Colligative properties: vapor pressure lowering, boiling point elevation, freezing point depression, osmotic pressure and colloids

The last topic of the course is a group of properties of solutions: the lowering of the solvent’s vapor pressure, the rise of the boiling point, the lowering of the freezing point and the osmotic pressure. All of them are called colligative properties, because they depend not on what the solute is, but on how many particles there are. A mole of sugar and a mole of urea lower the freezing point of water by the same amount. A mole of salt lowers it twice as much, because it splits into two ions.

Golden rules:

1. Count particles. For electrolytes multiply by the van ‘t Hoff factor $i$.
2. Molality in the ΔT formulas, molarity in the osmotic pressure.
3. ΔT is a difference. Freezing point = freezing point of the pure solvent − ΔTf.
4. From the measured colligative effect, go back to the number of moles, and from there to the molar mass.

Vapor pressure lowering: Raoult’s law

A nonvolatile solute takes the place of some of the solvent molecules at the surface and reduces vaporization. Raoult’s law:

$P_{\text{solvent}} = x_{\text{solvent}} \cdot P^\circ_{\text{solvent}} \qquad \Delta P = x_{\text{solute}} \cdot P^\circ$

$P^\circ$ is the vapor pressure of the pure solvent. A solution that obeys this law exactly is called an ideal solution; dilute solutions made of similar molecules are approximately ideal.

Example 1. The vapor pressure of a 1.00 m sucrose solution at 25 °C ($P^\circ = 23.76$ mmHg):

1 kg water = 55.51 mol;  sucrose 1.00 mol
x(water) = 55.51 / 56.51 = 0.9823
P = 0.9823 × 23.76 = 23.34 mmHg   (ΔP = 0.42 mmHg)

Two volatile components

If both components are volatile, Raoult is applied to each one separately; the total vapor pressure is found with Dalton’s law:

$P_{\text{tot}} = x_A P^\circ_A + x_B P^\circ_B$

The mole fraction in the vapor is $y_A = P_A / P_{\text{tot}}$. The more volatile component is richer in the vapor than in the liquid. In a benzene-toluene mixture, when the benzene fraction in the liquid is 0.400, it is 0.691 in the vapor; condensing this vapor and vaporizing it again (fractional distillation) separates the two liquids.

Boiling point elevation and freezing point depression

Because the vapor pressure of the solution is lower, it must be heated more to reach the external pressure: the boiling point rises. The solute particles also make it harder for the solvent to form an ordered crystal: the freezing point is lowered.

$\Delta T_b = i\,K_b\,m \qquad \Delta T_f = i\,K_f\,m$

• $m$: molality (mol solute / kg solvent).
• $K_b$: the boiling point elevation constant (ebullioscopic constant), $K_f$: the freezing point depression constant (cryoscopic constant). They depend only on the solvent.
• $i$: the van ‘t Hoff factor (1 if not an electrolyte).

Example 2. 5.00 g of urea ($\mathrm{CO(NH_2)_2}$, 60.06 g/mol) in 100.0 g of water:

n = 5.00 / 60.06 = 0.08325 mol      m = 0.08325 / 0.1000 kg = 0.8325 m
ΔTf = 1.86 × 0.8325 = 1.55 °C   → freezing point −1.55 °C
ΔTb = 0.512 × 0.8325 = 0.426 °C → boiling point 100.43 °C

Spreading salt on roads, putting antifreeze in a car radiator and adding salt to ice when making ice cream are applications of freezing point depression.

PITFALL · molarity versus molality. The ΔT formulas want molality. If the mass of the solvent is given, compute the molality directly; if the molarity is given, convert first (last week).

PITFALL · taking ΔT as the result. For the question “what is the freezing point?” the answer is not $\Delta T_f$ but $T_f^\circ - \Delta T_f$. In benzene the pure solvent freezes at 5.5 °C; if $\Delta T_f$ is 1.2 °C, the solution freezes at 4.3 °C.

Molar mass from colligative measurements

The recipe: measured ΔT → molality → moles (× kg of solvent) → molar mass (grams ÷ moles). Solvents with a large $K_f$ like benzene and camphor are preferred because the ΔT to be measured is large.

Osmotic pressure

A semipermeable membrane lets solvent molecules through and does not let solute particles through. If there are different concentrations on the two sides of the membrane, the solvent flows from the dilute side to the concentrated side: osmosis. The pressure that must be applied to the concentrated side to stop this flow is called the osmotic pressure:

$\Pi = i\,M\,R\,T$

The formula resembles the ideal gas equation ($\Pi V = nRT$). Here $M$ is the molarity.

Example 3. 0.0100 M glucose, 25 °C:

$\Pi = 0.0100 \times 0.08206 \times 298.15 = 0.245\ \text{atm} = 186\ \text{mmHg}$

The freezing point of the same solution drops by only 0.0186 °C. Because the osmotic pressure gives a much larger and easily measured signal, the molar mass of large molecules like proteins and polymers is found with an osmometer.

In biology:

• Isotonic: the same osmotic pressure as the cell; the cell does not change (physiological saline, 0.9% NaCl).
• Hypertonic: the outside is more concentrated; water leaves the cell and the cell shrinks (salted meat, pickles).
• Hypotonic: the outside is more dilute; water enters the cell, and the cell swells and may burst.

Reverse osmosis: if a pressure larger than the osmotic pressure is applied to the concentrated side, the solvent flows in the opposite direction. It is used to produce drinking water from seawater.

Electrolyte solutions and the van ‘t Hoff factor

Electrolytes split into ions, and the number of particles increases. $i$ accounts for this:

$i = \frac{\text{moles of particles in solution}}{\text{moles of formula units dissolved}}$

Example 4. For 0.100 m $\mathrm{CaCl_2}$, $\Delta T_f = 3 \times 1.86 \times 0.100 = 0.558$ °C. Three times that of glucose at the same molality.

Experimental $i$ values are generally smaller than ideal, especially as the concentration and the ion charges increase. Oppositely charged ions briefly form ion pairs in the solution and behave like a single particle. In very dilute solutions $i$ approaches the ideal value. For weak electrolytes $i$ is between 1 and the ideal value and can be used to compute the percent ionization.

TIP · colligative ranking. In questions like “which has the lowest freezing point / the highest boiling point / the largest osmotic pressure?” find the product $i \times m$ for each solution. The largest product is the largest effect.

Colloidal mixtures

By particle size, mixtures are divided into three:

The Tyndall effect: colloid particles scatter light, and the beam of light is visible from the side (headlight in fog, a sunbeam in a dusty room). It is not seen in true solutions; that is why it is the simplest way to tell a colloid from a solution.

Types of colloids are named by the dispersed phase and the dispersing medium:

Colloids are generally stable thanks to the charges on their surfaces: particles with the same charge repel each other and do not merge and settle. Adding an electrolyte can neutralize these charges and precipitate the colloid (coagulation); the formation of deltas at river mouths is related to the salt in seawater precipitating the mud colloid. Soaps and detergents keep oil in water as colloid particles called micelles.

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