Environmental Engineering Reference
In-Depth Information
The formal definition for d
G
is
i
μ
i
d
n
i
.
d
G
=−
S
d
T
+
V
d
P
+
(2.55)
Therefore, we have
+
+
μ
i
=
S
d
T
V
d
P
n
i
d
0.
(2.56)
i
This is called the
Gibbs-Duhem relationship
. At constant temperature and pressure
the above equation reduces to
n
i
d
μ
i
=
0.
(2.57)
i
This equation is useful in estimating the variation in the chemical potential of one
component with composition if the composition and variation in chemical potential
with that of the second component are known. By dividing the equation throughout
with the total number of moles,
n
i
, we can rewrite the above equation in terms of
mole fractions,
x
i
d
μ
i
.
2.5.2 S
TANDARD
S
TATES FOR
C
HEMICAL
P
OTENTIAL
Let us examine the chemical potential of an ideal gas. At a constant temperature
T
,
V
n
i
d
P
i
,
(
d
μ
i
)
T
=
(2.58)
where
P
i
is the
pressure exerted by the component i
(
partial pressure
) in a constant
volume
V
. Note that for an ideal gas
V/n
i
=
μ
i
at any
T
can now be
determined by integrating the above equation if we choose as the lower limit a starting
chemical potential (
(RT/P
i
),
0
i
)
at the temperature
T
and a reference pressure
P
i
, that is,
μ
RT
ln
P
i
P
i
.
0
μ
i
= μ
i
+
(2.59)
Thus, the chemical potential of an ideal gas is related to a physically real measurable
quantity, namely, its pressure. Whereas both
i
and
P
i
are arbitrary, they may not be
chosen independent of one another, since the choice of one fixes the other automat-
ically. The standard pressure chosen in most cases is
P
i
=
0
μ
1 atm. Thus for an
ideal
gas
the chemical potential is expressed as
0
i
(T
;
P
i
=
μ
i
= μ
1
)
+
RT
ln
P
i
.
(2.60)
The chemical potential for an
ideal solution
is analogous to the above expression for
the ideal gas. It is given by
0
i
(T
;
x
i
=
μ
i
= μ
+
·
1
)
RT
ln
x
i
,
(2.61)
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