Modification of Cubic Equations of State for Polar Compounds

Seminar by:Hitanshu Sachania12BCH025

Guided by:Dr. Milind JoshipuraAssociate Professor

DEPARTMENT OF CHEMICAL ENGINEERING INSTITUTE OF TECHNOLOGY

NIRMA UNIVERSITY

Equations of State

Equations of the form f(P,V,T) = 0 are known as

equations of state.

Such equations yield the volumetric properties of

the system and hence, describe its P-V-T behaviour.

“Volumetric Properties” are essential parts of

calculations used to compute thermodynamic

properties like internal energy (U), enthalpy (H),

entropy (S), etc.

During process plant design these aid the engineer in

sizing of pipelines and process equipment like: [5]

Distillation Columns

Reaction Tanks

Heat Exchangers, etc.

Cubic Equations of State

Cubic equations of state are a genus of equations

of state. They are in a cubic power of molar

volume (V). CEOS in the form of compressibility

factor Z, can be written as:

Z = Zrepulsive + Zattractive

Cubic equations of state are valid only for a

hypothesized gas known as an “Ideal Gas”.

Pragmatic gases manifest a non-ideal behaviour

and to study their P-V-T behaviour i.e. to determine

their volumetric properties, cubic equations of

state need to be modified.

An ideal gas is such a gas which has no

intermolecular interaction since all of its

constituent molecules are separated infinitely from

each other.

Van der Waals Cubic Equation of State

227

64C

C

RTa

P

1

8C

C

RTb

PWhere and

The concept of separation of repulsive forces due to

size of molecules and the cohesive forces due to

molecular attraction proposed by van der Waals, even

today remains as the basis of theories about the

prediction of fluid properties.

Real GasesPolarity in a molecule arises

due to the combined effect of

electronegativity difference of

the constituent atoms and the

spatial arrangement of the

molecule.

H2O is polar since it has a

net dipole moment of the

two Oδ- - Hδ+ dipoles.Non

-Ioni

c Co

mpo

unds

Polar

Non-Polar

CH4 is non-polar due to a null net dipole moment

since all four Cδ- - Hδ+ dipoles are oriented towards

the vertices of a tetrahedron.

There are three different types of intermolecular

interactions:

Dipole – Dipole: When a polar compound interacts

closely with another polar compound, the electric

fields of the two dipoles overlap and hence, a

net

mutual attractive force is experienced by the

two molecules.

Dipole – Induced Dipole: When a polar compound

comes in a close proximity to a non-polar

compound, the permanent dipole of the polar

molecule induces an instantaneous dipole in the

non-polar molecule resulting in the two molecules

experiencing a net attractive force.

Induced Dipole – Induced Dipole: Even when two

non-polar molecules come close enough to each

other, they induce evanescent dipoles into each

other which too results in a net attractive force

between the two molecules.

However, intermolecular interactions are not always

attractive; they may be repulsive as well. This can

easily be understood with the help of Lennard-Jones

Potential.

Lennard – Jones Potential[1]

12 6

( ) 4U LJr r

Where r = intermolecular separation distance, ε =

maximum negative potential’s magnitude or the

depth of potential well and σ = separation distance

at which the intermolecular interaction between the

two molecules is zero.

Schematic of Intermolecular Potential Energy for a pair of Non-ionic Molecules[1]

The first term in the LJ potential equation i.e. the r-

12 term represents repulsive interactions while the

second term i.e. the r-6 term represents attractive

interactions.

For practical purposes intermolecular interactions

possess considerable magnitudes at separations up

to ten times the molecular diameter. ε and σ are

characteristic parameters corresponding to the

substance.

General Methodology for Modification of Cubic Equations of State [3]

Cubic equations of state are generally modified in

three different ways:

Modification of the temperature dependent

function α(TR) in the attractive term of the EOS.

Volume Translation: Modification of the volume

dependence of the attractive pressure term.

Use of a 3rd substance dependent parameter.

Three Parameter Cubic Equations of State

(C3EoS) A major limitation of the two parameter cubic

equations of state (C2EOS) is the prediction of the

same critical compressibility factor zc for all

substances.

Clausius’ Modified CEoS:

This equation is based on the notion that at lower

temperatures form clusters with a strong mutual

attraction in lieu of moving freely. [2,3]

The a/V2 term in VdW CEoS is too small at low

temperatures and hence, doesn’t represent the

intermolecular attraction accurately.

To let zc be substance dependent, Clausius in his

EoS included a 3rd putative substance dependent

parameter (c) in the attractive volume term.

Clausius’ three parameter cubic equation of state

laid the radix for the genesis of a new class of

CEoS, the C3EoS.

Facets of Parameters in C3EoS

It’s easier to understand C3EoS parameters with

reference to the equation proposed by Schmidt and

Wenzel:

Where a(T) is the cohesive energy parameter and b

is the volumetric parameter.

For constant u and w, this equation becomes a

C2EoS.

When one of u or w is fixed or if there exists a

relationship between the two, the equation

becomes a C3EoS.

The parameters of Schmidt and Wenzel equation

can be obtained by:

1. Imposing the thermodynamic condition:

at Tc, Pc, Vc

2. Imposing the mathematical constraint:

Relationship between u and w CEOS

u = 0, w = 0 Van der Waals

u = 1, w = 0 Redlich/Kwong , Soave/Redlich/Kwong

u = 2, w = -1 Peng/Robinson

w = 0 Fuller, Usdin and McAuliffe

w = u2/4 Clausius (VT - VDW)

u + w = 1 Heyen , Schmidt and Wenzel, Harmens and Knapp , Patel and Teja

u - w = 3 Yu et al. , Yu and Lu

u - w = 4 Twu et al.

w = 2(u+2)2/(9 - u -1) VT – SRK

w = (u-2)2/(8-1) VT –PR

Table 3.1 - Systematics of C2EOS and C3EOS [2]

The C3EOS pertaining to the relation u + w = 0

stand out to best predict the PVT behaviour of both

non-polar and polar substances within a range of

critical compressibility factor (zc) values. [2]

Modification of

Various modifications of function of the reduced

temperature for the SRK and PR EOS have been

proposed for better predictions of vapour pressure of

polar compounds.

Expression for Model

Redlich and Kwong

Wilson

Selected models of in Cubic Equations of State [3,4]

Expression for Model

Heyen

Boston and Mathias

Usdin and McAuliffe

Joshipura

Twu

Soave

The variation of some of these with disparity in temperature[3]:

Cubic Plus Association Equation of State[13]

xAi is obtained from:

Where ΔAiBj, the association solidity between the site A of molecule and site B of molecule is given by:

CPA EoS surmounts other equations in determining values of properties like density, heat capacity, enthalpy, Joule-Thomson coefficient and velocity of sound for reservoir fluids which contain polar substances like water and methanol. [13]

SummaryThere are various methods for the modification of

CEoS to transpire.Though varied in manner, all such methods have a

ubiquitous idea as their base principle.The principle is the inclusion of additional

parameters and/or manipulation of the equation’s mathematical construct, so as to render the equations substance dependent.

No single equation has proved to be superior since, all of them have their own domain of supreme performance.

Moreover, molecular equations of state developed recently have superseded CEoS.

References1. "Lennard-Jones Potential," [Online]. Available:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mecha

nics/Atomic_Theory/Intermolecular_Forces/Lennard-

Jones_Potential. [Accessed August 2014].

2. W.-R. Ji and D. Lempe, "A systematic study of cubic three-

parameter equations of state for deriving a structurally optimized

PVT relation," Fluid Phase Equilibria, pp. 85-103, 1998.

3. J. O. Valderrama, "The State of the Cubic Equations of State,"

Industrial & Engineering Research, pp. 1603 - 1618, 20 March

2003.

4. M. H. JOSHIPURA, "PREDICTIONS OF VAPOR PRESSURES OF TEN

IONIC LIQUIDS USING PATEL TEJA EQUATIONS OF STATE,"

International Journal of Research in Engineering & Technology

(IJRET), pp. 49-54, July 2013.

5. "Chapter 2: Volumetric Properties of Real Fluids," [Online].

Available: http://nptel.ac.in/courses/103101004/downloads/chapter-

2.pdf. [Accessed 23 August 2014].

6. Heyen G. A Cubic Equation of State with Extended Range of

Application. In Chemical Engineering Thermodynamics by

Newman,175 1983.

7. Boston, J. F.; Mathias, P. M. Phase Equilibria in a Third- Generation

Process Simulator. Presented at the 2nd International Conference on

Phase Equilibria and Fluid Properties in the Chemical Industry,

Berlin, Germany, Mar 17-21, 1980.

8. Usdin, E.; McAuliffe, J. C. A One-Parameter Family of Equations of

State. Chem. Eng. Sci. 1976, 31, 1077.

9. Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V.

An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev.

1949, 44, 233.

10. Wilson, G. M. A New Expression for the Excess Free Energy of

Mixing. J. Am. Chem. Soc. 1964, 86, 127.

11. Soave, G. Application of a Cubic Equation of State to Vapor-Liquid

Equilibria of Systems Containing Polar Compounds. Inst. Chem. Eng.

Symp. Ser. 1979, 56, 1.2/1.

12. Twu, C. H. A Modified Redlich-Kwong Equation of State for Highly

Polar, Supercritical Systems. In Proceedings of the International

Symposium on Thermodynamics in Chemical Engineering and

Industry; Academic Periodical Press: Beijing, China, 1988; p 148.

13. C. Lundstrøm, M. L. Michelsen, G. M. Kontogeorgis, K. S. Pedersen and

H. Sørensen, "Comparison of the SRK and CPA equations of state for

physical properties of water and methanol," FLUID PHASE EQUILIBRIA,

pp. 149-157, 2006.

14. W. Yan, G. M. Kontogeorgis and E. H. Stenby, "Application of the CPA

equation of state to reservoir fluids in presence of water and polar

chemicals," FLUID PHASE EQUILIBRIA, pp. 75-85, 2009.