Van't Hoff Factor

To quantify exactly how much a solute's association or dissociation shifts the observed (abnormal) molar mass away from its true value, van't Hoff introduced a factor, denoted $i$ and now called the van't Hoff factor. It is defined as the ratio of the solute's true (normal) molar mass to its observed (abnormal) molar mass -- and, equivalently, as the ratio of the observed colligative property to the colligative property calculated assuming no association or dissociation:

For example, the estimated van't Hoff factor for an acetic acid solution in benzene is about $i=0.5$ (consistent with the dimerisation of section 9.11.1, which roughly halves the true particle count), while for a sodium chloride solution in water it is about $i=2$ (consistent with complete dissociation into two ions).

Relating $i$ to the degree of dissociation or association. If a solute dissociates into $n$ ions/species per formula unit, its degree of dissociation is

If instead $n$ solute molecules associate together into a single aggregate, the degree of association is

Incorporating $i$ into the four colligative-property equations. Once association or dissociation is accounted for, the equations of sections 9.9.1-9.9.4 generalise by simply inserting a factor of $i$:

• Relative lowering of vapour pressure: $\dfrac{P^\circ_{solvent}-P_{solution}}{P^\circ_{solvent}} = i\,\dfrac{n_{solute}}{n_{solvent}}$
• Elevation of boiling point: $\Delta T_b = i\,K_b\,m$
• Depression of freezing point: $\Delta T_f = i\,K_f\,m$
• Osmotic pressure: $\pi = i\,\dfrac{w_{solute}}{M_{solute}}\cdot\dfrac{RT}{V}$

Interpreting the three regimes of $i$:

• $i=1$: the solute neither dissociates nor associates; the molar mass calculated from the colligative property matches the true (actual) molar mass exactly.
• $i<1$: the solute associates in solution; the observed (abnormal) molar mass comes out greater than the true molar mass.
• $i>1$: the solute dissociates in solution; the observed (abnormal) molar mass comes out less than the true (normal) molar mass. …