Page 210 - Advanced Thermodynamics for Engineers, Second Edition

9.4 MIXTURES OF IDEAL GASES 197

But, for the mixture as a whole
pV ¼ n<T
Hence, by dividing Eqn (9.50) by Eqn (9.2)

p a n a
¼ ¼ x a ; (9.55)
p n
giving
p a ¼ x a p; and in general p i ¼ x i p (9.56)

and, also
n n n
X X X
p i ¼ x i p ¼ p x i ¼ p (9.57)
i¼1 i¼1 i¼1
Often mixtures are analysed on a volumetric basis, and from the volumetric results it is possible to
obtain the partial pressures and molar fractions of the components. Normally volumetric analyses are
performed at constant pressure and temperature. Then, considering the ith component

n i <T
V i ¼ ; (9.58)
p

and the total volume of the mixture, V,is
n<T
V ¼ : (9.59)
p
Hence, from Eqns (9.58) and (9.59)
V i n i
¼ ¼ x i ; (9.60)
V n
If the gas composition is given in volume percentage of the mixture, the molar fraction is

V i ð%Þ
x i ¼ ; (9.61)
100
and the partial pressure is

V i ð%Þ
p i ¼ p: (9.62)
100
The terms relating to the energy of a mixture can be evaluated from statement (3) of the
Gibbs–Dalton laws. If e m ¼ molar internal energy, and h m ¼ molar enthalpy, then for the mixture

n
X
E ¼ ne m ¼ n i e m;i ; (9.63)
i¼1