Page 210 - Advanced thermodynamics for engineers

196 CHAPTER 9 THERMODYNAMIC PROPERTIES OF IDEAL GASES

2. The total pressure exerted by a mixture is the sum of the pressures exerted by the
individual components as each occupies the whole volume of the mixture at the same
temperature.
3. The internal energy, enthalpy and entropy of the mixture are respectively equal to the sums of the
internal energy, enthalpy and entropy of the various components as each occupies the whole
volume at the temperature of the mixture.

9.4.3 MIXTURE RELATIONSHIPS
The total mass, m, of a mixture can be related to the mass of constituents by
n
X
m ¼ m a þ m b þ m c þ . ¼ m i (9.49)
i¼1
From the ideal gas law (Eqn (9.2))
m
pV ¼ mRT ¼ m w RT ¼ n<T
m w
and hence for the individual constituents
pV a ¼ m a R a T ¼ n a <T;

pV b ¼ m b R b T ¼ n b <T; (9.50)

pV c ¼ m c R c T ¼ n c <T; etc:
In Eqn (9.50) V a ¼ volume of constituent a at the pressure and temperature of mixture, and V b and
V c are similar volumes for constituents b and c. But the total volume, V, is given by
n
X
V ¼ V a þ V b þ V c þ . ¼ V i ; (9.51)
i¼1
n
X
and therefore n ¼ n a þ n b þ n c þ . ¼ n i (9.52)
i¼1
n a
Let ¼ x a ¼ molar fraction of a in the mixture: (9.53)
n
Similarly, n b /n ¼ x b and n c /n ¼ x c , etc, with n i /n ¼ x i .

n
X
Therefore x a þ x b þ x c þ . ¼ x i ¼ 1 (9.54)
i¼1
It is possible to develop the term for the partial pressure of each constituent from statement (2) of
the Gibbs–Dalton laws. Then, for constituent a occupying the total volume of the mixture at the
pressure and temperature of the mixture

p a V ¼ n a <T (9.50)