Modified van der Waals Equation of State
SIAVASH H. SOHRAB
Robert McCormick School of Engineering and Applied Science
Department of Mechanical Engineering
Northwestern University, Evanston, Illinois 60208
UNITED STATES OF AMERICA
http://www.mech.northwestern.edu/dept/people/faculty/sohrab.html
Abstract: - A modified form of the van der Waals equation of state is presented that is valid for general values of
the critical compressibility factor. The resulting modified law of corresponding states is shown to lead to closer
agreement with the experimental data. The model also suggests a finite value of vacuum pressure p v, such that
the pressure of the anti-matter pam and the matter pm fields will be bonded from below and above by the pressure
of the white-hole pWH and the black-hole pBH, 0  p WH  pam  p v  pm  p BH   , that constitute two
singularities of the field.
Key-Words: -Thevan der Waals equation of state. Nature of anti-matter.
1 The Law of Corresponding States
The classical form of the van der Waals equation of
state
RT
a
p
 2
vb v
(1)
involves two constants a and b who’s value are
obtained on the basis of the criteria
 p / v T  0
,
  p / v 
2
2
T
0
(2)
that leads to the classical law of corresponding
states [1]
pr 
8Tr
3
 2
3v r  1 v r
(3)
Since the criteria (2) only involves derivatives of
pressure, one can consider a modified form of the
van der Waals equation of state, involving three
instead of two parameters, expressed as [7]
p
RT
a
 2 c
vb v
(4)
without influencing the values of the important
constants a and b. Applying the criteria (2), along
with the conditions Tr  pr  vr  1 at the critical
point lead to the modified form of the law of
corresponding state
pr 
1  Tr
9
3
 2  Zc  

Zc  vr  1/ 3 8v r
8
(5)
Therefore, the inclusion of the third constant allows
for the specification of the actual value of the
critical compressibility factor Zc of various nonideal fluids. The importance of Zc as the possible
third parameter in the law of corresponding state has
been emphasized by Bejan [1]. This is because
generally Zc of most fluids are different from Zc =
3/8 corresponding to the van der Waals fluid.
The law of corresponding state is expressed by
the generalized compressibility chart Z = pv/RT as a
function of pr with different values of Tr as a
parameter for 30 non-ideal fluids by Nelson and
Obert [2]. The data corresponding to average value
of Zc for 30 fluids is reproduced for fixed values of
reduced temperature Tr = 1.1, 1.2, 1.3, 1.5 by the
solid lines shown in Fig.1. The actual experimental
data for different fluid fall almost exactly on the
curves shown in Fig.1 and are most dramatically
shown in Fig.3.12 of Moran and Shapiro [3]. To
test the prediction of the van der Waals equation of
state (3), at any fixed value of Tr, for various pr the
experimental value of Z is obtained from the data in
Fig.1 [2] and used in prvr = ZrRTr to determine a
corresponding value of vr. The values of vr and Tr
are then used in (3) to calculate a value of pr and the
2
resulting calculated data points are also shown as
solid circles in Fig.1. The critical properties of
ethylene with Zc = 0.276 are used for these
calculations in view of available experimental data
at all Tr values shown in Fig.1 [3].
Z
1.5
1.3
1.2
T r = 1.1
The agreement of the modified theory in Fig.2 is
seen to be somewhat closer than that of the classical
theory shown in Fig.1.
Further systematic
examination of the modified form of the van der
Waals equation of state (4) for various fluids with Zc
substantially different from the average value
such as water, ammonia, hydrogen,
Zc  0.275
neon with Zc (0.229, 0.242, 0.305, 0.311) will be
interesting.
2 The Finite Pressure of Vacuum
It is noted that in the limit vr   , the modified
form of the van der Waals equation of state (4) leads
to negative values of pressure since the inequality
pr
Fig.1 Comparisons between experimental data
(solid lines) and the calculations (solid circles)
based on the van der Waals equation of state.
Next, to test the predictions of the modified form
of the van der Waals equation of state (4), one
introduces a constant value of the critical
compressibility factor Zc  0.275 that is based on
the data given in Table 6.2 of Bejan [1, 4]. Again,
at any fixed Tr, for various values of pr the
corresponding experimental values of
Z are
obtained from the data in Fig.1 [2] and the results
are used in prvr = Zr RTr to evaluate vr. The values
of (vr, Tr) are then introduced in (4) to calculate the
predicted value of pr.
The results of such
calculations based on critical properties of ethylene
at the reduced temperatures Tr = 1.1, 1.2, 1.3, and
1.5 are shown in Fig.2.
Zc < 3/8 = 0.375
appears to hold for most fluids [1]. However,
because thermodynamic pressure may be viewed as
the volumetric energy density of the field [5],
negative values of pressure are expected to be nonphysical. This in turn suggests that pr in (4) may in
fact be a gage pressure prg rather than an absolute
pressure pra that are related to the finite absolute
vacuum pressure prv by the expression
p rg  p ra  p rv , such that (4) becomes
prg  pra  prv 
1
Zc
 Tr
9
3
 2  Zc   (7)

8
 vr  1/ 3 8vr
Now, in the limit vr   since pra  0 , one
obtains from (7) a finite and positive vacuum
pressure of
p rv 
Z
(6)
3/8
1  0
Zc
(8)
1.5
1.3
1.2
T r = 1 .1
pr
Fig.2 Comparisons between experimental data
(solid lines) and the calculations (solid circles)
based on the modified van der Waals equation of
state.
A finite vacuum pressure is in harmony with the
well-known finite value of the zero-point energy [5].
It is then reasonable to suggest that the Dirac Sea
associated with anti-matter should correspond to
pressures lower than that of the vacuum [6]. Hence
the pressure of matter pm and anti-matter pam fields
will be respectively larger and smaller than vacuum
pressure pv [6]
0  p WH  pam  p v  pm  pBH  
(9)
3
and ultimately limited by the pressures of white hole
p WH  0 and black hole pBH   that are the two
singularities of the field. Under such a model, the
conservation of physical space i.e. vacuum [6-8],
requires a symmetry between matter particles and
their conjugate antimatter particles such that their
interactions leads to mutual annihilation and
generation of vacuum, thereby resolving the
flatness-paradox of cosmology.
Also, the
conservation of angular momentum requires that the
spin of matter versus their conjugate anti-matter
particles be reversed, thereby naturally accounting
for the time-reversal paradox.
REFERENCES
[1]
[2]
[3]
Bejan,
A.,.
Advanced
Engineering
Thermodynamics, Wiley, 1988.
Nelson, L. C., and Obert, E. F., Generalized
pvT properties of gases, Trans. ASME 76,
1057-1066 (1954).
Moran, M. J., and Shapiro, H. N.,
Fundamentals
of
Engineering
Thermodynamics, Wiley, 1988.
[4]
[5]
[6]
[7]
[8]
Reid, R. C., Prausnitz, J. M., and Sherwood,
T. K., The Properties of Gases and Liquids.
3rd ed., McGraw Hill, New York, 1977.
Sohrab, S. H., A scale-invariant model of
statistical mechanics and modified forms of
the first and the second laws of
thermodynamics. Rev. Gén. Therm. 38, 845854 (1999).
Sohrab, S. H., Stochastic definitions of Planck
and Boltzmann constants and quantum theory
of gravitation. Bull. Am. Phys. Soc. 46, No.2,
160 (2001).
Sohrab, S. H., Modified form of the van der
Waals equation of state Bull. Am. Phys. Soc.
48, No.1, 432 (2003).
Sohrab, S. H., Some thermodynamic
considerations on the physical and quantum
nature of space and times WSEAS
Transactions on Mathematics Issue 4, Vol.3,
764-772 (2004).