Freezing Point Depression

ΔTf = Kf × m × i: dissolved solute lowers a solvent's freezing point in proportion to the solution's molality.

\Delta T_f = K_f \cdot m \cdot i

delta Tf = Kf x molality x i

What it means

Freezing point depression is a colligative property: it depends on how many solute particles are present, not on what they are. Each mole of dissolved particles per kilogram of solvent drops water's freezing point by 1.86 °C, and electrolytes multiply the effect by i, the number of ions each formula unit releases. That is why a 0.1 m NaCl solution freezes noticeably lower than a 0.1 m sugar solution.

Variables

Symbol	Name	Unit	Meaning
$ΔTf$	Freezing point depression	°C	The magnitude by which the freezing point drops, reported as a positive number of degrees.
$Kf$	Freezing point depression constant	°C·kg/mol	The solvent's cryoscopic constant; for water it is 1.86 °C·kg/mol.
$m$	Molality	mol/kg	The solution's concentration in moles of solute per kilogram of solvent.
$i$	van't Hoff factor	dimensionless	The number of particles each solute unit produces in solution — 1 for nonelectrolytes, 2 for NaCl, 3 for CaCl2, ideally.

Assumptions

The solution is dilute, so molality closely tracks particle concentration.
The solute is nonvolatile and does not react with the solvent.
The van't Hoff factor i equals the ideal ion count only for complete dissociation; real values may be lower.
Kf is a property of the solvent — 1.86 °C·kg/mol applies to water only.

Worked example

Problem

Calculate the freezing point of a 0.100 m aqueous NaCl solution (assume complete dissociation, i = 2; Kf for water = 1.86 °C·kg/mol).

Given:

Molality m = 0.100 mol/kg
Kf for water = 1.86 °C·kg/mol
NaCl gives 2 ions, so i = 2

Write ΔTf = Kf × m × i.
Substitute: ΔTf = (1.86 °C·kg/mol)(0.100 mol/kg)(2).
Multiply the first pair: 1.86 × 0.100 = 0.186.
Multiply by i: 0.186 × 2 = 0.372 °C.
Subtract from water's freezing point: 0.00 °C − 0.372 °C = −0.372 °C.

ΔTf = 0.372 °C; the solution freezes at −0.372 °C

The salt lowers water's freezing point by about a third of a degree — the same colligative principle behind salting icy roads and making ice cream.