Physical Chemistry of Foods

Question 2

An oil-in-water emulsion, with an oil volume fractionðjÞof 0.2, contains a certain
fatty acid (HAc); total concentration is 0.01 mole per liter emulsion. The following
data are provided: the association constant for dimerization of the acid in the oil
phaseKA¼ 104 L?mol
^1 ; the partitioning constant between oil and waterKN¼1;
the dissociation constant of the acid in waterKD¼ 1 : 25? 10
^5 mol?L
^1 ; the pH is
4.90. What is the proportion of the acid in the aqueous phase and what is its
(apparent) activity coefficient in the oil?

Answer

If dimerization in the oil and dissociation in the water phase would not be taken in to
account, a partitioning constant of unity would imply the concentrations to be equal
in both phases, and the quantity in the oil phase would bejtimes the overall
concentration, i.e., 2 mmol per liter emulsion. However, the situation is as follows:

OIL









WATER

HAcÐ^12 ðHAcÞ 2


Ð








HAcÐHþþAc

½HAcŠ¼f½ðHAcÞ 2 Š=KAg^0 :^5

½HAcŠ¼KD½HAcŠw

½HAcŠw¼½HþŠ½Ac
Š=KD

Let us call½HAcŠo¼x;½Ac
Š¼y, and½ðHAcÞ 2 Š¼z. BecauseKN¼1, we have that
½HAcŠw also equals x. This means that x¼½HþŠy= 1 : 25? 10
^5 , and since
½HþŠ¼ 10
pH¼ 1 : 26? 10
^5 , we obtainy&x. The total concentration in the water
phase is thus 2x. In the oil,x^2 ¼z=KA, and thusz¼ 104 x^2. The total concentration
of the fatty acid in the oil is nowxþ 2? 104 x^2 ; the factor 2 is needed becausezrefers
to dimers. Taking into account thatj¼ 0 :2, we obtain for the total amount of fatty
acid in a liter emulsion

1 : 6 xþ 0 : 2 xþ 0 : 4? 104 x^2 ¼ 0 : 01

which yields x¼ 0 :00137 mole per liter. We now calculate that the total
concentration in the oil ðxþ 2? 104 x^2 Þ amounts to 0.0389, and since the
concentration of nondimerized acid is 0.00137, the apparent activity coefficient
¼ 0 :035. The amount of fatty acid in the oil phase is 0.0078 moles, i.e., 78%of the
total. Note that in this case the overall partitioning ratio as well as the apparent
activity coefficient in the oil depend on the concentration.

Note We have tacitly assumed that all activity coefficients equal unity.

This may be reasonable, except for Ac
(see Section 2.3); taking its activity

coefficient into account would cause the total concentration in the water

phase to be somewhat higher. (We do not need an activity coefficient for

Hþ, since pH is a measure of activity, not concentration.)