Raoult's Law Calculator
Where the colligative properties come from — a solute takes up surface without contributing vapour.
The formula
P = P0 * x_solvent ; lowering = P0 * x_solute
The origin of the colligative properties
A non-volatile solute occupies part of the surface without contributing any vapour of its own, so the vapour pressure falls in proportion to the fraction of the surface it takes up. Everything else follows: the boiling point rises because the solution must be heated further before its vapour pressure reaches the surrounding air, and the freezing point falls for the matching reason on the solid side.
Ideal, and when it is not
Raoult's law assumes solute and solvent molecules attract each other exactly as strongly as they attract their own kind. Real mixtures deviate in both directions — ethanol and water notoriously so, which is why they form an azeotrope and cannot be separated completely by distillation.
Which denominator?
Most of the confusion in solution work is not arithmetic but bookkeeping. Molarity divides by the volume of the finished solution; molality divides by the mass of the solvent; percent by mass divides by the mass of the solution again. The three agree closely in dilute water and diverge sharply in concentrated acid, which is why a bottle labelled 98 % is also labelled 18.4 M without contradiction.