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THERMODYNAMICS - THEORY

    The Ideal-gas 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 ideal-gas 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 = Ru/M

where
      Ru = universal gas constant, 8.314 kJ/(kmol-K)
      M = molar mass, the mass of one mole of a
            substance in grams

The ideal-gas equation of state can also be expressed as

      PV = mRT or PV = nRuT

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

      P1v1/T1 = P2v2/T2

     
    Equations of State for a Non-ideal Gas

   

The ideal-gas 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/v2)(v-b) = RT

      a = 27 R2(Tcr)2/(64Pcr)
      b = RTcr/(8Pcr)

where
       Tcr = critical temperature
       Pcr = 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.

Beattie-Bridgeman Equation of State:

The Beattie-Bridgeman equation of state was proposed in 1928. It has five experimentally determined constants.

      

Per unit mass Per unit mole
v (m3/kg) (m3/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 m3/kmol, T is in K and Ru is equal to 8.314 (kPa-m3) / (Kmol-K). The Beattie-Bridgeman equation of state is valid when ρ < 2.5 ρcr (critical density).

     
Gas Ao a Bo 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, CO2 507.2836 0.07132 0.10476 0.07235 660,000
Helium, He 2.1886 0.05984 0.01400 0.0 40
Hydrogen, H2 20.0117 -0.00506 0.02096 -0.04359 504
Nitrogen, N2 136.2315 0.02617 0.05046 -0.00691 4,2000
Oxygen, O2 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
     
   

Benedict-Webb-Rubin Equation of State:

Benedict, Webb, and Rubin raised the number of experimentally determined constants in the Beattie-Bridgeman 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 m3/kmol, T is in K and Ru is 8.314 (kPa-m3) / (Kmol-K). This equation of state is accurate when ρ < 0.8 ρcr.

     
Gas a A0 b B0 c×10-4 C0×10-5 α ×105 γ
n-Butane, C4H10 190.68 1021.6 0.039998 0.12436 3205 1006 110.1 0.0340
Carbon dioxide, CO2 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, CH4 5.00 187.91 0.003380 0.04260 25.78 22.86 12.44 0.0060
Nitrogen, N2 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 = vactual / videal

The compressibility factor (Z) is a measure of deviation from the ideal-gas 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 ideal-gas 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:

       PR = P/Pcr and TR= T/Tcr

where
       PR = the reduced pressure
       TR = 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 (PR << 1).
  • The gases can be assumed as an ideal gas with good accuracy regardless of pressure for high temperature (TR > 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 = (|vtable - videal|/vtable) (100%)