This document describes the Elliott-Suresh-Donohue (ESD) equation of state, which is a modification of the Peng-Robinson equation of state. The ESD equation adds parameters to account for the shape of non-spherical molecules and their interactions. It can better estimate specific volume for a variety of substances, including those with hydrogen bonding. The document provides background on equations of state, the development of the Peng-Robinson and ESD equations, and defines terms like compressibility factor. It concludes by comparing specific volume calculations using the ESD and Peng-Robinson equations to experimental values for water and methane.

1. 1
EQUATION OF STATE ELLIOTT - SURESH - DONOHUE: ESTIMATION OF
SPECIFIC VOLUME BY A MATHEMATICAL MODEL
Sebastián Ramírez Meza(seramirezme@unal.edu.co), Hawer Nicolás Rodríguez
Villamil(hnrodriguezv@unal.edu.co), Camilo Alfonso Valencia Mejia
(caavalenciam@unal.edu.co).
Universidad Nacional de Colombia, Sede Bogotá, Facultad de Ingeniería, Departamento de Ingeniería Química y Ambiental. 2016.
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ABSTRACT
The principal aim of this article is to explain the modification that Elliott, Suresh and
Donohue made to Peng-Robinson’s equation of state in the repulsive and attractive
term due to the Van der Waals force acting between molecules. This modification is
very useful in the associated fluids, the most common substances presenting this
type of interaction represented by hydrogen bonds, permanent dipole and instant
dipole. Moreover, this new equation allow the addition of a new term “c” that take into
account the shape and the repulsion for the first term and a constant “q” that
considers shape and interactions in the attractive term (second in the equation),
providing a better value in the estimation of a specific volume. This equation could
be applied for pure substances and in mixture too.
___________________________________________________________________
1. INTRODUCTION
The thermodynamic properties
describe the behavior of a substance,
doing it a quantitative measure by a
relation of temperature, pressure and
volume in a mathematical model which
is known as a equation of state (EoS),
thus enabling a comparison between
differents states in which the
properties vary for some kind of
process.
The base of all the EoS´s that currently
are stipulated is the ideal gas
equation, that provides good
approaches in the calculation of a
specific volume (henceforth specific
volume will be the relation between the
total volume and “n”, the moles of the
substance, or a part of the total volume
occupied by a pure component and its
corresponding number of moles).
However, this equation makes some
mistakes in extreme high pressures
and temperature. For that reason, is
necessary to obtain a correction, which
is the essential base to improve the
estimate’s accuracy using another
EoS’s taken from differents research
projects. Those projects change or add
some terms, allowing us to extend the
EoS’s use to high values of pressures
or temperatures, non-spherical
molecules, polar and nonpolar
molecules, ranges of critical
properties, associated and
nonassociated fluids and by last, in the
mixture; through Van der Waals
(1873), Redlich-Kwong (1949), Soave
(1972), Peng-Robinson (1976) till
Elliott-Suresh-Donohue (1990) and
some more.
It have been developed four differents
groups of EoS’s: cubic equations, virial

2. 2
equations, equations based on
molecular simulation and equations
from the chemical theory. The exposed
equation in this documents is a cubic
equation with 3 roots and each one
gets an specific meaning.
Nomenclature:
P Pressure
T Temperature
R Universal gas constant
V Specific volume
b Volume excluded by molecules
a Attractive term of the
Van der Waals
𝑇𝑐 Critical temperature
𝑃𝑐 Critical pressure
𝜔 Acentric factor
𝑇𝑟 Reduced temperature
𝛼 Dimensionless parameter in PR-
EOS
k Chemical association constant
Z Compressibility factor
c Shape factor
n Reduced density
2. THEORETICAL FRAMEWORK
This work is based on a correction
made by Elliott, Suresh and Donohue
in the repulsive term of the Peng-
Robinson equation, adding a
parameter “c” that represents the
shape of non-spherical molecules and
is included in the 𝑍 𝑟𝑒𝑝
term of a
generalized model. At the same time
𝑍 𝑟𝑒𝑝
term depends on
corresponding states for an expression
using the compressibility factor. In the
same way this equation developes a
change in the attractive term too, but
the most important alteration in the PR
equation is that with the obtained
corrections it it is possible to apply the
new EoS’s to a variety of substances,
and not only with pure fluids [1].
𝑃𝑉
𝑅𝑇
= 1 + 𝑍. 𝑎𝑡𝑟𝑎𝑐
+ 𝑍. 𝑟𝑒𝑝𝑢𝑙
+ 𝑍. 𝑎𝑠𝑠𝑜
(1)
Equation (1) will be complete with
each term defined for the ESD
equation.
2.1 Van der Waals EoS
In real life, the gas does not act as an
ideal gas, and the fundamental
equation involve considerable
mistakes at high pressures (𝑃 >
𝑃 𝑎𝑡𝑚)To be more accurate to the
real model, in 1873 Van der Waals
proposed a constant “a” and “b” [3].
Thus, “b” represents the excluded
volume between molecules which
centers can not be shortened less than
a distance “d”; and “a” represents the
attraction forces between the
molecules located away from the edge
of a container [2]. as seen in the
following equation:
𝑃 =
𝑅𝑇
(𝑉−𝑏)
−
𝑎
𝑣2
(2)
𝑎 =
27𝑅2 𝑇𝑐
2
64𝑃𝑐
(3)
𝑏 =
𝑅𝑇𝑐
8𝑃𝑐
(4)
Constants “a” and “b” can be
computed in this equation, analysing
the thermodynamics properties in their
critical point, placed at C (the
representation for critical point) using
the P-V graph. At this point the slope
of the pressure as a function of volume
takes the exact value is zero, letting us

3. 3
easily computing the constants“a” and
“b” [2].
Figure 1: P-V graph, C is the critical
point, T1 y T2 are isotherms, any point
(1-5) is a state in the M-L-V region.
2.2 Peng-Robinson EoS (1976)
This equation is quite important in
Natural Gas industry and Oil Industry
because of the given precision for
liquids densities in hydrocarbons but is
not sufficiently accurate to apply to
another real life models.
. Usually, this equation has a pretty
good approximation near the critical
point to determine the compressibility
factor and density. This model takes
into account three properties of the
equation which are: critical pressure,
critical temperature and acentric factor,
the last determines a parameter of
cohesion (k) taking into account a
correction in the relationship with the
reduced temperature. PR equation
keeps the same “a” and “b” constants
of van der Waals, also generated with
the clearance volume of a cubic
equation, providing high accuracies
with the liquid- vapor equilibria.
Besides of that, this equation is widely
known in thermodynamics because it
is useful with nonpolar substances and
could be applied to mixtures. The
Equation is given by:
𝑃 =
𝑅𝑇
𝑉 − 𝑏
−
𝛼(𝑇𝑟, 𝜔) ∗ 𝑎
𝑣 2 + 2𝑏𝑣 − 𝑏 2
In which:
𝑘 = 0,37464 + 1,54226𝜔
+ 0,26992𝜔2
𝛼 = (1 + 𝑘(1 − 𝑇𝑟0,5
))
2
𝑎 = 0,45724
𝑅2
𝑇𝑐2
𝑃𝑐
𝑏 = 0,0778
𝑅𝑇𝑐
𝑃𝑐
2.3 Effects of hydrogen bonds:
It is have been used mathematical
models with predictions based on the
correspondence of states such as the
state equation of Peng Robinson and
Soave equation to accurately predict
the behavior of vapor-liquid equilibria
for mixtures non-associative, polar or
less polar substances. [1].
However, for mixtures such as
immiscible liquid-liquid mixtures or
mixtures of solid type where through
inhibitors such as methanol (MeOH),
ethanol (EtOH) or water equilibria
liquid-gas phase are created, it is
common to find hydrogen bonding
present between the solutes, these
bonds within the system cause the
characteristical behavior of its
molecules: they can self-associate
(forming trimers, tetramers, etc.),
mutually associate (forming chains

4. 4
solvation with other species, hydrates)
and in some cases they form new
pseudo-species as clathrates, where
different configurations between the
hydrogen bonds around the domains
of the more nonpolar structures [13].
Image 1: Segment of solvation chain
formed by hydrogen bonds.
In this way, considering the
associativity and non-associativity in
mixture products of the hydrogen
bonds and physical forces
respectively, we obtain an equation in
which the intermolecular forces and
separately the forces of repulsion and
attraction of involved molecules makes
a relation between the behavior of real
gas and ideal gas behavior, as follows:
Where superscript chem refers to the
effect of associativity caused by the
hydrogen bonds and the superscript
phys account the non-associated
contribution in the components [8].
The term 𝑍 𝑐ℎ𝑒𝑚
is an expression
based on the linear and infinite
association of the associate monomers
given by Vafaei Sefti et al; Dehaghani
association equation of state (EOS-
DA), suitably modified this expression
considering a linear finite association
of monomers , showing the following
relationship between the Z-factor of a
pure associative component [8]:
The first term of the equation (5.2) , is
responsible for the physical part of the
compressibility factor, they have the
contribution of the physical forces of
attraction and repulsion Van der Waals
into account.
The equation of state of Peng
Robinson takes account of these
physical interactions and provides
accurate values in nonpolar mixtures
that do not have associativity.
The equation of state Elliott-Suresh-
Donohue, as a modification of the
above named equation of Peng
Robinson also takes into account
these forces of repulsion and
attraction, and also considers the form
factor and its effect on the forces of
repulsion and attraction.
This effect it is so important because
nonspherical molecules considered.
where q is the parameter considered
for the effect of shape on the attractive
part of Z, Y= exp( 𝛽ε)-𝑘2, 𝛽=1/(kT), ε is
the depth of the square-well potential,
and 𝑧 𝑚, 𝑘1,𝑘2,𝑘3are constants in the
equation of state [1].
2.4 Compressibility factor
For the ideal gas equation, a factor
that related the real value of specific
volume and the ideal value (𝑣 𝑟𝑒𝑎𝑙

5. 5
and 𝑣 𝑖𝑑𝑒𝑎𝑙 respectively) of the same
property could be measured. It will be
expressed as the quotient 𝑍 =
𝑣 𝑟𝑒𝑎𝑙/𝑣 𝑖𝑑𝑒𝑎𝑙and represent the
variation between the real and the
ideal state [4]. This new property could
be calculated for any fluid after the
critical point because at this point, a
large number of different fluids
behaves in the same way in its
reduced temperature and reduced
pression [5]. A better approach for the
compressibility factor is given by the
corresponding states, this new term
have two ways to be calculated, by two
corresponding states (TCS) or three
corresponding states (ThCS). For
TCS, the reduced temperature and
pressure are used as corresponding
states, and by another hand the TrCS
is a function TrCS(𝑇𝑟, 𝑃𝑟, 𝜔), where 𝜔 is
the acentric factor, that represent the
shape, geometry and the polar
character for an specific substances
[6]. The function for 𝑍:
𝑍 = 𝑍(0)
+ 𝜔𝑍(1)
(7)
2.5 Elliott-Suresh-Donohue EoS
In 1990, Elliott, Suresh and Donohue
presented a new correction of the
Peng-Robinson EoS associating with
the forces of repulsion. Previous
modification in PR EoS were limited to
pure substances and have difficulty in
the calculation of a specific volume [7].
The added term “c” is a function of the
acentric factor, where some of the real
molecules effects like the shape,
polarity and the geometry are
quantified. However, the shape is the
main correction because in the ESD
EoS, both, the non-spherical form of
the molecules and the attraction/
repulsion that is exerted in the central
parts of the system (as a cylinder) [1]
are taken into account. This non-ideal
behavior is represented in the
compressibility factor as equation (1)
represents.
The ESD EoS is a cubic type for the
volume equation, and have function in
terms of the acentric factor. The Z part,
could be complete with an addition of
𝑍 𝑎𝑠𝑠𝑜
. It turn out to be really useful
because it represents the associating
contributions that is widely used in
mixtures and analysis for polymers.
For mixtures the equation is governed
by a mixing rules [8]. This mixing rules
change the way to calculated aný
proprietary. Now, the ESD EoS is
model like this:
𝑃𝑉
𝑅𝑇
= 1 +
4𝑐𝜂
1−1,9𝜂
−
𝑍𝑚𝑞𝜂𝑌
1+𝑘1 𝜂𝑌
(8)
𝑐 = 1 + 3,535𝜔 + 0,533𝜔2
(8.1)
𝜂 = 𝑏𝜌 (8.2)
𝑌 = 𝑒 𝛽𝜖
− 𝑘2 (8.3)
𝛽 = 1/𝐾𝑇 (8.4)
𝜖
𝐾
= 𝑇𝑐(
1+0,945(𝑐−1)+0,134(𝑐−1)2
1,023+2,225(𝑐−1)+0,732(𝑐−1)2
) (8.5)
𝑞 = 1 + 𝑘3(𝑐 − 1) (8.6)
From the six previous described
relations “c” is the representation of
shape in repulsive term, “q” have
implicitly the shape of a molecule and
the attractive term. Thus “𝑏” is defined
as in equation (4). 𝑘1and 𝑘2are
numerical constants with value of:
1,7745 and 1,0617, respectively. 𝑘3 is
found with the critical properties and
𝑌is the attractive energy parameter. K
is the Boltzmann’s constant (K=1,38 ∗

6. 6
10−23 𝐽
𝐾
. The model said that have
major accurrete with Tr>0,45.
3. RESULTS AND DISCUSSION
The main object for this article is the
calculation of an specific volume for
some pure substance. The water and
methane will be the substances for
apply the ESD equation. The values
for the temperature, pressure, critical
temperature and pressure, acentric
factor, molecular-weight and
experimental value for the volume, are
taken from the tables in the reference
[6]. For a good comparison, it will be
used the Peng-Robinson equatión and
de ESD equation plus the comparison
with the real volume.Added to this, the
rate will be represented in a graphs
which shows the difference between
the two methods. The coefficients of
each term in the cubic equation will
follow next structure:
𝑎𝑥3
+ 𝑏′𝑥2
+ 𝑐𝑥 + 𝑑 = 0 (8)
For the PR EoS, is used the next
clearance:
𝑎 = 𝑃,
𝑏′ = 𝑃𝐵 − 𝑅𝑇,
𝑐 = 𝛼 𝑎 − 3𝑃𝑏2
− 2𝑅𝑇𝑏
𝑑 = 𝑃𝑏2
+ 𝑅𝑇𝑏2
− 𝛼𝑎𝑏
The constant in the previous volume
are placed in the place that is
corresponding, show in equation (8).
And for ESD EoS is the clearance that
could be proven in the attached file in
the spreadsheet. Taking into account
the next constant for the methane and
water, respectability, are:
Table 1: Properties of methane
Table 3: Properties of wáter

7. 7
Figure 3: Error rate for specific
volumen of Methane.
Figure 4: Error rate for specific
volumen of Water.
4. Conclusions
As the graphs show, the comparison
between the specific volume leaves to
the PR Eos as a better approach for
the picked substances (methane and
water). But for high temperatures and
low pressures the error rate is the
smallest in both equation.
The ESD equation of state take a big
roll when the temperature increases,
but we can see that is not enough to
be a better aproximatión. The
calculation of an specific volume for
this both substances leave the ESD
EoS with a high error rate and do not
throws a secure value if this is
compared with other EoS.
The addition of a parameter “c”, take in
consideration more affectations in the
repulsive term, but for this particular
example it do not work effectively.
This two substances leave see that for
this ranges of pressures and
temperatures a non-polar o polar
molecule the value of a volume is
independent of this property. Which
leave conclude that in a higher
temperatures the Van der Waals
interaction take a less value and make
to the gas to tend into an ideal gas.
5. Recommendations
It is useful with associative
thermodynamics inhibitors like glycols
and alcohols because the number of
bonding sites they have in their
chemical shape also in the hydrate
formation conditions of a non ideal
liquid mixtures.
For liquid-vapor mixtures must be used
to manipulate the Peng-Robinson
equation to calculate through the roots
of the equation the compressibility
factor of the liquid phase (large root)
and the gas phase (low root).
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
150 175 200 225 250 275 300 350 400
%Error
Temperatura K
Methane at o,1 MPa
%Error-PR %&Error-real ESD
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
5,5
%Error
Temperatura (K)
Water 0,1 MPa
%Error - PR %Error- EDS

8. 8
6. Bibliography
Figura 1: Gráfico presión vs Volúmen.
Tomado de “Van der Waals, más que
una ecuación cúbica de estado”.
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Suresh Marc; D. Donohue. (1990). A
Simple Equation of State for
Nonspherical and Associating
Molecules. Industrial of engineering
chemistry research.
[2] Duque, P. Gracia-Fadrique,
J.(2014). Van der Walls, más que una
ecuación cúbica de estado.
Universidad Nacional Autónoma de
México.
[3] Rodríguez, J. Pardillo, E. (2002).
Características y aplicaciones de las
ecuaciones de estado en la ingeniería
química. Avances recientes. Parte 1.
Revista Facultad de ingeniería.
[4] K. Wark, D.E. Richards. (2001).
Thermodynamics . Madrid, España:
McGraw Hill. (6th Ed.)
[5] Smith, Van Ness. (2007)
Introduction To Chemical Engineering
Thermodynamics. USA: Mc Graw Hill
(7 th Ed.)
[4] K. Wark, D.E. Richards. (2001).
Thermodynamics . Madrid, España:
McGraw Hill. (6th Ed.)
[6] Van Wylen, G. Sonntag, R.
Borgnakke, C. Fundamentos de
termodinámica. Universidad de
Michigan.
[7] J. Richard Elliott, Carl T. Lira.
(1999). introductory chemical
engeneering thermodynamics .
London: Prentice Hall PTR.
[8] Amir Hossein Saeedi Dehaghani ,
Mohammad Hasan Badizad. (2016).
Thermodynamic modeling of gas
hydrate formation in presence of
thermodynamic inhibitors with a new
association equation of state . Iran:
Fluid phase equilibria.
[9]Mandy Klauck∗, Rico Silbermann,
Robert Metasch, Tatjana Jasinowski,
Grit Kalies, Jürgen Schmelzer. (2014).
VLE and LLE in ternary systems of two
associating components (water,
aniline, and cyclohexylamine) and a
hydrocarbon (cyclohexane or
methylcyclohexane) . Germany: Fluis
phase equilibria.
[10] Sebastian Giraldo. (2005).
Ecuaciones de estado.
[11] Hoyos B.(2003). Cálculo del
volumen específico de líquidos puros
con la ecuación de estado cúbica de
valderrama-patel-teja. Ingeniería e
investigación
[12] Paola R. Duque Vega, Jesus
Gracia-Fadrique. (2015). Van der
Waals, más que una ecuación cúbica
de estado. Mexico: Educación
Química.
[13] Antonio Sánchez Ruiz, Antonio
Heredia. Clatratos de agua y
biomoléculas. Take from:
http://www.encuentros.uma.es/encuent
ros54/clatratos.html

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