Chemistry Reference
In-Depth Information
expressed in terms of the activities of the molecule in the amorphous and crystalline
states,
a
A
and
a
C
, respectively:
G
A
=
C
T
RT
ln
a
A
a
C
Δ
=
:
(1.15)
G
A
=
C
T
Assuming activity coef
cients equal to 1,
Δ
can be expressed as
Δ
G
A
=
C
T
RT
ln
c
A
=
c
C
;
(1.16)
where
c
A
and
c
C
are the equilibrium solubilities of the API from the two solid forms.
Consequently, knowing the equilibrium solubility of the API in water at temperature
T
,
c
C
, and the free energy change
Δ
G
A
=
T
, one can estimate the expected value of
c
A
, the
supersaturation concentration. One can experimentally determine
Δ
G
A
=
C
T
by recognizing
that
G
A
=
C
T
H
A
=
C
T
S
A
=
C
T
Δ
Δ
T
Δ
;
(1.17)
S
A
=
C
T
are the enthalpy and entropy differences between the
amorphous and crystalline forms, respectively. Furthermore, it can be shown that
H
A
=
C
T
where
Δ
and
Δ
H
A
=
C
T
H
T
C
p
C
p
T
C
Δ
Δ
T
;
(1.18)
S
A
=
C
T
S
T
C
p
C
p
ln
T
C
Δ
Δ
=
T
;
(1.19)
and
S
T
Δ
H
T
=
T
C
Δ
;
(1.20)
where
T
C
is the melting temperature of the crystal,
H
T
Δ
is the heat of fusion of the
S
T
is the entropy of fusion of the crystal, and
C
p
and
C
p
are the heat capacities
of the amorphous and crystal forms, respectively. By experimentally measuring
C
p
and
C
p
using thermal analysis and
T
C
and
c
C
for the crystal at
T
, it is possible to
use Equation 1.16 to predict the value of
c
A
, the apparent equilibrium solubility of
the amorphous form that might be expected [30]. Recognizing that the activity of the
amorphous solid can be reduced by the presence of any residual absorbed water and that
many acidic and basic APIs can undergo some ionization in aqueous solution, a more
rigorous thermodynamic expression to predict solubility of amorphous solids under such
conditions has been reported [31]. As shown in Figure 1.16, the experimental maximum
supersaturation level
c
A
/
c
C
for indomethacin at 25
crystal,
Δ
C before the initiation of crystalliza-
tion is about 4.9. Using Equation 1.16 the value predicted at 25
°
C for indomethacin is on
average about 29.0, while correcting for water absorption and ionization the value is
reduced to 7.0, much closer to the experimental value of 4.9. Some uncertainty in the
agreement between predicted and experimental values of
c
A
/
c
C
comes from the fact that
crystallization can be triggered by nucleation events that are not easily predicted and
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