Osmosis and osmotic pressure

Water that crosses a skin

Drop a dried raisin into water. In a few hours it is plump. Sprinkle sugar over strawberries. After a while a pool of juice collects. In both cases water has crossed a thin skin, and the amount of dissolved matter decided which way.

Semipermeable membranes

A semipermeable membrane lets small solvent molecules pass. It stops larger solute particles. There are natural ones: the skin of a raisin, the thin lining of an egg, and the membrane around every living cell. There are also artificial ones: cellophane, and a thin layer of copper(II) hexacyanoferrate(II), $\text{Cu}_2[\text{Fe(CN)}_6]$, in the pores of porcelain.

Osmosis

Osmosis is the net flow of solvent through a semipermeable membrane, from the pure solvent into the solution. It also flows from a dilute solution into a more concentrated one.

Why does it happen? Solvent molecules cross the membrane in both directions. On the solution side, solute particles take up some of the space near the membrane. Fewer solvent molecules hit it from that side. So more solvent molecules cross towards the solution than away from it.

Picture 1 shows the start. Water crosses into the sugar solution, so the level on that side rises. The column of liquid grows taller and taller, and its weight presses back on the membrane. In picture 2 the pressure of the column of height $h$ stops the net flow. In picture 3 we get the same result without any rise: we push on the solution with a piston until the levels stay equal.

Osmotic pressure

The osmotic pressure, $\pi$, is the extra pressure that must be put on the solution to stop the solvent from flowing in. It is a colligative property. It depends on how many solute particles are in a given volume of solution, and not on what they are.

For a dilute solution, van’t Hoff found that $\pi$ follows a law like the one for an ideal gas:

Here $n_2$ is the moles of solute in a volume $V$ of solution. We divide by $V$. The concentration is $C = n_2/V$, in $\text{mol/L}$, so

The temperature $T$ is in kelvin. With $R = 0.0821\ \text{L atm K}^{-1}\text{mol}^{-1}$ the pressure comes out in atm. With $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ and $C$ in $\text{mol/L}$ it comes out in kPa, because $1\ \text{J/L} = 1\ \text{kPa}$.

The solute particles in the solvent behave a little like the molecules of a gas. This is why the law looks the same.

Worked example: how high does the water climb?

A solution of a non-electrolyte has $C = 0.050\ \text{mol/L}$ at $298\ \text{K}$.

1. In atm. $\pi = 0.050\ \text{mol/L} \times 0.0821\ \text{L atm/(K mol)} \times 298\ \text{K} = 1.22\ \text{atm}$.
2. In kPa. $\pi = 0.050\ \text{mol/L} \times 8.314\ \text{J/(K mol)} \times 298\ \text{K} = 123.9\ \text{kPa}$.
3. The column. A column of water of height $h$ presses with $\rho g h$. So $h = \dfrac{123\,900\ \text{Pa}}{1000\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2} = 12.6\ \text{m}$.

A solution as weak as $0.050\ \text{mol/L}$ could push water up a tube $12.6\ \text{m}$ tall. That is a big pressure from very little solute.

A tool for very large molecules

Proteins and polymers have molar masses of tens of thousands of grams per mole. They also dissolve poorly. So the molality is tiny, and the other colligative effects are too small to measure.

Worked example: a protein. A student dissolves $0.40\ \text{g}$ of a protein in water to make $100\ \text{mL}$ of solution. At $298\ \text{K}$ the osmotic pressure is $0.150\ \text{kPa}$.

We want $M_2$. From $\pi V = \dfrac{w_2}{M_2} R T$ we get

The units are $\text{g}$, $\text{J/(K mol)}$, $\text{K}$, $\text{kPa}$ and $\text{L}$, and they combine to give $\text{g/mol}$.

Now compare. The pressure of $0.150\ \text{kPa}$ holds up a column of water $15\ \text{mm}$ high. We can read that easily. The same solution in $100\ \text{g}$ of water would lower the freezing point by about $0.0001\ \text{K}$. No thermometer can see that.

So osmotic pressure is the best method for large molecules for three reasons:

• It is measured at room temperature, so a delicate protein is not heated or damaged.
• It uses the molarity, which is easy to find, and not the molality.
• It gives a large effect even for a very dilute solution.

Isotonic, hypertonic and hypotonic

Two solutions with the same osmotic pressure at the same temperature are isotonic. There is no net flow between them. A solution with the higher pressure is hypertonic to the other, and the one with the lower pressure is hypotonic.

Think of a red blood cell:

• In an isotonic solution it stays the same.
• In a hypotonic solution, such as pure water, water rushes in. The cell swells and can burst.
• In a hypertonic solution, water leaves. The cell shrinks.

This is why a drip given in a hospital must be isotonic with blood. Blood has an osmotic pressure of about $7.7\ \text{atm}$ at $310\ \text{K}$. A solution of $5\ \text{g}$ of glucose in $100\ \text{mL}$ has $C = 0.28\ \text{mol/L}$ and $\pi = 0.28\ \text{mol/L} \times 0.0821\ \text{L atm/(K mol)} \times 310\ \text{K} = 7.1\ \text{atm}$, which is close.

The same idea explains other things. Salt draws water out of bacteria, so salted fish and pickles keep. A person who drinks sea water loses water, because sea water is hypertonic to the cells.

Reverse osmosis

If we push on the solution with a pressure greater than $\pi$, the solvent flows the other way, from the solution into the pure solvent. This is reverse osmosis.

Sea water has an osmotic pressure of roughly $25$ to $30\ \text{atm}$. A pump presses sea water against a membrane that lets water through and holds back the salts. The pressure must be higher than $\pi$, or no fresh water comes out. Plants on coasts that are short of fresh water use this method, and so do many home water purifiers.

Practise with new numbers

Try it in Python

The program finds the osmotic pressure of a solution, and then the molar mass of a large molecule from a measured pressure.

Common mistakes

• Using the temperature in degrees Celsius. Change it to kelvin.
• Leaving the volume in millilitres. The concentration in $\pi = CRT$ is in mol/L.
• Using the molality in $\pi = CRT$. The law uses the molarity. For a dilute solution the two are close.
• Pushing on the wrong side in reverse osmosis. The extra pressure goes on the solution.
• Thinking that the solute crosses the membrane. Only the solvent does.

Remember these

• Osmosis is the flow of solvent through a semipermeable membrane towards the more concentrated solution.
• $\pi = CRT$, with $C$ in mol/L and $T$ in kelvin.
• Isotonic solutions have the same $\pi$. A cell swells in a hypotonic solution and shrinks in a hypertonic one.
• Reverse osmosis needs a pressure greater than $\pi$ on the solution.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Pure water and a salt solution are separated by a semipermeable membrane. In which direction does the water flow?

2. A solution of $0.12\ \text{mol/L}$ of a non-electrolyte is at $298\ \text{K}$. What is its osmotic pressure in atm?

3. Why is osmotic pressure better than the fall of the freezing point for a protein?

4. A plant cell is placed in a strong salt solution. What happens to it, and why?