Physical Chemistry of Foods

electroneutrality forces an equal transport of Naþ. This is depicted below,
where m means molar concentration. The vertical line depicts the
membrane.

Ions Naþ PEz Cl Naþ Cl

Initial concentration m 1 m 1 =z 0 m 2 m 2

Equilibrium concentration m 1 þxm 1 =zx m 2 xm 2 x

The equilibrium condition is that the ion activity product of NaCl

aðNaþÞ 6 aðClÞ¼½NaþŠ 6 ½ClŠ 6 ðg+Þ^2

is equal on both sides of the membrane. (See Section 2.3.1 for the theory of
electrolyte solutions.) Assuming dissociation of the salt to be complete and
g+to be equal on both sides, the condition would become

ðm 1 þxÞx¼ðm 2 xÞ^2 or x¼

m^22

m 1 þ 2 m 2

Calculated examples are in Table 6.2. The first conclusion to be drawn
is that the charge of the polyelectrolyte is compensated in two ways, i.e., by
an excess of counterions and by a depletion of coions. It is further seen that
the absoluteðxÞand relative amounts of NaClðx=m 2 Þtransferred increase
with increasing salt concentration. The ratio of concentrations of Naþon
either side of the membrane is closer to unity for higher salt concentration,

TABLE6.2 The Donnan Effecta

m 2 xx=m 2 ðm 1 þxÞ=ðm 2 xÞ DO

0.3 0.056 0.19 4.33 0.20
1 0.33 0.33 2.00 0.34
3 1.29 0.43 1.33 0.44
10 4.76 0.48 1.10 0.49
100 49.75 0.50 1.01 0.51

aCalculations of the change in concentrationx(millimolar) of coions and counterions between

a compartment (1) containing 0.1%of a polyelectrolyte ofM¼ 105 Da (i.e., 0.01 mmolar) and
valencez¼100. Compartment 1 initially containedm 1 ¼1 mmolar Naþ; compartment 2
m 2 mmolar NaCl.ðm 1 þxÞ=ðm 2 xÞis the ratio of the equilibrium concentration of Na in
compartment 1 to that in 2; the concentration ratio of the Clions is the reciprocal of the values
in this column.DOis the difference in osmolality (millimolar) between the two compartments.