Modern Engineering Thermodynamics

386 CHAPTER 11: More Thermodynamic Relations
Substituting this formula for a and b back into the van der Waals equation and dividing both sides by p c gives
p/p c = p R = 8T R /ð3v R − 1Þ − 3/v 2 R
which is the type of dimensionless equation of state that he was seeking. Unfortunately, the molecular interactions
in real substances are more complex than the elementary corrections used in the van der Waals equation,
so it does not work well over large pressure and temperature ranges.
A serious flaw in van der Waals’s law of corresponding states is that it breaks down at low pressures, where ideal
gas behavior is expected. Further, his equation of state (Eq. (3.44)) has the curious property that (from the
preceding equation for b) theratiop c v c /(RT c ) = 3/8 = 0.375, whereas for an ideal gas, pv/(RT) is always equal
to unity. This similarity led many researchers to investigate pv/(RT) data, and this grouping is now called the
compressibility factor Z, as
Z = pv/ðRTÞ (11.39)
where Z = 1 for an ideal gas. Figure 11.4 shows that the experimental data for the compressibility factor for
many different gases fall together when Z is plotted against reduced pressure p R while holding the reduced temperature
T R constant. Figures 11.4, 11.5, 11.6, and 11.7 constitute a set of compressibility charts that can be used to
solve real gas compressibility factor problems.
The van der Waals equation predicts that Z c = p c v c /(RT c ) = 0.375, but many experiments on a large number of
substances have shown that 0.23 ≤ Z c ≤ 0.375. So Z c is not the same for all substances. Though many people
tried to correlate Z with p or T for various substances, it was not until 1939 that H. C. Weber correlated Z with
p R and T R and thus produced the first generalized compressibility chart of the form Z = Z(p R , T R ). There was,
however, a problem with this chart, in that lines of constant reduced specific volume could not be added
because the v R data were inconsistent. In 1946, Gouq-Jen Su solved this problem, as shown in Figure 11.5, by
choosing the product v R Z C as a “pseudo” reduced specific volume v′ R , defined as
v R ′ = v R Z c = ðv/v c ÞZ c = v/v′ c
where a new critical state specific volume v c ′ has been defined as
v c ′ = v c /Z c = RT c /p c
This change produced a much better correlation of the experimental data and lines of constant v′ R could be accurately
added to the chart. The resulting Z = Zðp R , T R , v′ R Þ plot is now called the generalized compressibility chart and is
shown in Figures 11.5, 11.6, and 11.7. Values for p c and T c for various substances can be found in Table C.12.
1.1
Compressibility factor, Z = pv/RT
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.00
1.10
1.20
T R = T/T c = 2.00
1.30
1.50
Legend
Methane Iso-pentane
Ethylene N-heptane
Ethane Nitrogen
Propane Carbon dioxide
N-butane Water
Average curve based on
data on hydrocarbons
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Reduced pressure, p R = p/p c
FIGURE 11.4
A generalized compressibility chart for various gases. (Sources: Reprinted with permission from Su, G.-J. “Modified law of corresponding
states for real gases.” Ind. Eng. Chem. 38 (8), 1948, p. 804. Also reprinted with permission of the publisher from Reynolds, W. C.,
Perkins, H. C., 1977. Engineering Thermodynamics. McGraw-Hill, New York.)