Biomedical Engineering Reference
In-Depth Information
a
is the nonrandom factor, which is similar to nonrandom, two-liquid model. Notice that
the first two terms are the same as UNIQUAC model; therefore, this model is known as the
UNIQUAC-NRF model. The combinatorial activity coefficient is expressed as
n
*
Φ
*
Φ
zq
∑
(
)
+
θ
i
i
i
i
ln
=
ln
+
ln
+ −
l
x l
q
γ
i γi,combinatorial
,
i
j
j
i
x
2
x
*
Φ
i
i
j
=
1
i
(5.112)
n
n
n
a a
a a
−
1
2
a a
a a
∑
(
)
∑
1
ij
ji
+
1
+
ln
a
−
a
+
1
−
ln
θθ
ln
lk kl
ll kkl
θ
θ
ii
j
ij
j
k l
ii
jj
j
j
=
≠
1
k
k i
=
≠
1
l
=
≠
≠
i
l k
l
i
Charging of species in the solution does not only influence geometric distribution of com-
ponents in the solution, but more importantly, the electrostatic interaction contributes to
excess enthalpy, which is denoted as long-range interaction contribution, whereas short-
range contribution can be taken from the residual part of the UNIQUAC.
E
E
E
g
RT
g
RT
g
RT
=
+
(5.113)
residual
long-range
short-range
Hence, the total activity coefficient of species
i
is
ln
γ
i
= ln
γ
i,
long-range
+ ln
γ
i,
short-range
+ ln
γ
i,
γi,combinatorial
(5.114)
The electrostatic contribution to the excess Gibbs energy was suggested by Fowler and
Guggenheim electrostatic interactions [60]
2 1 2
/
Az I
bI
ln
= −
i
(5.115)
γ
i
,
long-range
1 2
/
1
+
I
is the ionic strength of the mixture, which is expressed in terms of molarity of species
i
.
n
1
2
∑
(5.116)
I
=
m z
i
2
i
i
=
1
z
is the absolute charge number of ionic species and
b
is a constant that depends on sizes of
the components. For the protein with a size of 4 nm,
b
was taken as 15 (kg mol
-1
)
1/2
[61] and
A
is the Debye-Huckel constant for a given system:
A
= 1.327757 × 10
5
d
0.5
/(
DT
)
1.5
(5.117)
For water,
A
= 1.131 + 1.335 × 10
-3
× (
T
- 273.15) + 1.164 × 10
-5
× (
T
- 273.15)
2
(5.118)
