Modern Engineering Thermodynamics

3.9 Thermodynamic Equations of State 83
3.9.4 Real Gases
The Clausius equation of state accounts for the volume actually occupied by the gas molecules themselves. If we
let b represent the specific volume of the molecules themselves, then the Clausius equation of state is
pv− ð bÞ = RT (3.43)
In 1873, van der Waals included a second correction factor to account for the forces of molecular attraction.
These forces produce a net decrease in the observed pressure that is inversely proportional to v 2 .Thevander
Waals equation of state has the form
p + a
v 2 ðv − bÞ = RT (3.44)
The values of the molecular coefficients a and b in Eqs. (3.43) and (3.44) can be found in Table C.15 of
Thermodynamic Tables to accompany Modern Engineering Thermodynamics.
Other important real gas equations of state that are commonly used in engineering analysis are the Dieterici
equation,
pv− ð b
Þ = RT exp½− a/ ðRTvÞŠ (3.45)
and the Berthelot equation,
pðv − bÞ = RT − a
v − b
T v 2
(3.46)
But perhaps the most useful, best known, and most accurate equations of state for real gases are those of Beattie
and Bridgeman,
p =
1 − ε
v 2 ðv + BÞRT − A (3.47)
v 2
where
and Redlich and Kwong,
A = A 0 ð1 − a/v
Þ, B = B 0 ð1 − b/vÞ, and ε ¼ c
vT 3
pv− ð bÞ = RT − a
pffiffiffi
v − b
v T v + b
where A 0 , B 0 , a, b, and c are constants, whose values for various gases can be found in Table C.15.
A more general form for a real gas equation of state is a power series expansion such as
(3.48)
pv = RT + A v + B v 2 + C v 3 + … (3.49)
where A, B, C, … are all empirically determined functions of temperature. These equations are called virial expansions,
and the temperature dependent coefficients A, B, C, … are called the virial coefficients.
EXAMPLE 3.8
When an artillery cannon using a nitrocellulose propellant is
fired, a maximum temperature of 2830°C ismeasuredin
the breech behind the moving projectile. The density of the
propellant gases at this temperature is 200. kg/m 3 , and
the molecular mass of the propellant gases is 23.26 kg/kgmole.
The volume occupied by the molecules of the propellant gases
is b =0.960× 10 –3 m 3 /kg (Figure 3.23). Determine the maximum
pressure in the breech as the cannon fires.
FIGURE 3.23
Example 3.8.
(Continued )