Freezing-Point Depression with the van 't Hoff Factor

Colligative properties count particles. That is the whole content of the word — from the Latin for "bound together", meaning the effect depends on how many things are dissolved and not at all on what they are. A mole of glucose and a mole of urea depress water's freezing point identically. A mole of sodium chloride depresses it roughly twice as far, because sodium chloride does not dissolve as NaCl; it dissolves as Na⁺ and Cl⁻, and the solvent counts two.

The van 't Hoff factor i is the honest bookkeeping for that. It is the number of particles a formula unit actually delivers into solution: 1 for sugar and alcohol, 2 for NaCl and KBr, 3 for CaCl₂ and Na₂SO₄, 4 for AlCl₃ and K₃PO₄. Multiply the ordinary depression law by it and you have ΔT_f = i·K_f·b. Road salt at 1.0 mol/kg drops water's freezing point by 2 × 1.86 × 1.0 = 3.7 C°; calcium chloride at the same molality manages 3 × 1.86 = 5.6 C°, which is why it is the one spread when the forecast is genuinely cold.

The measured i is almost always short of the ideal integer, and that shortfall is the point of the experiment. A 0.10 molal NaCl solution behaves like about 1.87 particles, not 2.00. Oppositely charged ions spend part of their time paired up, and a pair depresses the freezing point once rather than twice. The gap widens as concentration rises and widens faster for highly charged ions — Debye and Hückel built their 1923 theory of strong electrolytes on exactly this discrepancy. So an i of 1.87 is not a failed measurement of 2; it is a successful measurement of ion pairing.

Jacobus van 't Hoff took the first Nobel Prize in Chemistry in 1901 for this family of relations, having noticed that dilute solutions obey laws with the same shape as the gas laws. The same i appears in the boiling-point elevation, in osmotic pressure as Π = iMRT, and in the vapour-pressure lowering all three descend from. If you correct one for dissociation and forget the others, the set stops agreeing with itself.