deviations from ideality at high concentration, especially if the solute
molecules are large.
The effect of concentration is often expressed in avirial expansion. For
the osmotic pressure it reads
P¼RTðmþBm^2 þCm^3 þÞð 2 : 18 Þwheremis in mol?m^3 .Bis called the second virial coefficient,Cthe third
virial coefficient, and so on. In first approximation,Bis due to both solvent
quality and volume exclusion effects and it can either be negative or, more
likely, positive;Cis only caused by volume exclusion and would be zero if
solvent and solute molecules have equal size. Figure 2.5 illustrates some
trends. It is seen that for a not too high concentration the second virial
coefficient may suffice (i.e.,P=mis linear withm), but this is rarely true for
macromolecular solutes.
The same relation can be used for the chemical potential of the solvent
(more precisely for m 1 m 1 , which equals zero form¼0) and for all
FIGURE2.6 Effect of sucrose on the relative solubility of lactose in water at 50 8 C.
After results by T. A. Nickerson and E. E. Moore.J. Food Sci. 37 (1972), 60.