RK-, SRK-, & SRK-PR-TYPE EQUATION OF STATE FOR HYDROCARBONS, BASED ON SIMPLE MOLECULAR PROPERTIES

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1 Journal of Applied Chemical Science International 2(2): 65-74, 2015 RK-, SRK-, & SRK-PR-TYPE EQUATION OF STATE FOR HYDROCARBONS, BASED ON SIMPLE MOLECULAR PROPERTIES KAMAL I AL-MALAH * Department of Chemical Engineering, King Faisal University, Hofuf, Saudi Arabia [KIAM] [*For Correspondence: almalak61@hotmailcom] ABSTRACT A database made of 429 different hydrocarbons was utilized to fit their critical pressure and temperature as a function of molecular weight and carbon atomic fraction The critical parameters and, appearing in Redlich-Kwong, Soave-Redlich-Kwong and Redlich-Kwong-Peng-Robinson equation of state (EoS), were replaced by two curve-fitted equations The afore-mentioned original equations were tested and contrasted versus their counterparts as far as predicting the molar volume of both liquid and vapor at a given pair of pressure and temperature for a given hydrocarbon Using either an empirical or a reference datum for the estimated molar volume, it was found that the substituted equation did predict the volumetric properties with accuracy as good as the original equation The replacement will ease the calculation of volumetric properties of liquid and vapor with a reasonable accuracy It was found that, in general, all models, the original and the substituted/modified equations, predicted well, with a good accuracy (ie, a percent relative error (PRE) < 10%) The predicted values almost coincided Moreover, none of the models, neither the original nor its counterpart equation, would predict, with a good accuracy (ie, PRE <10%), if the hydrocarbon molecular size was greater than C 7 On the other hand, for a hydrocarbon molecular size less than C 7, the best three models that predicted, with a good accuracy (ie, PRE < 10%) were SRK-type, RK-type, and SRK Finally, at the critical condition (ie, T r =10 and P r =10), all models failed to predict the critical molar volume ; ie, PRE values > 10% Keywords: Van der waals, equation of state, redlich-kwong, soave-redlich-kwong, carbon content, molecular formula, hydrocarbons, EoS INTRODUCTION An equation of state (EoS) is a thermodynamic equation describing the state of matter under a given set of physical conditions It is a constitutive equation that provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, and volume Equations of state are useful in describing the properties of fluids, mixtures of gases, liquids and solids van der Waals is the pioneer in postulating an equation of state for a fluid van der Waals EoS was derived by Johannes Diderik van der Waals in 1873 (van der Waals, 1873), based on a modification of the ideal gas law, who received the Nobel Prize in

2 Introduced in 1949 (Redlich and Kwong, 1949), the Redlich Kwong (RK) equation of state was considered a significant improvement over other equations of the time It is still of interest primarily due to its relatively simple form While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor-liquid equilibria However, it can be used in conjunction with separate liquid-phase correlations for this purpose The Redlich Kwong equation was mainly coined to predict the properties of small, non-polar molecules in the vapor phase, which it generally does well However, it had been subject to various attempts to refine and improve it In 1972, Soave (1972) replaced the 1/ term of the Redlich Kwong equation with a function α(t,ω) involving the temperature and the acentric factor (the resulting equation is also known as the Soave Redlich Kwong equation, SRK) The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for such materials In 1975 (Redlich, 1975), Redlich himself published an equation of state adding a third parameter, in order to better model the behavior of both long-chain molecules, as well as more polar molecules His 1975 equation took advantage of computer calculations, which was not available at the time the original equation was published Schmidt and Wenzel (1980) pointed out that the SRK (or, RKS) equation yields a critical compressibility factor Z c =0333, whereas experimental compressibility factors Z c range between 02 and 03, which means that SRK equation overestimates the saturated liquid density and critical volume Frey, et al, (2007) examined the molar volumes, as predicted by the cubic volumetric equations of state (PR, SRK and TST), over a range of reduced temperatures and pressures from T r = and P r = and compared them to reference molar volume values for water, carbon dioxide, hydrogen sulfide, ammonia, nitrogen, oxygen, argon and the first eight normal paraffin alkanes They found a similarity of molar volume residuals amongst the examined substances for predictions given by the SRK equation of state, indicating that volume translation could be used to improve the accuracy of these molar volume predictions The untranslated volume (V UT ) is replaced by the translated volume (V T ) such that V UT = V T c; c is the magnitude of volume translation They found that the functional form of the volume translations needed was the same for many different polar and non-polar fluids and highly dissimilar fluids such as water, ammonia, argon, and many simple hydrocarbons required volume translations that were alike in both shape and magnitude Cismondi and Mollerup (2005) indicated that the combination of SRK PR equation was found to be the best among the different three-parameter cubic equations of state, found in literature The possibility of this combined equation, connecting the SRK and PR EoS, was explored in greater detail than previous studies for the modeling of P T properties of pure fluids They proposed two parameters δ 1 and b being constants for each pure fluid, while a simple temperature-dependent attractive parameter a, which is adjusted to reproduce the acentric factor Hence, their proposed parameterization procedure requires Tc, Pc, Zc, and ω for each fluid Kontogeorgis and Economou (2010) used a methodological approach based on the excess Gibbs energy and activity coefficient expressions derived from cubic equations of state (EoS) for analyzing and understanding the capabilities and limitations of those classical models They have showed that cubic EoS of vdw-type have a functional form similar to well-known polymer models and provide quantitatively an adjustment factor to account for size/free-volume (combinatorial) effects, dominant in mixtures of alkanes of different size and other asymmetric systems; hence, cubic EoS could be extended to include fluid polymers In this work, an attempt for replacing the critical pressure and temperature parameter of a hydrocarbon substance, appearing in a typical equation of state, will be carried out Such fingerprint critical parameters of a hydrocarbon substance will be expressed in terms of its molecular weight and carbon content fraction 66

3 MODEL DEVELOPMENT Four hundred and twenty nine (429) hydrocarbon compounds were used in the non-linear regression process for finding the best fit for their critical properties The database of hydrocarbon compounds includes the following categories: 1 Normal paraffin: Example: n-alkane 2 Non-normal paraffin: Example: Iso-alkane, methyl-alkane, ethyl-alkane, & methylethyl-alkane 3 Naphthene: The major structure is saturated ring; example: Cyclo-alkane 4 Olefin: Contains a single C=C double bond; example: Alkene, methyl-alkene, ethylalkene, & di-methyl-alkene 5 Diolefin: Contains two C=C double bonds; example: Alkadiene, methyl-alkadiene, and ethyl-alkadiene 6 Cyclic olefin: Contains a single C=C double bond within the otherwise saturated ring; example: Cyclo-alkene, methyl-cycloalkene, & ethyl-cyclo-alkene 7 Alkyne: Contains a triple bond between carbons; example: Acetylene, methyl acetylene, pentyne, and hexyne 8 Aromatic: Contains a single ring; example: benzene, toluene, & xylene 9 Aromatic with attached olefin side chain: Example: Styrene, ethenyl-benzene, and propenyl-benzene 10 Aromatic with multiple rings directly connected by C-C bonds between the rings: Example: bi-phenyl and 1-methyl-2- phenylbenzene 11 Aromatic with multiple rings connected through other saturated carbon species: Example: Di-phenyl-methane and 1,1-diphenyl-dodecane 12 Aromatic with multiple rings connected through other carbon species with triple bond: Example: Di-phenyl-acetylene 13 Aromatic with multiple condensed rings: Example: naphthalene, pyrene, methylnaphthalene, and nonyl-naphthalene 14 Aromatic with attached saturated rings: Examples: 1,2,3,4-tetra-hydro-naphthalene and 1-methyl-2,3-dihydro-indene 15 Aromatic with attached unsaturated (but not aromatic) rings: Example: Indene and 1- methyl-indene The detailed components of each hydrocarbon category were listed in (Al-Malah, 2013) It should be noticed that the attractive term a and the co-volume repulsive term b appearing in equations of state, like: Van der Waals (vdw), Peng-Robinson (PR), Peng-Robinson-Stryjek- Vera (PRSV), Relich-Kwong (RK), and Soave- Redlich-Kwong (SRK), do contain and, respectively Hence, it is worth estimating such two quotients based on molecular properties of a hydrocarbon molecule The carbon atomic fraction (X) and molecular weight (Y) were chosen as the independent variables and both quotients, defined by Z 1 and Z 2, respectively, represent the dependent variable from regression point of view: (, )= = = (1) (, )= = = (2) For example, given methane (CH 4 ), then its C frac will be 1/(1+4)=020 Moreover, its MW is simply equal to = 16 The results of non-linear regression for both equations (1) and (2), with 95% confidence interval, are: (, )= = ( ) ( ) ( ) (3) The goodness of fit for Eq (3) is given by the R- square as 0999 and adjusted R-square as 0999 (, )= = ( ) ( ) ( ) (4) 67

4 The goodness of fit for Eq (4) is given by the R-square as and adjusted R-square as Al-Malah To demonstrate the validity of the proposed equations, will replace and will replace in the following three models of EoS; namely, Redlich-Kwong (RK), Soave-Redlich-Kwong (SRK), & Redlich- Kwong-Peng-Robinson (RKPR) Using three models of EoS will be a sufficient evidence that such a replacement of quotients by their equivalence via equations (3) and (4) is reasonable and justifiable Redlich-Kwong (RK) Equation of State = (5) = (6) = (7) Where Redlich-Kwong (RK)-Type Equation of State = (8) Where = = = = (9) = = 7913 (10) = (11) = = (12) = (13) Soave-Redlich-Kwong (SRK) Equation of State = (14) Where = ( ) (15) = (16) = w 0176 w (17) Soave-Redlich-Kwong (SRK)-Type Equation of State = (18) 68

5 Where = ( ) = [1 + (1 )] (19) = [1 + (1 )] 7913 (20) = [1 + (1 )] (21) = = (22) = (23) = w 0176 w (24) Redlich-Kwong-Peng-Robinson (RK-PR) Equation of State = = (25) = + ( ) + ( ) (26) = ( ) ( ) (27) = 1 + {2(1 + )} / + / (28) = / (29) = Ω = = (30) = { } / / / (31) = Ω = (32) = ( + ) + ( + ) + ( + ) (33) = ( ) + ( ) + ( ) (34) Redlich-Kwong-Peng-Robinson (RK-PR)-Type Equation of State = = (35) 69

6 = + ( ) + ( ) (36) = ( ) ( ) (37) = 1 + {2(1 + )} / + = / / (38) (39) = Ω = = (40) = { } / / / (41) = Ω = (42) = 7913 (43) = ( + ) + ( + ) + ( + ) (44) = ( ) + ( ) + ( ) (45) SOLUTION OF THE CUBIC EQUATIONS OF STATE Regarding the cubic in molar volume, equations of state (see equations (5), (8), (14), (18), (25), and (35)); each equation was put in the polynomial form where it shows the coefficients associated with each order in molar volume, For example, Redlich-Kwong (RK) equation of state (Eq (5) can be put in the form of: ( + ) = 0(46) If the absolute values of pressure and temperature (P and T) are given for a pure substance, then Eq (46) will be a non-linear algebraic equation in,which will have, in general, three roots for the molar volume at the given pressure and temperature MATLAB code, in the form of an m-file, was written to utilize the MATLAB builtin root-finding algorithm for finding the roots for such an equation Two roots were chosen such that the imaginary part is zero, which correspond to the volume of a gas and volume of liquid Any additional parameters appearing in the main EoS, like a and b, have to be defined by some additional algebraic equations other than the main equation itself Fig (1) shows the MATLAB m- file, written for Redlich-Kwong (RK) EoS, which is composed of the following steps: 1 Retrieve the critical properties of hydrocarbons from Excel file called HC429Critxls 2 Search for a particular component by either name or chemical formula 3 Select one component at a time out of many search results 4 Enter the operating pressure and temperature 5 Define a, b, and any other parameters appearing in EoS (see equations (5), (8), (14), (18), (25), and (35) 6 Use MATLAB built-in root finding function 7 Display the results Finally, the percent relative error, PRE is defined as: = / / 100% (47) 70

7 clear all; Pure Comp = dataset('xlsfile','hc429critxlsx'); myvar= cellstr (Pure Comp, 'Compound'); myvar2= cellstr (Pure Comp, 'Formula'); Search Key= str2double(input('input "1" to search by Compound Name "2" to search by Formula: ', 's')); If Search Key==1 Comp Name= input ('Input the name of the Compound, say methane: ', 's'); Match Start = regexpi(myvar, Comp Name); elseifsearchkey==2 FormName=input('Input the chemical formula, say CH4: ', 's'); matchstart = regexpi(myvar2, FormName); else display('you did not enter 1 or 2'); return end % match Start = regexpi(myvar, 'methane'); tempindex=1; flag=0; myindx(1,1)=1; foridx=1:size(matchstart,1) if cell2mat(match Start(idx,1))>=1 myindx(tempindex,1)=idx; tempindex=tempindex+1; flag=1; end end if flag==0 display ('No Entry was Found!; Good Luck and Try Again') return end for idex2=1:size(myindx,1) value=myindx(idex2,1); display([value,purecompcompound(value),purecompformula(value)]) end CompNum= str2double (input('input the number for the chosen compound, say "1" for methane: ', 's')); Tabs= str2double (input ('Input the absolute temperature in Kelvin ', 's')); Pabs= str2double (input ('Input the absolute pressure in atm ', 's')); Pcr=Pure Comp Pcr (Comp Num); Tcr=Pure Comp Tcr (Comp Num); Tr=Tabs/Tcr; Pr=Pabs/Pcr; Rgas= ; aterm=(042748*rgas^2*tcr^25)/(pcr*tabs^05); bterm=008664*rgas*tcr/pcr; Vols=roots ([Pabs, - Rgas* Tabs, (aterm - Pabs*bterm*bterm - Rgas*Tabs*bterm), -aterm*bterm]); % Finds all roots GasVol=max(Vols(imag(Vols) == 0)); % finds largest real root for the gas volume LiqVol=min(Vols(imag(Vols) == 0)); % finds smallest real root for the liquid volume display(['the material is:',pure Comp Compound (Comp Num)]); display(['tr=',num2str(tr),' Pr=',num2str(Pr)]) display(['vg=',num2str(gasvol),' L/mol']) display(['vl=',num2str(liqvol),' L/mol']) Fig 1 The m-file code for retrieving critical data from Excel sheet, searching and selecting the requested compound by either name or formula, entering operating pressure and temperature, defining EoS parameters, using MATLAB built-in root finding function, and displaying the results 71

8 Table 1 The prediction of molar volume of both liquid and vapor (evaluated at the normal boiling point of the hydrocarbon) as given by the original and modified equation Either an empirical or a reference datum was chosen to calculate the percent relative error associated with each estimated volumetric propertyvg: Volume of a gas; VL: Volume of a liquid; and Vc: Critical volume of a fluid PRE represents the percent relative error: PRE = {Absolute (Estimated Reference)/Reference} 100% The adopted colorcoding, for the associated PRE, is bright green for the lowest PRE; pink for 10% <PRE<100%; and red for PRE>100% Component RK volume (L/mol) (PRE) RK-type volume (L/mol)(PRE) SRK volume (L/mol) (PRE) SRK-type volume (L/mol)(PRE) RK-prvolume(L/mol) (PRE) RK-PR-type volume (L/mol)(PRE) Empiricalª, (Reference * ) volume (L/mol)(PRE) n-propane (C 3H 300 K, atm (Tr= Pr=023484) V G =20851(24 %) V G =20964(294 %) V G =20646 (138 %) V G =2077(199 %) V G =21036(329 %) V G =21115(368 %) n-propane (C 3H 300 K, atm (Tr= Pr=023484) V L=010141(126 %) V L= (66 %) V L= (93 %) V L= (87 %) V L=21036(2235 %) V L=010135(125 %) n-butane (C 4H K, 1 atm (Tr= Pr= ) V G =216591(007 %) V G =216745*(000 %) V G =216028(033 %) V G =216198(025 %) V G =216667(0036 %) V G = (0034 %) * n-butane (C 4H K, 1 atm (Tr= Pr= ) V L=010628(105 %) V L= (088 %) V L= (79 %) V L= (13 %) V L=010463(88 %) V L= (23 %) 00962ª n-pentane (C 5H 3092 K, 1 atm (Tr=065831Pr= ) V G =245214(03 %) V G =245413(038 %) V G =244258(009 %) V G =244483*(000 %) V G =245105(025 %) V G = (034 %) * n-pentane (C 5H 3092 K, 1 atm(tr=065831pr= ) V L=013442(135 %) V L=012592(64 %) V L=012979(962 %) V L=01218 (29 %) V L=013342(127 %) V L=012326(41 %) 01184ª n-hexane (C 6H 3419 K, 1 atm (Tr=067382Pr= ) V G =270634(041 %) V G =27091(051 %) V G =269216(012 %) V G =269535*(00 %) V G =270298(028%) V G =27058(039 %) * n-hexane (C 6H 3419 K, 1 atm (Tr=067382Pr= ) V L=016486(173%) V L=015657(114%) V L=015744(1200%) V L=014973(65 %) V L= (182 %) V L=015612(110 %) 01406ª n-heptane (C 7H K, 1 atm (Tr= Pr=003704) V G =293658(051 %) V G =29396(062 %) V G =291804(012 %) V G =29216*(00 %) V G =29305 (030%) V G = (040 %) 29216* n-heptane (C 7H K, 1 atm (Tr=068779Pr=003704) V L=019605(204%) V L=018883(160%) V L=018557(1400%) V L= (99 %) V L=019892(222 %) V L=01905 (170 %) 01628ª Toluene (C 7H 8)@ K, 1 atm (Tr=064851Pr= ) V G =305937(030 %) V G =306107(035 %) V G =304833(006 %) V G =305028*(000 %) V G =305832(026 %) V G =306003(032 %) * Toluene (C 7H K, 1 atm (Tr=064851Pr= ) V L=01376(164%) V L=012916(93 %) V L=013266(1223%) V L=012476(55 %) V L=013661(156 %) V L=012618(675 %) 01182ª n-octane (C 8H K, 1 atm (Tr=070113Pr= ) V L= (245 %) V L=022268(204 %) V L=02161(168%) V L=020893(129 %) V L=024152(305%) V L= (259 %) 0185ª n-nonane (C 9H K, 1 atm (Tr=071299Pr= ) V L= (280 %) V L=025791(245 %) V L=024692(192%) V L=023996(158 %) V L=029512(424 %) V L=028803(390 %) 02072ª Naphthalene (C 10H 8)@ K, 1 atm (Tr=065618Pr= ) V G =391654(042 %) V G =392239(057 %) V G =390019*(000 %) V G =390684(017 %) V G =391184(030 %) V G = (045 %) * Naphthalene (C 10H 8)@ K, 1 atm (Tr=065618Pr= ) V L=017775(204 %) V L=017938(215 %) V L=017018(153 %) V L=017114(159 %) V L=017239(168 %) V L=01774(202 %) 01476ª n-octadecane (C 18H 38)@ K, 1 atm (Tr=079093Pr=00835) V G =455818(11 %) V G = (17 %) V G =44744(075 %) V G =450823*(000 %) V G =454319(077 %) V G =456816(13 %) * n-octadecane (C 18H 38)@ K, 1 atm (Tr=079093Pr=00835) V L=069696(539 %) V L= (380 %) V L=060846(344 %) V L=054733(209 %) V L= (154 %) V L=456816(9989 %) Butane (C 4H K, 3747 atm (Tr=1 Pr=1) V c=030471(195 %) V c=018008(294 %) V c=030471(195 %) V c= (294 %) V c= (1391%) V c= (1353%) 0255 Benzene (C 6H K, 4834 atm (Tr=1 Pr=1) V c= (165 %) V c=018379(314 %) V c=031227(165 %) V c=018379(314 %) V c=063137(1356%) V c= (1293%) 0268 n-hexane (C 6H 5074 K, 2973 atm(tr=1pr=1) V c=045267(230 %) V c=04668(268 %) V c= (245 %) V c=04668(268 %) V c=094626(1571%) V c=09490(1571%) 0368 ªBased on the empirical value of molar volume of liquid at its normal boiling point, reported by Mackay et al (2005) Based on the values reported by Poling et al (2001) *The referencev G is taken as that of the model which gives the most accurate (ie, the lowest PRE value) predicted V L Al-Malah 72

9 RESULTS AND DISCUSSION Hydrocarbons, with similar and different molecular weight (ie, size) and carbon atomic fraction (ie, carbon to hydrogen ratio), were specifically selected to examine the applicability of the suggested model(s) Low-, medium-, and high-molecular weight hydrocarbons were examined The normal boiling point of a hydrocarbon was chosen, because there exists an empirical (ie, curve-fitted, based on experimental data) value for the liquid molar volume given byle Bas method at normal boiling point (Mackay, et al, 2006; Poling, et al, 2001) As quoted by Poling, et al (2001) that the liquid molar volume evaluated at the normal boiling point by the simple methods of Schroeder or LeBas can be used with errors generally less than 5% The liquid molar volume either is directly reported in (Mackay, et al, 2006), or can be calculated by the group/atom contribution method of Le Bas (Poling, et al, 2001) On the other hand, the reference V G is taken as that of the model which gives the most accurate (ie, the lowest PRE value) predicted V L For the critical molar volume of a pure fluid, such a value is reported by Poling, et al, (2001) Table 1 shows the estimated molar volume of both the liquid and vapor for a given substance at a given pressure and temperature Notice that the adopted color coding, for the associated PRE value, is bright green for the lowest PRE; pink for 10% < PRE <100%; and red for PRE>100% The following trends can be observed as far as the accuracy (manifested via the magnitude of PRE) of the model is concerned: 1 In general, all models, the original and the substituted/modified equations, predict well, with a good accuracy (ie, a PRE value less than 10%) The predicted values almost coincide 2 In general, none of the models, including the original and the substituted/modified equations, will predict, with a good accuracy (ie, a PRE value less than 10%), if the hydrocarbon molecular size is greater than C 7 3 On the other hand, for a hydrocarbon molecular size less than C 7, the best three models that predict, with a good accuracy (ie, a PRE value less than 10%)are SRKtype, RK-type, and SRK 4 At the critical condition (ie, T r =10 and P r =10), all models failed to predict the critical volume (V c ); ie, PRE values > 10% CONCLUSION The replacement of critical pressure and temperature of a hydrocarbon, like and ratios, appearing in RK, SRK, and RK-PR, by itsmolecular properties; namely, the molecular weight and carbon atomic fraction, was successfully done The replacement will ease the calculation of volumetric properties of liquid and vapor with a reasonable accuracy (ie, PRE <10%), taking into account the following limitations: 1 In general, all models, the original and the substituted/modified equations, predict well, with a good accuracy (ie, PRE <10%) The predicted values almost coincide 2 In general, none of the models, including the original and the substituted/modified equations, will predict, with a good accuracy (ie, PRE<10%), if the hydrocarbon molecular size is greater than C 7 3 On the other hand, for a hydrocarbon molecular size less than C 7, the best three models that predict, with a good accuracy (ie, PRE <10%) are SRK-type, RK-type, and SRK 4 At the critical condition (ie, T r =10 and P r =10), all models failed to predict the critical volume (V c ); ie, PRE values > 10% COMPETING INTERESTS Author has declared that no competing interests exist REFERENCES Al-Malah K Prediction of normal boiling points of hydrocarbons using simple molecular properties J of Advanced Chemical Engineering 2013;3:1-9 Doi:104303/jace/

10 Cismondi M, Mollerup J Development and application of a three-parameter RK PR equation of state Fluid Phase Equilibria 2005;232:74 89 DOI:101016/jfluid Frey K, Augustine C, Ciccolini RP, Paap S, Modell M, Tester J Volume translation in equations of state as a means of accurate property estimation Fluid Phase Equilibria 2007;260: DOI:101016/jfluid Kontogeorgisa GM, Economoub IG Equations of state: From the ideas of van der Waals to association theories J of Supercritical Fluids 2010;55: DOI:101016/jsupflu Mackay D, Shiu WY, Ma KC, Lee SC Handbook of Physical-Chemical Properties and Environmental Fate for Organic Chemicals, 2 nd edition Taylor & Francis, New York; 2006 Al-Malah Poling BE, Prausnitz JM, O Connell JP The Properties of Gases and Liquids, 5 th edition McGraw-Hill, New York; 2001 Redlich O On the Three-Parameter Representation of the Equation of State Industrial & Engineering Chemistry Fundamentals 1975;14(3): DOI:101021/i160055a020 Redlich O, Kwong JNS On the thermodynamics of solutions Chem Rev 1949;44(1): DOI:101021/cr60137a013 PMID: Schmidt G, Wenzel H A modified van der waals Type Equation of State Chem Eng Sci 1980;35: Soave G Equilibrium constants from a modified redlich kwong equation of state Chem Eng Sci 1972;27: Waals, JD van der On the continuity of the gas and liquid state Doctoral Dissertation, Leiden; 1873 Copyright International Knowledge Press All rights reserved 74