Modern Engineering Thermodynamics

12.2 Thermodynamic Properties of Gas Mixtures 407
With the exception of system mass, the extensive properties of a system are not generally conserved in any
thermodynamic process, so that their mixture values are not normally equal to the sum of their constituent
values. The changes in extensive or intensive properties of a mixture with composition must always be determined
experimentally. However, the extensive properties of gases are mathematically homogeneous functions
of the first degree in mass. 2 For example, the total volume V m of a homogeneous mixture of gases can be
written as a function of the mixture total pressure p m , mixture temperature T m , and the mass composition of
the mixture m 1 , m 2 , … , m N as
V m = V m ðp m , T m , m 1 , …, m N Þ
and, when the mixture pressure and temperature are held constant, this is a homogeneous function of the
first degree in the masses m i . This means that, if all the remaining variables (the m i ) are multiplied by an
arbitrary constant λ, then the mixture volume also is multiplied by λ, or
λV m = V m ðp m , T m , λm 1 , …, λm N Þ
where λ is an arbitrary variable. Differentiating this equation with respect to λ while holding the pressure and
temperature constant and setting λ = 1gives
V m j pm,T m
= ∂V
m
m 1 + +
∂V m
m N = ∑ N
∂m 1
∂m N
i=1
m i^v i (12.5)
where
^v i =
∂V
m
(12.6)
∂m i p m,T m,m j
here ^v i is defined to be the partial specific volume of gas i in the mixture and V i = m i^v i = n i^v i is the partial
volume of gas i in the mixture. 3 Equation (12.5) leads us to a third common composition measure, the
volume fraction:
The volume fraction ψ i of gas i in a mixture of gases is
ψ i = V i
V m
(12.7)
Even though pressure is not an extensive property, the implicit function theorem from calculus tells us that, if
∂V m /∂p m ≠ 0 in Eq. (12.5), we can write the total pressure p m of a homogenous mixture of N gases as a function
of the mixture volume V m , mixture temperature T m , and the mass composition of the mixture m 1 , m 2 ,…, m N as
p m = p m ðV m , T m , m 1 , m 2 , :::, m N Þ
and, when the total volume and temperature of the mixture are constant, this too is a homogenous function of
the first degree in the masses m i . Following the development of Eq. (12.5), we can write
∂p m
p m = ∑m i
∂m i
V m,T m ,m j
= ∑ N
i=1
m i^p i = ∑p i = p 1 + p 2 + + p N (12.8)
where ^p i = ð∂p m /∂p i Þ V m,T m,m j
is the partial specific pressure of gas i in the mixture and p i = m i ð∂p m /∂p i Þ V m,T m,m j
is the
partial pressure of gas i in the mixture. Equation (12.8) provides our fourth common composition measure,
the pressure fraction:
The pressure fraction π i of gas i in a mixture of gases is
π i = p i
p m
(12.9)
2 See, for example, Kestin, J., 1979. A Course in Thermodynamics. Hemisphere Publishing Corporation, McGraw-Hill Book Company,
New York, vol. 1, pp. 326–327.
3 The concept of “partial properties” was introduced by Lewis, G. N., 1907. A new system of thermodynamic chemistry, in Proceedings
of the American Academy, 43, p. 273.