Abnormal Molar Masses

All four colligative properties studied so far — relative lowering of vapour pressure, elevation of boiling point, depression of freezing point, and osmotic pressure — depend only on the number of solute particles present in solution, not on their nature. Each one therefore lets us work back to the molar mass of the solute. But this back-calculation quietly assumes that the solute stays as the same particles in solution as it was as a solid: that it neither breaks apart nor sticks together. When that assumption fails, the molar mass we obtain from experiment does not match the true (formula-based) molar mass. Such a value is called an abnormal molar mass.

Why the particle count changes in solution

The whole idea rests on a simple fact: colligative properties count particles. If a dissolved substance produces more particles than we assumed, or fewer, every colligative property shifts accordingly, and the molar mass we deduce is thrown off.

There are two ways this happens.

Dissociation (ionic solutes give more particles). Ionic compounds split into ions when they dissolve in water. Dissolving one mole of KCl (74.5 g), for instance, releases one mole of $\text{K}^+$ and one mole of $\text{Cl}^-$ — two moles of particles from what looks like one mole of solute. Ignoring inter-ionic attractions, one mole of KCl in 1 kg of water would raise the boiling point by roughly $2 \times 0.52\ \text{K} = 1.04\ \text{K}$, twice what a single mole of particles would give. If we were unaware of the dissociation, we would wrongly conclude that the observed effect came from 2 mol of "KCl particles", and so calculate the mass of one mole as about half the real value — near 37.25 g instead of 74.5 g.

Association (some molecules join together). The opposite can also occur: molecules can combine into larger aggregates in solution, cutting the particle count. A classic case is ethanoic (acetic) acid in benzene. Because benzene has a low dielectric constant, two ethanoic acid molecules pair up through hydrogen bonding into a dimer:

The two molecules are held together by a pair of O–H⋯O hydrogen bonds, forming a closed ring:

If every acid molecule dimerised, the number of independent particles would halve, and so would $\Delta T_b$ and $\Delta T_f$. A colligative measurement would then report only half the expected effect — and the molar mass calculated from it would be about twice the true value.

The van't Hoff factor, $i$

To handle both situations with a single quantity, van't Hoff (1880) introduced a correction factor $i$, called the van't Hoff factor, which measures the extent of dissociation or association. It can be written in three equivalent ways:

Here the abnormal molar mass is the experimentally measured one, and the calculated colligative property is the value we would predict assuming the non-volatile solute is neither associated nor dissociated.

Reading these definitions tells us at a glance what is happening in solution:

• Dissociation produces more particles than assumed, so the observed colligative property is larger and $i > 1$. For KCl in water, $i$ is close to 2.
• Association produces fewer particles, so the observed colligative property is smaller and $i < 1$. For ethanoic acid in benzene, $i$ is nearly 0.5.

The degree of dissociation or association depends on concentration; measured $i$ values for common strong electrolytes at different dilutions (and their limiting values for complete dissociation) are summarised in the accompanying van't Hoff factor table.

As the printed values in the table show, dilution pushes $i$ toward its

complete-dissociation limit: for KCl, NaCl and MgSO$_4$ the measured $i$

approaches 2 as the solution becomes very dilute, and for K$_2$SO$_4$

(three ions per formula unit) it climbs toward 3 — exactly as expected.

Colligative properties corrected with $i$

Since $i$ captures the real number of particles, we simply insert it into each colligative-property equation. This gives the corrected relations that apply to dissociating or associating solutes.

Relative lowering of vapour pressure of the solvent:

$\frac{p_1^{\,\circ} - p_1}{p_1^{\,\circ}} = i\,\frac{n_2}{n_1}$ …