Answer
The sucrose solution has anawof 0.94 according to Figure 8.1. A mole fraction of
0.05 implies per 100 moles 5 6342 ¼1710 g of sucrose and 95 618 ¼1710 g water,
i.e., a mass fraction of 0.5. Hence 50 g of protein per kg solution corresponds to 100 g
per kg water, or (100/20¼) 5 mol protein per (1000/0.018¼) 55,555 mol water. Since
10 mol Naþis present per mol protein, the total solute amounts to 55 mol, yielding a
mole fraction of water in the protein solution of 0.999. However, a protein is a very
large molecule and we should consider the volume ratio solute/solventq. Assuming
the mass density of the protein in water to be 1400 kg?m^3 , we deriveq&800, and
insertion into Eq. (8.4) withb¼1 yieldsaw¼0.9986. By applying the Ross equation
(8.5),awof the mixture would be 0.94 6 0.9986¼0.9387. Another nonideality must
be nonsolvent water for sucrose. Figure 8.7b would yield for a molecule of 342 Da a
quantity of about 0.4 g per g of protein. Assuming this value to hold for the protein
involved (which is by no means certain), we arrive at 0.4 650 ¼20 g nonsolvent
water per 500 g, or 4%of the water. The quantity (1aw) should then be multiplied
by 1.04. The water activity would then become (11.04 6 0.06) 6 0.9986¼0.9363.
This would mean that (1aw) would be increased by only about 6%due to the
addition of a fairly large quantity of protein, and the greater part of this effect would
be due to nonsolvent water.
8.4 REACTION RATES AND WATER CONTENT
Most, though not all, changes or reactions occurring in foods proceed
slower at a smaller water content. Often, water activity is considered to be
the key variable, but the situation may be far more complicated.
Unfortunately, reliable quantitative theory is not available. We therefore
can only give some general considerations and examples on physical
changes, chemical reactions, and microbial growth.
8.4.1 Physical Changes
Important examples are loss or uptake of water, gases, and volatiles, and
crystallization, often resulting in changes in mechanical and some physical
properties. An example of crystallization is given for sucrose in Figure 8.6a.
At 6%water, it takes about 2 days before sucrose crystallization becomes
manifest; at 3%water, it takes about 500 times longer. Water content thus
has a very large influence. This is via its effect onmolecular mobility, i.e., the
diffusion coefficient of the molecules considered. This was discussed to some
extent in Chapter 5, especially Section 5.3.2 and Figure 5.16a. The smaller
the mass fraction of waterw, the smaller the effective diffusion coefficient
Deff. The smallerw, the steeper this dependence and the more difficult it is to