Physical Chemistry of Foods

colligative properties. From the results depicted in Figure 2.3, it can be
derived that for sucrose solutions a second virial coefficient suffices up to a
mole fraction of about 0.015 (about 25%w/w sucrose).
If we have more than one solute, the situation becomes far more
complicated. For two solutes (subscripts 2 and 3) we have in principle

P¼RTðm 2 þm 3 þB 2 m^22 þB 3 m^23 þB 23 m 2 m 3 þÞð 2 : 19 Þ

We now must also take into account a second virial coefficient for the
interaction between both solutesB 23. If both solutes are very similar in
chemical nature,B 23 may be negligible. As an example we take the effect of
sucrose on lactose. Figure 2.6 shows that high concentrations of sucrose
considerably lower the solubility of lactose. This is to be expected, since (a)
at high sucrose concentrations the activity coefficient of sucrose is greatly
increased (see Figure 2.3); (b) because of the similarity between both sugars
we may expect that also the activity coefficient for lactose is greatly
increased; and (c) since the solubility is to be expressed as an activity, and
activity equals concentration times activity coefficient, a higher activity
coefficient means a lower solubility in terms of concentration. In Table 2.2
some calculations are given. Column (2) gives the solubility of lactose; in (3)
the mole fraction of lactoseþsucrose is given; in (4) a presumed activity
coefficient is given, based on the assumption that all sugar behaves as if it
were sucrose (thus derived from Figure 2.3); and finally the product of (2)
and (4) should be the solubility of lactose in terms of activity and thus be
constant, if the assumption is correct. It is seen that up to about 1 kg sucrose
per kg water, this is nearly correct (the variation is probably within
experimental error), but for higher concentration it is not so. Naturally, for
solutes that differ considerably in chemical structure, much larger deviations

TABLE2.2 Effect of Sucrose on the Solubility of Lactose

(1) Sucrose

concentration,

kg/kg water

(2) Lactose

solubility,

kg/kg water

(3) Mole

fraction

of sugar

(4) Activity

coefficient ð 2 Þ 6 ð 4 Þ

0 0.45 0.023 1.26 0.57

0.40 0.38 0.042 1.55 0.59

0.66 0.34 0.050 1.73 0.59

1.00 0.28 0.063 1.98 0.55

1.50 0.21 0.083 2.28 0.48

2.32 0.19 0.167??