11.4 Colligative Properties

Solution Concentrations and Colligative Properties

Colligative properties are solution properties that depend on how many solute particles are dissolved, not on what those particles are. This means that 1 mole of sugar and 1 mole of any other nonvolatile, non-dissociating solute will affect a solvent in the same way. The four main colligative properties are vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

These show up everywhere: salt lowering the freezing point of road ice, antifreeze raising the boiling point of engine coolant, and osmosis moving water through cell membranes. To work with them, you first need two concentration measures that are especially useful for colligative calculations.

Mole Fraction and Molality Calculations

Mole fraction ($X_i$) is the ratio of moles of one component to the total moles in the solution. For a two-component system:

$X_1 = \frac{n_1}{n_1 + n_2}$

The mole fractions of all components always add up to 1, so $X_1 + X_2 = 1$.

Molality ($m$) is the number of moles of solute per kilogram of solvent (not solution):

$m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}$

Why use molality instead of molarity? Molality doesn't change with temperature. Molarity depends on volume, and volume expands or contracts as temperature changes. Molality depends on mass, which stays constant.

Effects of Solute Concentration

Vapor Pressure Lowering

When you dissolve a nonvolatile solute in a solvent, the solution's vapor pressure drops below that of the pure solvent. Solute particles take up space at the liquid's surface, so fewer solvent molecules can escape into the gas phase.

Raoult's law describes this quantitatively:

$P_\text{solution} = X_\text{solvent} \cdot P^\circ_\text{solvent}$

Since $X_\text{solvent}$ is always less than 1 when solute is present, $P_\text{solution}$ is always less than $P^\circ_\text{solvent}$.

Boiling Point Elevation

A lower vapor pressure means the solution needs a higher temperature to boil (the point where vapor pressure equals atmospheric pressure). The increase is:

$\Delta T_b = K_b \cdot m$

$K_b$ is the molal boiling point elevation constant, a value specific to each solvent. For water, $K_b = 0.512 \, ^\circ\text{C/m}$. So a 1 m aqueous sugar solution boils at about 100.512 °C instead of 100 °C.

Freezing Point Depression

Solute particles disrupt the orderly crystal structure that forms when a solvent freezes, so the solution must be cooled to a lower temperature before it solidifies:

$\Delta T_f = K_f \cdot m$

For water, $K_f = 1.86 \, ^\circ\text{C/m}$. This is why salt on icy roads works: dissolving salt in the thin water layer on ice lowers its freezing point, causing the ice to melt.

Osmotic Pressure

Osmotic pressure ($\Pi$) is the pressure needed to stop solvent from flowing through a semipermeable membrane into a more concentrated solution:

$\Pi = MRT$

Here $M$ is molarity, $R$ is the gas constant (0.0821 L·atm/mol·K), and $T$ is absolute temperature in Kelvin. Even small concentrations of solute can produce surprisingly large osmotic pressures.

Equations for Colligative Effects

Here are all four equations collected in one place:

Pay attention to which concentration unit each equation requires. If a problem gives you molarity but you need molality, you'll need the solution's density to convert.

The van 't Hoff factor ($i$)

For electrolytes (ionic compounds that dissociate in water), you multiply by the van 't Hoff factor $i$, which represents the number of particles one formula unit produces. For example, NaCl dissociates into $\text{Na}^+$ and $\text{Cl}^-$, so $i = 2$. The adjusted equations look like:

$\Delta T_b = i \cdot K_b \cdot m$

$\Delta T_f = i \cdot K_f \cdot m$

$\Pi = i \cdot MRT$

A common mistake: forgetting to apply $i$ for ionic solutes. A 1 m NaCl solution has roughly twice the colligative effect of a 1 m sugar solution because NaCl produces twice as many particles.

Solution Properties and Phase Behavior

Solubility is the maximum amount of solute that dissolves in a given amount of solvent at a specific temperature. A solution at this limit is called saturated.

Phase diagrams map out which physical state (solid, liquid, gas) a substance occupies at different temperatures and pressures. For solutions, the key takeaway is that the liquid region on the phase diagram expands: the boiling point shifts higher and the freezing point shifts lower compared to the pure solvent. This visual shift is a direct result of the colligative properties described above.

Distillation and Osmosis

Distillation Process and Applications

Distillation separates the components of a liquid mixture based on differences in their boiling points (volatility). The basic process works in three steps:

1. Heat the mixture until the more volatile component vaporizes first.
2. Condense that vapor by cooling it, collecting the purified liquid (called the distillate).
3. Collect the residue, the less volatile component(s) left behind in the original flask.

Two common types:

• Simple distillation works well when the components have very different boiling points (at least ~25 °C apart). Example: separating water from dissolved salt.
• Fractional distillation uses a fractionating column to achieve better separation when boiling points are closer together. This is how crude oil is separated into gasoline, diesel, kerosene, and other fractions at a refinery.

Other applications include desalination (purifying seawater) and producing distilled spirits.

Osmosis in Industry and Nature

Osmosis is the net movement of solvent molecules through a semipermeable membrane from a region of lower solute concentration to a region of higher solute concentration. The membrane lets solvent through but blocks solute. This process is spontaneous and continues until the osmotic pressure difference is balanced.

Industrial applications:

• Reverse osmosis applies external pressure greater than $\Pi$ to force solvent backward through the membrane, from the concentrated side to the dilute side. This is widely used for water purification and desalination.
• Forward osmosis uses a natural osmotic gradient to drive separation, applied in wastewater treatment and some food processing.

Biological examples:

• Cell membranes regulate water flow by osmosis, which is critical for maintaining cell shape. A red blood cell placed in pure water will swell and burst because water rushes in; in a very salty solution, it shrivels as water flows out.
• Plant roots absorb water from soil through osmosis, driving water transport up to the leaves.
• Kidneys use osmotic gradients to filter blood and control the body's water balance.