reverses and net <water> transfer favors B to A, toward the more dilute reservoir. Raising <pressure> changes the total <water potential>, rather than changing the membrane into a one-way gate.
\Image[/osmosis-concentration-pressure-cases.png]
{title=Stipulated <concentration> and <pressure> conditions determine net <water> transfer}
{description=Original computed comparison for ideal retained <solute> at $300\,\mathrm K$, with A at $10\,\mathrm{mol\,m^{-3}}$ and B at $30\,\mathrm{mol\,m^{-3}}$. The horizontal axis is $P_B-P_A$; the vertical axis is $\Psi_A-\Psi_B$. Three marked cases show forward transfer, balance and reversed transfer. These are instantaneous reservoir conditions, not a time course or measured biological data; slope does not specify permeability or speed.}
Fixed <concentrations> require large maintained reservoirs or an instantaneous comparison. In a closed <membrane osmometer>, entering <water> causes <dilution> of the receiving <chemical solution> and changes its <volume>; the final balance must use final <concentrations>. A supported <cell wall> provides another way to develop the opposing <pressure>, as explained under <turgor pressure>.
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