Modern Engineering Thermodynamics

12.2 Thermodynamic Properties of Gas Mixtures 409
Table 12.3 Total, Molar Specific, and Partial Molar Specific Properties
of a Mixture of Gases
Total Property
of the Mixture
V m = ∑ N
n i^v i
i=1
U m = ∑ N
n i^u i
i=1
H m = ∑ N
i=1
n i^hi
S m = ∑ N
n i^s i
i=1
Molar Specific Property
of the Mixture
v m = V m /n m = ∑ N
u m = U m /n m = ∑ N
h m = H m /n m = ∑ N
s m = S m /n m = ∑ N
ni
nm
i=1
ni
nm
i=1
ni
nm
i=1
ni
nm
i=1
^v i = ∑ N
i=1
^u i = ∑ N
i=1
^
h i = ∑ N
^s i = ∑ N
i=1
i=1
χ
^v i = ∂V m
i^v i ∂n i
χ
^u i = ∂U
m
i^u i ∂n i
^hi
χ
= ∂H
m
i^hi ∂n i
χ
^s i = ∂S
m
i^s i ∂n i
Partial Molar Specific Property
of Gas i in the Mixture
pm,Tm,nj
pm,Tm,nj
pm,Tm,nj
pm,Tm,nj
Table 12.4 Specific Heats of a Mixture and the Partial Specific Heats
of the Gases in the Mixture
Specific Heat of the Mixture
Partial Specific Heat of Gas i in the Mixture
c vm = ∂u
m
=
∂T m
∑ N
w
^c i^c vi = ∂^u
i
vi ∂T m vm
vm i=1
c pm = ∂h
m
=
∂T m
∑ N
w
^c i^c pi = ∂^h
i
pi ∂T m pm
vm
c vm = ∂u
m
∂T m
c pm = ∂h
m
∂T m
vm
vm
i=1
= ∑ N
i=1
= ∑ N
i=1
χ
^c vi = ∂^u i
i^c vi ∂T m vm
!
χ i^c pi
^c pi = ∂^h i
∂T m
vm
therefore, when either w i , χ i , ψ i ,orπ i is multiplied by 100, it represents the composition percentage of gas
i present on a mass, molar, volume, or pressure basis, respectively. Note, however, that w i and χ i do not have
the same numerical values; therefore, a mass based percentage analysis of the composition depends on which fractional
base is used in its determination.
If we consider the mixture to be a unique substance, then we can compute its equivalent molecular mass M m from
Eqs. (12.1), (1.9), and (12.4) as
M m = m m
n m
= 1
n m
∑ N !
m i = ∑ N
n i M i
n
=
m
∑ N
χ i M i (12.11)
i=1 i=1 i=1
and using Eqs. (12.2), (1.9), and (12.3) as
M m = m m
n m
=
m m
∑ N
m i
i=1
M i
! =
∑ N
i=1
1
! =
ðm i /m m Þ
M i
1
∑ N
w i
i=1
M i
! (12.12)
Using Eqs. (12.11) and (12.12), we can now easily convert back and forth between mass and mole
fractions with
w i = m i
m m
=
n iM i
n m M m
= χ i
M i
M m
(12.13)