# Summary Notes for Chapter 13

```To MEEG344 students
Summary Notes for Chapter 13: Gas Mixtures
Prof. Lian-Ping Wang, University of Delaware
A. General definitions and notations
The composition of a gas mixture can be specified either by mole fractions yi or by mass fractions mfi of
the components,
Ni
,
Nm
yi 
mi
;
mm
mfi 
where the subscript “i” denotes a property associated with the ith component in the mixture and the
subscript “m” for the mixture as a whole. N represents the mole number and m the mass. By definition,
y
i
 mf
 1,
i
 1.
Since mi  N i M i ( M i is the molar mass for the i component), the molar mass of the mixture can be
defined, similar to a single-component gas, as
th
Mm 
mm
  yi M i ,
Nm
and the specific gas constant of a mixture can be calculated, similar to a single-component gas, as
Rm 
Ru
,
Mm
where Ru = 8.314 kJ/kmol.K is the universal gas constant. In terms of M m , mass and mole fractions can
be related as
mfi  yi
Mi
.
Mm
Our overall goal is to find ways to compute the properties of the mixture in terms of known properties of
the components (when each exists alone as a pure substance).
B. The equation of state for a mixture
The equation of the state for an ideal-gas mixture remains unchanged if the mixture mole number N m
and the mixture specific gas constant Rm are used, namely,
PmVm  N m RuTm  mm RmTm .
The equation of the state for a real-gas mixture can be formally written as,
PmVm  Z m N m RuTm  Z m mm RmTm .
The question is how to determine the apparent compressibility factor Z m for the mixture. We first observe
that the following statements are true at least for an ideal-gas mixture,
Pm 
N m RuTm

Vm
Vm 
N m RuTm

Pm
N R T
i
u m
Vm
N R T
i
Pm
u m

N i Ru Tm
  Pi (Tm ,Vm )
Vm
 Dalton' s law

N i RuTm
  Vi (Tm , Pm )
Pm
 Amagat' s law
where the component pressure Pi (Tm ,Vm ) is the pressure that one would realize if the ith component were
to exist alone at Tm and Vm ; the component volume Vi (Tm , Pm ) is the volume that the ith component
would take if it alone exists at Tm and Pm . For an ideal-gas mixture, the component pressure is equal to
the partial pressure ( yi Pm ) , and the component volume is equal to the partial volume ( yiVm ) .
If we assume that the above laws are approximately valid for a real-gas mixture, then we have two ways
of determining the compressibility factor of a mixture:
PmVm
Vm
V
  Pi (Tm ,Vm )
  yi Z i (Tm , m )
N m RuTm
N m RuTm
mi
PV
Pm
Based on the Amagat' s law Z m  m m  Vi (Tm , Pm )
  yi Z i (Tm , Pm )
N m RuTm
N m RuTm
Based on the Dalton' s law Z m 
The third way is to treat the mixture as if it is a pure substance with the following pseudocritical
temperature and pseudocritical pressure:
Z m  Z (TR 
Tm
P
, PR  ' m )
'
Tcr , m
Pcr , m
wher e Tcr' , m   yiTcr ,i and
Pcr' , m   yi Pcr ,i
 Kay' s Rule
This third method is known as the Kay’s rule.
C. Properties of an ideal-gas mixture
Here u m , hm , Cv , m , and C p , m are only function of the mixture temperature Tm and, therefore,
um   mfi ui (Tm ), hm   mfi hi (Tm ), Cv,m   mfi Cv,i (Tm ),
or for molar analysis,
um   yi ui (Tm ), hm   yi hi (Tm ), Cv,m   yi Cv,i (Tm ),
C p,m   mfi C p,i (Tm )
C p,m   yi C p,i (Tm )
The mixture entropy change, however, should be determined, as
Tm , 2 C p ,i (T )

y P 
y P 
sm   mfi  
dT  Ri ln i , 2 m, 2    mfi si0 (Tm, 2 )  si0 (Tm,1 )  Ri ln i , 2 m, 2 
yi ,1Pm,1 
yi ,1Pm,1 
Tm ,1 T


where the subscripts “1” and “2” refer to the initial and final state, respectively. For molar analysis,
Tm , 2 C p ,i (T )

y P 
y P 
sm   yi  
dT  Ru ln i , 2 m, 2    yi si0 (Tm, 2 )  si0 (Tm,1 )  Ru ln i , 2 m, 2  .
yi ,1Pm,1 
yi ,1Pm,1 
Tm ,1 T


The important point is that the component entropy change is computed with the mixture temperature Tm
and the component partial pressure ( yi Pm ) (i.e., the spirit of Dalton’s law).
D. Properties of a real-gas mixture
In this case, the departures from the ideal gas behavior need be determined. For example, the enthalpy
change of a real-gas mixture can be determined as
hm   mfi hi   mfi hi (Tm )ideal  RiTcr ,i Zh,i (Tm, 2 , Pm, 2 )  Zh,i (Tm,1, Pm,1 )
and


hm   yi hi   yi  hi (Tm ) ideal  RuTcr ,i Zh,i (Tm, 2 , Pm, 2 )  Zh,i (Tm,1, Pm,1 )
And similarly, the entropy change can be determined as
sm   mfi si   mfi si ideal  Ri Z s,i (Tm, 2 , Pm, 2 )  Z s,i (Tm,1, Pm,1 )
and
sm   yi si   yi si ideal  Ru Z s ,i (Tm, 2 , Pm, 2 )  Z s,i (Tm,1, Pm,1 )
The important point to note is that the component departure factors should be evaluated at the mixture
temperature Tm and the mixture pressure Pm (i.e., the spirit of Amagat’s law). The ideal-gas contributions
are computed following Part C above.
Alternatively, the mixture may be treated as a pseudo pure substance with pseudocritical properties given
by the Kay’s rule.
```