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We continue our study of colligative properties of solutions initiated inmath-ph/0407034. We focus on the situations where, in a system of linear size$L$, the concentration and the chemical potential scale like $c=\xi/L$ and$h=b/L$, respectively. We find that there exists a critical value $\xit$ suchthat no phase separation occurs for $\xi\le\xit$ while, for $\xi>\xit$, the twophases of the solvent coexist for an interval of values of $b$. Moreover, phaseseparation begins abruptly in the sense that a macroscopic fraction of thesystem suddenly freezes (or melts) forming a crystal (or droplet) of thecomplementary phase when $b$ reaches a critical value. For certain values ofsystem parameters, under ``frozen'' boundary conditions, phase separation alsoends abruptly in the sense that the equilibrium droplet grows continuously withincreasing $b$ and then suddenly jumps in size to subsume the entire system.Our findings indicate that the onset of freezing-point depression is in fact asurface phenomenon.Comment: 27 pages, 1 fig; see also math-ph/0407034 (both to appear in JSP

Colligative Properties of Solutions: II. Vanishing Concentrations