An Extension to the Corresponding States Principle - Trad April 2012

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ANEXTENSIONTO THECORRESPONDINGSTATES PRINCIPLE

PREDICTION AND CORRELATION OF THERMOPHYSICAL PROPERTIESUSING THE CORRESPONDING STATES PRINCIPLE

G = G(0) (Tc,Pc) +ω G(1) (Tc,Pc,ω) +ξ G(2) (Tc,Pc,ω,ξ)

Iván Jesús Castilla-Carrillo

e-mail:[email protected] [email protected]

mailto:[email protected]

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AN EXTENSION TO THE CORRESPONDING

STATES PRINCIPLE

Iván Jesús Castilla - Carrillo

© 2010 Iván Jesús Castilla – Carrillo

For the translation:

© 2010 Felipe Riancho-Seguí

No total or partial reproduction of this work is allowed or its computerized treatment, nor its transmission in any way, is it electronic, mechanical, by photocopy, or any other method without previous written authorization of the copyrighters.

First edition: 2012

Iván Jesús Castilla-Carrillo, Mérida, Yucatán, México April, 2012.

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P R E F A C E

I really do not remember when I dedicate myself to study and to understand the “Corresponding States Principle” (CSP), also called Theorem of Corresponding States or Law of Corresponding States. It was not during my first years of attending the National Polytechnic Institute, since at that time, equations such as the Van der Waals were too complicated for me and the BWR equation, I believed it was impossible to apply. The only equation that to me it was reasonable to use with the aid of a calculator was the ideal gases.

During my time as a scholar at the Mexican Petroleum Institute (IMP), from 1976 to 1977, I worked under the direction of PhD. Raúl Acosta García using the BWRS equation for the prediction of cryogenic fluids vapor-liquid equilibrium. It was then I became familiar with the multi-parametric equations of state, mathematical modeling and scientific literature.

In 1978, I worked in the late company Bufete Industrial in the physical properties project and it seems that it was here where I started becoming interested on the Principle of Corresponding States. My boss and friend, Ing. Manuel Del Villar Casillas allowed me to use the Lee-Kesler equation for prediction of physical properties and vapor-liquid equilibrium of simple and normal fluids. However, it did not work for polar fluids which are of the most importance on secondary petrochemical. It is when; I believe arises my interest to make the CSP works for abnormal or polar fluids.

The information I am presenting in this Part 1 was obtained or collected during 9 months prior to May, 1983. I remember in those days we did not have personal computers and there was no internet network. I used to process my programs on a HP-3000 minicomputer that was installed on the third floor of building N° 8 which comprised the Superior School of Chemical Engineering and Extracting Industries (ESIQUIE) at the National Polytechnic Institute (IPN).

The PhD Mateo Gómez-Nieto (†), my thesis director and friend and I, spent many hours reading and trying to understand the specialized literature on the theme.

Our workshops were fun since we had them at a known coffee shop named Sanborns, a few blocks away of the IPN Campus in San Pedro Zacatenco. For very long hours we had coffee, made commentaries and argued on the interpretation of the Principle of Corresponding States. He always cautioned me about scientific literature, do not believe everything that I read, many specialists in the matter do not really understand what the Principle of Corresponding States is really about, and they write and publish mathematical models that do not work and only lead you to confusion.He also recommended me to maintain the mathematical models as simple as possible and not to fall in redundancies and over parameterized models. This is an error found frequently in the scientific community. Researchers think that the larger and more complicated models work better and this is definitely untrue. The more correlational parameters, the more correction terms, have no sense and they will not make our models better if they are not based upon correct observations and measurements and overall in the comprehension and understanding of reality. “EVEN THOUGH YOU ARE NO LONGER WITH US, I THANK YOU MATEO (†) FOR MAKE ME THINK DIFFERENT”


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DIVERSION

1910 Physics Nobel prize, Johannes Diderik van der Waals(1837-1923)

Biography

Johannes Diderik van der Waals was born on November, 1837 in Leyden, The Netherlands, the son of Jacobus Waals and Elizabeth van den Burg. After having finished elementary education ay his birthplace he became a schoolteacher. Although he had no knowledge of classic languages, and thus was not allowed to take academic examinations, he continued studying at Leyden University in his spare time during 1862-65. In this way he also obtained teaching certificates in mathematics and physics.

In 1864 he was appointed teacher at a secondary school in Deventer; in 1866 he moved to The Hague, first as a teacher and later as Director of one of the secondary schools in that city.

New legislation whereby university students in science were exempted from the conditions concerning prior classical education enabled Van der Waals to sit for university examinations. In 1873 he obtained his doctor’s degree for a thesis entitled Over de Continuïteit van den Gas - en Vloeistoftoestand (About the continuity of the gas and liquid state), which put him at once in the foremost rank of physicists. In this thesis


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he put forward an “Equation of State” embracing both the gaseous and the liquid state; he could demonstrate that these two states of aggregation not only mix each other in a continuous manner, but that they are in fact of the same nature. The importance of this conclusion from Van der Waals’ very first paper can be judged from the remarks of James Clerk Maxwell in Nature, “that there can be no doubt that the name of Van der Waals will soon be among the foremost in molecular science” and “it has certainly directed the attention of more than one inquirer to the study of the Low-Dutch language in which it is written” (Maxwell probably meant to say “Low German” which would also be incorrect since Dutch is a language in its own right). Subsequently, numerous papers on this and related subjects were published on the Proceedings of the Royal Netherlands Academy of Sciences and in the Archives Néerlandaises, and they were also translated into other languages.

When in 1876, the new Law on Higher Education was established which promoted the Athenaeum Ilustre of Amsterdam to university status, Van der Waals was appointed Professor of Physics. Together with Van’t Hoff and Hugo de Vries, the geneticist, he contributed to the fame of the University, and remained faithful to it until his retirement, in spite of tempting invitations elsewhere.

The immediate cause of Van der Waals’ interest on the subject of his thesis was R. Clausius treatise considering heat as a phenomenon of motion, which led him to look for an explanation for T. Andrews’ experiments (1869) revealing the existence of “critical temperatures” on gases. It was Van der Waals’ genius that made him see the necessity of taking into account the volume of molecules and intermolecular forces (“Van der Waal’s forces as they are now generally called) in establishing the relationship between the pressure, volume and temperature on gases and liquids.

A second great discovery – arrived after much arduous work – was published in 1880 when he enunciated the Law of Corresponding States. This showed that if pressure is expressed as a simple function of the critical pressure, volume as one of the critical volume, and temperature as one of the critical temperature, a general form of the equation of state is obtained which is applicable to all substances, since the three constants a, b, and R in the equation where expressed in the critical quantities of a particular substance, will disappear. It was this Law who served as a guide which ultimately led to the liquefaction of hydrogen by J. Dewar in 1898 and of helium by H. Kamerlingh Onnes in 1908. The latter, who in 1913 received the Nobel Prize for his low temperature studies and his production of liquid helium, wrote “that Van der Waals’ studies have always been considered as a magic wand for carrying out experiments and that the Cryogenic Laboratory at Leyden has developed under the influence of his theories”.

Ten years later, in 1890, the first treatise on the “Theory of Binary Solutions” appeared in the Archives Néerlandises – another great achievement of Van der Waals. By relating his equation of state with the Second Law of Thermodynamics, in the form first proposed by W. Gibbs treatises on the equilibrium of heterogeneous substances, he was able to arrive at a graphical representation of his mathematical formulations in the form of a surface


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which he called “Psi surface” in honor of Gibbs, who had chosen the Greek letter Psi as a symbol for the free energy, which he realized was significant for the equilibrium. The theory of binary mixtures gave rise to numerous series of experiments, one of the first being carried out by J.P. Kuenen who found characteristics of critical phenomena fully predictable by the theory. Lectures on this subject were subsequently assembled in the Lehrbuch der Thermodynamik (textbook of thermodynamics by Van der Waals and Ph. Kohnstamm.

Mention should be made of Van der Waals’ thermodynamic theory of capillarity, which in its basic form first appeared in 1893. In this, he accepted the existence of a gradual, though very rapid, change of density at the boundary layer between liquid and vapor – a view which differed from that of Gibbs, who assumed a sudden transition of the density of the fluid into that of vapor. In contrast to Laplace, who had earlier formed a theory on these phenomena, van der Waals also held the view that the molecules are in permanent, rapid motion. Experiments with regard to phenomena in the vicinity of the critical temperature decided in favor of Van der Waals’ concepts.

Van der Waals was the recipient of numerous honors and distinctions, of which the following should be particularly mentioned: He received an honorary doctorate of the University of Cambridge; was made honorary member of the Imperial Society of Naturalists of Moscow; the Royal Irish Academy and the American Philosophical Society; corresponding member of the Institut de France and the Royal Academy of Sciences of Berlin; associate member of the Academy of Sciences of Belgium; and foreign member of the Chemical Society of London; the National academy of Sciences of the U.S.A. and of the Accademia dei Lincei of Rome.

In 1864, Van der Waals married Anna Magdalena Smit, who died early. He never married again. They had three daughters and one son. The daughters were Anne Madeleine who, after her mother’s early death, ran the house and looked after her father; Jaqueline Elizabeth who was a teacher of history and a well-known poetess; and Johanna Diderica who was a teacher of English. The son, Johannes Diderik Jr., was professor of Physics at Groningen University from 1903 to 1908, and subsequently succeeded his father in the Physics Chair of the University of Amsterdam.

Van der Waals’ main recreations were walking, particularly in the country, and reading. He died in Amsterdam on March 8, 1923.

From Nobel Lectures, Physics 1901-1921, Elsevier Publishing Company, Amsterdam, 1967

This autobiography/biography was written at the time of the award and first published in the books series Les Prix Nobel. It was later edited and republished in Nobel Lectures. To cite this document, always make mention of the source as shown above.


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Copyright © The Nobel Foundation 1910 TO CITE THIS PAGE: MLA style: "J. D. van der Waals - Biography". Nobelprize.org. 19 Jan 2012 http://www.nobelprize.org/nobel_prizes/physics/laureates/1910/waals-bio.html


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DIVERSION

Kenneth S. Pitzer (1914–1997)

Biography

Kenneth S. Pitzer was born on January 6, 1914 in Pomona, California, U.S.A. His father, Russell K. Pitzer, was a lawyer, orange grower and banker. His mother, Flora Sanborn Pitzer, was teacher of mathematics. His father contributed to the development of superior education institutions such as the Claremont Colleges, including the Harvey Mudd College, the Pitzer College, and what it is now known as Claremont McKenna College.

Pitzer graduated in chemistry in 1935 at the California Institute of Technology. In his first year he began research and investigation work along with on the field of reactions of oxide-reduction on solutions of silver salts. Pitzer published his first independent paper in 1935 in relation to the crystalline structure of a perrenato salt, following the suggestion of Linus Pauling.

After graduation, Pitzer began his graduate career at the University of California at Berkeley finishing his doctorate degree in just two years. He gained a position as instructor in the same University in 1937. By 1945, he became a full tenure professor In spite of several interruptions due to military service. During World War II, he worked on


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military investigation at the Maryland Laboratory of Investigation where he worked as technical director during 1943 and 1944.

After the war, Pitzer returned to the University of California at Berkeley, leaving it almost immediately to start a long career of public work and administrative work. In 1949, he became the first director of investigation at the Atomic Energy Commission, which under his leadership began to finance basic investigation. He returned to the University of California at Berkeley in 1951 and was appointed dean of the Chemical College until 1960.

In 1961, Pitzer became the third president of Rice University in Houston, Texas, U.S.A. In those days, Rice was a regional technological institute and Pitzer contributed to convert it into a university with national recognition. This process included the racial integration; recruiting a new body of faculty professors; adding academic programs and building new buildings.

Subsequently, in 1968, Pitzer became president of Stanford University. He drove the University through the turbulent period of the end of that 1960 decade. In 1971 he returned to his role as professor of chemistry to the University of California at Berkeley.

Throughout his administrative career, Pitzer supported in a masterly manner an investigation program. His work included the utilization of quantic mechanics and statistical mechanics to explain the thermodynamic and conformational properties of the molecules. He was a pioneer of the theory of the quantic dispersion to describe the chemical reactions and contributed to the statistical theory of liquids, solids and solutions.

Pitzer was a prolific contributor to scientific literature, publishing more than 334 papers, half of them in the period from1935 to 1960. While working as president of Rice he worked only with doctorate graduate students and published around 30 papers, abandoning research while being president at Stanford. At his return to the University of California at Berkeley at the age of 57, he re-started his investigations again and continued to work on them even after retiring from teaching in 1984. In this latter part of his career he published 140 papers. He is considered the founder of the modern theoretical chemistry at the University of California at Berkeley where the Pitzer Center for Theoretical Chemistry was created in 1999. Pitzer married Jean Mosher (Jean Pitzer) and had three children: Anne E. Pitzer, Russell M. Pitzer y John S. Pitzer.

Professor Pitzer retired from the University of California at Berkeley in 1985, but continued his investigations about thermodynamics and quantic theory until his death of a heart failure on December 26, 1997 at the age of eighty-three. His beloved wife, Jean, passed away on 22 April 2000. They are survived by three children, Ann, Russell and John.

Bibliography: Information obtained through internet.


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AN EXTENSION TO THE CORRESPONDINGSTATES PRINCIPLEPREDICTION AND CORRELATION OF TERMOPHYSICAL PROPERTIES USING THE CORRESPONDING STATES PRINCIPLE.

PART 1Extracts of my professional thesis to obtain the degree of Industrial Chemical Engineer at the Superior School of Chemical Engineering and Extractive Industries at the National Polytechnic Institute, México, D. F.Individual professional thesis developed under direction and guidance of PhD. Mateo Gómez-Nieto.Comments and actualizations not on the original thesis work are included.


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INDEX OF THEMESGLOSSARY 15EXTRACT 19I. INTRODUCTION. 23II. THE TWO PARAMETER CORRESPONDING STATES PRINCIPLE (CSP). 27III. THE THREE-PARAMETER CORRESPONDING STATES PRINCIPLE (CSP). 31

1. CSP proposed by Meissner y Seferian. 312. CSP proposed by Riedel. 323. CSP proposed by Pitzer. 33

a. Modification by Lee-Kesler. 35b. Modification by Teja. 36c. Modification by Castilla-Carrillo. 37

IV. THE FOUR-PARAMETER CORRESPONDING STATES PRINCIPLE (CSP). 451. CSP proposed by Eubank-Smith. 452. CSP proposed by Thompson. 473. CSP proposed by Halm-Stiel. 494. CSP proposed by Harlacher. 515. CSP proposed by Passut. 536. CSP proposed by Tarakad. 557. CSP proposed by Castilla-Carrillo. 578. CSP proposed by Wilding-Rowley. 62

V. USES AND APPLICATIONS OF CSP. 63VI. OBSERVATIONS. 65VII. RECOMMENDATIONS. 68BIBLIOGRAPHY

1. REFERENCES ON THE CORRESPONDING STATES PRINCIPLE. 692. REFERENCES ON EXPERIMENTAL VALUES OF THERMOPHYSICAL PROPERTIES. 723. RECOMMENDED LECTURES. 72

INDEX OF TABLESTable 1.Deviations of Lee-Kesler vapor pressure equation for linear n-alkanes from C1 to C20.

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Table 2.Comparison of deviations for Lee-Kesler and Castilla-Carrillo vapor pressure equations for linear n-alkanes from C1 to C20.

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Table 2.1Used data and comparison of deviations for Lee-Kesler and Castilla-Carrillo vapor pressure equations for 98 fluids y 5931 points.

414243

Table 3.Average deviations in vapor pressure predictions of different four-parameter CSP with respect to experimental data.

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Table 4.Correlation parameters used in the models of Thompson, Halm-Stiel, Passut and Harlacher.

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Table 5.Correlation of parameters used in the model of Castilla-Carrillo.

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SYMBOLS

GLOSSARY

a1(i), a2

(i), a3(i) - Coefficients for equation 76.

a, b - Constants for Van der Waals’ equation.a, b - Slope and intercept of equation 44.A, B, C - Main molecular inertia moments.B, C - Constants of ecuación 94.B, C, D - Constants of Frost-Kalkwarf equation.Bn, Cn - Constants of Frost-Kalkwarf equation for n-paraffins.B - Second virial coefficient.B* - Reduced second virial coefficient.C - Necessary specific constant in definition of Eubank-Smith fourth parameter.G - Any correlatable property by the corresponding states principle.G(0) - Generalized or universal function that considers the fluid as a simple fluid.G(1) - Generalized or universal function to correct molecular size-shape

deviations.G(2) - Generalized or universal function to correct molecular polarity deviations.G(3) - Generalized or universal function to correct inseparable size-shape-polarity

deviations (according to the four-parameters CSP proposed by Thompson).G(1)’, G(3)’, G(3)” - Generalized or universal functions necessary in Harlacher model.G(r) - Generalized or universal function for evaluation of properties or behavior of

a reference fluid (r).G(r1) - Generalized or universal function for evaluation of properties or behavior of

a reference fluid (r1).G(r2) - Generalized or universal function for evaluation of properties or behavior of

a reference fluid (r2).G(1)(ω), G(1)

(ω),- Generalized or universal function proposed in this work to correct the

deviations for molecular size-shape. This function changes with each value of ω or ω.

G(2)(ω,ξ) - Generalized or universal function proposed in this work to correct the deviations for molecular polarity. This function changes with each vale of ω and ξ.

M - Molecular weight.n - Constant in the fourth parameter definition of Eubank and Smith (n=5/3).P - Absolute pressure.P’ - Vapor pressure.P* - Fourth parameter proposed by Eubank and Smith.Pa - Parachor.Pc - Critical pressure.Pr - Reduced pressure.Prh - Homomorph reduced pressure.Pr’ - Reduced vapor pressure.Pr’b - Reduced vapor pressure at the normal boiling point.Pr’(n) - Reduced vapor pressure of a normal fluid.Pr’(0) - Reduced vapor pressure of a fluid considered as a simple fluid.


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Pr’(1) - Generalized or universal function of correction for molecular size-shape deviations for the reduced vapor pressure.

Pr’(2) - Generalized or universal function of correction for molecular polarity deviations for the reduced vapor pressure.

Pr’(1)(ω) - Generalized or universal function of correction proposed in this work for molecular size-shape deviations for the reduced vapor pressure. It changes with the molecular size-shape of each normal or polar fluid.

Pr’(2)(ω, ξ) - Generalized or universal function of correction proposed in this work for molecular size-shape-polarity deviations for the reduced vapor pressure. It changes with the molecular size-shape-polarity of each abnormal or polar fluid.

R - Constant of the ideal gases.R - Geometrical radius of gyration proposed by Thompson.T - Absolute temperature.Tc - Critical temperature.Tr - Reduced temperature.Trh - Homomorph reduced temperatura.Trb - Reduced temperature of normal boiling point.V - Volume.V0 - Hypothetical molar volume at absolute zero degrees (cm3/gmol)Vc - Critical volume.Vr - Reduced volume.X - 1/Trb.Y - Atmospheric pressure/Pc.Z - Compressibility factor.Z(0) - Compressibility factor considered as a simple fluid.Z(1) - Generalized or universal function of correction for molecular size-shape

deviations for the compressibility factor.Z(2) - Generalized or universal function of correction for molecular polarity

deviations for the compressibility factor.Zc - Critical compressibility factor.


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GREEK LETTERS

αc - Third parameter proposed by Riedel.α1, α2 - Correction factors for the B and C constants B y C of the Frost-Kalkwarf

equation utilized in the Passut model.γ - Surface tension.Є, σ - Parameters of Stockmayer potential intermolecular function.κ - Fourth parameter proposed by Passut.μ - Dipolar moment.μr - Reduced dipolar moment.π - 3.1415926536ρl - Density of a saturated liquid.ρv - Density of saturated vapor.σo - Hypothetical surface tension at absolute zero degrees (dines/cm).τ - Fourth parameter proposed by Thompson.Ф - Fourth parameter proposed by Tarakad.χ - Fourth parameter proposed by Halm-Stiel.ω - Pitzer’s acentric factor.ω - True acentric factor calculated for polar substances.ωh - Homomorph acentric factor.ω(r) - Acentric factor of an r referenced fluid.ω(r1) - Acentric factor of an r1 referenced fluid.ω(r2) - Acentric factor of an r2 referenced factor.

SUBSCRIPTS

c - Property at the critical point.calc - Calculated value.exp - Experimental value.h - Homomorph property.l - Saturated liquid property.n - Lineal paraffin property.rb - Reduced property at the normal boiling point.v - Saturated vapor property.o - Evaluated property at absolute zero degrees.

SUPERINDEX

‘ - Vapor pressure.(0), (1) … - Universal or generalized functions utilized in the correlations of

corresponding states.


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FUNCTIONS

f( ) - Functions in general.θ(Tr) - Pr/Tr – 1 + 2 Ln Trθ(0.7) - θ (Tr) evaluated at Tr=0.7ψ(Tr) - 1 – (Ln Tr)/TrΨ(0.7) - ψ(Tr) evaluated at Tr=0.7


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SUMMARY

In 1873, J. D. van der Waals (55) discovered the Corresponding States Principle (CSP) that says:

G = G(Tc,Pc) or also thatG = G(Tr,Pr) sinceTr = T/Tc Pr = P/Pc

Where:G - Any correlatable property using the (CSP).Tc - Critical Temperature.Pc - Critical Pressure.Tr - Reduced temperature.Pr - Reduced pressure.

However, the van der Waals CSP (two parameters CSP) only works for some noble gases as well as methane.

In 1955, Pitzer and his colleagues extended the application of CSP to substances different than noble gases through the inclusion of the acentric factor to correct deviations due to molecular size-shape.

G= G(0)(Tr,Pr) + ωG(1)(Tr,Pr)

ω = -log P’r (Tr=0.7) - 1.0

Where:G - Any correlatable property using the CSP.Tr - Reduced temperature.Pr - Reduced pressure.ω - Acentric factor.logP’r (Tr=0.7) - Base 10 logarithm of experimental reduced vapor pressure at reduced

temperature of 0.7.G(0)(Tr,Pr) - Property calculated considering fluid as simple fluid.G(1)(Tr,Pr) - Correction due to a molecular size-shape.

In the Pitzer’s three-parameter CSP, the molecular size-shape are appropriately characterized but the correction function is the same for all substances, which means it doesn’t change with the molecular size-shape. This deficiency can be clearly appreciated in the deviations presented in his model when applied to normal fluids of high molecular weight.


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In 1983, Castilla-Carrillo (58) proposed the following modification to the Pitzer’s three parameter CSP:

G= G(0)(Tr,Pr) + ωG(1)(Tr,Pr,ω)

ω = -log P’r (Tr=0.7) - 1.0

Where:G - Any correlatable property using the CSP.Tr - Reduced temperature.Pr - Reduced pressure.ω - Acentric factor characterizes the molecular size-shape.logP’r (Tr=0.7) - Base 10 logarithm of the reduced vapor pressure at reduced

temperature of 0.7.G(0)(Tr,Pr) - Calculated property considering the fluid as a simple fluid.G(1)(Tr,Pr,ω) - Correction function that changes due to molecular shape- size.

In the three-parameter CSP proposed by Castilla-Carrillo in 1983 (58), the molecular size-shape is appropriately characterized by the acentric factor and the correction function changes with the molecular size-shape.

In the same work, Castilla-Carrillo (58) proposed the following four-parameter CSP:

G= G(Tr,Pr)(0) + ωG(1)(Tr,Pr,ω) + ξG(2)(Tr,Pr,ω,ξ)

ω = -log P’r (Tr=0.7) - 1.0

ξ = B [1+(1+4C/B2)0.5]/2

B = 1.037824ω – 0.09573304

C = log P’r(Tr=0.6)/1.272854 – 0.005396275ω2 + 1.337863ω +1.215762

ω = ω - ξ

The restrictions must meet the values of ω y ξ are:

0 <= ω<= ω0 <= ξ <= ωω = ω + ξ

Where:G - Any correlatable property using the CSP.Tr - Reduced temperature.Pr - Reduced pressure.ω - Pitzer’s acentric factor.logP’r (Tr=0.7) - Base 10 logarithm of experimental reduced vapor pressure at

reduced temperature of 0.7.


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G(0)(Tr,Pr) - Calculated property considering the fluid as a simple fluid.G(1)(Tr,Pr,ω) - Correction due to molecular size-shape that changes with the

molecular size-shape.ξ - Polar factor proposed in my thesis.logP’r (Tr=0.6) - Base 10 logarithm of the experimental reduced vapor pressure at

reduced temperature of 0.6.G(2)(Tr,Pr,ω,ξ) Correction due to molecular size-shape-polarity that changes with

the molecular size-shape-polarity.

For the case of normal fluids it is assumed,ξ = 0ω = ω

In the four-parameter CSP proposed by Castilla-Carrillo (58), to predict the behavior of simple, normal and polar fluids, the behavior of simple fluid is provided by the van der Waals CSP. The third-parameter is used for characterization of molecular size-shape and the correction function changes with the third-parameter; this means changes with the molecular size-shape. The fourth parameter is used for the characterization of polarity of the molecules and the correction function changes with the third and fourth-parameter; this means changes with the molecular size- shape- polarity.

The only additional necessary information is an experimental vapor pressure point to a reduced temperature of 0.6

For the specific case of vapor pressure, the proposed four-parameter CSP takes the following form:

Ln Pr’ = Ln Pr’(0)(Tr) + ω Ln Pr’(1)(Tr, ω) + ξ Ln Pr’(2)(Tr, ω, ξ)

Where the values of ω y ξ were defined before and the generalized or universal functions are:

Ln Pr’(0)(Tr) = -5.928773 (1/Tr -1) -1.018383 Ln Tr + 0.1346956 (Tr7 -1)

Ln Pr’(1)(Tr,ω) = - (14.91911 + 2.568562 ω) (1/Tr -1)- (12.60737 + 4.373356 ω) Ln Tr+ (0.4271343 + 0.5203998 ω) (Tr7 -1)

Ln Pr’(2) (Tr,ω , ξ) = - (10.76377 + 45.56516 ω - 4.557136 ξ) (1/Tr -1)- (8.137270 + 50.35548 ω - 7.992805 ξ) Ln Tr+(0.6392783 - 1.691165 ω - 0.9771521 ξ) (Tr7 -1)

The proposed four-parameter CSP allows to predict the behavior of simple, normal and polar fluids with deviations below 1%.


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We note that for simple fluids the CSP proposed by Castilla-Carrillo (58) is reduced to 2 parameters CSP proposed by van der Waals in 1873 (55).For normal fluids is reduced to three-parameters CSP proposed by Pitzer in 1955 (6,36) but with a correction function that changes molecular size-shape.For abnormal or polar fluids, is necessary the full fourth-parameter CSP.


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I. INTRODUCTION.In the design of process equipment and plants, engineers are often found with the need for accurate values of thermophysical and transport properties for the fluids of process. These properties are essential for determining the size of equipment, power requirements, separation ratios and operating conditions. The reliability of these design calculations are always influenced by the accuracy of the data used during execution.With the arrival of computers new design concepts have been created. The use of optimization techniques for process design is widespread. These techniques require reliable data in a wide range of compositions and operating conditions. Small tolerances required in the final design, create the need for thermophysical property data and transport as accurately as possible.The most desirable method for obtaining design data is the experimental one. However, the number of industrially important chemical compounds is quite large. Experimental determinations of the interest properties for each one of the compounds over the entire PVT region could never be completed.The situation is even worse for the case of mixtures. To get an idea, let us consider the binary systems important paraffinic hydrocarbons in the set C1 to C10. The "API Data Book" of 1970 (2) lists 150 paraffins in this set. In order to document the behavior of vapor-liquid equilibrium of all binary systems at 10 different pressures, for 10 different temperatures, 20 million of experimental determinations are required.Fortunately, these extensive determinations are neither necessary nor desirable. Using knowledge on physical chemistry, molecular physics and mathematical techniques, engineers can predict or correlate data for a wide variety of systems, based on experimental data of some known systems. For this reason, the new experimental data should be obtained so as to allow the development and extension of correlations for predicting properties over wide ranges of temperature and pressure, applicable to new classes of compounds.

The thermophysical and transport correlations may be classified in the following basic categories:

1. Specific correlations.Which apply only to a property and one type of compound. These are obtained by fitting data to mathematical equations that lacks theoretical foundation. The true predictive power of these correlations is limited. Usually very little theoretical knowledge is required to develop these correlations and are rather considered as means of data storage and interpolation. Extrapolation outside of the experimental data used for its development is not advisable.

2. Generalized correlations.In these, the available experimental data for some classes of compounds are described for characteristic parameters, also called correlation parameters. The same correlation parameters can be used to estimate more than one property. Examples of these are: The critical temperature (Tc); the Critical pressure (Pc) and the Critical volume (Vc), among others. The correlation parameters are identified as those quantities for which variations from compound to compound may be related to the structure and nature of the


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compound. Ideally, the number of correlation parameters required must be small and all should be mutually independent.

3. Correlations based in the functional groups method .These consist in the definition of groups of atoms necessary for the composition of a molecule. Then you get the value of the contribution of each group to the property of interest by using experimental data of known molecules. Finally, the molecule in study is assembled with the calculated values of properties of the known groups. It really seems to be the solution to all problems, but until today it doesn’t work; its deviations are too large to be used in engineering design where data with deviations within the experimental error are needed. Experience has shown that value obtained from a CH3- functional group, is not the same in ethane (CH3-CH3) than it is in methanol (CH3-OH). The most accurate functional group correlations depends on an experimental value related to the desired property. Example, the best correlations for critical temperature depends on the normal boiling point temperature.

4. Correlations based on artificial intelligence algorithms.With the arrival of personal computers becoming more powerful every day, and less expensive, it is now possible the use of models called “artificial intelligence models”. They so called because they use one scheme of “learning” or “training”; another one of “validation” and finally the one for “prediction”. They work according to what you propose, like “functional groups”; “families of compounds” or whatever you are interested into. There are several algorithms, "back propagation", "forward propagation", "ARTMAP fuzzy logic" and others. You do not need a mathematical model and appear to be the solution to all problems. This will be discussed in Part 2 of this monograph.

5. Correlations based on molecular similarity.Is about to correlate the properties of the compound of interest using the known properties of similar compounds. Theoretically, if we have the properties of n-eicosane and n-triacontane, we can get the properties of all n-alkanes from C21 to C29. It seems a great idea and if working, would be a particularization of the CSP.

Once above these ratings will take care of our case that is the development of generalized correlations.The development of generalized correlations of physical properties requires knowledge about the molecular structure (size-shape) and about the nature of the intermolecular forces (polarity and hydrogen bonding) of substances to be included in the correlation, since these factors govern the observed macroscopic properties. Its use, however, is very simple and they can be used even with a manual calculator or a programmable cell phone.For its wide range in terms of easy application and coverage, generalized correlations have gathered great attention from physicists, chemists and chemical engineers. Many generalized correlations using different characterization parameters have been proposed. These are based upon the powerful correlational framework called Corresponding States Principle (CSP) also called corresponding states law. Some authors also name it corresponding states theorem since they consider it to be “a non-evident but demonstrable reality”.The Van der Waal’s state equation (55) is the first and simplest form of the corresponding states principle (CSP). With only two parameters: critical temperature (Tc) and critical pressure (Pc) its


25

correlative and predictive capacity is limited. It is only applicable to some noble gases (Ar, Kr and Xe) and to methane (CH4)The inclusion of a third parameter, Zc de Meissner and Zeferian, αc from Riedel or the acentric factor from Pitzer (ω), the capacity of CSP increases considerably. Its use is extended to non-polar and lightly polar compounds. As the most hydrocarbons are included in this classification, the three-parameter CSP is a powerful tool widely accepted and used in the petroleum industry. In its macroscopic form, the three-parameter CSP is able to accurately predict the thermo-physical and transport properties required for the sizing and design the necessary equipment in the processes and unit operations of the industry. The three-parameter CSP solves the problem of prediction and correlation of Thermophysical properties for non-polar and slightly polar compounds, however, for the case of polar and hydrogen bonding compounds (water, alcohols, aldehydes, ketones, and others) the three-parameter CSP shows significant deviations. Despite the efforts made, not found procedures and techniques to predict or correlate the behavior of these substances using the three-parameter CSP. It is then; at this point is a clear need to add a fourth parameter to the CSP.Numerous attempts to introduce a fourth parameter to CSP have been done. These attempts have not been entirely successful and the theories upon which are based are not completely understood nor accepted. In 1983, I proposed a four parameter PEC (58) for prediction of vapor pressures of simple, normal and polar fluids. To date, May 2012, the deviations obtained have not been matched or improved.At that time, my purpose was to develop the necessary parameters to extend the CSP to polar and hydrogen bonding compounds. To achieve this, I had to redefine the role of the correction function for three-parameter CSP, to propose a fourth parameter and the shape of the correction function for the four-parameter CSP. These parameters were developed with emphasis on the characterization of the effects of shape-size and polarity. I also emphasized to develop parameters that must be obtained with a minimum of experimental information. I also developed the idea of a correction function that change with the molecular shape-size for the three-parameter CSP and a correction function that change with the molecular size-shape-polarity for fourth-parameter CSP. Thus redefining the conceptual form of the CSP.

My four-parameter CSP was laid aside since I dedicated myself to work for a living and to learn other matters but I never lost sight the advances and evolution of CSP.

With the arrival of internet and the personal computers, I decided to show my fourth-parameter CSP. It was then I decided to publish my original thesis work:

http://www.scribd.com/doc/72775877/UNA-EXTENSION-DEL-PRINCIPIO-DE-ESTADOS-CORRESPONDIENTES-AN-EXTENSION-OF-THE-CORRESPONDING-STATES-PRINCIPLE


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This, with the purpose of allowing the scholars and researchers to see that it is possible to develop models that work and avoid working focused on wrong thesis and ideas. I also decided to update these developments to offer them at no cost to the engineering and scientist community of the entire world for its academic and personal use with nonprofit. However, individuals and companies that derive their income by charging consulting; sale simulators or usage of simulation programs and require of physical properties or are dedicated to sale physical property data, should contact me to obtain a written permission before including or referencing my correlations in their activities. This permit will be granted at no cost and the collected data will be treated confidentially and will be used exclusively for statistical and control purposes.

The present work purpose is then:1. To provide the knowledge of the proposed concepts included in my thesis, since I am

sure it will be of utility to specialized scholars and scientists in this area.2. To realize a revision and actualization to compare them with models developed for the

last 28 years by different groups of researchers.3. To donate the use of this model to community of Engineers and Scientists that do not

profit through their use.4. To promote the use of this model at industrial level and to grant the corresponding

permits to those who request it.

Correlations and ideas appearing herewith are appropriately registered at the international bureaus of industrial property, intellectual property and author’s copyright. Its commercial exploitation without the corresponding written authorization will result in legal sanctions that apply in each case.


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II. THE TWO PARAMETER CORRESPONDING STATES PRINCIPLE (CSP).

The corresponding states principle was discovered in 1873 when Van der Waals (55) proposed an empirical modification to the equation of state of the ideal gas to interrelate the state properties pressure, volume and temperature of fluids;

(P + a/V2) (V-b) = RT . . . [1]

Where:P - Absolute pressure.V - Molar volume.T - Absolute temperature.a, b - Specific constants for each substance.R - Ideal gas constant.

Constant “a” varies according to attractive intermolecular forces of each substance. Constant “b” is a measure of the “excluded” volume, occupied by the molecules, which is not available for the molecular movement.

But, how do we get to the theorem of corresponding states?

Let us clear P from the equation [1]

P = RT/(V-b) – a/V2 . . . [2] Let us apply the equation at the critical point;

Pc = RTc/(Vc-b) – a/Vc2 . . . [3]

Where:Pc - Critical pressure.Vc - Critical volume.Tc - Critical temperature.a, b - Specific constants for each substance.R - Ideal gas constant.

Considering that the critical isotherm has an inflection point with zero slope at the critical point:

(∂P/∂V)Tc = 0 . . . [4]

(∂2P/∂V2)Tc = 0 . . . [5]


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Applying conditions [4] and [5] to equation [3] we get:

- RTc/(Vc-b)2 + 2a/Vc3 = 0 . . . [6]

2RTc/(Vc-b)3 – 6a/Vc4 = 0 . . . [7]

Resolving simultaneously equations [6] y [7] we get:

b = Vc/3 . . . [8]

a = 9RTcVc/8 . . . [9]

Substituting [8] y [9] at [3] the critical compressibility factor is obtained:

Zc = PcVc/(RTc) = 3/8 . . . [10]

Equation [10] tells us that the van der Waals equation of state predicts a critical compressibility factor of 0.375 for all substances. This is incorrect because is a higher value than the reported experimental values and reflect of approximate nature of his equation of state.

Substituting [8] and [9] in [1];

(P+(9RTcVc/8)/V2) (V-Vc/3) = RT . . . [11]

Dividing both sides between Pc y Vc:

(P/Pc + (9RTcVc/(8Pc))/V2) (V/Vc-1/3) =RTc/(PcVc) . . . [12]

Substituting Zc = PcVc/(RTc) y re-arranging:

(P/Pc+9(1/Zc)(Vc/V)2/8)(V/Vc-1/3) = (T/Tc)(1/Zc) . . . [13]

Multiplying both sides by Zc;

(Zc(P/Pc)+9(Vc/V)2/8)(V/Vc-1/3) = T/Tc . . . [14]


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Defining the reduced conditions as the relation between the temperature, pressure or volume of the system and the corresponding one in the critical point:

Pr = P/Pc . . . [15]Tr = T/TcVr = V/Vc

Where:Pr - Reduced pressure.Tr - Reduced temperature.Vr - Reduced volumen.

Substituting deffinitions [15] in [14];

(ZcPr+9/(8Vr2))(Vr-1/3) = Tr . . . [16]

Since Zc has a unique value of 3/8 for all substances, equation [16] is a generalized function which contains only 2 independent variables.

As of this moment, the two-parameter CSP takes the following form:

G = G(Tc,Pc) or also . . . [17]G = G(Tr,Pr) sinceTr = T/Tc Pr = P/Pc

Where:G - Any correlatable property using the Corresponding State Principle (CSP).Tc - Critical temperature.Pc - Critical pressure.Tr - Reduced temperature.Pr - Reduced pressure.T - Temperature.P - Pressure.

The importance of equation [16] resides in the formal establishment of the two-parameter CSP which in its more general form tells us:

“ALL SUBSTANCES AT THE SAME CONDITIONS OF REDUCED TEMPERATURE AND PRESSURE WILL HAVE THE SAME REDUCED VOLUME”.

In other words, equation [16] establishes that the reduced volume is only a function of reduced temperature and of reduced pressure. There are other opinions about the shape of the function since this, from its origin is approximate.


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Once formally established, CSP began to be utilized to predict and correlate the behavior of substances. Soon, practicing demonstrated that this works with good precision for very few substances. In 1939, Pitzer(35) explained the limited predictive capacity of CSP. In the first part of his work, demonstrated that the two parameter CSP, equation [17], only works for spherical molecules such as Argon, Krypton and Xenon. For substances that follow the two parameter CSP behavior, he proposed the name of “perfect liquids”. In the second part of his work, Pitzer (35) explained why substances that have more complex molecules deviate from this behavior and propose for them the name of “imperfect liquids”.

It is from this moment (1939) that from a scientific point of view, establishing the need to add additional CSP characterization parameters if it is wished to apply it to more complex fluids than Argon, Krypton y Xenon.


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III. THE THREE-PARAMETER CORRESPONDING STATES PRINCIPLE (CSP).

When considering different substances (more complex) than Argon, Krypton and Xenon results that the behavior established by the equation [17], is inadequate for the describing their properties.The physical properties of the interest substances cannot be correlated only with Tc and Pc.In order to try and extend the application of the corresponding states principle (CSP) to more complex substances, attempts were made to add more characterization parameters to the basic parameters Tc and Pc already existing.

1. CSP proposed by Meissner y Seferian.One of the first CSP extensions was proposed by Meissner y Seferian(28). They observed that the critical compressibility factor should be identical for all substances if the reduced volume was in reality a two parameter function, but not being the case, the consideration of a unique compressibility factor utilized on van der Waals equation to reach the formal CSP proposition, is basically incorrect. It is for this reasson, they proposed the compressibility factor at the critical point as a third characterization parameter to be used in conjunction with the critical temperature and critical pressure.

Zc = PcVc/(RTc) . . . [18]

Where:Zc - Compressibility factor at the critical point.Pc - Critical pressure.Vc - Critical volumen.R - Constant volume.The ideal gas.Tc - Critical volumen.

Later on, Lydersen, Greenkorn y Hougen(27), used the critical compressibility factor as a third-parameter to develop generalized correlations. These correlations are presented in a tabular form as well as graphics and include the densities calculation and thermodynamic derived properties, based upon the formal extension proposition:

G = G(Tc,Pc,Zc) or else . . . [19]G = G(Tr,Pr,Zc) sinceTr = T/Tc Pr = P/Pc


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The work of Lydersen, Greenkorn and Hougen(27) extends the application of the corresponding state principle and improves the prediction of the previous correlations, but the fact of using the critical compressibility factor as a third parameter has its drawbacks; 1. The critical compressibility does not change regularly with the molecular size-shape and

polarity therefore they are not properly characterized.2. The inherent experimental error during the critical volume determination.

The introduction of Zc as third parameter attempts to correct the three-parameter CSP deviations due to molecular size-shape and polarity, because in the development of their correlations, Lydersen, Greenkorn y Hougen (27) used indiscriminately all kinds of “imperfect liquids”, this is non-polar substances, polar ones and hydrogen bonding. This increases the generality of the method but decreases accuracy. The accuracy of the Lydersen, Greenkorn and Hougen (27) method is good in the critical region, but decreases away from it.

2. CSP proposed by Riedel.Riedel (41), proposed a third parameter based on the slope of the reduced vapor pressure curve at the critical point.

αc = dLn Pr’/dLnTr | Tr=Pr=1 . . . [20]

Where:αc - Compressibility factor at the critical point.Pr’ - Reduced vapor pressure.Tr - Reduced temperature.Pr - Reduced pressure.

With the extension proposed by Riedel(41), the corresponding states principle (CSP) takes the following form:

G = G(Tc, Pc, αc) or else . . . [21]G = G(Tr,Pr, αc) sinceTr = T/Tc Pr = P/Pc

Riedel developed tabular correlations and graphics for the prediction of vapor pressures, vaporization enthalpies, surface tension and thermal conductivities as functions of these three-parameters (41, 42, 43, 44). These tables were made for hydrocarbons and were not included polar and hydrogen bonding compounds. Although these tables are less general than the tables from Lydersen, Greenkorn and Hougen (27), they are more accurate for non-polar compounds.


33


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3. CSP proposed by Pitzer.Pitzer et al (36, 37) developed a third parameter that according to their work, only takes in consideration the deviations due to molecular size-shape. He called this parameter the acentric factor, which was defined in terms of the deviation of base 10 logarithms from the reduced vapor pressure calculated by the two parameter CSP (simple fluid) with respect to the experimental vapor pressure at a reduced temperature of 0.7.

ω = log Pr’(0) (Tr=0.7)-log Pr’exp (Tr=0.7) . . . [21.1]

The two-parameter CSP predicts a reduced vapor pressure for simple fluids of 0.1 at a reduced temperature of 0.7 and the equation takes the form that we all know:

ω = -log Pr’exp (Tr=0.7) – 1 . . . [22]

Where:ω - Acentric factor.Pr’exp - Experimental reduced vapor pressure.Tr - Reduced temperature.

Pitzer selected the vapor pressure to define his third parameter because the effects of molecular interaction are more pronounced in the change of phase of vapor-liquid.

With Pitzer third parameter, CSP takes the following form:G = G(Tc, Pc, ω) or else . . . [23]G = G(Tr,Pr, ω) sinceTr = T/Tc Pr = P/Pc

Curl and Pitzer (6) and Pitzer et al (36, 37, 38, and 39) developed extensive tabular correlations for the prediction of fugacity, enthalpies, entropies, compressibility factors and vapor pressures. Correlations have the form of a term that considers a two parameter CSP plus the acentric factor multiplying to a term that is considered as the contribution or correction due to the molecular size-shape.

For the case of the prediction of the compressibility factor, the correlation takes the following form:

= Z(0)(Tr,Pr)+ ωZ(1)(Tr,Pr) . . . [24]


35

Where:Z(0)(Tr,Pr) - Universal or generalized compressibility factor, the fluid being under

study as a simple fluid.ω - Pitzer acentric factor, ec. [22].Z(1)(Tr,Pr) - Universal or generalized function for the correction of the deviations

due to molecular size-shape.

Pitzer also defined a new name for the “perfect liquids” mentioned in his previous work, he called them “simple fluids” and the “imperfect liquids” whose deviations to the behavior of the simple fluid is attributed to its molecular size-shape he called them “normal fluid”. These names and definitions are still valid and its use is common today.

To have a definition of the normal fluids, Curl y Pitzer based upon Riedel observations (42), presented the following equation:

σoVo2/3 /Tc = 1.86 + 1.18ω . . . [25]

Where:σo - Hypothetical superficial tension at absolute zero degrees (dina/cm).Vo

2/3 - Hypothetical molar volume at absolute zero degrees cm3/mol).

The hypothetical superficial tension and hypothetical molar volume necessary in the equation [25] are calculated using an experimental superficial tension point and another of any available Tr density and the tables presented by Curl y Pitzer(6). If the value obtained at evaluating the right side of the equation [25] has a maximum deviation of 5% with respect to the predicted one at the left side the fluid may be considered as a “normal fluid”.

Curl y Pitzer correlations (36, 37, 38, 39 and 6) for vapor pressures, enthalpies of vaporization, vaporization entropies, fugacity, enthalpies and entropies have been extended at lower reduced temperatures by Carruth y Kobayashi (5) due to its ample acceptance in the oil industry. The correlation for compressibility factors has been extended to very high reduced temperatures and pressures.Besides the wide acceptance of the correlations from Curl and Pitzer (6) and Pitzer et al (36, 37, 38 and 39) the third parameter proposed by Pitzer has shown to be useful in other correlations. For example, el acentric factor has been used in the correlation and prediction of the constants of many equations of state.

A revision of acentric factors based on the original definition by Pitzer was developed by Passut y Danner (33).


36

a. Modification by Lee-KeslerLee and Kesler (25) modified the Pitzer three-parameter CSP as follows:

Z = Z(0) + ω/ω(r) (Z(r)-Z(0)) . . . [25.1]

Where:Z - Compressibility factor predicted by the Lee-Kesler three-parameter CSP.Z(0) - Compressibility factor for the fluid considered as a simple fluid.ω - Pitzer’s acentric factor.ω(r) - Pitzer’s acentric factor for a reference fluid (r) of non-spherical

molecules. A value of 0.3978 was granted, corresponding to the acentric factor of n-octane.

Z(r) - Experimental compressibility of the referenced fluid (r).

The three-parameter modification to the CSP proposed by Lee-Kesler, equation [25.1] provides more exact predictions tan Pitzer’s model, since it effects an interpolation when the fluid has an acentric factor comprised in the 0 < ω < ω(r) interval and extrapolations for every fluid whose acentric factor is ω > ω(r).Lee and Kesler (35) developed an analytical representation of the tabular correlations developed by Pitzer and his collaborators.

Yuh-Jen and Lu (57) developed a tabular correlation for the compressibility factor , apparently more precise than the one proposed by Lee-Kesler (25).

It’s very important to mention that Lee-Kesler did not use equation [25.1] in the development of their correlation for vapor pressure.


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b. Modification by Teja.Teja (51) proposed a modification to the three-parameter CSP that uses two non-spherical reference fluids (r1) y (r2) and has the following form.

Z = Z(r1) + (ω- ω(r1)) /( ω(r2) - ω(r1)) (Z(r2) -Z(r1)) . . . [25.2]

Where:Z - Compressibility factor predicted by Teja three-parameter CSP.Z(r1) - Compressibility factor for the r1 reference fluid.ω - Pitzer acentric factor of the interest fluid or under study.ω(r1) - Pitzer acentric factor for a reference fluid (r1) on non-spherical

molecules.ω(r2) - Pitzer acentric factor for a reference fluid (r2) of non-spherical

molecules.Z(r1) - Experimental compressibility factor for the fluid of reference (r1).Z(r2 Experimental compressibility factor of the fluid of reference (r2).

The fluid of reference (r1) must have a minor acentric factor than the fluid of interest or under study. The fluid of reference (r2) must have a major acentric factor than the fluid of interest or under study.

In such a way that ω(r1)<ω<ω(r2).

So far I do not find in the open literature recommendations about generalized values of some property for the (r1) and (r2) fluids.

The correlative and predictive of Teja model (51) appears to be good according to Sorner work (66) who used it to correlate vapor pressures of 15 different compounds, including methane; ethane; propane; n-butane; neopentane; refrigerants and carbon tetrachloride. The average of absolute deviation was of 0.46%.

From my point of view, the Teja model (51) is rather a particularization of the CSP for families of compounds, than a generalization of the three-parameter CSP.


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c. Modification by Castilla-Carrillo.In May 1983, I wrote the following (58):

It is necessary to eliminate or to minimize at maximum the deviations resulting from the three-parameter CSP when it is applied to normal fluids with a larger molecular weight to avoid the carrying on the deviations obtained by the three-parameter CSP to the four-parameter one.

This idea is based on own observations about the deviations reported by the equation of vapor pressure developed by Lee-Kesler (25) using the Pitzer three-parameter CSP (36, 37,38 y 39) when applied to heavier normal fluids.These deviations can be clearly appreciated in page 35 of (58), which I am copying and adapting as follows:

Component No. CarbonAtoms

Acentricfactor

AAD%Lee-Kesler

Methane 1 0.0077 0.66Ethane 2 0.0958 1.00n-propane 3 0.1511 1.18n-butane 4 0.1985 1.14n-pentane 5 0.2526 2.40n-hexane 6 0.3008 2.41n-heptane 7 0.3509 1.94n-octane 8 0.3974 2.41n-nonane 9 0.4517 1.64n-decane 10 0.5011 1.50n-undecane 11 0.5539 1.81n-dodecane 12 0.6073 2.14n-tridecane 13 0.6614 2.62n-tetradecane 14 0.7150 3.11n-pentadecane 15 0.7708 3.63n-hexadecane 16 0.8260 4.17n-heptadecane 17 0.8847 5.90n-octadecane 18 0.9361 6.46n-nonadecane 19 0.9892 7.87n-eicosane 20 1.0471 9.10

Table 1. Deviations presented by de Lee-Kesler equation for vapor pressure prediction for C1 to C20 linear hydrocarbons.

AAD% = 100/N Σ abs [(P’r calc – P’r exp)/P’r exp]


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My observations are as follows:1. The acentric factor increases as the size of the lineal chain increases and

consequently the molecular weight, which makes us conclude that the effects of size-shape are appropriately characterized by it; however, the deviations also increase as the molecular size and weight increases.

2. If the acentric factor correctly characterizes the molecular size-shape, the correction function is the only responsible for the deviations.

In his original work Pitzer (36, 37) proposes a three-parameter CSP for normal fluids and show us an equation that has been inadvertently overviewed by all, and is the solution to this problem:

Z = Z(0) + ω(∂Z(1)/∂ω) . . . [25.3]

Which later Pitzer converted it in the expression we know:

Z = Z(0) + ωZ(1) . . . [25.4]

Equation 25.3 is an exact mathematical expression and its deviation should be 0% for normal fluids, if they follow the three-parameter CSP proposed by Pitzer and if the function (∂Z(1)/∂ω) is expressed in the right way.

Equation 25.4 is a simplification and a way to express equation 25.3 but is not the right one since it presents deviations shown on Table 1.

Based on observations 1 and 2 on deviations shown on table 1, the equation [25.3] from the work of Pitzer and the modifications proposed by Lee-Kesler (25) and Teja (51), I proposed in 1983 (58) the following equation:

Z = Z(0) + ωZ(1)(ω) . . . [25.5]

Where:Z - Compressibility factor predicted by Teja’s three-parameter CSP for

the fluid under study.Z(0) - Compressibility factor for the simple fluid. Is the same one from

Pitzer proposal for a three-parameter CSP, or the same one proposed by Van der Waals.

ω - Pitzer acentric factor.Z(1)(ω) - Correction for molecular size-shape deviations. This function is

different for each acentric factor because is a function of this.


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Applying the proposed equation [25.5] to calculate vapor pressures, we have:

Ln Pr’ = Ln Pr’(0)(Tr)+ ωLn Pr’(1)(Tr,ω) . . . [25.6]

Where:

Ln Pr’(0)(Tr) = -5.928773 (1/Tr -1) -1.018383 Ln Tr + 0.1346956 (Tr7 -1)

. . . [25.7]

ω - Pitzer’s acentric factor calculated according to equation (22).

Ln Pr’(1) (Tr,ω) = - (14.91911 + 2.568562 ω) (1/Tr -1)- (12.60737 + 4.373356 ω) Ln Tr+ (0.4271343 + 0.5203998 ω) (Tr7 -1)

. . . [25.8]

The three-parameter CSP I proposed in equations [25.5, 25.6, 25.7 y 25.8] is an improvement to proposed by Pitzer (35, 36, 37, 38 and 39). The improvement consists in that the correction function changes with the molecular size-shape. In other words, the correction function is not only one as suggested by Pitzer, or interpolations between 2 fluids (one spherical and another one of reference (r)) as suggested by Lee-Kesler (25) nor interpolations either between 2 non-spherical reference fluids as suggested by Teja (51 y 52).

It is a continuous correction function that changes with the molecular size-shape and it works very well. Maybe it lacks some mathematical refinements but this is the general form.

The percentage of average absolute deviations (AAD) gotten in the calculation of the reduced vapor-pressure of n-alkanes from C1 to n-C20 can be appreciated in Table2. In all cases the proposed model (58) was more accurate than the Lee-Kesler (25).



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In my original work (58), table 1 (pages 39 and 40) the following deviations can be appreciated:

Component No. CarbonAtoms

Acentric factor

AAD%Lee-Kesler

AAD%Castilla-Carrillo

Methane 1 0.0077 0.66 0.38Ethane 2 0.0958 1.00 0.66n-propane 3 0.1511 1.18 0.73n-butane 4 0.1985 1.14 0.47n-pentane 5 0.2526 2.40 0.76n-hexane 6 0.3008 2.41 0.68n-heptane 7 0.3509 1.94 0.41n-octane 8 0.3974 2.41 0.88n-nonane 9 0.4517 1.64 0.44n-decane 10 0.5011 1.50 0.56n-undecane 11 0.5539 1.81 0.39n-dodecane 12 0.6073 2.14 0.33n-tridecane 13 0.6614 2.62 0.41n-tetradecane 14 0.7150 3.11 0.36n-pentadecane 15 0.7708 3.63 0.42n-hexadecane 16 0.8260 4.17 0.31n-heptadecane 17 0.8847 5.90 0.33n-octadecane 18 0.9361 6.46 0.31n-nonadecane 19 0.9892 7.87 0.53n-eicosane 20 1.0471 9.10 0.57

Table 2.Deviations from the models by Lee-Kesler and Castilla-Carrillo for vapor pressure prediction of C1 to C20 linear hydrocarbons. Data for critical temperature (Tc), critical pressure (Pc) and experimental points for reduced vapor pressure at Tr=0.7 were taken from the work of Gomez-Nieto and Papadopoulos (13). The Pitzer acentric factor (ω) was calculated using its original definition Eq. [22].

The most precise generalized equation available in 1983, for calculation or prediction of vapor pressure, is the Lee-Kesler (25). The percentage average absolute deviations (AAD%) ranges from 0.66% for methane to 9.10% for n-eicosane, while the proposed model deviations ranges from 0.38% for methane to 0.57% for n-eicosane.

The AAD% for reduced vapor pressure calculation or prediction to 98 simple and normal fluids including some noble gases, n-alkanes, iso-alkanes, cyclo-alkanes, n-alkenes, iso-alkenes, n-alkenes, alkynes and aromatics for 5,931 experimental data points, is 0.76%.In all cases the proposed model (58) was more accurate than the model from Lee-Kesler (25).


Used data and deviations are shown in Table 2.1


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Component TcK

PcAtm

ωFac Ac

IntervalTr

No. ofPoints

AAD%L-K

AAD%Prop

Ref.

Argon 150.60 48.00 -0.0017 0.56-1.0 171 0.27 0.25 101, 102Krypton 209.40 54.17 -0.0013 0.55-1.0 95 0.48 0.33 101, 102Xenon 289.75 57.64 0.0030 0.56-1.0 74 0.22 0.18 101, 102Methane 191.04 46.06 0.0077 0.47-1.0 182 0.66 0.38 101, 102Ethane 305.44 48.20 0.0958 0.43-1.0 138 1.00 0.66 101, 102n-Propane 369.98 42.01 0.1511 0.45-1.0 160 1.18 0.73 101, 102n-Butane 425.18 37.47 0.1985 0.46-1.0 104 1.14 0.47 101, 102n-Pentane 465.79 33.31 0.2526 0.44-1.0 127 2.40 0.76 101, 102n-Hexane 507.87 29.94 0.3008 0.48-1.0 152 2.41 0.68 101, 102n-Heptane 540.18 27.00 0.3509 0.50-1.0 155 1.94 0.41 101, 102n-Octane 569.37 24.54 0.3974 0.47-1.0 133 2.41 0.88 101, 102n-Nonane 593.80 22.60 0.4517 0.53-0.76 51 1.64 0.44 101, 102n-Decane 616.10 20.70 0.5011 0.54-0.77 50 1.50 0.56 101, 102n-Undecane 636.00 19.18 0.5539 0.55-0.78 50 1.81 0.39 101, 102n-Dodecane 653.90 17.83 0.6073 0.56-0.80 52 2.14 0.33 101, 102n-Tridecane 670.10 16.64 0.6614 0.57-0.81 45 2.62 0.41 101, 102n-Tetradecane 684.90 15.58 0.7150 0.58-0.82 42 3.11 0.36 101, 102n-Pentadecane 698.20 14.64 0.7708 0.59-0.83 41 3.63 0.42 101, 102n-Hexadecane 710.40 13.79 0.8260 0.60-0.84 47 4.17 0.31 101, 102n-Heptadecane 721.30 13.14 0.8847 0.60-0.84 31 5.90 0.33 101, 102n-Octadecane 731.20 12.31 0.9361 0.61-0.85 31 6.46 0.31 101, 102n-Nonadecane 740.30 11.67 0.9892 0.62-0.86 31 7.87 0.53 101, 102n-Eicosane 748.70 11.09 1.0471 0.63-0.87 31 9.10 0.57 101, 1022-Methyl propane 409.20 36.36 0.1787 0.46-0.68 40 2.02 0.37 101, 1022-Methyl butane 460.56 33.48 0.2288 0.47-0.70 51 2.88 0.73 101, 1022,2-Dimethyl propane 433.00 31.74 0.2060 0.59-0.70 21 1.08 0.14 101, 1022-Methyl pentane 498.70 29.98 0.2723 0.48-0.72 45 1.41 0.55 101, 1023-Methyl pentane 504.00 31.40 0.2827 0.48-0.71 45 2.64 1.07 101, 1022,2-Dimethyl butane 491.14 30.94 0.2235 0.47-0.70 51 2.34 0.68 101, 1022,3-Dimethyl butane 500.52 31.43 0.2510 0.48-0.71 44 2.55 0.65 101, 1022-Methyl hexane 532.20 26.99 0.3178 0.49-0.73 51 1.69 0.64 101, 1023-Methyl hexane 535.42 28.05 0.3264 0.50-0.73 51 2.38 0.47 101, 1023-Ethyl pentane 541.10 29.21 0.3168 0.49-0.73 49 2.62 0.59 101, 1022,2-Dimethyl pentane 519.76 28.00 0.3005 0.49-0.73 49 2.39 0.99 101, 1022,3-Dimethyl pentane 537.87 29.28 0.3011 0.49-0.72 51 2.84 0.77 101, 1022,4-Dimethyl pentane 522.27 27.10 0.3064 0.49-0.73 50 2.64 0.64 101, 1023,3-Dimethyl pentane 536.52 30.19 0.2774 0.48-0.72 50 2.80 1.50 101, 1022,2,3-Trimethyl butane 533.66 29.93 0.2452 0.47-0.71 50 2.87 0.83 101, 1022-Methyl heptane 556.96 24.69 0.4038 0.51-0.75 51 3.23 1.49 101, 1023-Methyl heptane 564.02 25.42 0.3722 0.51-0.74 51 1.70 0.38 101, 1024-Methyl heptane 562.01 25.33 0.3729 0.51-0.74 31 1.51 0.42 101, 102

Table 2.1 Used data and deviations by the Lee-Kesler (25) and proposed models (58) for vapor pressures of 98 fluids and 5931 points.


43

Componente TcK

PcAtm

ωFac Ac

IntervaloTr

No.Puntos

AAD%L-K

AAD%Prop

Ref.

3-Ethyl hexane 566.60 26.22 0.3598 0.50-0.74 31 1.52 0.45 101, 1022,2-Dimethyl hexane 550.27 25.32 0.3412 0.50-0.74 31 1.56 0.37 101, 1022,3-Dimethyl hexane 564.97 26.35 0.3412 0.50-0.73 31 1.42 0.55 101, 1022,4-Dimethyl hexane 554.56 25.43 0.3378 0.50-0.74 31 0.99 0.93 101, 1022,5-Dimethyl hexane 549.07 24.86 0.3663 0.50-0.74 31 2.15 1.07 101, 1023,3-Dimethyl hexane 563.81 26.81 0.3198 0.50-0.72 31 2.92 1.01 101, 1023,4-Dimethyl hexane 568.53 27.24 0.3517 0.50-0.73 31 2.80 1.06 101, 1022-Methyl, 3-ethyl pentane 568.30 27.24 0.3298 0.50-0.72 31 1.86 0.29 101, 1023-Methyl, 3-ethyl pentane 577.71 28.84 0.3095 0.49-0.72 31 2.92 1.60 101, 1022,2,3 Trimethyl pentane 566.68 27.86 0.2904 0.49-0.72 31 2.46 0.63 101, 1022,2,4 Trimethyl pentane 543.64 25.64 0.3087 0.49-0.73 51 2.33 0.56 101, 1022,3,4 Trimethyl pentane 567.91 27.46 0.3120 0.49-0.73 51 2.01 0.27 101, 1022,3,3 Trimethyl pentane 567.12 29.34 0.3077 0.67-0.72 17 1.22 0.53 101, 102Etene 283.10 50.30 0.0843 0.42-1.00 82 0.97 0.92 101, 102Propene 365.00 45.60 0.1419 0.44-1.00 61 1.09 0.91 101, 1021-Butene 419.60 39.70 0.1902 0.43-1.00 61 1.31 0.68 101, 1022 cis-Butene 435.20 40.90 0.2020 0.46-0.68 44 1.86 0.90 101, 1022 trans-Butene 430.20 41.20 0.2083 0.46-0.68 46 4.02 1.60 101, 1021-Pentene 464.20 34.95 0.2407 0.47-0.70 39 2.81 0.87 101, 1022 cis-Pentene 474.80 35.95 0.2494 0.47-0.72 45 1.84 0.23 101, 1022 trans-Pentene 473.90 35.88 0.2483 0.47-0.72 45 1.85 0.25 101, 1021-Hexene 503.80 31.22 0.2856 0.48-0.71 44 2.21 0.36 101, 1021-Heptene 537.50 28.11 0.3311 0.47-0.73 51 2.08 0.39 101, 1021-Octene 566.80 25.50 0.3785 0.51-0.74 49 1.57 0.54 101, 102Propadiene 385.86 52.37 0.1845 0.45-0.67 31 5.47 4.60 101, 1021,2-Butadiene 450.98 45.36 0.1986 0.45-0.67 31 2.87 1.17 101, 1021,3-Butadiene 425.20 42.80 0.1934 0.38-1.00 60 2.47 0.83 101, 1021,2-Pentadiene 491.92 38.87 0.2390 0.47-0.70 31 1.73 1.11 101, 1021,3 cis-Pentadiene 485.71 38.54 0.2717 0.47-0.70 31 3.24 1.49 101, 1021,3 trans-Pentadiene 485.62 38.19 0.2457 0.47-0.70 31 2.19 0.64 101, 1021,4 Pentadiene 458.00 36.90 0.2556 0.42-0.70 43 3.60 1.67 101, 1022,3 Pentadiene 492.12 39.42 0.2838 0.48-0.70 31 2.39 0.63 101, 102Etine 309.65 61.60 0.1823 0.62-1.00 42 1.32 0.49 101, 102Propine 391.75 47.58 0.2577 0.47-0.81 33 3.00 1.27 101, 1021-Butine 436.63 43.86 0.2661 0.44-0.69 45 3.42 0.61 101, 1022-Butine 471.33 47.25 0.2487 0.51-0.68 27 2.34 0.41 101, 1021-Pentine 474.76 37.78 0.3042 0.48-0.70 28 2.83 0.85 101, 1022-Pentine 504.06 39.62 0.2833 0.47-0.70 28 2.77 0.75 101, 102Cyclopropane 401.70 57.00 0.1153 0.45-0.70 14 2.27 1.48 101, 102Cyclobutane 464.40 50.29 0.1579 0.43-0.62 22 2.86 0.58 101, 102Cyclopentane 512.10 44.60 0.1972 0.44-1.00 58 2.23 0.50 101, 102

Table 2.1 (Continuation).Used data and deviations by the Lee-Kesler (25) and proposed models (58) for vapor pressures of 98 fluids and 5931 points.


44

Componente TcK

PcAtm

ωFac Ac

IntervaloTr

No.Puntos

AAD%L-K

AAD%Prop

Ref.

Methyl cyclopentane 534.20 37.44 0.2238 0.46-0.70 47 1.58 0.72 101, 102Ethyl cyclopentane 570.80 33.56 0.2643 0.48-0.70 51 2.07 0.38 101, 102Cyclohexane 553.20 39.80 0.2123 0.51-1.0 108 1.29 0.93 101, 102Methyl cyclohexane 570.90 34.18 0.2447 0.47-0.70 51 2.57 0.38 101, 102Ethyl cyclohexane 603.40 30.90 0.2992 0.49-0.72 38 2.57 0.59 101, 102Cycloheptane 593.20 36.30 0.2970 0.57-0.73 16 3.29 2.17 101, 102Cyclooctane 626.10 33.07 0.3740 0.59-0.75 16 4.72 3.56 101, 102Benzene 562.20 48.50 0.2132 0.49-1.0 143 1.20 0.65 101, 102Toluene 593.50 41.36 0.2590 0.47-1.0 108 1.79 0.80 101, 102Ethyl benzene 621.10 36.31 0.2886 0.48-0.70 51 2.47 0.70 101, 102o-Xylene 632.10 36.83 0.2990 0.48-0.70 51 1.69 0.79 101, 102m-Xylene 620.10 36.01 0.3151 0.49-0.71 51 2.25 0.67 101, 102p-Xylene 618.20 35.01 0.3119 0.49-0.71 51 2.57 0.42 101, 102Naphthalene 749.70 39.10 0.2909 0.48-0.70 47 1.98 1.08 101, 1021-Methyl naphthalene 769.30 34.39 0.3519 0.49-0.72 48 2.54 0.21 101, 1022-Methyl naphthalene 764.30 34.39 0.3488 0.49-0.72 48 1.75 0.76 101, 102Table 2.1 (Continuation)

Used data and deviations by the Lee-Kesler (25) and proposed models (58) for vapor pressures of 98 fluids and 5931 points. Data for critical temperature (Tc), critical pressure (Pc) and experimental points for reduced vapor pressure at Tr=0.7 were taken from the work of Gomez-Nieto and Papadopoulos (13). The Pitzer acentric factor (ω) was calculated using its original definition Eq. [22].

Based on deviations from tables 2 y 2.1 we can conclude that:

1. The proposed model meets the accuracy requirements by designers of new products, designers of plants and new processes.

2. Estimated or correlated data values can be used in equipment sizing.3. The three-parameter CSP does not add or carry deviations to the four-parameter

CSP.


45

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LEFT IN BLANK)


46

IV. THE FOUR-PARAMETER CORRESPONDING STATES PRINCIPLE (CSP)1. CSP proposed by Eubank y Smith.

One of the first four-parameter CSP was developed by Eubank y Smith (9). They developed compressibility factors and enthalpies correlations at the vapor phase. The method was developed as an extension of the Pitzer et al (36 y37). Correlations developed by Curl y Pitzer (36, 37, 38 y 39) were used to take into account the spherical molecules properties (simple fluid) and the deviations of size-shape (normal fluid). Due to that the polar molecules exhibit much larger acentric factors tan the one that could indicate its size-shape measure, the homomorph idea was used. This idea was originally proposed by Bondi and Simkin (4). The acentric factor of polar material is taken as the acentric factor of its homomorph hydrocarbon. For example, the acentric factor of ethylic alcohol is that of the propane, which its homomorph hydrocarbon is. Also, the critical properties of the homomorph are taken as the critical properties of the fluid under study.The fourth parameter, utilized to take into account the effects of polarity, was determined as of the reduced dipolar moment which is defined in terms of the Stockmayer intermolecular potential function.

μr = μ/(Єσ3)1/2 . . . [26]

Where:

μr - Reduced dipolar moment.Μ - Dipolar moment.Є, σ - Constants of the Stockmayer intermolecular potential function.

The fourth parameter was defined as:

P* = C μrn . . . [27]

Where:

P - Fourth parameter proposed by Eubank y Smith.C - Specific constant for each substance.μr - Reduced dipolar moment. n - Exponent equal to 5/3.

For the case of compressibility factor, we have:

Z = Z(0)(Trh,Prh) + ωh Z(1)(Trh,Prh) + P*Z(2)(Tr,Pr) . . . [27.1]

Trh = T/TchPrh = P/PchTr = T/TcPr = P/Pc


47

Where:

Z - Compressibility factor of the fluid under study or of interest.Trh - Homomorph reduced temperature.Prh - Homomorph reduced pressure.Tch - Homomorph critical temperature.Pch - Homomorph critical pressure.Z(0) - Homomorph compressibility factor of the fluid under study or of interest,

considered as a simple fluid.ωh - Homomorph acentric factor.Z(1) - Function to correct the deviations due to molecular size-shape effects

presented by the homomorph compressibility factor.P* - Fourth parameter proposed by Eubank-Smith.Z(2) - Function to correct deviations due to molecular polarity effects presented

by the compressibility factor.Tr - Reduced temperature.Pr - Reduced pressure.Tc - Critical temperature.Pc - Critical pressure.

Eubank y Smith’s CSP has the following form:

G = G(Tch, Pch, ωh, P*,Tc, Pc) or else . . . [28]G = G(Trh, Prh,ωh, P*,Tr, Pr ) sinceTrh = T/Tch Prh = P/PchTr = T/TcPr = P/Pc

Eubank y Smith proposed extension represents an advance in the application of the CSP to polar substances, but it has the following drawbacks:

1. Arbitrariness of constant C, completely destroys the rigorous theoretical image created while using the reduced dipolar moment.

2. The dipolar moment in itself is not completely able of characterize the polar behavior.

3. No previsions were made for inorganic substances that have a difficult to find homomorph, which limits the correlation to the polar substances that have a homomorph hydrocarbon.

4. It is a CSP that uses six characterization parameters and even so, it does not work.

In spite of these inconveniences, Eubank y Smith’s (9) work has the importance to be the first formal intent of extending the three-parameter CSP proposed by Pitzer, to four-parameters in order to try to include polar substances within the same correlational framework.


48

2.CSP proposed by Thompson.The best fundamentally based four-parameter CSP is the one proposed by Thompson (53), who observed that the third parameter proposed by Pitzer (35,36,37,38 y 39) To characterize the behavior of the normal fluids, could not be utilized for abnormal or polar fluids since it was not developed for that task. It was developed to characterize the deviations of molecular size-shape, dominant in the normal fluids and thus it does not work to characterize the deviations presented in the abnormal or polar fluids. In other words, Pitzer’s acentric factor calculated for abnormal or polar fluids, includes besides the deviations caused by the molecular size-shape, deviations caused by molecular polarity. In an attempt to correlate these deviations, Thompson (53) developed a third parameter in terms of the molecular structure of the substances to which he called the “true acentric factor”.The “true acentric factor” was defined as a function of the “molecular modified geometrical radius of gyration”, which was expressed for molecules that have a three dimensional configuration as:

R = (2π(ABC)1/3/M)1/2 . . . [30]

And for two dimensional molecules as:

R = (2π(AB)1/2/M)1/2 . . . [31]

Where:

R - Geometrical radius of gyration in angstroms.A,B,C - Principal moments of inertia calculated using only the molecular

configuration.M - Molecular weight.

In the case of molecules with more than one structural configuration, the inertia moments were taken as an average value. Later on, Thompson developed a ω, acentric factor plot as a geometrical ratio function for normal fluids, obtaining thus a correspondence between the radius of gyration and the “true acentric factor” for the abnormal or polar fluids. This relation is represented by the following equations:

ω = 0.01533 R + 0.00767 R for 0 <R< 3.5ω = 0.115 R – 0.1885 for 3.5 <R< 6.0ω = 0.6775 + 0.04225 R – 2.58 / R for 6.0 <R

. . . [32]Once developed, the third parameter in function, just of the molecular size-shape effects, the fourth parameter, to characterize the effects of polarity was defined as the difference between the acentric factor value by Pitzer (36,37) and the “true acentric factor”.


49

τ = ω – ω . . . [33]

where:τ - The fourth parameter proposed by Thompson to characterize the

molecular polarity.ω - Pitzer’s acentric factor.ω - “True acentric factor” developed by Thompson.

In accordance with equation [33], the Pitzer acentric factor calculated for polar substances is the sum of two contributions: one contribution for size-shape and another for polarity. For non-polar substances equation [33] is reduced to:

ω = ω . . . [34]

With the addition of the fourth parameter proposed by Thompson(53), CSP takes the following form:G = G(Tc, Pc,ω, τ) or else . . . [35]G = G(Tr,Pr,ω, τ) sinceTr = T/TcPr = P/Pc

To demonstrate the predictive capacity of the developed parameters, Thompson generalized the Frost-Kalkwarf(11) equation for the prediction of vapor pressures and developed an expression for the critical compressibility factor calculation. Both equation have the following form:

G = G(0) + ωG(1) + τ G(2) + ωτ G(3) . . . [36]

Where: G - Any correlatable property using the CSP.G(0) - Property of the interest fluid or under study considered as a simple fluid.ω - “True acentric factor” developed by Thompson.G(1) - Function for correction of the deviations due to molecular size-shape.τ - Fourth parameter developed by Thompson.G(2) - Function for the correction of deviations due to molecular polarity.G(3) - Function for the correction of deviation of inseparable size-shape-polarity.

The fourth parameter CSP proposed by Thompson is today the best intent to correlate the deviations by molecular size-shape and the deviations by molecular polarity but it presents the following drawbacks:


50

1. The presence of a cross term that talks about inseparable effects of size-shape-polarity, contradicts the definition expressed by equation [33].

2. Data getting and processing data for molecular configurations is more complicated and costly than the experimental measurement of the interest property.

3. Thompson model predictions for vapor pressure are not good.

3. CSP proposed for Halm-Stiel.In an attempt to characterize the behavior for polar compounds, Halm-Stiel (15) Added an extra parameter to Pitzer correlation (35 ,36 ,37 and 39). The extra or fourth parameter was developed to correct the deviations presented by Pitzer CSP when applied to polar substances and was defined in a similar way to the acentric factor, being the basic difference of base 10 logarithm of experimental reduced vapor pressure less base 10 logarithm of the reduced vapor pressure that predicts Pitzer CSP for normal fluids at a Tr = 0.6, this is:

χ = log10 Pr’ exp (Tr=0.6) – log10Pr’(n)(Tr=0.6) . . . [37]

But:

– log10Pr’(n)(Tr=0.6) = 1.57ω + 1.552 . . . [38]

Then:

χ = log10 Pr’ exp (Tr=0.6) + 1.57ω + 1.552 . . . [39]

Where:

χ - Fourth parameter proposed by Halm-Stiel to characterize the behavior of the abnormal or polar fluids.

log10 Pr’ exp (Tr=0.6) - Base 10 logarithm of the experimental reduced vapor pressure at a reduced temperature of 0.6.

log10Pr’(n)(Tr=0.6) - Base 10 logarithm of the reduced vapor pressure calculated with Pitzer CSP at a reduced temperature of 0.6.

ω - Pitzer acentric factor.

With Halm- Stiel proposed fourth parameter, the CSP takes the following form:

G = G(Tc, Pc, ω, χ) or else . . . [40]G = G(Tr, Pr, ω, χ) sinceTr = T/TcPr = P/Pc


51

For the case of vapor pressure, Halm-Stiel proposed the following equation:

log10 Pr’= log10Pr’(0) + ω log10 Pr’(1) + χlog10 Pr’(2) . . . [41]

Where:

log10 Pr’ - Base 10 logarithm of the reduced vapor pressure of the fluid under study or of interest predicted by the CSP of Halm-Stiel.

log10 Pr’(0) - Base 10 logarithm of the reduced vapor pressure of the fluid under study or of interest predicted in the three-parameter CSP by Pitzer, considering the fluid as a simple fluid.

ω - Pitzer acentric factor.log10 Pr’(1) - Correction function that considers the fluid under study or of interest

as normal fluid.χ - Fourth parameter proposed by Halm-Stiel To characterize the behavior

of the abnormal or polar fluids.log10 Pr’(2) - Correction function that considers the fluid under study or of interest

as an abnormal or polar fluid.

Halm-Stiel suggested that the polarity correction is necessary only at reduced temperatures below de 0.7 consequently, the term log10Pr’(2) should be only used under these conditions. At reduced temperatures higher than 0.7, the correlation of Halm-Stiel is identical to Pitzer.Halm-Stiel (15, 16, 17) developed tabular correlations for the vaporization entropy, liquid densities, saturated vapor and virial coefficients. Yuan y Stiel (56) developed a correlation to calculate calorific capacities at the saturation zone.Stipp, Bai y Stiel (47) developed tabular correlations to calculate compressibility factors in the gaseous and liquid regions.Hung-Stiel (25) developed a correlation to calculate the second virial coefficient of polar fluids.Kalback-Starling (24) developed a generalization of the equation of state by Lee-Kesler for the calculation of the compressibility factor in liquid phase, vapor or gas for polar fluids.These correlations are complex and have terms of mayor order in ω y χ.For the case of the second virial coefficient we have:

B Pc/(RTc) = f(0)(Tr) + ω f(1)(Tr) + χ f(2)(Tr) + χ2f(3)(Tr) + ω6 f(4)(Tr) + ωχ f(5)(Tr). . . [42]

Many of the terms on equation [42] are irregular and cannot be represented analytically in a simple form. The most important failure is that the χ parameter is incongruent in itself, since even inside the same group of compounds presents inconsistencies as in the case of alcohols. Alcohols have approximately the same dipolar moment (1.7 +/- 0.03 Debyes), while χ has values of


52

0.037 for methanol, 0.003 for ethanol and -0.057 for n-propanol.

Also, a very polar component might have an χ = 0, parameter, while a non-polar compound might have an χ different than zero. Finally the χ parameter is highly sensible to small errors in the vapor pressure data utilized to calculate it.The complexities and inconsistencies due to the lack of a correlational adequate surrounding are very clear in the four-parameter CSP by Halm-Stiel.

4. CSP proposed by Harlacher.Harlacher (18,19) revised the method of trying to provide a simplified calculation procedure and found a correspondence between the parachor defined by Sugden (48) and the Thompson radius of gyration.

The parachor was defined by Sugden (48) as:

Pa = γ1/4 (M/(ρl - ρv)) . . . [43]

Where:Pa - Parachor.γ - Superficial tensión.M - Molecular weight.ρl - Density of a saturated liquid.ρv - Density of saturated vapor.

Later on, Quayle (40) developed a group contribution method for the calculation of the parachor. This procedure eliminates the need for equation (43) but it does not make differences between isomers. For easy to calculate the parachor, Harlacher decided to use it to characterize the deviations for the molecules size-shape.

ω = a Pa + b . . . [44]

The fourth parameter was defined by equation [45] proposed by Thompson:

τ = ω – ω . . . [45]

Harlacher substituted equations [44] y [45] in equation [46] proposed also by Thompson.

G = G(0) + ωG(1) + τ G(2) + ωτG(3) . . . [46]

Arranging:


53

G = G(0) + a Pa G(1)’ + ωG(2) + ωPaG(3) – Pa2 G(3)’’ . . . [47]

G(1)’= G(1) - G(2)

G(3)’= a G(3)

G(3)’’= a2 G(3)

Where:G - Any correlatable property using CSP.G(0) - Universal function that considers the fluid under study as a simple fluid.Pa - Parachor.ω - Pitzer acentric factor.a, b - Are slope and intercept respectively of equation 44.G(1)

G(2)

G(3)

G(1)’

G(3)’’

-----

Generalized or universal correction functions necessary in the Harlacher model.

According to Harlacher the corresponding states principle has the following form:

G = G(Tc, Pc, ω, Pa) or else . . . [48]G = G(Tr, Pr, ω, Pa) sinceTr = T/TcPr = P/Pc

Harlacher (18, 19) developed correlations to calculate vapor pressures, critical compressibility factors and densities of saturated liquid and vapor using the proposed extension. All his correlations are more complicated and less accurate than those of Thompson, therefore are not used.

5. CSP proposed by Passut.Passut (31, 32) utilized the turn ratio proposed by Thompson directly as a third parameter. Because of the fact that the radius of gyration is strictly defined from structural considerations, the presence of polar effects theoretically does not affect its magnitude. This provides a possibility for the separation and correlation of deviations caused by the size-shape and polarity.Passut selected the Frost-Kalkwarf (11) equation as the base to define his fourth parameter. The reduced form of Frost-Kalkwarf equation is:

Ln Pr’ = B (1-1/Tr) + C ln 1/Tr + D (Pr’/Tr2 -1) . . . [49]


54

Where:

Pr’ - Reduced vapor pressure.Tr - Reduced temperature.B, C - Frost-Kalkwarf equation constants specific for each substance.D - Constant with a unique value of 0.4218

Passut eliminated the constant C forcing equation [49] to go through the normal boiling point.

C = (Ln Prb’ – B (1-1/Trb) – 0.4219 (Prb’/Tr2 -1)) / Ln 1/Trb . . . [50]

Subscript rb represents variables reduced in the normal boiling point.For normal paraffins, Passut represented constants B y C in a generalized form, with less than 1% of deviation by the equations:

Bn = 4.6776 + 1.8324R – 0.03501 R 2 . . . [51]

Cn = 0.7751 Bn – 2.6354 . . . [52]

The subscript n means normal paraffins. For all other compounds, the predicted constants by equations [51]and [52] are corrected by association effects.

B = Bn + α1 . . . [53]

C = Cn + α2 . . . [54]

α1 y α2 are correction factors by association for Bn y Cn. Substituting equations [53] y [54] in equation [49] we have:

Ln Pr’ = (Bn + α1) (1-1/Tr) + (Cn + α2) ln 1/Tr + D (Pr’/Tr2 -1) . . . [55]

Then Passut defined his fourth parameter to which he called association factor as:

K = ((1-1/Tr) + α2 ln 1/Tr) / (1-1/Tr + ln 1/Tr) . . . [56]

Combining equations [55] and [56], K is written as:

K = [Ln Pr’ – Bn (1-1/Tr) – Cn Ln 1/Tr - D(Pr’/Tr2 -1)] / [1 – 1/Tr + Ln 1/Tr] . . . [57]

Applying equation 57 to the normal boiling point, the K definition becomes:

K = [Ln Y – Bn (1-X) – Cn Ln X – 0.4218 (Y/X2 -1)] / [1 – X + Ln X] . . . [58]


55

Where:X = 1/TrbY = Pr’b

Constants of equation [49] for any fluid are given by:

B = 4.7016 + 1.9639R - 0.04971R2 + 0.4976K . . . [59]C = 0.7272 + 1.7163R - 0.05925R2 + 0.3906K . . . [60]

According to Passut, CSP has the following form:

G = G(Tc, Pc, R,K) or else . . . [61]G = G(Tr, Pr, R, K) sinceTr = T/TcPr = P/Pc

Using parameters Tc, Pc, R y K, Passut generalized the Frost-Kalkwarf equation with excellent results.He also utilized the proposed parameters to generalize the second virial coefficient of polar compounds, but the precision of the correlation obtained is not good.The principal inconvenient of Passut model is that there is not a direct significance for the K parameter, but is more a fitting parameter.The polar contribution to the configurational properties, it was thought to be characterized by the deviations in the individual coefficient of the Frost-Kalkwarf (11).These coefficients are parameters for the adjustment of curves and do not have any physical significance. Furthermore, the K parameter assumes positive and negative values. To correlate the behavior of simple fluids as Argon and Krypton, four parameters are necessary when is well known that two-parameter CSP can describe adequately the behavior of these.Likewise, substances like carbon tetrachloride, where the polar effects are definitively absent, have a K value different from zero. Finally, all the hydrocarbons other than the normal paraffins, require four-parameters, as is well known that these can be correlated by the three-parameter CSP.

6. CSP proposed by Tarakad.Tarakad (49, 50) proposed an extension to CSP using the turn ratio proposed by Thompson (53) directly as the third parameter to characterize the molecular size-shape effects. As a fourth parameter he proposed the deviation presented by the second virial coefficient of the polar substances with respect to the second virial coefficient of the normal fluids at a reduced temperature of 0.6.


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Tarakad characterized the behavior of the simple fluids using the first term of the correlation by Tsonopoulos (54) for the calculation of the second virial coefficient:

B*simple fluid = B Pc/(R Tc) simple fluid = 0.1445 -0.3300/Tr – 0.1395/Tr2

- 0.0212/Tr3 – 0.000607/Tr8 . . . [62]Where;

B* - Second reduced virial coefficient.Pc - Critical pressure.R - Constant of the ideal gases.Tc - Critical temperature.Tr - Reduced temperature.

Para characterize deviations for molecular size-shape, Tarakad included a correction term that utilizes the radius of gyration proposed by Thompson (53):

B*correction= B Pc/(R Tc) correction= ( - 0.00787 + 0.0812/Tr2 – 0.0646/Tr6)R

size-shape size-shape

- ( 0.00347/Tr2 – 0.000149/Tr7)R 2 . . . [63]

Where:B* - Second reduced virial coefficient.Pc - Critical pressure.R - Constant of the ideal gases.Tc - Critical temperature.Tr - Reduced temperatura.R - Radius of gyration proposed by Thompson, expressed in Angstroms.

Tarakad made notice that the normal fluid concept proposed by Pitzer (36,37) for the fluids that follow the behavior of the three-parameter corresponding states principles is not adequate when the radius of gyration is utilized as the third parameter and introduced the concept of standard fluid:

B*standard fluid = B*

simple fluid + B* correcction . . . [64] size-shape

A fluid can be considered as standard if deviates a maximum of 5% of the behavior defined by equation [64] at reduced temperatures of 0.75. It is the clarification that the definition given by equation [64] does not contemplate the so called quantum fluids.Once the behavior of the standard fluid is established, the fourth parameter was defined as:

Ф = - [B*total - B*standard]Tr=0.6 . . . [65]

By definition the value of Ф is zero for all the fluids that obey the standard fluid behavior.


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The correction function developed for polar fluids has the shape:

B*polar = - (0.028/Tr) Ф . . . [66]

correction

The generalized equation to predict the behavior of any simple, standard or polar liquid is expressed as:

B* = B* simple fluid + B* size-shape + B* polar . . . [67]

correction correction

According with Tarakad the four-parameter CSP has the following form:

G = G(Tc, Pc, R, Ф) or else . . . [68]G = G(Tr, Pr, R, Ф) sinceTr = T/TcPr = P/Pc

Tarakad demonstrated the correlative ability of his model through the second virial coefficient of simple, standard and polar fluids but the dependency of his model from the radius of gyration and experimental second virial coefficient at TR=0.6 makes his four-parameter CSP not to be used.


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7. CSP proposed by Castilla-Carrillo.Analyzing the literature, I realized that Thompson four-parameter CSP (53) is the most convincing from the theoretical point of view; but the Halm-Stiel (15) is the most often used by the easy to calculate its fourth parameter. So, why not develop a four-parameter CSP that have both benefits ?

According with my original idea that if the three-parameter CSP correction function must change with the molecular size-shape, then the correction function for the fourth-parameter CSP must change with the molecular size-shape-polarity.

Putting these ideas and observations in the same context, we obtain the following equation:

G = G(0) + ωG(1)(ω) + ξ G(2)(ω,ξ) . . . [82]

Where:G - Any correlatable property using the CSP.G(0) - Property of the fluid under study considered as a simple fluid.ω - True acentric factor proposed by Thompson.G(1)(ω) - Function of correction to take into account the molecular size-shape.

This function varies according with the molecular size-shape.ξ - Fourth parameter proposed in my thesis work.G(2)(ω,ξ) - Function of correction to take into account the molecular polarity. This

function varies in accordance to the molecular size-shape-polarity.

Applying equation [82] to calculate or predict vapor pressures, results:

Ln Pr’ = Ln Pr’(0) + ω Ln Pr’(1)(ω) + ξLn Pr’(2)(ω,ξ) . . . [83]

Knowing the experimental values of Pr’ to Tr=0.7, Pitzer’s acentric factor is calculated according to its original definition:

ω = -log P’r (Tr=0.7) - 1.0

With the experimental value of Pr’ to Tr=0.6, lets calculate the polar factor proposed in this work:

ξ = B [1+(1+4C/B2)0.5]/2 . . . [84]

B = 1.037824ω – 0.09573304 . . . [85]

C = log P’r(Tr=0.6)/1.272854 – 0.005396275ω2 + 1.337863ω +1.215762. . . [86]


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Finally we calculate the contribution by size-shape.

ω = ω– ξ . . . [87]

Another way of obtaining the third and fourth parameters is establishing a two equations system with two unknown quantities (With the experimental values of Pr’ a Tr=0.6 y Tr=0.7) and equations 83, 25.7, 25.8 y 91. Later we will solve by iteration using some numerical method.

In both cases restrictions to be complied with the values of ω and ξ are:

0 <= ω<= ω0 <= ξ <= ω . . . [88]ω = ω + ξ

Once the third and fourth parameters are defined, the correction function proposed for the deviations of size-shape-polarity for the correlation of the vapor pressure, has the following form:

Ln Pr’(2) (ω , ξ) = - (10.76377 + 45.56516 ω- 4.557136 ξ) (1/Tr -1)- (8.137270 + 50.35548 ω- 7.992805 ξ) Ln Tr+(0.6392783 - 1.691165 ω- 0.9771521 ξ) (Tr7 -1)

. . . [91]

Equations 25.7, 25.8 y 91 were utilized along with equation 83 for the prediction of vapor pressures of simple, normal and polar fluids. Table 3, compares deviations of the proposed model with some already mentioned.


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Component AAD%Thompson

AAD%Halm-Stiel

AAD%Harlacher

AAD%Passut

AAD%Castilla-Carrillo

Ammonia 10.71 1.11 1.57 12.81 0.54Acetone 5.53 0.49 3.22 2.08 0.36Diethyl ether 9.66 0.55 1.37 0.78 0.40Acetic acid 2.21 1.03 5.69 4.92 0.62Phenol 4.59 0.98 2.17 6.99 0.72Phosgene 8.41 0.98 1.26 21.73 0.98Methyl fluoride 4.76 0.99 1.75 2.78 0.46Ethyl fluoride 4.87 0.56 1.42 1.43 0.54Methyl Chloride 17.77 1.89 0.89 4.23 0.83Ethyl Chloride 9.06 0.76 0.73 1.29 0.43Chloroform 11.94 1.25 1.13 2.36 1.10Fluorobenzene 7.73 0.63 1.64 0.86 0.36Chlorobenzene 11.74 1.18 1.47 6.06 0.86Piperidine - 1.26 - - 0.80Aniline 3.6 0.61 4.90 5.60 0.53Chlorhydric acid 8.47 0.93 0.75 1.28 0.44Sulphur dioxide 10.69 0.77 0.96 2.86 0.55Water 10.28 1.08 2.89 13.94 0.43Methanol 2.82 0.72 1.86 3.86 0.30Ethanol 3.83 0.92 1.27 1.88 0.301-propane 7.37 2.09 3.96 3.23 0.361-butane 8.93 2.21 4.92 5.13 0.381-pentane - 1.79 - - 0.89

Table 3. Average deviations in vapor pressure predictions of different four-parameter CSP with respect to experimental data.

The averge absolute deviation percentage (AAD%) obtained in calculation of the reduced vapor pressure of 23 polar fluids for a total of 1,287 experimental points, is of 0.57%.

AAD% = 100/N Σ abs [(P’r calc – P’r exp)/P’r exp

As it may be observed, in all the cases the proposed model was more accurate than all those published in the literature . The original table is located on page 59 of my original thesis work (58).


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The correlation parameters utilized in the Thompson, Halm-Stiel, Passut y Harlacher models are:

Component Tc0K

PcAtm

ω RAngst

χ Pa К

Ammonia 403.50 111.30 0.2727 0.8533 0.0194 64 6.9221Acetone 508.10 46.38 0.3076 2.7404 0.0123 161 3.1677Diethyl éter 466.74 35.90 0.2805 3.1395 -0.0007 210 1.3295Acetic acid 594.80 57.00 0.4415 2.5950 0.0402 131 7.2091Phenol 692.40 60.40 0.4468 3.5496 -0.0061 222 4.3831Phosgene 455.20 56.15 0.1942 2.8269 -0.0009 152 2.8269Methyl fluoride 315.80 58.00 0.2152 1.4186 0.0222 82 4.2800Ethyl fluoride 375.31 49.72 0.2160 2.1758 0.0116 122 2.3345Methyl chloride 418.30 65.80 0.1421 1.4500 0.0062 111 2.7997Ethyl chloride 460.40 52.00 0.1903 2.2800 0.0038 152 1.3946Chloroform 534.60 54.15 0.2197 3.1779 -0.0015 190 -0.4105Fluorobenzene 559.80 44.60 0.2487 3.3454 0.0019 215 -0.3441Chlorobenzene 634.40 44.50 0.2388 3.5684 -0.0007 245 -0.8499Piperidine 588.00 44.00 0.2727 -- -0.0003 -- --Aniline 696.80 52.60 0.3973 3.3926 0.0086 234 3.3293Chlorhydric acid 324.60 81.60 0.1242 0.2989 0.1080 71 5.4689Sulphur dioxide 430.70 77.80 0.2561 1.6379 0.0031 127 4.8707Water 647.31 218.17 0.3438 0.6150 0.0230 51 9.4339Methanol 512.64 79.91 0.5647 1.5360 0.0382 88 14.296Etanol 513.92 60.58 0.6463 2.2495 0.0058 127 14.7941-propanol 536.71 50.92 0.6220 2.7359 -0.0475 165 13.1171-butanol 562.98 43.55 0.5905 3.2250 -0.0783 203 10.5691-pentanol 584.90 38.30 0.6091 -- -0.0678 -- --

Tabla 4. Data for critical temperature (Tc), critical pressure (Pc), and the necessary experimental reduced vapor pressure points to Tr=0.6 and Tr=0.7 were taken from the works of Gómez-Nieto and Papadopoulos (13). Pitzer acentric factor (ω) and Halm-Stiel fourth parameter were calculated using its original definition eqs. [22] and [39] The radius of gyration, the Parachor (Pa) and the Harlacher association factor (К) were calculated or taken from the tables authors present in their original works.


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Correlation parameters utilized in the Castilla-Carrillo model equations [25.7], [25.8] y [91] are the following:

Component Tc0K

PcAtmosphere

ω ξ ω=ω - ξ

Ammonia 403.50 111.30 0.2727 0.2376 0.0351Acetone 508.10 46.38 0.3076 0.2490 0.0586Diethyl ether 466.74 35.90 0.2805 0.1727 0.1078Acetic acid 594.80 57.00 0.4415 0.4297 0.0118Phenol 692.40 60.40 0.4468 0.3434 0.1034Phosgene 455.20 56.15 0.1942 0.0529 0.0963Methyl fluoride 315.80 58.00 0.2152 0.1986 0.0166Ethyl fluoride 375.31 49.72 0.2160 0.1638 0.0522Methyl chloride 418.30 65.80 0.1421 0.0727 0.0694Ethyl chloride 460.40 52.00 0.1903 0.0986 0.0917Chloroform 534.60 54.15 0.2197 0.0662 0.1535Fluorobenzene 559.80 44.60 0.2487 0.1504 0.0983Chlorobenzene 634.40 44.50 0.2388 0.1201 0.1187Piperidine 588.00 44.00 0.2727 0.1656 0.1071Aniline 696.80 52.60 0.3973 0.3267 0.0706Chlorhydric acid 324.60 81.60 0.1242 0.0904 0.0338Sulphur dioxide 430.70 77.80 0.2561 0.1645 0.0916Water 647.31 218.17 0.3438 0.3086 0.0352Methanol 512.64 79.91 0.5647 0.5387 0.0260Ethanol 513.92 60.58 0.6463 0.5754 0.07091-propanol 536.71 50.92 0.6220 0.4594 0.16261-butanol 562.98 43.55 0.5905 0.2940 0.29651-pentanol 584.90 38.30 0.6091 0.3888 0.2203

Tabla 5. The necessary data of critical temperature to (Tc), critical pressure (Pc) and the experimental reduced vapor pressure points to Tr =0.6 y Tr=0.7 were taken from the works of Gómez-Nieto y Papadopoulos (13). The Pitzer acentric factor Pitzer (ω) was calculated using his original definition, equation [22]. The polar parameter (ξ), was calculated according to definition equations [84], [85], [86] y [87].

According to my work, the four-parameter CSP takes the following form:

G = G(Tc, Pc, ω, ξ) or else . . . [92]G = G(Tr, Pr, ω, ξ) sinceTr = T/TcPr = P/Pc


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8. CSP proposed by Wilding-Rowley.Wilding (59) and Wilding-Rowley (60) extended the application of Lee-Kesler model (ELK) to polar fluids through the introduction of polarity factor β and replacing the Pitzer acentric factor ω with a factor of size-shape α.The necessary parameters are those already required by the two-parameter CSP, critical temperature (Tc) and critical pressure (Pc). The radius of gyration (R), to take in consideration the geometrical deviations of the CSP and a density of liquid at a known condition to calculate the fourth parameter that takes in consideration the molecular polar and association effects. Separating in this manner the CSP deviations for simple fluids from deviations for size-shape and polarity.To evaluate the departure functions reference fluids used by Lee-Kesler are used and water as reference polar fluid.

Fort the case of compressibility factor, we have:

Z(Tr,Pr)= Z0(Tr,Pr)+ α Z(1)(Tr,Pr)+ βZ(2)(Tr,Pr) . . . [93]

Z(1)= (Z1−Z0)/α1 . . . [94]

Z(2)= [ (Z2-Z0) – α2/α1 (Z1-Z0) ] / β2 . . . [95]

β2 = 1 . . . [96]

The third parameter was defined as:

α = -7.706x10-4 + 0.0330R + 0.01506R2 – 9.997x10-4R3 . . . [97]

Where:R is the radius of gyration measured in Angstroms.

The fourth parameter was defined as:

β = [Z-Z0 – α(Z1-Z0)/α1] / (Z2-Z’2) . . . [98]

Z2’ = Z0 + α2(Z1-Z0)/α1 . . . [99]

The values of Z0 and Z1 are obtained from the modified BWR equation proposed by Lee-Kesler (25) and the values of Z2 are obtained from the Keenan equation (61). The results obtained for non-polar fluids are equivalent to the ones obtained by the three-parameter method of Lee-Kesler (25). The accuracy of the results for polar fluids are not acceptable to be used in process simulation or in process equipment sizing.

According with Wilding-Rowley (59 y 60) the four-parameter CSP has the following form:

G = G(Tc, Pc, α, β) o else . . . [100]G = G(Tr, Pr, α, β) sinceTr = T/TcPr = P/Pc


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V. USES AND APPLICATIONS OF CSP.The Corresponding States Principle (CSP) is not only used for the prediction of thermodynamic predictions, it could be said that it has unlimited applications and that it can be used almost for anything, referring by this, to unitary processes and operations within Chemical Engineering. As examples we have:

Helfand y Rice (20), developed in a general way the basic theory about the utilization of the CSP for the prediction of transport properties.

Hirschfelder, Curtiss, Bird y Spotz (21,22) developed basic correlations with theoretical fundament, in terms of reduced variables with the parameters of intermolecular potential Є and σ for the prediction of viscosities, diffusivities and thermal conductivities in the region of the ideal gas.

Damasius y Thodos (7), prepared correlations of corresponding states for obtention of the necessary parameters in the correction of the Enskog dense gas for the calculation of mixture viscosities of hydrocarbons and quantic gases.

Teja y Rice (52), developed a generalized method for prediction of liquid mixtures viscosities.

Abe y Nagashima (1), developed a CSP for the prediction of viscosities of melted salts of pure and mixed alkaline halides.

Dean y Stiel (8), Shimotake y Thodos (46), y Giddings (12), utilized the concept of residual properties of transport in the preparation of reduced correlations for the prediction of high pressure viscosities. Owens y Thodos (29,30), utilized a similar approach for the prediction of thermal conductivity of monoatomic gases and Shaefer y Thodos (45) did the same for the case of the diatomic gases.

Frisch, Bak y Webster (10), offered an interesting discussion about the possibility of developing a CSP for the prediction of constants of reaction velocity.

Guggenheim (14), Utilized the CSP to predict the behavior of solid Argon.

Bake, Erdelyi y Kedues (3), proposed a reduced equation of state for metals.

Paulatis y Eckert (34), developed a generalized model for the prediction of thermodynamic properties of mixtures for liquid metals.

The works mentioned previously show the practically unlimited potential of the CSP in the prediction of the substances behavior although actually there is no theory formally established that can support this supposition.


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VI. OBSERVATIONS. There have been many more attempts to extend the CSP application to complex, polar and hydrogen bonding molecules. In Part 1, I have shown the main ones or the most significant ones, according to their publication date and correlational framework.

In 1968, Leland and Chappelear (26), discussed the use of Zc compared with the use of parameters derivative from the vapor pressure.

In 1981, Nishiumi and Robinson (65), expressed the compressibility factor for a fluid in terms of Pitzer’s acentric factor, ω, and a fourth parameter ψE, obtained from the second virial coefficient data at low reduced temperatures. Their calculations shown that their predictions are good for polar substances in liquid and gaseous phases. However, the properties of substances like methanol and ethanol were not able to be correlated.

In 1985, Wu and Stiel (63), developed a four-parameter CSP using Pitzer ω acentric factor as a third parameter and a fourth parameter, Y, calculated from P-V-T data.

In 1987, Valderrama and Cisternas (62), compared the ω acentric factor and the Zc critical compressibility factor, as to which is the best third parameter and selected the critical compressibility factor as the best third parameter. With the Zc as a third parameter, they developed successful correlations for the prediction of volumetric properties of pure compounds and vapor-liquid equilibrium for certain types of fluids. They finally concluded that more than three-parameters are necessary to generalize equations of state that work for simple, normal and polar fluids.

In 1994, Golobic and Gaspersic (64), introduced 2 new parameters for the calculation of volumetric properties and 2 additional parameters for the calculation of pressures and other thermodynamic properties.

In 1996, Sorner M. (66), proposed a four-parameter CSP for the prediction of thermodynamic properties of refrigerants. Her third parameter is based on considerations of molecular geometry and her fourth parameter is calculated using dipolar moment and molecular polarizability. The reported results are not better than those provided by the three-parameter Teja model using 2 freons as reference fluids.

In 2005, Sun y Ely (67), in a very intense work proposed a four-parameter CSP that seems as an extension of the Wilding y Rowley (59,60 ) model, since utilizes the simple fluid just like they did, two non-spherical reference fluids, while, Wilding y Rowley only used one and the water as a polar reference fluid. The state equation to correlate the behavior of all the fluids that participate was developed by them on a previous work that utilizes fitting parameters. The third parameter used is Pitzer acentric factor and as a fourth parameter, they defined polarity factor obtained by fitting. The results obtained seem to be good for the properties and compounds


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used in the fitting, but the correlative capacity for other properties and predictive for different substances to the treated ones is null.

The last four-parameter CSP that from my point of view presents a new idea is the work of Wilding-Rowley (59,60) but the results obtained are not good.

Through these pages I have proved with results the following:1. The Pitzer three-parameter CSP (36, 37, 38 y 39) really works for simple and normal

fluids with molecules not so long or that its shape is not too different to the spherical one.

2. The Pitzer three-parameter CSP needed a correction function that varies with the size-shape. This is, the correction function for the n-pentane is not the same that the one for the neopentane or for the n-eicosane. The correction function changes with the molecular size-shape. This can be appreciated on table 2 of this monograph where the results of the deviations obtained are of less tan 1%. To these results no statistic test was applied to eliminate compounds with possible errors in their measurements.

3. The Lee-Kesler (25) and Teja (51) Works, are proofs of the necessity for a correction function for the three-parameter CSP that changes with the size-shape.

4. The Eubank-Smith approach (9) that the acentric factor should not be used for polar substances and in its place the homomorph factor should be used for the calculation of the shape-size contribution since this correct and works. But the balance of his correlation is not correct.

5. The Thompson approach (53) that the Pitzer (35,36,37,38 and 39) acentric factor calculated for polar substances, are two contributions; size-shape and 2. Polarity, is correct. I am not quite sure that it is a sum, since the correlations where it works are logarithmic.The concept of the “true acentric factor” is also correct and it Works but not the correlation that he proposed. The correction function to size-shape and polarity are not the adequate ones.

6. My own results demonstrate that the Pitzer acentric factor (35,36) calculated for polar substances are two contributions; 1. Size-shape and 2. Polarity. What I can not assure is that it is a sum; this is, I am not sure that ω = ω + ξ for other properties because the proposed vapor pressure equation is logarithmic. The relation between ω y ξ may be different if different models other than the logarithmic ones are used. I realize this when I tried to correlate the Zc values for n-alkanes from C1 to C-20, results were not so good. Possible explanations for this are: 1. The Vc values are subject to much


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uncertainty for being an extensive property, difficult to measure. 2. That the lineal model does not work. Also, it may be both things. This correlation appears in my original thesis work (58) and the case is treated in the Part 2 of this monograph.

7. There are two ways of solving the problem of size-shape-polarity:1. Through the calculation of the contribution to the size-shape using the radius of gyration by Thompson; the Parachor or the acentric factor of the homomorph and from there apply the equation ξ = ω - ω and thus calculate the polar factor to be applied to our correlations. 2. Utilize in additional point of vapor pressure as suggested by Halm-Stiel (15,16 y 17) and me (58).In 1982 when I prepared this work, the possibility of calculate the radius of gyration for new or unknown substances was unthinkable.Nowadays, I would prefer the first method and the second as an alternative option.

8. Wilding-Rowley work (59,60) is a proof that the four-parameter CSP needs correction functions that vary with the polarity of the substances.

9. Sun and Ely work (67) is definitely very intense and he devoted many computational resources and hours. However, is only a demonstration that CSP needs correct characterization parameters and adequate correction functions. From my perspective adds nothing new to the CSP.

10. The four-parameter model presented by Castilla-Carrillo(58) in 1983 offers better predictions for vapor pressures than all the models encountered in all the open literature available until today, April, 2012.

The problem, however, is not solved yet, isolated and repetitive efforts are being done. The same mistakes are made once and again in different dates and in different parts of the world. Non trustable experimental data and manipulated statistical analysis are common problems encountered in the scientific literature.

CSP was discovered in 1873 by J. D. van der Waals (55) and it was until 1955 when it could be applied to more complex substances than the so called simple fluids (Argon, Krypton, Xenon and Methane). This is, it took 82 years since its discovery.

From 1955 through 1975, it was not possible to improve the CSP predictions from Pitzer (36,37) three-parameter. Twenty years had to go by until Lee-Kesler (25) accomplished it. This time was too long because by then the necessary mathematical tools were available even with electronic data processing. Even so, the problem of the normal fluids was not totally resolved due to that


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the correction functions were not the adequate ones to correlate with good precision the generality of the normal fluids.

The four-parameter CSP that pretends to include the abnormal or polar fluids, has not had better luck. In 1961, Eubank y Smith (9) proposed the first four-parameter model and it was until 1983 that the problem is visualized already resolved with the proposal of my model. It took 22 years and in my opinion it is a too long period.

Supposing that the problem of thermophysical property predictions of pure simple, normal and polar fluids is solved, it still lacks the difficult problem of the mixtures. Because there is not a set of mixing rules for a mixture that contains simple, normal and polar fluids.

The previous reasons, demonstrate that the efforts made, a lot or too little, good or bad, have not given positive results in the development of the necessary correlations for the utilization of the CSP.


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VII. RECOMMENDATIONSThe CSP has a lot to offer to engineering and science, it is necessary, however, to analyze and put under one context the following: 1. Critical properties.

It is necessary to standardize and if possible, to obtain more precise experimental values at the critical point, both at the critical point of Tc, Pc y Vc as to the different predicting properties. Many of the reported values are inferred and therefore divergent and subject to significant errors that impair the development of our correlations.

2. Experimental data.There are reliable experimental data but are not available to all at no cost. It is necessary to implement procedures that allow reliable data available at no cost for all of those that are providing something good for science.

3. Characterization parameters.Which is the best set of characterization parameters: Tc and Pc, Tc and Vc or maybe Pc y Vc?Which are the indicated ones for size-shape?All parameters for size-shape are correlated, why then use two or three characterization parameters for size-shape?Which ones are the indicated ones for polarity?(Dipolar moment, quadrupolar moment, octupolar moment, multipolar moment, molecule polarizibility, etc.)

4. We need generalized models for all thermophysical properties. That are flexible and can be applied to Argon, n-Octane, n-eicosane, water, i-propanol, carbon dioxide, etc. The curves must be able to be to fit all these components individually to then proceed to generalize them. Actually each group of researchers uses the models that they think are correct by families of substances and this only complicates the work.BWRS equation is superior to the BWR and BWR-32 is better than BWRS?In a generalized context, which model must be used?It does not matter how big or how many parameters are. Theoretically, there is only one work to do at once and then obtain benefits for a lifetime.If the molecule is simple like that of argon y methane, some or many parameters will be zero and if the molecule is complicated as in the case of the freons or water, all parameters will have a value.

The last recommendation and perhaps the most important one is to think before acting, make a correct analysis of the problems to solve. Optimization and data fitting are tools that do not solve problems by themselves.

I conclude Part 1 of this monograph saying that:In the universe there is nothing messy or chaotic and much less erratic, things that happen are causalities of physical laws well established.The disordered and chaotic is our understanding and knowledge about reality, the universal order and the laws governing them and CSP is a proof of that.

This reminds me of Albert Einstein when he said:“God does not play to dice”


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BIBLIOGRAPH

REFFERENCES ABOUT THE CORRESPONDING STATES PRINCIPLE.

1. Y. Abe y A. Nagashima, “THE PRINCIPLE OF CORRESPONDING STATES FOR ALKALI HALIDES”, J. Chem. Phys., 75, 3977 (1981).

2. API “TECHNICAL DATA BOOK, PETROLEUM REFINING”, Second Edition, The American Petroleum Institute, Washington D. C. (1970).

3. D. L. Beke, G. Erdelyi, y F. J. Kedues, “THE LAW OF CORRESPONDING STATES FOR METALS”, J. Phys. Chem. Solids, 42, 163 (1981).

4. A. Bondi, y D. J. Simkin, “HEATS OF VAPORIZATION OF HYDROGEN-BONDED SUBSTANCES”, Aiche J., 3, 473 (1957).

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LIQUIDS”23. Lin Hung-Huei y L. I. Stiel, “SECOND VIRIAL COEFFICIENTS OF POLAR GASES FOR A

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27. A. L. Lydersen, R. A. Greenkorn, y O. A. Hougen, “GENERALIZED THERMODYNAMIC PROPERTIES OF PURE FLUIDS”, College of Engineering, University of Wisconsin Eng. Sta., Report No. 4 (Oct., 1955).

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32. C. A. Passut y R. P. Danner, “DEVELOPMENT OF A FOUR-PARAMETER CORRESPONDING STATES METHOD: VAPOR PRESSURE PREDICTION”, AICHE Simp. Ser., 70, 30.

33. C. A. Passut y R. P. Danner, “ACENTRIC FACTOR, A VALUABLE CORRELATING PARAMETER FOR THE PROPERTIES OF HYDROCARBONS”, Ind. Eng. Chem. Proc. Des. Dev., 12, 365 (1973).

34. M. E. Paulatis y C. A. Eckert, “A PERTURBED HARD-SPHERE, CORRESPONDING STATES FOR LIQUID METAL SOLUTIONS”, AICHE J., 27. 418 (1981).

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60. W. V. Wilding y R. L. Rowley, “A FOUR-PARAMETER CORRESPONDING-STATES MESTHOD FOR THE PREDICTION OF THERMODYNAMIC PROPERTIES OF POLAR AND NONPOLAR FLUIDS”, Int. J. Thermophys. 7, 525 (1986).

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63. G. Z. A. Wu y L. I. Stiel, “A GENERALIZED EQUATION OF STATE FOR THE THERMODYNAMIC PROPERTIES OF POLAR FLUIDS”, AIChE J., 31 (10), 1632-1644 (1985).

64. I. Golobic y B. Gaspersic, “A GENERALIZED EQUATION OF STATED FORPOLAR ANDNON-POLAR FLUIDS BASED ON FOUR-PARAMETER CORRESPONDING STATES THEOREM”, Chem. Eng. Com., 130, 105-126 (1994).


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65. H. Nishiumi y D. B. Robinson, “COMPRESIBILITY FACTOR OF POLAR SUBSTANCES DASED ON A FOUR-PARAMETER CORRESPONDING STATES PRINCIPLE”, J. Chem. Eng. J., 14 (4), 259-266 (1981).

66. M. Sorner, “CORRESPONDING STATES CORRELATIONS FOR THE PREDICTION OF THERMODYNAMIC PROPERTIES OF REFRIGERANTS”, Ph. D. Thesis, Chalmers University of Technology, Goteborg, Sweden (1996).

67. L. Sun y J. F. Ely, “A CORRESPONDING STATES MODEL FOR GENERALIZED ENGINEERING EQUATIONS OF STATE”, Int. J. of Therm., Vol. 26, No. 3, 705-728 (2005).

REFFERENCES ABOUT EXPERIMENTAL VALUES.

1. API “TECHNICAL DATA BOOK, PETROLEUM REFINING”, Second Edition, The American Petroleum Institute, Washington D. C. (1970).

2. M. A. Gomez-Nieto and C. G. Papadopoulos, “THE VAPOR PRESSURE BEHAVIOR OF POLAR AND NONPOLAR SUBSTANCES”, Northwestern University, Rept. Of Chem. Eng. Dept. (1976).

3. K. R. Hall, “VAPOR PRESSURE OF CHEMICALS”, Landolt-Bornstein, Group IV, Vol. 20, Springer-Verlag, 1999.

4. N. B. Vargaftik, “HANDBOOK OF PHYSICAL PROPERTIES OF LIQUID AND GASES”, Sec. Ed., Hemisphere Pub. Corp., 1975.

5. R. C. Wilhoit and B. J. Zwolinski, “PHYSICAL AND THERMODYNAMIC PROPERTIES OF ALIPHATIC ALCOHOLS”, Jour. Of Phys. And Chem. Ref. Data, Vol. 2, 1973, Sup. 1.

RECOMMENDED READINGS

1. A. Bondi, “Physical Properties of Molecular Cristals”, John Wiley & Sons, 1968.2. J. H. Dymond and E. B. Smith, “THE VIRIAL COEFFICIENTS OF GASES”, Clarendon Press,

1969.3. W. J. Lyman, W. F. Reehl and D. H. Rossenblatt, “HANDBOOK OF CHEMICAL PROPERTY

ESTIMATION METHODS”, McGRAW-HILL, 1981.4. B. Poling, J. M. Prasusnitz and J. P. O’Connel, “THE PROPERTIES OF GASES AND

LIQUIDS”,Fifth Ed., McGRAW-HILL, 2001.5. R. C. Reid, J. M. Prasusnitz and T. K. Sherwood, “THE PROPERTIES OF GASES AND

LIQUIDS”,Third Ed., McGRAW-HILL, 1977.6. R. C. Reid, J. M. Prasusnitz and B. E. Poling, “THE PROPERTIES OF GASES AND

LIQUIDS”,Fourth Ed., McGRAW-HILL, 1987.7. W. C. Reynolds, “THERMODYNAMIC PROPERTIES IN SI”, Stanford University, 1979.8. Z. Sterbaceck, B. Biskup and P. Tausk, “CALCULATION OF PROPERTIES USING

CORRESPONDING STATES METHODS”, Elsevier Scientific, 1979.9. S. Sugden, “THE PARACHOR AND VALENCY”, The Mayflower Press, 1930.

10. H. W. Xiang, “THE CORRESPONDING-STATES PRINCIPLE AND ITS PRACTICE”, Elsevier, 2005.

11. R. C. Reid, T. K. Sherwood, “THE PROPERTIES OF GASES AND LIQUIDS”, Second Ed., McGRAW-HILL, 1966.

12. R. C. Reid and T. K. Sherwwod, “THE PROPERTIES OF GASES AND LIQUIDS: Their Estimation and Correlation”, McGRAW-HILL, 1958.

13. A. Hinchliffe, “MOLECULAR MODELING FOR BEGINNERS”, Sec. Ed., Wiley & Sons, 2008.


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ANEXTENSION OF THE CORRESPONDING STATES PRINCIPLE

PREDICTION AND CORRELATION OF THERMOPHYSICAL PROPERTIES USING THE CORRESPONDING STATES PRINCIPLE.

PART 2Investigation and development of the four-parameter corresponding states principle developed in the thesis work that I presented to obtain the degree of Industrial Chemical Engineer at the Superior School of Chemical Engineering and Extractive Industries of the National Poly-technical Institute, MEXICO,D.F.


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IN PREPARATION

MEANWHILE, BE HAPPY.


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