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Modifying the Redlich-Kwong-Soave Equation of State James H 127 Modifying the Redlich-Kwong-Soave Equation of State James H. McNeill Department of Natural Sciences, Middle Georgia State University, Macon, GA 31206 James E. Bidlack Department of Biology, University of Central Oklahoma, Edmond, OK 73034 Abstract: The Redlich-Kwong-Soave equation of state for real gases and liquids has been PRGL¿HGLQRUGHUWRLPSURYHWKHPDWFKRIFDOFXODWHGOLTXLGPRODUYROXPHVZLWKPHDVXUHGGDWDE\ allowing the b-parameter to be a function of the temperature and molar volume of real gases and liquids. Molecular dynamics simulations of the noble gas krypton, using the hard-sphere potential for a large number of krypton atoms, were performed to evaluate how the b-parameter varies with gas molar volume. The results of these simulations were applied when modifying the original 5HGOLFK.ZRQJ6RDYHHTXDWLRQRIVWDWH,QWKHPRGL¿HG5HGOLFK.ZRQJ6RDYHHTXDWLRQRIVWDWH the aSDUDPHWHULVNHSWFRQVWDQW$OVRLQWKLVPRGL¿HGYHUVLRQWKHPHDVXUHGFULWLFDOPRODUYROXPH is employed as well as the measured critical temperature and critical pressure values in parameter determination at the critical point. From this determination of parameters for a number of real gases, the same parameters can be approximately calculated using the measured critical compressibility factor avoiding numerical procedures. This results in an equation of state that matches closely with the measured critical point for many gases as well as improving the match between the model and measured liquid molar volumes. moles of gas. Introduction v = V/n (3) 7KH¿UVWHTXDWLRQRIVWDWHGHYHORSHGWRPRGHO real gases at relatively high temperatures and However, at low temperatures and high low pressures is the ideal gas equation: pressures, real gases deviate from the ideal gas H[SUHVVLRQJLYHQLQ(TXDWLRQ$WVX൶FLHQWO\ PV = nRT (1) low temperatures and high pressures, any real gas will condense into the liquid phase and an P is the pressure of the gas, V is the volume of equilibrium situation takes place between the the gas, n is the number of moles of the gas, T is liquid and gas phases of any pure substance. the temperature of a gas in absolute temperature 7KH¿UVWDWWHPSWWRPRGHOUHDOJDVHVDQGOLTXLGV scale, and R is the ideal gas constant. If one was accomplished by van der Waals using the divides through the ideal gas equation with the following mathematical relation. number of moles of gas, one has the general 2 2 expression below in terms of gas molar volume P = nRT/(V – nbvdW) – avdW n /V (4) v. In the van der Waals expression, Equation Pv = RT (2) 4, bvdW is the molar volume of space occupied by individual gas atoms, in the case of a noble The gas molar volume v is the unit volume gas, or gas molecules which is much less than occupied by one mole of a gas and is equal to the volume of the gas itself at relatively low the gas volume V divided by the number n of pressures. It is this space occupied by gaseous Proc. Okla. Acad. Sci. 98: pp 127 - 138 (2018) 128 J.H. McNeill and J.E. Bidlack particles which prevent any real gas volume JDV WKH ¿UVW DQG VHFRQG GHULYDWLYH RI WKH YDQ going to zero at extremely large pressures. The der Waals expression, Equation 5, with respect avdW term represents the weak attractive forces to the molar volume v are both equal to zero. amongst gaseous atoms or molecules which is responsible for the liquid phase at temperatures Pv) = 0 (At T = Tc, P = Pc and v = vc) (6) below the critical temperature value Tc of any 2 2 pure gas. Above the critical temperature value, Pv ) = 0 (At T = Tc, P = Pc and v = vc) (7) it is impossible to liquefy any gas, because only below the critical temperature both the liquid By using Equations 6 and 7 along with the and vapor phases coexist. In theory, above following expression one can determine the the critical temperature, the average kinetic numerical values of avdW and bvdW: HQHUJ\RIJDVDWRPVRUPROHFXOHVLVVX൶FLHQWWR 2 overcome the weak attractive forces that occur Pc = RTc/(vc – bvdW) – avdW/vc (8) when two gas atoms or molecules come into close contact (Barrow, 1979). Above the critical :KHQYDQGHU:DDOV¿UVWVWXGLHGUHDOJDVHVLQ temperature value, it is impossible to liquefy the early part of the 20th century, it was practically any gas due to the average kinetic energy of gas impossible to measure the critical molar volume atoms or molecules successfully overcoming the vc of a liquid with the technology available at weak attractive forces that occur when two gas that time. When solving for parameters avdW and atoms or molecules come into close contact. If bvdW, they were determined to be the following one substitutes Equation 3 into Equation 4, then two functions of the measured critical pressure the van der Waals equation of state can be more and critical temperature values. conveniently represented as a function of gas 2 2 molar volume v instead. avdW = (27/64) R Tc / Pc (9) 2 P = RT/(v – bvdW) – avdW/v (5) bvdW = RTc/(8Pc) (10) Unlike the ideal gas equation, one must And the estimated critical molar volume vc is experimentally determine the numerical values three times the numerical value of the van der of parameters avdW and bvdW in the van der Waals b-parameter. Waals expression. The exact values of these two parameters depend upon the chemical vc = 3bvdW (11) composition of any pure gas. To determine WKHLUYDOXHVRQHKDVWRXVHGL൵HUHQWLDOFDOFXOXV 7KH¿UVWHUURULQWKHYDQGHU:DDOVH[SUHVVLRQ DQGWKHGH¿QLWLRQRIWKHFULWLFDOSRLQWRIDUHDO is that it overly estimates the values of most gas. The critical point takes place when a measured critical molar volumes determined real gas is at its critical temperature value Tc with today’s technology. Also, it over-estimates and its pressure value P is equal to its critical the pressures of most real gases as compared pressure value Pc. The critical pressure value Pc to high pressure data measured at temperatures is that value the equilibrium vapor pressure of above the critical temperature value where a pure liquid approaches to in the limit of the condensation is unable to take place. To critical temperature, and likewise in the limit improve this model, Redlich and Kwong later of the temperature approaching the critical on developed the following expression. temperature value at the critical pressure value, the liquid molar volume v of the liquid P = RT/(v – b) – a /{T1/2 [v(v + b)]} (12) approaches its critical molar volume vc. Or, at the critical temperature and pressure, the molar Division by the square root of the absolute volume of the liquid is equal to that of its gas temperature T compensates for the increase molar volume instead of being less than the gas of overcoming weak intermolecular forces of molar volume. At the critical point of any pure attraction with an increase of temperature. With Proc. Okla. Acad. Sci. 98: pp 127 - 138 (2018) 0RGL¿HG5HGOLFK.ZRQJ6RDYH(TXDWLRQ 129 the inclusion of the b parameter in the Redlich- by the van der Waals equation. Another problem Kwong equation of state for real gases and with the Redlich-Kwong function is that with liquids, Equation 12, in the subtraction term, the a parameter divided by the square-root of the Equation 12 can be rearranged to similar format absolute temperature scale over estimates the of the original van der Waals equation into the weak attractive forces amongst gas atoms below following equation. the critical temperature value, especially as the temperature drops below the freezing point of P = RT/(v – b) – {a/[T1/2(1 + b/v) ]} / v2 (13) any substance. Thus, by the Redlich-Kwong equation of To avoid this problem for low temperatures state, the van der Waals avdW parameter is the concerning the Redlich-Kwong equation, Soave following function of temperature and molar PRGL¿HG WKH 5HGOLFK.ZRQJ HTXDWLRQ LQ WKH volume. following expression referred to as the Redlich- Kwong-Soave equation of state (Prausnitz et al., 1/2 avdW = a/[T (1 + b/v) ] (14) 1998). Thus, in view of Equation 13, the decrease P = R T / (v – b) – a(T,Z) / [v (v + b)] (18) in molar volume allows the van der Waals avdW parameter approach zero in the limit of zero For this expression, a(T,Z) is the next function PRODUYROXPHEHFDXVHDWVX൶FLHQWO\ORZPRODU of the absolute temperature T, the measured volumes, atoms and molecules then would begin critical pressure value Pc, the measured critical to overlap their electron clouds. This equation temperature value Tc, and the measured acentric will then model decreasing attractive forces of factor Z of a real gas or liquid: gaseous atoms and molecules in the limit of zero 2 2 molar volume along with the positive term in the a(T,Z) = [0.42748 R Tc / Pc]{1 + [0.480 + original van der Waals equation of state. Again 2 1/2 2 XVLQJWKHGL൵HUHQWLDOFDOFXOXVJLYHQLQ(TXDWLRQV 1.574Z – 0.176Z ][1 – (T/Tc) ]} (19) 6 and 7, the a and b parameters and the critical molar volume are approximately equal to the However, the b parameter and the critical following expressions given below to a limited molar volume vc in the Redlich-Kwong-Soave QXPEHURIVLJQL¿FDQW¿JXUHV HTXDWLRQRIVWDWHDUHVWLOOGH¿QHGE\(TXDWLRQV 16 and 17. Thus, the calculated critical 2 2 a = 0.42748 R Tc /Pc (15) molar volumes are larger than that observed H[SHULPHQWDOO\%\GH¿QLWLRQ 3UDXVQLW]HWDO b = 0.08664 R Tc /Pc (16) 1998), the measured acentric factor Z is equal to the following equation involving the observed vc = 0.38473 b (17) equilibrium liquid-vapor pressure value Pequil, at an absolute temperature value equal to 70% of :KHQ XVLQJ GL൵HUHQWLDO FDOFXOXV IRU WKH the measured critical temperature value Tc of Redlich-Kwong equation, numerical techniques any pure substance, and the measured critical must be utilized in a computer program in order pressure value Pc: to evaluate the fractions given in Equations 15 to 17.
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