THERMODYNAMICS  THEORY



The Idealgas Equation of State



Any equation that relates the pressure, temperature,
and specific volume of a substance is called the equation of state. The
following equation is the idealgas equation of state. A gas that obeys
this relation is called an ideal gas.
Pv = RT
R is the gas constant, which
is determined from
R = R_{u}/M
where
R_{u} = universal gas constant,
8.314 kJ/(kmolK)
M = molar mass, the mass of one mole
of a
substance in grams
The idealgas equation of state can also be expressed as
PV = mRT or PV = nR_{u}T
where
m = mass of the gas
n = mole of the gas
For a fixed mass system (m = constant), the properties of an ideal gas
at two different states can be related as
P_{1}v_{1}/T_{1} =
P_{2}v_{2}/T_{2}






Equations of State for a Nonideal Gas



The idealgas equation of state is very simple, but its application
range is limited. The following three equations which are based on assumptions
and experiments can give more accurate result over a larger range.
Van der Waals Equation of State:
The Van der Waals equation of state was proposed in 1873, and
it states that
(P + a/v^{2})(vb) = RT
a = 27 R^{2}(T_{cr})^{2}/(64P_{cr})
b = RT_{cr}/(8P_{cr})
where T_{cr} = critical temperature
P_{cr} = critical pressure
Van der Waals equation of state is the first attempt to model the behavior
of a real gas. However, it is only accurate over a limited range.
BeattieBridgeman Equation of State:
The BeattieBridgeman equation of state was proposed in 1928. It has
five experimentally determined constants.

Per unit mass 
Per unit mole 
v (m^{3}/kg) 
(m^{3}/kmol) 
u (kJ/kg) 
(kJ/kmol) 
h (kJ/kg) 
(kJ/kmol) 
Properties Per Mass and Per Mole


The properties with a bar on top are molar basis.
The five constants can be found in the table below where P is in kPa, is in m^{3}/kmol, T is in K and R_{u} is equal to 8.314 (kPam^{3}) / (KmolK). The BeattieBridgeman equation of state is valid when ρ < 0.8 ρ_{cr }(critical
density).




Gas 
A_{o} 
a 
B_{o} 
b 
c 
Air 
131.8441 
0.01931 
0.04611 
0.001101 
43,400 
Argon, Ar 
30.7802 
0.02328 
0.03931 
0.0 
59,900 
Carbon dioxide, CO_{2} 
507.2836 
0.07132 
0.10476 
0.07235 
660,000 
Helium, He 
2.1886 
0.05984 
0.01400 
0.0 
40 
Hydrogen, H_{2} 
20.0117 
0.00506 
0.02096 
0.04359 
504 
Nitrogen, N_{2} 
136.2315 
0.02617 
0.05046 
0.00691 
42,000 
Oxygen, O_{2} 
151.0857 
0.02562 
004624 
0.004208 
48,000 
Source: Gordon J. Van Wylen and Richard E. Sonntag, Fundamentals of
Classical
Thermodynamics, 3rd ed., p46, table 3.3 





BenedictWebbRubin Equation of State:
Benedict, Webb, and Rubin raised the number of experimentally determined
constants in the BeattieBridgeman Equation of State to eight in 1940.
The constants appearing in the equation are given in the table below
where P is in kPa, is
in m^{3}/kmol, T is in K and R_{u} is 8.314 (kPam^{3}) / (KmolK).
This equation of state is accurate when ρ < 2.5 ρ_{cr}.




Gas 
a 
A_{0} 
b 
B_{0} 
c×10^{4} 
C_{0}×10^{5} 
α ×10^{5} 
γ 
nButane, C_{4}H_{10} 
190.68 
1021.6 
0.039998 
0.12436 
3205 
1006 
110.1 
0.0340 
Carbon dioxide, CO_{2} 
13.86 
277.30 
0.007210 
0.04991 
151.1 
140.4 
8.470 
0.0054 
Carbon monoxide, CO 
3.71 
135.87 
0.002632 
0.05454 
10.54 
8.673 
13.50 
0.0060 
Methane, CH_{4} 
5.00 
187.91 
0.003380 
0.04260 
25.78 
22.86 
12.44 
0.0060 
Nitrogen, N_{2} 
2.54 
106.73 
0.002328 
0.04074 
7.379 
8.164 
12.72 
0.0053 
Source: Kenneth Wark, Thermodynamics, 4th ed., p.141. 





Compressibility Factor

Generalized Compressibility Chart
Click to view large chart 

The compressibility factor (Z) is a dimensionless ratio
of the product of pressure and specific volume to the product of gas
constant and temperature.
Z = (Pv) / (RT)
or
Z = v_{actual} / v_{ideal}
The compressibility factor (Z) is a measure of deviation from the idealgas
behavior. For ideal gas, Z is equal to 1. Z can be either greater or
less than 1 for real gases. The further away Z is from unity, the more
the gas
deviates
from the idealgas behavior. 



The Generalized Compressibility Chart
Click to
view Movie (84 kB) 

The generalized compressibility chart is developed to be used for all gases. They are plotted as a function of the reduced pressure and reduced temperature, which are defined as follows:
P_{R} = P/P_{cr} and
T_{R}= T/T_{cr}
where
P_{R} = the reduced pressure
T_{R} = the reduced temperature




Percentage of error involved in assuming steam to be an ideal gas
Click to view Movie (68 kB) 

From the generalized compressibility chart, the following
observations can be made.
 The gases behave as an ideal gas regardless of temperature for very
low pressure (P_{R} << 1).
 The gases can be assumed as an ideal gas with good accuracy regardless
of pressure for high temperature (T_{R} > 2).
 In the vicinity of the critical point, the gases deviate from ideal
gas greatly.
The animation on the left shows
the
error involved in assuming steam to be an ideal gas. The red region
where steam can be treated as an ideal gas has error of less than 1%.
percentage of error = (v_{table}  v_{ideal}/v_{table})
(100%) 


