Modern Engineering Thermodynamics

408 CHAPTER 12: Mixtures of Gases and Vapors
Table 12.1 summarizes the four composition measures thus far defined.
Dividing both sides of Eq. (12.5) by the mixture mass m m produces the specific volume of the mixture as
v m = V m
m m
= ∑ N
i=1
m i
m m
^v i = ∑w i^v i
where w i is the mass fraction m i /m m . The concepts of total, specific, and partial specific properties can
be extended to other extensive properties, as shown in Table 12.2. On a molar basis, total, molar specific, and
partial molar specific properties are defined in Table 12.3.
Finally, the constant volume and constant pressure specific heats for the mixture are defined as
c vm = ∂u
m
and c pm = ∂h
m
∂T m v m
∂T m
These specific heats can also be written on a molar basis and in terms of the partial specific heats of the component
gases, as shown in Table 12.4.
The mass, mole, volume, and pressure composition fractions have the characteristic that they always sum to
unity; that is,
v m
∑ N
w i = ∑ N
χ i = ∑ N
Ψ i = ∑ N
π i = 1:0 (12.10)
i=1 i=1 i=1 i=1
Table 12.1 Four Composition Measures
Name of Composition Fraction
Defining Equation
Mass fraction w i
Mole fraction χ i
Volume fraction ψ i
Partial pressure fraction π i
w i = m i /m m
χ i
= n i /n m
ψ i = V i / V m
π i = p i /p m
Table 12.2 Total, Specific, and Partial Specific Properties of a Mixture of Gases
Total Property of the Mixture
V m = ∑ N
m i^v i
i=1
U m = ∑ N
m i^u i
i=1
H m = ∑ N
m i^hi
i=1
S m = ∑ N
m i^s i
i=1
Partial Specific Property
Specific Property of the Mixture of Gas i in the Mixture
v m = V m /m m = ∑ N
mi ^v i = ∑ N
w
^v
mm i^v i = ∂V
m
i ∂m i pm,Tm,mj
i=1 i=1
u m = U m /m m = ∑ N
mi ^u i = ∑ N
w
^u
mm i^u i = ∂U
m
i ∂m i pm,Tm,mj
i=1
i=1
h m = H m /m m = ∑ N
mi ^h
mm i = ∑ N
^hi
w
= ∂H
m
i^hi ∂m i pm,Tm,mj
i=1
i=1
s m = S m /m m = ∑ N
mi ^s i = ∑ N
w
^s
mm i^s i = ∂S
m
i ∂m i
i=1
i=1
pm,Tm,mj
CRITICAL THINKING
Is it possible to define a partial temperature of gas i in a mixture of N gases? Since the mixture temperature varies inversely
with the system mass in most equations of state for gases, is temperature a homogeneous function of the first degree in the
masses m i ? Could the partial temperature of gas i in the mixture be the temperature exhibited by gas i when it alone occupies
the volume of the mixture Vm at the pressure of the mixture p m ? (See Problem 67 at the end of this chapter.)