Physical Chemistry of Foods

6.4 MORE CONCENTRATED SOLUTIONS

So far, we essentially considered interactions between one polymer molecule
and the solvent. In this section, mutual interactions between polymer
molecules come into play. Nevertheless, the solutions remain fairly dilute,
i.e., at most a small percentage of polymer.

6.4.1 Nonideality

Nonideality of solutions is discussed in Section 2.2.5. It can be expressed as
the deviation of the colligative properties from that of an ideal, i.e., very
dilute, solution. Here we will consider the virial expansion of osmotic
pressure. Equation (2.18) can conveniently be written for a neutral and
flexible polymer as

P¼RT

1

Vp

jþ

b

2 Vs

j^2 þ

1

3 Vs

j^3 þ



ð 6 : 11 Þ

whereVpis the molar volume of the polymer (in m^3 ?mol^1 ),Vsthe molar
volume of the solvent,jthe net volume fraction of polymer present (i.e.,
concentration in kg per m^3 divided by the mass density of the polymer), and
b is the excluded volume parameter defined in Section 6.2.1. By
determination of osmotic pressure over a range of concentrations, the
number-average molar mass andbcan be derived by use of Eq. (6.11).
Some calculated results are in Table 6.3. It is seen that the nonideality
(i.e., the magnitude of the second and third virial terms in comparison to the
first virial term) may be very large, especially for largen: a largenimplies a
small first virial term, and the second and third terms are independent ofn.
If the polymer behaves like an ideal random chainðb¼ 0 Þ, the second virial
term equals zero. Forb<0, it is even negative, but then the solubility of the
polymer is quite small (Section 6.5.1).
Even far stronger nonideality may occur forpolyelectrolytes, especially
at low ionic strength. The chain is much more expanded (highb). Moreover,
the condition of electroneutrality implies that the polymer is accompanied
by additional ions. The second virial term then approximately equals

z^2

4 mcVp^2

j^2 ð 6 :11aÞ

where z is the valence of the polymer andmc is the concentration of
(monovalent) counterions (in mol?m^3 ). Results are also given in Table 6.3.
It is seen that forj¼ 0 :003 andmc¼1, the second virial term gives rise to
an osmotic pressure over 3 barð 126 RT&310 kPa& 3 :1 barÞ. These results