W. Richard Bowen and Nidal Hilal 4

50 2. MEASUREMENT OF PARTICLE ANd SURFACE INTERACTIONS
Considering equation (2.36) as an example, the equilibrium constant
for the reaction may be written as:
K
1 �
ˆ
�
s
[ COO ][ H ]
[ COOH]
(2.38)
where [H � ] s is the hydrogen ion concentration at the BSA surface, which
can be determined from:
� + ⎛�ze
o
[ H ] s � [ H ] bulk exp⎜
� ⎞
⎝⎜
kT ⎠⎟
(2.39)
with the bulk hydrogen ion concentration being found from the pH of
the dispersion.
Now, let
R
K1
[ H ]
� �
s
⎛
ˆ
[ COO ] ⎞
⎜
⎜˜
⎝⎜
[ COOH]
⎠⎟
then, the fraction of carboxyl groups ionised, X COO, will be:
X
COO ˆ
ˆ
COO
COOH COO R
[ ]
�
�
ˆ
[ ] � [ ] 1�R
(2.40)
(2.41)
The number of surface charges generated on the BSA surface owing to
the ionisation of the carboxyl groups can then be found via:
Z ˆ � X ˆ � Total number of carboxyl groups on BSA surface (2.42)
COO COO
Similar calculations can be performed for the other amino acids and
the total charge number due to the acid–base equilibria on the BSA surface
can then be found.
ZAB � ∑ �� � ∑ ��
(2.43)
To solve these equations, the pK a values of the amino acid groups in
their environment at the BSA surface need to be known. The pK a values
for the amino acid groups on BSA are available in the literature [51, 52].
If this data were not available for the specific protein being investigated,
general data is available in the literature for the intrinsic pK a values of
the ionisable amino acid groups found in proteins [52]. These pK a values
can be substantially different from the pK a values of the free amino acids.
The above equations have shown how the complex acid–base equilibria
of the amino acid surface groups may be described, but the average