Modern Engineering Thermodynamics

12.3 Mixtures of Ideal Gases 413
Then, Eq. (12.6) can be used to find
^v i =
∂V
m
= R ∂m i p m,T m,m j
M i
Tm
p m
= R iT m
p m
= v i (12.16)
where v i is the specific volume of gas i at the pressure and temperature of the mixture. Similarly, using the
appropriate equations for the partial specific internal energy ^u i , enthalpy ^h i ,andentropy^s i , we can show that,
for a mixture of ideal gases with constant specific heats, the partial specific properties of gas i in the mixture are
the same as the corresponding specific properties; that is,
and
where T 0 and p 0 are arbitrary reference state values.
^u i = c vi ðT m − T 0 Þ
^h i = c pi ðT m − T 0 Þ
^s i = s i = c pi lnðT m /T 0 Þ − R i lnðp m /p 0 Þ
These relations were also discovered experimentally in the 19th century and are now known as the Gibbs-Dalton
and Amagat laws. In 1801, John Dalton (1766–1844) carried out a series of experiments that led him to conclude
that the total pressure p m of a mixture of ideal gases was equal to the sum of the partial pressures of the
individual component gases in the mixture, where the partial pressure p i of gas i in a mixture of ideal gases is
the pressure gas i would exert if it alone occupied the volume of the mixture at the temperature of the mixture.
This is known as Dalton’s law of partial pressures, and it can be written as
where
p m = ∑ N
i=1
p i = p 1 + p 2 + p 3 + + p N (12.17)
p i = m iR i T m
V m
(12.18)
Later, Emile Amagat (1841–1915) discovered experimentally that the total volume V m of a mixture of ideal
gases was equal to the sum of the partial volumes V i of the individual component gases in the mixture, where
the partial volume V i of gas i in a mixture of ideal gases is the volume gas i would occupy if it alone was at
thepressureandtemperatureofthemixture.ThisisknownasAmagat’s law of partial volumes, anditcanbe
written as
where
V m = ∑ N
i=1
V i = V 1 + V 2 + V 3 + + V N (12.19)
V i = m iR i T m
p m
or v i = R
iT m
p m
(12.20)
WHAT IS DALTON’S LAW OF PARTIAL PRESSURES?
Dalton’s law of partial pressures states that the partial pressure p i of ideal gas i in a mixture of ideal gases is equal to the
pressure gas i would exert if it alone occupied the volume of the mixture at the temperature of the mixture.
WHAT IS AMAGAT’S LAW OF PARTIAL VOLUMES ?
Amagat’s law of partial volumes states that the partial volume V i of ideal gas i in a mixture of ideal gases is equal to the
volume gas i would occupy if it alone was at pressure and temperature of the mixture.