diagram into various domains. For a supersaturation ratiob <1, the
solution is undersaturated and no crystallization will occur. Forb>1.3, in
the domain designated ‘‘labile,’’ crystallization occurs spontaneously. In the
metastable domain, crystallization does not occur within a reasonable time,
and in the intermediate domain crystallization depends on the time available
and on the purity of the solution.
The simplest interpretation is that forb>1.3 homogeneous nucleation
occurs, that in the intermediate domain heterogeneous nucleation occurs,
and that in the metastable domain virtually no catalytic impurities are
present. The results for sucrose in Table 14.2 agree well with homogeneous
nucleation occurring at a supersaturation ratio of about 1.3. It is seen that
an undercooling by about 23 K will be needed to obtain crystals rapidly and
reliably. This is often not suitable, and to ensure fast crystallization at lower
supersaturation, especially in the metastable domain, seeding with small
crystals is generally applied. Remember that sucrose crystals tend to grow
quite fast even at low supersaturation, which means that nucleation is often
the limiting step for fast crystallization.
We now turn tolactose, where the situation is more complicated. As
mentioned, several sugars, including lactose, occur in two forms, the
anomers denotedaandb. In solution, the two are in equilibrium with each
other, i.e.,a>b. The equilibrium ratio varies with temperature:R¼[b]/[a]
¼ 1.64 0.0027 (T273.15). The reaction leading to equilibrium—
generally calledmutarotation—is rather slow, the first-order rate constant at
308 C being about 1 h^1. Botha-lactose, as a monohydrate, and anhydrous
FIGURE15.17 Solubility and supersaturation of (a) sucrose and (b) lactose.Xis
concentration in kg per kg water,bis supersaturation ratio.