1. 
Abstract— The aim of this paper is to review the Equations of
state and pay particular attention to phase equilibria modeling.
A number of equations of state will be considered and the
strengths, weaknesses and their applicability will be assessed.
The culminated intention is to use the Peng Robinson equation
of state to compute the thermodynamic interactions of volatile
organic compounds in biodiesel polymers. Thermodynamic
models are used to simulate vazspor liquid equilibria, and this
knowledge on phase equilibrium behavior plays a fundamental
role in the prediction of the solubility of the components found
in biodiesel. This aids in the separation process of the
individual components, normally referred to as volatile organic
compounds. It is crucial to use models in the determination of
these equilibriums since actual measurements are costly and
lingering.
Keywords— Activity coefficients, biodiesel, phase equilibrium,
Peng Robinson, Wong Sandler.
I. INTRODUCTION
New legislation on environmental conservation has forced
industries to closely monitor any effluence emitted to the
environment. Actual measurements of the levels of pollution
of such effluence may be time consuming and costly. This is
where thermodynamic models, which calculate phase
equilibrium data, come-in to help simulate actual situations.
A.Brief History on the development of Equation of states.
In 1662, the noted Irish physicist and chemist Robert Boyle
performed a series of experiments employing a J-shaped glass
tube, which was sealed on one end.[2] Mercury was added to
the tube, trapping a fixed quantity of air in the short, sealed
end of the tube. The volume of gas was measured as additional
mercury was added to the tube. The pressure of the gas could
be determined by the difference between the mercury level in
the short end of the tube and that in the long, open end.
S.Ramdharee is with the Department of Chemical Engineering, Faculty of
Engineering and the Built Environment, University of Johannesburg,
Auckland Park, Johannesburg 2028 (e-mail: sashay.ramdharee@sasol.com;
sashayr007@gmail.com )
E. Muzenda is with the Department of Chemical Engineering, Faculty of
Engineering and the Built Environment, University of Johannesburg,
Doornfontein, Johannesburg 2028; phone: 0027-11-5596817; fax: 0027-11-
5596430; e-mail: emuzenda@uj.ac.za )
Through these experiments, Boyle noted that the gas volume
varied inversely with the pressure. In mathematical form, this
can be stated as:
constant=pV
(1)
p = Pressure; V = Volume.
In 1787, Charles postulated that the volume of a gas is
proportional to its temperature at isobaric conditions. In 1801,
Dalton introduced the concept of partial pressures and
recognized that the total pressure of a gas is the sum of the
individual (partial) contributions of its constituents. In 1802,
Gay-Lussac helped to define the universal gas constant “R”.
Gay-Lussac showed that a single constant applied to all gases,
and calculated the “universal” gas constant.
In 1873, Van der Waals quantitative approach proposed the
continuity of gases and liquids that won him a Nobel Prize and
it is for this reason that he has provided the most important
contribution to EOS development. In 1875 Gibbs, an
American mathematical physicist, made the most important
contribution to the thermodynamics of equilibrium in what has
been recognized as a monumental work.
In 1927, Ursell proposed a series solution (polynomial
functional form) for EOS:
...+
d
+
cb
+1=P 32
VVV
 (2)
This is known as the virial EOS. The virial EOS has better
theoretical foundation than any other EOS. However, cubic
EOS (as vdW’s) need only two parameters and has become
more widespread in use.
In 1949, Redlich & Kwong introduced a temperature
dependency to the attraction parameter “a” of the vdW EOS.
In 1955, Pitzer introduced the idea of the “acentric factor” to
quantify the non-sphericity of molecules and was able to relate
it to vapor pressure data. In 1972, Soave modified the Redlich
& Kwong EOS by introducing Pitzer’s acentric factor.
In 1976, Peng and Robinson proposed their EOS as a result of
a study sponsored by the Canadian Gas Commission, in which
Review of the Equations of state and their
applicability.
Sashay Ramdharee, Prof. Edison Muzenda

2. the main goal was finding the EOS best applicable to natural
gas systems [2].
Since then, there has not been any radical improvement to
SRK and PR EOS, although a great deal of work is still
underway.
B. The definition of an Equation of state.
An Equation of State (EOS) is a semi-empirical functional
relationship between pressure, volume and temperature of a
pure substance. More specifically, an equation of state is a
thermodynamic equation describing the state of matter under a
given set of physical conditions. Nowadays, most applicable
EOS’s are semi-empirical, in the sense that they have some
hypothetical basis but their parameters (ak) must be adjusted.
Furthermore, equations of state are generally developed for
pure substances. Their application to mixtures requires an
additional variable (composition) and hence an appropriate
mixing rule.
The functional form of an EOS can be expressed as:
0=)n1,=k,aT,V,f(P, pk (3)
ak = EOS parameters. The number of parameters (np)
determines the category/intricacy of the EOS. For instance, 1-
parameter EOS are those for which np = 1, and 2-parameter
EOS are those for which np = 2. The higher the “np” number,
the more complex the EOS. Also, in general terms, the more
complex the EOS, the more accurate it is. At present, there is
no single equation of state that accurately predicts the
properties of all substances under all conditions. [1]
Since the time of the ideal gas law (ideal gas EOS), a great
number of equations of state have been proposed to describe
real gas behavior. Since the introduction of the van der
Waals EOS, many cubic EOS’s have been proposed like the
Redlich and Kwong EOS (RK EOS) in 1949, the Peng and
Robinson EOS (PR EOS) in 1976, to name only a few. Most
of these equations retain the original van der Waals repulsive
term 2
b)-(V
RT
, modifying only the denominator in the
attractive term. Conversely, many of those have not passed the
test of time. In the petroleum business, the most common
modern EOS are the Peng Robinson EOS (PR EOS) and
Soave-Redlich-Kwong EOS (SRK EOS).[3]
The main purpose for the development of the EOS approach
was for generating of data which included volumetric, thermo
physical data, and to help perform vapor/liquid equilibrium
calculations. [3]
Some of the properties derived from an EOS include:
 Densities (vapor and liquid),
 Vapor pressures of pure components,
 Critical pressures and temperatures for the mixture,
 Vapor-Liquid equilibrium (VLE) information,
 Thermodynamic properties (ΔH, ΔS, ΔG, ΔA).
The advantage of the equations of state method is its
applicability over wide ranges of temperature and pressure to
mixtures of diverse components, from the light gases to heavy
liquids. They can be used for the representation of vapor-
liquid, liquid-liquid and supercritical fluid phase equilibria and
they can be also applied to the gas, liquid and supercritical
phases without encountering any conceptual difficulties. Many
equations of state have been proposed in the literature with
either an empirical, semi-empirical or theoretical basis.
II.DISCUSSION
A.Van der Waals Equation of state
Most of the EOS being used widely today for practical design
purposes has been derived from vdW EOS.
The contributions of vdW EOS can be summarized as follows:
 It laid foundations for modern cubic EOS.
 It radically improved predictive capability over ideal
gas EOS.
 It formulated the Principle of Corresponding States.
 It was the first to predict continuity of matter between
gas and liquid.
In his PhD thesis in 1873, vdW accounted for the non-zero
molecular volume and non-zero force of attraction of a real
substance. Until his publications, the accepted description of
gas behavior stemmed from the Ideal Gas Law that proposed a
relationship between the pressure, volume, and temperature of
a closed system. This equation, however, conformed to
experimental data only at low pressure and high temperatures.
It does so for two reasons: the ideal gas law does not take into
consideration a particle's finite volume or inter-particle
interactions. Therefore, the need to account for the gas
particle's volume became clear to Mr. Van der Waals.
One of the first things vdW recognized is that molecules must
have a finite volume. At the same time, he modified the
pressure term to acknowledge the fact that molecules do
interact with each other though cohesive forces. [4] The
prediction of liquid behavior is more accurate because volume

3. approaches a limiting value, b, at high pressures, and can be
illustrated as:
b=V(p)limp  (4)
When particles are in close proximity at high pressure, they are
very likely to interact with each other in some shape or form
due to attractive and repulsive electrostatic forces. Coulomb’s
law governs the forces between isolated charged particles. Van
der Waals hypothesized that the molecules experience a
cohesive force that attracts them to each other. It is for this
reason that the observed pressure is reduced, since if the
particles attracted each other they would hit the walls with less
momentum and thus less force. More particles near the walls
would mean more interactions and so this correction factor for
pressure must be dependent upon the concentration of gas
(N/V) in some manner. The exact form of this correction factor
comes from the assumption that particle interactions occur in
pairs. The vdW EOS gives a simple qualitatively accurate
relation between pressure, temperature and molar volume.
Finally, with both corrections to the ideal gas law taken into
account, the Van der Waals equation of state can be written as:
2
V
a
-b)-RT/(v=P (5)
P = absolute pressure, v = molar volume, T = absolute
temperature, a = attraction parameter, R = universal gas
constant, b = repulsion parameter.
The Van der Waals equation gives a qualitative description of
the vapor and liquid phases, but it is rarely sufficiently
accurate for phase equilibria calculations. The Van der Waals
equation has been superseded by a large number of other more
accurate equations of state.
B.Redlich Kwong Equation
In 1949, after the van der Waals equation was invented, the
Redlich-Kwong equation of state was conceived. It was
formulated by Otto Redlich and Joseph Neng Shun Kwong in
1949. [5] The Redlich Kwong is a cubic equation of state
loosely modeled after the van der Waals equation. The
Redlich-Kwong equation of state can be used for mixtures by
applying mixing rules to the equation of state parameters. It
offered remarkable success in improving the Van der Waals
equation with a better description of the attractive term.
Redlich and Kwong developed an adjustment in the van der
Waal’s attractive pressure term (a/V^2) which could
considerably improve the prediction of the volumetric and
physical properties of the vapor phase. The Redlich-Kwong
equation has been used extensively, and has been the basis for
several successful modified forms of it.
The advantages of the Redlich-Kwong equation are that it is a
relatively simple equation, and that it does not require much
information about the substance or the components of a
mixture. The disadvantages are that the equation cannot be
used for liquid phase systems or vapor-liquid equilibria, but
only for gases. [6] The basic form of the Redlich-Kwong
equation is:
b))+v(vT(
a
-
b)-(v
RT
=P (6)
The constants a and b can be given in terms of critical
conditions, Tc and Pc, and the universal gas constant, R.
)/PT*(R0.42748=a c
2.5
c
2
(7)
/PRT*0.08664=b cc
(8)
The Redlich-Kwong equation is better than the van der Waals
equation above the critical point. It is almost always more
accurate than the van der Waals equation and often more
accurate than some equations with more than two parameters.
The equation can also be written as:




bN)+(VT
Na-
bN)-(V
NRT
=P 1/2
2
(9)
Naturally, Redlich and Kwong did not have the last word on
possible improvements to the vdW EOS. The Redlich-Kwong
EOS, as shown here, is no longer used in practical
applications. Research continued and brought with it new
attempts to improve the RK EOS. After more than two
decades, a modified RK EOS with very good potential was
developed. This lead to the development of the Soave-RK
EOS.
C.Soave-Redlich Kwong Equation
In 1972, Soave proposed an important modification to the
Redlich-Kwong Equation of State (EOS). A new concept for
fluid characterization was being discussed. In contrast to the
original Redlich-Kwong equation, Soave’s modification fitted
the experimental (vapor-liquid) curve well and it was able to
predict the phase behavior of mixtures in the critical region. It
is generally more accurate than the van der Waals equation
and the ideal gas equation at temperatures above the critical
temperature. Between the time of vdW EOS and Redlich-
Kwong’s, a new concept for fluid characterization was being
discussed. All modifications to the vdW EOS had focused on
the temperature dependency of the attractive parameter. Soave
expanded this by proposing a two-variable dependency for
“a”:
)a(T,a  (10)
This was in fact very different as “a” was expressed not only
as a function of temperature, but also as a function of the shape

4. (sphericity) of the molecules (through  , Pitzer’s acentric
factor). [7] Pitzer had introduced the concept of acentric
factor in 1955. Pitzer’s acentric factor is a measure of the
configuration and sphericity of the molecule. Like all cubic
equations of state, the Redlich-Kwong-Soave EOS is also
explicit in pressure. The Soave-Redlich-Kwong single
equation of state can accurately describe both the liquid and
vapor phase. The Redlich–Kwong equation has undergone
many revisions and modifications, in order to either improve
its accuracy in terms of predicting gas-phase properties of
more compounds, as well as in better simulating conditions at
lower temperatures, including vapor-liquid equilibria.
b)+v(v
a
-
b)-(V
RT
=P
 (11)
2
r )](T-S(1+[1= (12)
Where Tr is the reduced temperature,
0.176-1.574+0.48=S 2
 (13)
 = accentric factor,
The major drawback in the SRK EOS is that the critical
compressibility factor takes on the unrealistic universal critical
compressibility of 0.333 for all substances. [7] Consequently,
the molar volumes are typically overestimated, i.e., densities
are underestimated. . Another major drawback of the SRK
EOS was the poor liquid density prediction.
D.Peng Robinson Equation
The Peng-Robinson equation was developed in 1976 to handle
both vapor and liquid properties near the equilibrium point.
The Peng-Robinson EOS has become the most popular
equation of state for natural gas systems in the petroleum
industry. During the decade of the 1970’s, D. Peng was a PhD
student of Prof. D.B. Robinson at the University of Alberta
(Edmonton, Canada). The Canadian Energy Board sponsored
them to develop an EOS specifically focused on natural gas
systems. [9] For the most part the Peng-Robinson equation
exhibits performance similar to the Soave equation, although it
is generally superior in predicting the liquid densities of many
materials, especially non-polar ones. A slightly better
performance around critical conditions makes the PR EOS
somewhat better suited to gas/condensate systems.
v))-(bb)+(v(v
a
-
b)-(v
RT
=P
b
 (14)
Peng and Robinson conserved the temperature dependency of
the attractive term and the acentric factor introduced by Soave.
However, they presented different fitting parameters to
describe this dependency, and further manipulated the
denominator of the pressure correction (attractive) term.
)/PT*(R0.45724=a c
2
c
2
(15)
2
c
)]
T
(-S(1+[1=
T
 (16)
0.26992-1.54266+0.37464=S 2
 (17)
/PRT*0.07780=b cc
(18)
Two years later, Peng and Robinson (1978) presented the
following modification to be used for ω > 0.49:
0.016660.1644-1.4850+0.3796=S 32
  (19)
This equation provides a more accurate vapor pressure
prediction for pure heavy hydrocarbons over wide range of
temperature.
Alternatively, the Peng-Robinson equation of state is given by:
b-2b1
a
-
)b-(1
NRT
=P 22
2





(20)
P = pressure,  = molar density, R = gas constant, T =
absolute temperature, a, b = coefficients that can be calculated
knowing the critical temperature Tc and pressure Pc of the gas
under pressure, and are given by the following relationships.
caa  (21)
)/PT*(R0.45724=a c
2
c
2
c
(22)
2
r )](T(1+[1=  (23)
T
T
T
c
r  (24)
/PRT*0.07780=b cc
(25)
2
0.26992-1.54226+0.37464=  (26)
Note that is the acentric factor and relates to the
compressibility of the gas. The Peng-Robinson and Soave-
Redlick-Kwong equations are widely used in industry. The
advantages of these equations are that they are easy to use and
that they often accurately represent the relation between
temperature, pressure, and phase compositions in binary and
multicomponent systems. These equations only require the
critical properties and the acentric factor for the generalized
parameters. Little computer resources are required and those
lead to good phase equilibrium correlation. However, the
success of these modifications is restricted to the estimation of
phase equilibria pressure. The calculated saturated liquid
volumes are not improved and they are invariably higher than
the measured data.

5. E. The General Equation of State.
First it is noted that the user-defined equations of state with,
T),(  and T),(Cp 
Noting that:
pT
T
v
Tv
p
h















 (27)
Where,

1
v
The equation of state for enthalpy therefore follows as:
dp
T
v
TvdTCh
p
p

















(28)
Note, however, that user-defined expressions for  and pC
must be thermodynamically consistent. Consistency requires
that mathematical properties for exact differentials be satisfied.
For example, suppose dz(x,y) is an exact differential defined
as:
NdyMdxyxdz ),( (29)
yx
z
M 







 (30)
X
y
z
N 







 (31)
Consistency then requires that:
YX
x
N
y
M















 (32)
By application of this concept it therefore follows that general
equations of state must obey:
p
p
T
v
Tv
Tp
C


























 (33)
F. The Modern cubic Equation of state.
Cubic equations of state are equations, which when expanded
have volume terms that are raised to the first, second, and third
power. Most commonly encountered phase equilibrium
calculations, such as vapor-liquid equilibria, involve only two
phases for which a cubic equation is suitable. Cubic equations
have the advantage that the three values of volume can be
obtained analytically without the need for an iterative solution
procedure. If we multiply the vdW EOS by
2
v and expand the
factorized product, the result is the vdW EOS expressed in
terms of molar volume, as follows:
0
ab
-v
a
v
RT
v 23




















b (34)
Furthermore, we can substitute the definition of
compressibility factor Z, into equation (34), as follows:
RT
PV
Z  (35)
A cubic polynomial in Z is obtained, as shown:
0
abP
-Z
aP
Z
bP
1Z 3
2
22
23



















RTTRRT
(36)
As we see, vdW EOS is referred to as cubic because it is a
polynomial of order 3 in molar volume (and hence in
compressibility factor Z). In general, any equation of state that
is cubic in volume (and Z) and explicit in pressure is regarded
as a cubic equation of state [10]. All cubic equations of state
have their foundation in Van de Waals EOS. The advantages
are outlined as follows:
 Simplicity of application
 Only a few parameters need to be determined
 Low computational overhead is required to
implement them.
Firstly it can be noted that the vdW cubic behavior is
qualitatively reasonable; and second, that it is capable of
describing the continuity between liquid and vapor.
Nevertheless, vdW cubic EOS has been proven not to be
quantitatively suitable for most engineering purposes. All of
the development in the field of phase behavior that has been
achieved today is due to the work of van der Waals. VdW
concepts were so far reaching that he won the Nobel Prize for
his equation.
The point is that van der Waals’ accomplishment in 1873
triggered a tremendous effort among scientists to make
modifications to his EOS which would remove from it large
disagreements with experimental data. This effort has not yet
ceased today and is not likely to stop in the near future. Much
of this struggle has focused on how to better model the
attractive parameter “a” and the repulsive term “b”, with the
hope that we can get better quantitative predictions.

6. G.Making a comparison of the PR, SRK and RK equations of
state.
It is essential for an engineer to be able to decide which of the
methods will render a favorable result and also give a true
reflection of the calculated data used to describe a system.
Over the years the EOS’s have been tested and some
deductions have been made with regards to where they operate
best. [9] These include the following:
Redlich Kwong EOS:
 Satisfactory for gas phase fugacity calculation @ Pr <
Tr/3.
 Satisfactory for enthalpy and entropy departure
calculations.
 Better when used in conjunction with a correlation for
liquid phase behavior.
 Ordinarily good for gas phase properties.
 Poor for liquid phase properties.
Soave Redlich Kwong & Peng Robinson EOS
 PR obtains better liquid densities than SRK.
 Provide similar purposes as the Redlich Kwong EOS.
 Overall, PR does a better job for gas and condensate
systems than SRK.
 If polar systems, SRK always makes a better
prediction.
III. CONCLUSION
The discharge of volatile organic compounds occurs on a daily
bases and has a major effect on the environment, as stated
previously. Therefore it is of pronounced importance to
reduce the amount of discharge that currently exists. By
finding a better solvent for the absorption of volatile organic
compounds and also making a contribution to existing
knowledge of the calculation methods used to predict vapor
liquid equilibria, we will in turn make an effort to solve this
worldwide phenomenon. In essence, using equations of state
to predict VLE data will hopefully provide answers which
could be used as design bases for the absorption process to
eliminate or control the release of toxic VOC’s into the
atmosphere. The most popular cubic EOS, which time has
proven to be most reliable are:
 Redlich-Kwong EOS,
 Soave-Redlich-Kwong EOS (very popular among
chemical engineers),
 Peng-Robinson EOS (very popular among petroleum
and natural gas engineers).
ACKNOWLEDGMENT
The authors wish to acknowledge fellow group members for
constructive discussions during this work in particular during
manuscript preparation.
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S. Ramdharee: The author was born in 1984 in Newcastle, Kwa-Zulu Natal,
South Africa. This author became a member of ECSA(Engineering Council
of South Africa) in 2010. He successfully completed a NDip:Chemical
Engineering at Durban University of Technology, Kwa-Zulu Natal, South
Africa in 2007. After which he pursued a BTech: Chemical Engineering at
the same institution in 2008, graduating Cum Laude, with the Deans Merit
award for being the top student in year 2008, and also received the award for
Mathematics, Statistics and Physics Year 2008. Currently he is studying
towards a MTech:Chemical Engineering at the University of Johannesburg,
Gauteng, South Africa.
He has worked at African Amines : JUNIOR PRODUCTION ENGINEER;
Karbochem Ltd: JUNIOR PROJECTS ENGINEER. International Furan
Technology: CHEMICAL ENGINEER; Sasol Synfuels: SENIOR AS-BUILT
AUDITOR; Sasol Synfuels in Secunda, Mpumalanga, South Africa:
PROCESS TECHNOLOGIST.