Page 212 - Advanced thermodynamics for engineers

198 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

and
n
X
H ¼ nh m ¼ n i h m;i (9.64)
i¼1
If the molar internal energy, and molar enthalpy of the mixture are required then
n
n
1 X X
e m ¼ n i e m;i ¼ x i e m;i ; (9.65)
n
i¼1 i¼1
and
n
n
1 X X
h m ¼ n i h m;i ¼ x i h m;i (9.66)
n
i¼1 i¼1
Neglecting motion, gravity, electricity, magnetism and capillary effects, then e m ¼ u m ,and
hence
X
u m ¼ x i u m;i (9.67)
The definition of enthalpy for an ideal gas is
h m ¼ u m þ<T: (9.68)
Thus, for the mixture

X
h m ¼ x i ðu m þ<TÞ
i
X X X X
¼ x i u m;i þ x i <T ¼ x i u m;i þ<T x i (9.69)
X
¼ x i u m;i þ<T:

9.4.4 SPECIFIC HEATS OF MIXTURES
Statement (3) of the Gibbs–Dalton law and the above expressions show that (Eqn (9.67))
X
u m ¼ x i u m;i
P
And (Eqn (9.69)) h m ¼ x i h m;i
du dh
By definition, the specific heats are c v ¼ and c p ¼ .
Thus dT dT

du m X du m X
c v;m ¼ ¼ x i ¼ x i c v;m i (9.70)
dT dT
i
and, similarly,

X
c p;m ¼ x i c p;m (9.71)
i