In the case of a single-phase, multicomponent open system, Gibbs energy of the phase not
only depends on pressure and temperature, but also on the number of moles of each component
existing in the phase, i.e.,
G
=
G
(
P, T, n
1
, n
2
, ..., n
k
)
(7.1-10)
Therefore, the total di
ff
erential of
G
is expressed in the form
dG
=
μ
∂G
∂P
¶
T,n
j
dP
+
μ
∂G
∂T
¶
P,n
j
dT
+
k
X
i
=1
μ
∂G
∂n
i
¶
P,T,n
j
6
=
i
dn
i
(7.1-11)
The subscript
n
j
in the
fi
rst two terms of Eq. (7.1-11) implies that the number of moles of
all components are held constant during di
ff
erentiation. For constant number of moles, Eqs.
(7.1-7) and (7.1-8) are still valid. On the other hand, the partial molar Gibbs energy is de
fi
ned
by
G
i
=
μ
∂G
∂n
i
¶
P,T,n
j
6
=
i
(7.1-12)
Thus, Eq. (7.1-11) becomes
dG
=
V dP
−
S dT
+
k
X
i
=1
G
i
dn
i
(7.1-13)
Historically, for a multicomponent system, the partial molar Gibbs energy has been called the
chemical potential
and designated by
μ
i
, i.e.,
G
i
=
μ
i
.
The enthalpy is related to the Gibbs energy by
H
=
G
+
TS
(7.1-14)
Di
ff
erentiation of Eq. (7.1-14) gives
dH
=
dG
+
T dS
+
S dT
(7.1-15)
Substitution of Eq. (7.1-13) into Eq. (7.1-15) yields
dH
=
T dS
+
V dP
+
k
X
i
=1
G
i
dn
i
(7.1-16)
The internal energy is related to the enthalpy by
U
=
H
−
PV
(7.1-17)
Di
ff
erentiation of Eq. (7.1-17) gives
dU
=
dH
−
P dV
−
V dP
(7.1-18)
The use of Eq. (7.1-16) in Eq. (7.1-18) results in
dU
=
T dS
−
P dV
+
k
X
i
=1
G
i
dn
i
(7.1-19)
170