Modern Engineering Thermodynamics

414 CHAPTER 12: Mixtures of Gases and Vapors
Finally, the thermodynamic description of a mixture of ideal gases was completed through the work of Josiah
Willard Gibbs (1838–1903), who generalized Dalton’s law to define all the partial properties (except volume) of
the components in the mixture to be equal to the values that those properties would have if each component
gas alone occupied the volume of the mixture at the temperature of the mixture and at the partial pressure of
that component. The Gibbs-Dalton ideal gas mixture law presumesthatnomolecularinteractionstakeplace
between the components of the mixture, because each component is presumed to behave as though the other
components were not present.
Undertheseconditions,weconcludethatalltheextensiveproperties of a mixture of ideal gases are conserved, and the
mixture value of any extensive property can be determined by summing the contributions made by each gas present in the
mixture.
Therefore, for ideal gases only,
V m = ∑ N
i=1
V i
= ∑ N
and the specific volume of this mixture can be calculated from
v m = V m
m m
= ∑ N
i=1
i=1
m i v i (12.21)
m i
v i = ∑ N
w i v i (12.22)
m m
where w i is the mass fraction of ideal gas i in the mixture and v i is the specific volume of that gas determined at
thepressureandtemperatureofthemixture(i.e.,v i = R i T m /p m ). Table 12.5 lists the mass and molar equations
for all the total and specific properties of a mixture of ideal gases.
The mass fraction w i ,themolefractionχ i ,thevolumefractionψ i, and the pressure fraction π i make four
composition measures that can be used to describe a mixture. However, for ideal gases, there is a simple
relation between these four quantities. From Eqs. (12.16) and (12.18), we can write the pressure and volume
fractions as
and
i=1
π i = p i
=
m iR i T m /V m
= w iR i
= w iM m
= n i
= χ
p m m m R m T m /V m R m M i n i
m
ψ i = V i
=
m iR i T m /p m
= w iR i
V m m m R m T m /p m R m
= w iM m
M i
= n i
n m
= χ i
Table 12.5 Mass and Molar Total and Specific Thermodynamic Properties of Ideal
Gas Mixtures
Total Property Mass Specific Property Molar Specific Property
V m = ∑ N
V i
= ∑ N
m i v i
i=1 i=1
U m = ∑ N
U i = ∑ N
m i u i
i=1 i=1
H m = ∑ N
H i = ∑ N
m i h i
i=1 i=1
S m = ∑ N
S i = ∑ N
m i s i
i=1 i=1
v m = ∑ N
w i v i
i=1
u m = ∑ N
w i u i
i=1
h m = ∑ N
w i h i
i=1
s m = ∑ N
w i s i
i=1
c vm = ∑ N
w i c vi
i=1
c pm = ∑ N
w i c pi
i=1
v m = ∑ N
χ i v i
i=1
u m = ∑ N
χ i u i
i=1
h m = ∑ N
χ i h i
i=1
s m = ∑ N
χ i s i
i=1
c vm = ∑ N
χ i c vi
i=1
c pm = ∑ N
χ i c pi
i=1