Put a raisin in water and it swells; put a slice of cucumber in salty brine and it goes limp. Both are osmosis: water moving through a membrane that lets water pass but not dissolved particles. Its pressure follows a law that looks just like the The Ideal Gas Law, and like Freezing Point Depression and Boiling Point Elevation, it depends only on the number of particles.
Key idea
Water flows through a semipermeable membrane toward the more concentrated solution (from fewer solute particles to more). The pressure needed to stop it is the osmotic pressure: $$\Pi = iMRT$$ with $M$ in mol/L, $R = 0.08206\ \text{L·atm/(mol·K)}$ and $T$ in kelvins.
Why it works
On the pure-water side, every molecule hitting the membrane is water and may pass through. On the solution side, some of the molecules hitting the membrane are solute particles that can't pass, so water crosses from that side less often. More water crosses toward the solution than back, until the solution's level rises enough that its extra pressure pushes water back just as fast. The dissolved particles behave much like a gas in a container of that volume, which is why $\Pi = (n/V)RT$, times $i$ to count the particles.
Worked example: A glucose solution
Find the osmotic pressure of $0.100\ \text{M}$ glucose at $25^\circ\text{C}$.
- Nonelectrolyte: $i = 1$; $T = 298.15\ \text{K}$.
- $$\Pi = (1)(0.100)(0.08206)(298.15) = 2.45\ \text{atm}$$
Answer: $2.45\ \text{atm}$: enough to hold up a column of water about 25 m tall.
Worked example: A salt solution
Find the osmotic pressure of $0.050\ \text{M}$ NaCl at $37^\circ\text{C}$.
- $i = 2$; $T = 310.15\ \text{K}$.
- $$\Pi = (2)(0.050)(0.08206)(310.15) = 2.55\ \text{atm}$$
Answer: $2.55\ \text{atm}$
Worked example: Which way?
A membrane separates $0.10\ \text{M}$ glucose (side A) from $0.10\ \text{M}$ NaCl (side B). Which way does water flow?
- Particles: A has $0.10\ \text{M}$; B has $2 \times 0.10 = 0.20\ \text{M}$ of ions.
- Water moves toward more solute particles: from A to B.
Answer: From A (glucose) into B (NaCl).
Common mistake
Thinking water flows toward the side with less solute, "to the emptier side". Water moves toward the side where it is less concentrated, which is the side with more solute, diluting it. Check: the raisin (sugary inside) swells in plain water because water moves in, toward the sugar.
Reverse osmosis turns this around: pushing on salt water with more than its osmotic pressure forces water out through the membrane, leaving the salt behind. Many desalination plants work this way.