Abnormal Molar Mass and van't Hoff Factor

The molar mass calculated from a measured colligative property occasionally disagrees, sometimes quite dramatically, with the molar mass expected from the solute's known chemical formula — a discrepancy called an abnormal molar mass. This happens because every colligative-property formula in this chapter was derived assuming that each formula unit of solute dissolved contributes exactly one independent particle to the solution; if that assumption fails, the measured colligative property reflects the true, altered number of particles actually present, while the formula-based molar mass still assumes the ideal one-particle-per-formula-unit count, and the two calculations disagree.

Two opposite kinds of failure are possible. An electrolyte — an ionic compound such as $\text{NaCl}$ or $\text{KCl}$ — dissociates in solution into more particles than the single formula unit suggests ($\text{NaCl} \to \text{Na}^+ + \text{Cl}^-$ gives two particles per formula unit), so the measured colligative effect is larger than expected, and the molar mass calculated from it (using the ordinary formula, which assumes only one particle) comes out lower than the true formula mass. Conversely, some solutes associate into aggregates in certain solvents — most classically, carboxylic acids such as acetic and benzoic acid, which dimerize through pairs of hydrogen bonds when dissolved in a non-polar solvent like benzene, so that two formula units behave, for colligative purposes, as a single particle. Association reduces the number of independent particles below the formula-unit count, so the measured colligative effect is smaller than expected, and the molar mass calculated from it comes out higher than the true formula mass.

This discrepancy is captured quantitatively by the van't Hoff factor, $i$, defined as the ratio of the actual (observed) colligative property to the value that would be expected if the solute neither dissociated nor associated:

$i = 1$ for a solute that neither dissociates nor associates; $i > 1$ for a dissociating electrolyte (more particles than formula units); and $i < 1$ for an associating solute (fewer effective particles than formula units). Every colligative-property equation in this chapter is corrected for a real, dissociating or associating solute simply by inserting $i$ as a multiplying factor:

$\frac{p_1^{\circ}-p_1}{p_1^{\circ}} = i\,\frac{n_2}{n_1} \qquad \Delta T_b = i\,K_b\, m \qquad \Delta T_f = i\,K_f\, m \qquad \pi = i\,CRT$ …